We propose the topological object, a gravitational knot, could exist in Newton's theory of gravitation by assuming that the Ricci curvature tensor especially the metric tensor consists of a scalar field i.e. a subset of the Ricci curvature tensor. The Chern-Simons action is interpreted as such a knot.

By assuming that the Ricci curvature tensor consists of a subset (scalar) field, we propose that Newton's second law of gravitation in (2+1)-dimensional space-time, a linear equation, could have hidden nonlinearity. This subset field satisfies a non-linear subset field theory where in the case of an empty space-time or the weak field, it reduces to Newton's linear theory of gravitation.

Probability, as manifested through entropy, is presented in this study as one ofthe most fundamental components of physical reality. It is demonstrated that thequantization of probability allows for the introduction of the mass phenomenon.In simple terms, gaps in probability impose resistance to change in movement,which observers experience as inertial mass. The model presented in the paperbuilds on two probability fields that are allowed to interact. The resultant prob-ability distribution is quantized, producing discrete probability levels. Finally, aformula is developed that correlates the gaps in probability levels with physicalmass. The model allows for the estimation of quark masses. The masses of theproton and neutron are arrived at with an error of under 0.04%. The masses ofsigma baryons are calculated with an error between 0.2% and 0.05%. The Wboson mass is calculated with an error of under 0.5%. The model explains whyproton is stable while other baryons are not, and it gives an explanation of theorigins and nature of dark matter. Throughout the text, the article illustratesthat the approach required to describe the nature of mass is incompatible withthe mathematical framework needed to explain other physical phenomena.

This paper is a continuation of [2]. Here, we discuss twisted branes, the free loop superspace, and, in particular, a deformation of the modal lightcone which allows us to model cobordisms of generically small, portable, locally closed systems.

We prove Payne's nodal line conjecture for any bounded simply connected, possibly non-convex, smooth boundary domain $Omega$ in Plane; Payne conjectured that any second Dirichlet eigenfunction of laplacian in any simply connected bounded domain in Plane can not have a closed nodal line.

In this paper, ER bridges are discussed as bordisms. We treat these bordisms as fibers, whose sections are holographically entangled to copies of $S^1$. Diffemorphisms of these fibers are discussed, as well as the implication of replacing $S^1$ with the supercircle, and the replacing its underlying algebra with a Lie superalgebra.

Miniature crycoolers are small refrigerators that can reach cryogenic temperatures in the range of 60K to 150K. They have the capability of accumulating a small temperature drop into a large overall temperature reduction. The cooldown time estimation is becoming more and more as a design parameter, certainly in hands-on applications. The various complicated physical processes involved in crycooler operation make it hardly possible to explicitly simulate the temperature time response. The numerical methods for solving a typical crycooler suffer from numerical instability,time step restrictions and high computational costs, among others. Since the operation of crycoolers involve processes in range of 15Hz−120Hz, actually solving the crycooler transient response would require different software tools to support the design and analysis of physical processes such as heat transfer, fluid dynamic, electromagnetic and mechanical. These processes would also require an excessive amount of calculations, incurring time consuming and precision penalty. In thisarticle we try to bridge the gap between the explicit impractical approach to steady state based approach. A framework developed in Python for calculating the cooldown time profile of anycrycooler based on a steady state database, is introduced, while utilizing a semi-analytic approach under various operating conditions. The cooldown time performance can be explored at various target and ambient temperature conditions, and also the effects of an external load, material properties or thermal capacitance on the overall cooldown time response. Two case studies based on linear and rotary crycoolers developed at Ricor are used for verfication, with a good agreement between the simulated and measured values.

By imposing a requirement for spatial isotropy, it is possible to find an algebra with a subalgebra structure having a pattern matching that of the bosons and three families of fermions of the standard model.

Rational numbers Q have much more structure beyond the ordered field structure which leads to Real Numbers as a metric completion.The modular group representation of continued fractions is used as a Number Theory "friendly" implementation of the real numbers, with a possible unification with p-adic numbers, beyond the "direct sum" adeles framework. This approach also allows to extend Fourier and Haar Wavelet Analysis, by including inversion as a geometric antipode. Other applications in Mathematical-Physics steam from the central role of the modular group: Belyi maps, Farey graphs and tessellations etc. which allow the study of important classes of numbers (algebraic, periods) in a systematic way. The presentation is a preliminary version the project, stating the motivation, goals and approach.

A restricted path integral method is proposed to simulate a type of quantum system or Hamiltonian called a sum of controlled few-fermions on a classical computer using Monte Carlo without a numerical sign problem. Related methods and systems of Monte Carlo quantum computing are disclosed for simulating quantum systems and implementing quantum computing efficiently on a classical computer, including methods and systems for simulating many-variable signed densities, methods and systems for decomposing a many-variable density into a combination of few-variable signed densities, and methods and systems for solving a computational problem via Monte Carlo quantum computing.

Electrostatic fields, cornerstone elements in understanding electrical phenomena, serve as key components in diverse scientific and engineering fields. This paper elucidates the concept of electrostatic fields, explores their properties, and outlines their broad applications. We start from the basics of electric charges and their interactions, leading us to the core principles of electrostatics. A deep dive into Coulomb's law is presented to scrutinize the behavior of electrostatic fields, along with the concept of electric potential and its relationship with the electric field. We underline the instrumental role of electrostatic field analysis in practical applications like electrical power systems, electronics, and telecommunications. Furthermore, we introduce techniques to tackle electrostatic field problems and showcase their applications in engineering and technology. By providing a comprehensive review of electrostatic fields, we aim to deepen understanding and propel further research into this vital domain of electromagnetism.

The intent of this paper is to provide a simple focus on that mathematical concept and solution, phenomenological velocity to shine light on aworthy topic for mathematicians and physicists alike. Phenomenological Velocity is essential to the formulation of a gestalt cosmology. The bibliography of this paper provides references to the extensive research that has been conducted by myself on the topic. I have performed conditional integrals of the phenomenological velocity in its most liberated standard-algebraic form, I have shown that the computational-phenomenological velocity satisfies its real-analytic solutionwhen not using the speed of light in scientific notation to get the computational version, thus demonstrating that it is a valid solution. So, phenomenological velocity has profound consequences to the foundations of physics as civilization moves into a galactic scale and information is communicated at the quantumlevel, because it is such a mathematical reality it ought not be ignored when considering topics from hidden dimensions (a real, algebraic technique) and relativity to gravity and dark matter. It gives us a new perspective on how weperceive the meaning of velocity itself with pragnanz, and thus with the new meaning, perspectives can change. I hope the reader will investigate the combined research I have performed on this topic, available by referencing the works in this bibliography to fully understand the nature of the arguments being made within. So, this points the right direction for future research, perhaps even withintent to encourage experimental design.

A particular class of real manifolds (Hermitian spaces) naturally model smooth, possibly complex n-spaces. We show how to realize such a space as a restriction of a super-smooth stack using a compass. We also discuss the classical relationship between iterated loop spaces and the configuration space of a particle.

In the context of QFT and Gauge Theory, the introduction of Natural Units, as a quantization in disguise'' combined with Buckingham's Pi-Theorem, provides a direct connection with de Rham Periods, as also hinted by Feynman amplitudes, Dessins d'Enfant and Belyi maps models of baryon modes etc.A program emerges: Physics Laws as Period Laws, and Alpha, as an element of the Pi-groups, a period. Our models of the Physical Reality emerge from the union of Cohomological Physics and Number Theory,helping us understand ``the unreasonable effectiveness of Mathematics''.An overview of the Network Model is included, with impacts to Sciences in general. Further prospects for understanding the fine structure constant are presented.

En este artículo, se plantea una nueva reinterpretación de la geometría curva del espacio-tiempo, donde de considera que el espacio-tiempo experimenta una contracción longitudinal. Este efecto, se manifiesta en cambios de la métrica del espacio-tiempo que determinan como se miden las distancias y los intervalos temporales en esa región. Es decir, una variación de dimensión en la escala, tamaño o longitud aparente del espacio-tiempo. Esta reinterpretación es compatible con las ecuaciones de campo de Einstein y Maxwell. La constante gravitacional universal GNewton, la constante de Hubble para la expansión acelerada del universo H(0) y la constante cosmológica asociada con la hipotética energía oscura Λ, se pueden obtener y aproximar mediante este nuevo enfoque, donde la masa del bosón de Higgs con sus privilegiadas y únicas características, juega un papel trascendental para dar respuesta a una multitud de preguntas abiertas en física y en la actualidad cosmológica moderna. La reinterpretación de la geometría curva por la contracción espacio-temporal, proporciona un nuevo marco para entender mucho mejor la gravedad. Al obtener valores muy aproximados de la constante de gravitación universal, es posible determinar la fuerza inversa a la gravedad responsable de la expansión acelerada del universo. Esto es posible gracias a la teoría de divergencia de Gauss, donde la distribución de carga determinada por la constante de Coulomb en el marco de la expansión multipolar definida por el electromagnetismo, constituye una analogía bastante sólida, siendo de forma inversamente proporcional a la gravedad, mediante la cuál es posible obtener y calcular con mucha precisión, el valor de la constante H(0) de Hubble. La constante cosmológica Λ, considerada como una posible energía oscura que impulsa la expansión acelerada del universo, también puede ser obtenida y explicada a través de este nuevo enfoque. La reinterpretación de la geometría curva de la gravedad, como una contracción espacio-temporal, afectaría a las propiedades de la expansión del espacio-tiempo, donde la interpretación de contracción del universo, que describe la Relatividad General, debe ser reinterpretada, comprendida y aceptada, como la gravedad misma a cualquier escala.

This article presents a new reinterpretation of the curved geometry of spacetime, where it is considered that spacetime undergoes longitudinal contraction. This effect is manifested in changes in the spacetime metric that determine how distances and temporal intervals are measured in that region. In other words, a variation in the scale, size, or apparent length of spacetime. This reinterpretation is compatible with Einstein's field equations and Maxwell's equations. The universal gravitational constant of Newton, G, the Hubble constant for the accelerated expansion of the universe, H(0), and the cosmological constant associated with hypothetical dark energy, Λ, can be obtained and approximated using this new approach, where the mass of the Higgs boson with its unique and privileged characteristics plays a crucial role in addressing numerous open questions in physics and modern cosmology. The reinterpretation of curved geometry through spacetime contraction provides a new framework for better understanding gravity. By obtaining very close values of the universal gravitational constant, it is possible to determine the inverse force to gravity responsible for the accelerated expansion of the universe. This is achievable through Gauss's divergence theorem, where the charge distribution determined by the Coulomb constant within the framework of multipolar expansion defined by electromagnetism constitutes a quite solid analogy, being inversely proportional to gravity. This allows for the precise calculation of the value of the Hubble constant, H(0). The cosmological constant Λ, considered as a potential dark energy driving the accelerated expansion of the universe, can also be obtained and explained through this new approach. The reinterpretation of the curved geometry of gravity as spacetime contraction would affect the properties of spacetime expansion, where the interpretation of the universe's contraction described by General Relativity must be reinterpreted, understood, and accepted as gravity itself at any scale.

We construct the geometric optical knot in 3-dimensional Euclidean (vacuum or weak-field) space using the Abelian Chern-Simons integral and the variables (the Clebsch variables) of the complex scalar field, i.e. the function of amplitude and the phase related to the refractive index. The result of numerical simulation shows that in vacuum or weak-field space, there exists such a knot.

In physics, a single center of gravity is assumed for forces. However, at least 3 fixed points π, π², π³are required as the center, orthograde for the 3 spatial dimensions. With this approach, the universe can be understood as a set of rational numbers Q. This is to be distinguished from how we see the world, a 3-dimensional space with time. Observations from the past is the subset Q⁺ for physics. A system of 3 objects, each with 3 spatial coordinates on the surface and time, is sufficient for physics. For the microcosm, the energy results from the 10 independent parameters as a polynomial P(2). For an observer, the local coordinates are the normalization for the metric. Our idea of a space with revolutions of 2π gives the coordinates in the macrocosm in epicycles. For the observer this means a transformation of the energies into polynomials P(2π). This is used to simulate the energies of a system. c can be calculated from the units meter and day.π/2 c m day = r_Earth²This formula provides the equatorial radius of the earth with an accuracy of 489 m. Orbits can be calculated using polynomials P(2π) and orbital times in the planetary system with P(8). A common constant can be derived from h, G and c with the consequence for H0:h G c⁵s⁸/m¹⁰ ( π⁴- π²- π^-1 - π^-3) ^1/2 H0_theory= π^1/23 h G c³ s⁵/m⁸A photon consists of 2 entangled electrons e^- and e⁺m_neutron / m_e=(2π)⁴ +(2π)³+(2π)²-(2π)¹-(2π)⁰-(2π)^-1+2(2π)^-2+2(2π)^-4-2(2π)^-6 +6(2π)^-8 = 1838.6836611 Theory: 1838.6836611 m_e measured: 1838.68366173(89) m_eFor each charge there is an energy C = -π+2π^-1- π^-3+2π^-5-π^-7+π^-9- π^-12Together with the neutron mass, the result for the proton is: m_proton=m_neutron + C m_e= 1836.15267363 m_eFine-structure constant:1/α= π⁴+ π³+ π²-1- π^-1 + π^-2- π^-3 + π^-7 - π^-9- 2 π^-10-2 π^-11-2 π^-12 = 137.035999107The muon and tauon masses as well as calculations for the inner planetary system are given.

Using non-commutative space-time quaternion algebra, we represent the generalization of one-dimensional and three-dimensional telegraph equations, which are widely applied to consider the propagation of an electromagnetic signal in communication lines, as well as to describe particle diffusion and heat transfer. It is shown that the system of telegraph equations can be represented in compact form as a single quaternion equation taking into account the space-time properties of physical quantities. The distinctive features of the one-dimensional and three-dimensional telegraph equations are discussed.

We present the theoretical description of plane turbulent wall-bounded flows based on the previously proposed equations for vortex fluid, which take into account both the longitudinal flow and the vortex tubes rotation. Using the simple model of eddy viscosity we obtain the analytical expressions for mean velocity profiles of steady-state turbulent flows. In particular we consider near-wall boundary layer flow as well as Couette, Poiseuille and combined Couette-Poiseuille flows. In all these cases the calculated velocity profiles are in good agreement with experimental data and results of direct numerical simulations.

We address the unsolved question of how best to estimate the collision entropy, also called quadratic or second order Rényi entropy. Integer-order Rényi entropies are synthetic indices useful for the characterization of probability distributions. In recent decades, numerous studies have been conducted to arrive at their valid estimates starting from experimental data, so to derive suitable classification methods for the underlying processes, but optimal solutions have not been reached yet. Limited to the estimation of collision entropy, a one-line formula is presented here. The results of some specific Monte Carlo experiments give evidence of the validity of this estimator even for the very low densities of the data spread in high-dimensional sample spaces. The method strengths are unbiased consistency, generality and minimum computational cost.

Regarding the interference of the two light beams, We usually think of it as interference between two light beams of the same wavelength and frequency. However , if the interferometer of the Michelson-Morley experimental device accurately records the interference of the two beams of the same wavelength, that is negative evidence for the correctness of the postulate 2 of Special relativity.

We describe two kinds of equations of motion in classical electrodynamics, there are dynamicallaws for the charges and the equations for the electromagnetic (EM) field.

We use an effective Schwinger-Keldysh field theory of long-range massless modes to derive the Navier-Stokes equations as an energy-momentum balance equation. The fluid will be invariant under the linear subgroup of the volume-preserving diffeomorphisms, which are the non-linear, time-independent spatial translations.

It is well known that memristors can be classified into the four classes according to the complexity of their mathematical representation. Furthermore, the four classes of memristors are used to qualitatively simulate many of the experimentally measured pinched hysteresis loops. In this paper, we define the 2-terminal devices, which do not belong to the above four classes of memristors, but have the same non-volatile property as the ideal memristor. We then study the non-volatile mechanism of these devices and the memristors that can retain the previous value of the state even when the driving signal is set to zero. We show that the ideal generic memristors and the generalized 2-terminal devices can have interesting applications: non-volatile multi-valued memories and two-element chaotic oscillators, if we remove the condition that no state change occurs after the zero driving signal. We also show that the 2-terminal devices and the four classes of memristors can exhibit a wide variety of pinched hysteresis loops similar to those measured experimentally. Furthermore, we show that a wide variety of Lissajous curves are possible, depending on whether the direction of the Lissajous curve is clockwise or counterclockwise and which quadrants the Lissajous curve passes through.

The Reality-Sucks theory can compute both, the dark matter and the dark energy, in the required range. The ratio between dark and ordinary matter is estimated between 83.0 % and 85.7 %. Dark matter and dark energy together constitute between 94.4 % and 95.8 % of the total mass-energy content. Similar results should be obtained using the Λ-CDM model.

Interaction theories are usually based on a elativistically invarieant Lagrange function. This function is generally known and accepted for the electromagnetic interaction. The variation of that Lagrangian leads to the system of the coupled Maxwell-Dirac equations. It contains a non-linear term. If you neglect this term, you obtain the well-known linear Dirac equation and rules for determining the correct values of the spectral lines of atoms. However, one cannot describe the radiation process and has to introduce the quantum hypothesis. But, if the non-linear term is also taken into account, there are solutions of the system what describe the emission of "quantum jumps" in space and time with correct frequencies. This is demonstrated in the presented work for hydrogen and helium atoms. It explains the entangled eigenfunctions in the context of a classical mear-field theory. Further problems like diffraction effects, photo effects and relativistic transformation of the field tensor are discussed. Aim of the work is a proposal of an alternative to the statistical interpretation of the quantum theory in context of a classical near-field theory.

We construct the geometric optical knot in 3-dimensional Euclidean (flat) space of the Abelian Chern-Simons integral using the variables (the Clebsch variables) of the complex scalar field, i.e. the function of amplitude and the phase, where the phase is related to the refractive index.

In this work, we solve the main mathematical puzzles of the UMMO file which contains severalthousand postal letters sent since the 1960s addressing many fields such as philosophy, mathematics, human sciences, biology, cosmology, theoretical physics, among others. "The UMMO affair" refers to more than 200 listed documents, representing at least 1300 typed pages (the "Ummite letters"), which are said to have been sent since 1966 to numerous recipients, in particular in Spain and France by editors - the Ummites - claiming to be extraterrestrials on an observation mission on Earth and which have four centuries of technological advance on the terrestrial human technologies. We demonstrate the importance of angular tetravalent logic in mathematics and theorical physics. As an example, we give a proof of Fermat’s last theorem using angular tetravalent logic, as suggested by the Ummites. Then, we will pierce the secrets of the universe always using the tetravalent logic, we will explain the reasoning which proves the existence of a twin universe and we give the mathematical formula for the folding of space-time which separates the two twin universes and finally we explain why the curvature of the universe is necessarily negative.

The object of this paper is the problem of the motion of three bodies subjected to the attraction of gravitation. In section 1, we write the equations of motion, then we give the ten first integrals of motion. We treat the equations of motion of the case of an artificial satellite around the earth in section 2, where we deduce the 3 laws of Kepler and we give the resolution of the equations of motion of the artificial satellite. In section 3, we consider the case of the motion of two bodies. Finally in section 4, we give details of the equations of motion of 3 bodies, we develop the inverse of the squares of the distances to the first order. We treat, by neglecting the mass of a body, the problem called the restricted movement of two bodies.

We treat the geometrical optics as an Abelian $U(1)$ local gauge theory the same as the Abelian $U(1)$ Maxwell's gauge theory. We propose there exists a knot in a 3-dimensional Euclidean (flat) space of the geometrical optics (the eikonal equation) as a consequence there exists a knot in the Maxwell's theory in a vacuum. We formulate the Chern-Simons integral using an eikonal. We obtain the relation between the knot (the geometric optical helicity, an integer number) and the refractive index.

A restricted path integral method is proposed to simulate a type of quantum system or Hamiltonian called a sum of controlled few-fermions on a classical computer using Monte Carlo without a numerical sign problem. Then a universality is proven to assert that any bounded-error quantum polynomial time (BQP) algorithm can be encoded into a sum of controlled few-fermions and simulated efficiently using classical Monte Carlo. Therefore, BQP is precisely the same as the class of bounded-error probabilistic polynomial time (BPP), namely, BPP = BQP.

We treat the geometrical optics as an Abelian $U(1)$ local gauge theory in vacuum curved space-time. We formulate the eikonal equation in (1+1)-dimensional vacuum centrally symmetric curved space-time using null geodesic of the Schwarzschild metric and obtain mass-the U(1) gauge potential relation.

The paper is concerned with the mathematical basics for the calculation of the Einstein tensor. Einstein's tensor is part of Einsteins field equations in General Relativity.

The quantification of Length and Time in Kepler's laws implies an angular momentum quantum, identified with the reduced Planck's constant, showing a mass-symmetry with the Newtonian constant G. This leads to the Diophantine Coherence Theorem which generalizes the synthetic resolution of the Hydrogen spectrum by Arthur Haas, three years before Bohr. The Length quantum breaks the Planck wall by a factor 10^61, and the associated Holographic Cosmos is identified as the source of the Background Radiation in the Steady-State Cosmology. An Electricity-Gravitation symmetry, connected with the Combinatorial Hierarchy, defines the steady-state Universe with an invariant Hubble radius 13.812 milliard light-year, corresponding to 70.793 (km/s)/Mpc, a value deposed (1998) in a Closed Draft at the Paris Academy, confirmed by the WMAP value and the recent Carnegie-Chicago Hubble Program, and associated with the Eddington number and the Kotov-Lyuty non-local oscillation. This confirms definitely the Anthropic Principle and the Diophantine Holographic Topological Axis rehabilitating the tachyonic bosonic string theory. The Holographic Principle uses the Archimedes pi-value 22/7. This specifies $G$, compatible with the BIPM measurements, but at 6 sigma from the official value, defined by merging discordant measurements.

We treat the geometrical optics as the classical limit of quantum electrodynamics i.e. an Abelian $U(1)$ local gauge theory in flat space-time. We formulate the eikonal equation in a (1+1)-dimensional Minkowskian (flat) space-time and we found that the refractive index as a function of the $U(1)$ gauge potential.

The main goal of this note is to explain that classical mathematics is a special degenerate case of finite mathematics in the formal limit p→∞, where p is the characteristic of the ring or field in finite mathematics. This statement is not philosophical but has been rigorously proved mathematically in our publications. We also describe phenomena which finite mathematics can explain but classical mathematics cannot. Classical mathematics involves limits, infinitesimals, continuity etc., while finite mathematics involves only finite numbers.

An algebra providing a possible basis for the standard model is presented. The algebra is generated by combining the trigintaduonion Cayley-Dickson algebra with the complexified space-time Clifford algebra. Subalgebras are assigned to represent multivectors for transverse coordinates. When a requirement for isotropy with respect to spatial coordinates is applied to those subalgebras, the structure generated forms a pattern matching that of the fermions and bosons of the standard model.

The refractive index-curvature relation is formulated using the second rank tensor of Ricci curvature as a consequence of a scalar refractive index. A scalar refractive index describes (an isotropic) linear optics. In (an isotropic) non-linear optics, this scalar refractive index is decomposed into a contravariant fourth rank tensor of non-linear refractive index and a covariant fourth rank tensor of susceptibility. In topological space, both a contravariant fourth rank tensor of non-linear refractive index and a covariant fourth rank tensor of susceptibility, are related to the Euler-Poincare characteristic, a topological invariant.

Consider the Navier-Stokes equation for a one-dimensional and two-dimensional compressible viscous liquid. It is a well-known fact that there is a strong solution locally in time when the initial data is smooth and the initial density is limited down by a positive constant. In this article, under the same hypothesis, I show that the density remains uniformly limited in time from the bottom by a positive constant, and therefore a strong solution exists globally in time. In addition, most existing results are obtained with a positive viscosity factor, but current results are true even if the viscosity factor disappears with density. Finally, I prove that this solution is unique in a class of weak solutions that satisfy the usual entropy inequalities. The point of this work is the new entropy-like inequalities that Bresch and Desjardins introduced into the shallow water system of equations. This discrepancy gives the density additional regularity (assuming such regularity exists first).

Viewing the Kepler's laws as Diophantine non-local equations introduces the action quantum and the Diophantine Coherence Theorem which generalizes the method of Arthur Haas, which anticipated the Bohr's radius. This leads to a Space quantum breaking the Planck wall by a factor 10^{61} and the associated Holographic Cosmos, identified as the source of the Background Radiation. An Electricity-Gravitation symmetry, connected with the Combinatorial Hierarchy, defines the steady-state Universe with invariant Hubble radius 13.81 Glyr, corresponding to 70.79 (km/s)/Mpc, a value anticipated since 1997 by the Three Minutes Formula, confirmed by the Eddington Number, the Kotov period and the recent Carnegie-Chicago Hubble Program. This specifies G, compatible with the BIPM measurements, and confirms definitely the Anthropic Principle.

In a two-dimensional space, a refractive index-curvature relation is formulated using the second rank tensor of Ricci curvature. A scalar refractive index describes an isotropic linear optics. In a fibre bundle geometry, a scalar refractive index is related to an Abelian (a linear) curvature form. The Gauss-Bonnet-Chern theorem is formulated using a scalar refractive index. Because the Euler-Poincare characteristic is the topological invariant then a scalar refractive index is also a topological invariant.

A polynomial power series is constructed for the one-sided step function using a modified Taylor series, whose derivative results in a new representation for Dirac $\delta$-function.

The discretization process of the full potential equation (FPE) both in the quasi-linear and in the conservation form, is addressed. This work introduce the rst stage toward a development of a fast and ecient FPE solver, which is based on the algebraic multigrid (AMG) method. The mathematical diculties of the problem are associated with the fact that the governing equation changes its type from elliptic (subsonic ow) to hyperbolic (supersonic ow). A pointwise relaxation method when applied directly to the upwind discrete operator, in the supersonic ow regime, is unstable. Resolving this diculty is the main achievement of this work. A stable pointwise direction independent relaxation was developed for the supersonic and subsonic ow regimes. This stable relaxation is obtained by post-multiplying the original operator by a certain simple rst order downwind operator. This new operator is designed in such a way that makes the pointwise relaxation applied to the product operator to become stable. The discretization of the FPE in the conservation form is based on the body-tted structured grid approach. In addition the 2D stable operator in the supersonic ow regime was extended to 3D case. We present a 3D pointwise relaxation procedure that is stable both in the subsonic and supersonic ow regimes. This was veried by the Von-Neumann stability analysis.

This article reports the development of an ecient, and robust full potential equation (FPE) solver for transonic ow problems, which is based on the algebraic multigrid (AMG) method. AMG method solves algebraic systems based on multigrid principles but in a way that it is independent on the problem's geometry. The mathematical diculties of the problem are associated with the fact that the governing equation changes its type from elliptic (subsonic ow) to hyperbolic (supersonic ow). The ow solver is based of the body-tted structured grid approach in complex geometries. We demonstrate the AMG performance on various model problems with dierent ow speed from subsonic to transonic conditions. The computational method was demonstrated to be capable of predicting the shock formation and achieving residual reduction of roughly an order of magnitude per cycle, both for elliptic and hyperbolic problems.

Einstein's theory of general relativity, which Newton's theory of gravity is a part of, is fraught with the problem of singularity that has been established as a theorem by Hawking and Penrose, the later being awarded the Nobel Prize in recent years. The crucial {\it hypothesis} that founds the basis of both Einstein's and Newton's theories of gravity is that bodies with unequal magnitudes of masses fall with the same acceleration under the gravity of a source object. Since, the validity of the Einstein's equations is one of the assumptions based on which Hawking and Penrose have proved the theorem, therefore, the above hypothesis is implicitly one of the founding pillars of the singularity theorem. In this work, I demonstrate how one can possibly write a non-singular theory of gravity which manifests that the above mentioned hypothesis is only valid in an approximate sense in the ``large distance'' scenario. To mention a specific instance, under the gravity of the earth, a $5$ kg and a $500$ kg fall with accelerations which differ by approximately $113.148\times 10^{-32}$ meter/sec$^2$ and the more massive object falls with less acceleration. Further, I demonstrate why the concept of gravitational field is not definable in the ``small distance'' regime which automatically justifies why the Einstein's and Newton's theories fail to provide any ``small distance'' analysis. In course of writing down this theory, I demonstrate why the continuum hypothesis as spelled out by Goedel, is undecidable. The theory has several aspects which provide the following realizations: (i) Descartes' self-skepticism concerning exact representation of numbers by drawing lines (ii) Born's wish of taking into account ``natural uncertainty in all observations'' while describing ``a physical situation'' by means of ``real numbers'' (iii) Klein's vision of having ``a fusion of arithmetic and geometry'' where ``a point is replaced by a small spot'' (iv) Goedel's assertion about ``non-standard analysis, in some version'' being ``the analysis of the future''. A major drawback of this work is that it can easily appear to the authorities of modern science as too simple to believe in. This is, firstly due to the origin of the motivations being rooted to the truthfulness of the language in which physics is written and secondly due to the lucidity of the calculations involved. However, at the same time, this work can also appear as a fresh and non-standard approach to do physics from its roots, where the problem of singularity is not even there to begin with. The credibility of this work depends largely on whether the reader is willing adopt the second mindset.

In this article we propose a computational algorithm written in Matlab as well as a mathematical formula to calculate the magnitude of the ripple voltage encountered in halfwave rectifying circuits. After rectification of a symmetric AC harmonic voltage signal, this ripple voltage remains, frequently measurable, superimposed on the resulting DC voltage signal. Both, the algorithm as well as the derived formula enable us to calculate the magnitude of the ripple voltage to a degree of precision any better than $1$ to $10^{6}$, i.e. 1 ppm. We conclude this article by comparing the accuracy of the proposed algorithm to simulated findings. The technique discussed here for calculating the magnitude of the remaining ripple voltage can easily be generalized and extended to the calculation of the magnitude of the ripple voltage encountered in full wave rectifying circuits.

Il manuale intende orire un percorso sul formalismo lagrangiano dei sistemi olonomi cercando il giusto equilibrio tra l'approccio sicoapplicativo ed il contesto astratto dell'ambiente geometrico. Agli ordinari argomenti collegati ai sistemi olonomi si aggiunge uno studio sui potenziali generalizzati, l'applicazione ai sistemi non inerziali ed un breve percorso sui sistemi anolonomi. <p> The handbook aims to oer a path on the Lagrangian formalism of holonomic systems by seeking the right balance between the physical-application approach and the abstract context of the geometric environment. In addition to the ordinary arguments related to holonomic systems, there is a study of generalized potentials, application to non-inertial systems and a short path on nonholonomic systems.

Partendo dai classici argomenti delle equazioni canoniche di Hamilton (ottenute tramite la trasformazione di Legendre) e dei campi vettoriali hamiltoniani, il manuale intende proseguire il percorso sul formalismo hamiltoniano presentando l'approccio variazionale collegato all'integrale di Hilbert e i campi di Weierstrass. In questo modo si ottiene l'invariante integrale di PoincaréCartan che caratterizza i sistemi hamiltoniani e si ha accesso alla teoria delle trasformazioni canoniche e delle funzioni generatrici. Si conclude presentando l'equazione di HamiltonJacobi e accennando alla denizione di sistema integrabile. <p> Starting from the classic arguments of Hamilton's canonical equations (obtained through Legendre's transformation) and Hamiltonian vector elds, the manual intends to continue the path on Hamiltonian formalism presenting the variational approach linked to the Hilbert's integral and the Weierstrass elds. In this way the invariant integral of Poincar'eCartan that characterizes Hamiltonian systems is obtained and one can access to the theory of canonical transformations and generating functions. We conclude by presenting the Hamilton-Jacobi equation and mentioning the denition of an integrable system.

We look at Lorentz transformations from the perspective of functional analysis and show that the theory of functional analysis so far has neglected a critical point by not taking into consideration inputs of functions when measuring distances in function spaces.

We consider some problems concerning the classical string and the quantum strings which can have the deep physical meaning. We show that there is the asymmetry of the action and reaction in the string motion for the string with the left end fixed and the right end being in periodic motion. We derive also the quantum internal motion of this system. The quantization of the string with the interstitial massive defect is performed. The classical motion of uniformly accelerated string and its relation the Bell paradox is considered. We discuss the relation of this accelerated string to the Bell spaceship paradox involving the Lorentz contraction. It is evident that the acceleration of the string can be caused by gravity and we show that such acceleration causes the different internal motion of the string. Gravity can be described by the string medium in the Newton model of gravity. We show, that in case of the string model of gravity the motion of planets and Moon are oscillating along the classical trajectories In the string model of hadrons the quarks are treated to be tied together by a gluon tube which can be approximated by the tube of vanishing width, or by string. We apply the delta-function form of force to the left side of the string and calculate the propagation of the pulse in the system.

The refractive index and curvature relation is formulated using the Riemann-Christoffel curvature tensor. As a consequence of the fourth rank tensor of the Riemann-Christoffel curvature tensor, we found that the refractive index should be a second rank tensor. The second rank tensor of the refractive index describes a linear optics. It implies naturally that the Riemann-Christoffel curvature tensor is related to the linear optics. In case of a non-linear optics, the refractive index is a sixth rank tensor, if susceptibility is a fourth rank tensor. The Riemann-Christoffel curvature tensor can be formulated in the non-linear optics but with a reduction term. The relation between the (linear and non-linear) refractive index and a (linear and non-linear) mass in curved space are formulated. Related to the Riemann-Christoffel curvature tensor, we formulate "the (linear and non-linear) generalized Einstein field equations". Sine-Gordon model in curved space is shown, where the Lagrangian is the total energy. This total energy is the mass of a kink (anti-kink) associated with a topological charge (a winding number). We formulate the relation between the (linear and non-linear) refractive index of the kink (anti-kink) and the topological charge-the winding number. Deflection of light is discussed in brief where the (linear and non-linear) angle of light deflection are formulated in relation with the mass (the topological charge, the winding number) of the kink (anti-kink).

I suggest a new approximate approach, the Multipoints Summation method, to solve non-linear differential equations analytically. The method connects several local asymptotic series. I present applications of the method to two examples of non-linear differential equations: saddle-node bifurcation and the non-linear differential equation of the pendulum. Explicit approximate solutions expressed in terms of elementary functions are obtained from an analysis of phase space. This approach may be also applied to other non-linear differential equations.

The noncommutative algebra of space-time sedeons is used for the generalization of the system of nonlinear self-consistent equations in hydrodynamic two-fluid model of vortex plasma. This system describes both longitudinal flows as well as the rotation and twisting of vortex tubes taking into account internal electric and magnetic fields generated by fluctuations of plasma parameters. As an illustration we apply the proposed equations for the description of sound waves in electron-ion and electron-positron plasmas.

The Fisher-KPP equation is a reaction-diffusion equation originally proposed by Fisher to represent allele propagation in genetic hosts or population. It was also proposed by Kolmogorov for more general applications. A novel method for solving nonlinear partial differential equations is applied to produce a unique, approximate solution for the Fisher-KPP equation. Analysis proves the solution is counterintuitive. Although still satisfying the maximum principle, time dependence collapses for all time greater than zero, therefore, the solution is highly irregular and not smooth, invalidating the traveling wave approximation so often employed.

The relation of an angle of deflection-mass is given where we replace mass with the mass of kink (anti-kink). The mass of kink (anti-kink), in turn, can be replaced with topological charge and winding number. Because mass is related with the refractive index, the angle of deflection can be formulated in relation with the decomposed form of refractive index.

The proposals that mass-energy is topological charge and topological charge is winding number are given. The refractive index-mass relation using the decomposed form of refractive index is showed where we replace mass with topological charge and winding number.

We assume that the curvature in the Atiyah-Singer index theorem is related with the Riemann-Christoffel curvature tensor where the Riemann-Christoffel curvature tensor is decomposed into the unrestricted electric (scalar) potential part and the restricted magnetic (vector) potential part. This decomposition is a consequence of the magnetic symmetry existence for the gauge potential in the geometrical optics.

Planck's constant, Planck length and the golden ratio can all be written as a simple equation. It is not yet clear if this is simply a mathematical coincidence or rather something with a deeper fundamental meaning. A few ideas will be suggested that might be physical in nature.

The problem of a flow with its velocity gradient being of \textit{real Schur form} uniformly in a cyclic box is formulated for numerical simulation, and a semi-analytic algorithm is developed from the precise structures. Computations starting from two-component-two-dimensional-coupled-with-one-component-three-dimensional initial velocity fields of the Taylor-Green and Arnold-Beltrami-Childress fashions are carried out, and some discussions related to turbulence are offered for the multi-scale eddies which, though, present precise order and symmetry. Plenty of color pictures of patterns of these completely new flows are presented for general and specific conceptions.

We reformulate Gauss-Bonnet-Chern theorem in relation with magnetic symmetry of geometrical optics. If Euler-Poincare characteristic is a topological invariant, should unrestricted electric potential of $U(1)$ gauge potential be a topological invariant?

The refractive index and curved space relation is formulated using the Riemann-Christoffel curvature tensor. As a consequence of the fourth rank tensor of the Riemann-Christoffel curvature tensor, the refractive index should be a second rank tensor. The second rank tensor of the refractive index describes a linear optics. In case of a non-linear optics, if susceptibility is a fourth rank tensor, then the refractive index is a sixth rank tensor. In a topological space, the linear and non-linear refractive indices are related to the Euler-Poincare characteristic. Because the Euler-Poincare characteristic is a topological invariant then the linear and non-linear refractive indices are also topological invariants.

The two-dimensional autonomous cellular neural networks (CNNs) having one layer or two layers of memristor coupling can exhibit many interesting nonlinear waves and bifurcation phenomena. In this paper, we study the nonlinear waves (solitons) in the one-dimensional CNN difference equations. From our computer simulations, we found that the CNN difference equations can exhibit many interesting behaviors. The most remarkable thing is that the first-order linear CNN difference equation can exhibit a train of solitary waves, if the initial condition is given by the unit step function. Furthermore, the second-order linear CNN difference equation can exhibit soliton-like behavior, if the initial condition is given by a pulse wave. That is, the solitary waves pass through one another and emerge from the collision. Furthermore, the solution exhibits the area-preserving behavior, and it returns exactly to its initial state (the recurrence of the initial state). In the case of the nonlinear CNN difference equation, we observed the following interesting behaviors. In the Korteweg-de Vries CNN difference equation, the three-dimensional plot of the interaction of the solitary waves looks like a chicken cockscomb. In the Toda lattice CNN difference equation, a train of solitary waves with a negative amplitude interact with a train of solitary waves with a positive amplitude, and they emerge from the collisions. Furthermore, after a certain period of time, the solution breaks down. In the Sine-Gordon CNN difference equation, the solution moves at constant speed, and it emerges from the collision. Furthermore, the solution returns the state which is roughly similar to the initial state. In the memristor CNN difference equations, the three-dimensional plots of solitary waves exhibit more complicated (chaotic or distorted) behavior.

We will consider the string, the left end of which is fixed at the beginning of the coordinate system, the right end is fixed at point l and mass m is interstitial between the ends of the string. We determine the vibration of such system. The proposed model can be also related to the problem of the Moessbauer effect, or, recoilless nuclear resonance fluorescence, being resonant and recoilfree emission and absorption of gamma radiation by atomic nuclei bound in a solid (Moessbauer,1958).

[Note: This is Henri Poincare's paper edited by Abdelmajid Ben Hadj Salem] <p></p> This article is a numerical version of the first chapter of the long paper of Henri Poincar\'e " The Three-Body Problem and the Equations of Dynamics " published by the celebrate journal \textit{Acta Mathematica} (Vol.13, n$^{\circ}1-2$, 1889), created by the Swedish mathematician Gösta Mittag-Leffler in 1882, and he was the Editor-in-Chief. The new version kept the original text with some minimal changes and adding the bibliography which summarizes all the references cited in the article.

We show that there exists a magnetic monopole in the $U(1)$ geometrical optics as a consequence of the magnetic symmetry in a $(4+d)$-dimensional unified space where the magnetic symmetry is a consequence of the extra internal symmetry. This magnetic symmetry restricts the gauge potential. The restricted (decomposed) gauge potential is made of the scalar potential as the unrestricted electric part and the vector potential as the restricted magnetic part. We also show that the refractive indices can be formulated in relation to the decomposed gauge potential. We treat the curvature in the curvature-refractive index relation of the $U(1)$ geometrical optics as an Abelian curvature form in the fibre bundle.

A two-component-two-dimensional coupled with one-component-three-dimensional (2C2Dcw1C3D) flow may also be called a real Schur flow (RSF), as its velocity gradient is uniformly of real Schur form. The thermodynamic and ‘vortic’ fine structures of 2C2Dcw1C3D flows are exposed and, in particular, the Lie invariances of the decomposed vorticity 2-forms of RSFs in d-dimensional Euclidean space E d for any interger d ≥ 3 are also proved. The two Helmholtz theorems of the complementary components of vorticity found recently in 3-space RSF is not coincidental, but underlied by a gen- eral decomposition theorem, thus essential. Many Lie-invariant fine results, such as those of the combinations of the entropic and vortic quantities, including the invariances of the decomposed Ertel potential vorticit 3-formsy (and their multiplications by any interger powers of entropy), then follow.

The Euler tetrahedron volume formula is used to define dimensionality of space and the Euclidean straight line and area. The double disk modul (DDM) is used to define trajectories on the metrical surfaces. The relations of generalized Lobachevskii geometry are derived and related to the Einstein equations. It is considered spherical and pseudospherical geometry, Riemann geometry, Lobachevskii and generalized Lobachevskii geometry, Poincare model, Beltrami model, gravity as deformation of space and Schwinger theory of gravity.

We propose the system of self-consistent equations for vortex plasma in the framework of hydrodynamic two-fluid model. These equations describe both longitudinal flows and the rotation and twisting of vortex tubes taking into account internal electric and magnetic fields generated by fluctuations of plasma parameters. The main peculiarities of the proposed equations are illustrated with the analysis of electron and ion sound waves.

This paper shows the exact integrability analysis of two classes of non-autonomous and nonlinear differential equations. It has been possible to recover some equations of general relativity from the first class of equations and consequently to compute their solution in fashion way. The second class is shown to include the Emden-Fowler equation and its integrability analysis, performed with the first integral theory developed by Monsia et al. [16] allowed to compute the exact solution of some subclasses of Emden-Fowler equations.

The paper focuses on the part of the Coulomb’s Law that is just a definition and provides one possible mechanism for operationally defining electric charge based on the concept of force(action at a distance). Then a derivation of Coulomb’s law from the definition is presented and the sign of the charges are defined. Finally, the paper concludes with a discussion on the conservation of charge.

In order to show some power of the division by zero calculus we will give several simple applications to physics. Recall that Oliver Heaviside: {\it Mathematics is an experimental science, and definitions do not come first, but later on.}

This paper investigates the onset of turbulence in incompressible viscous fluid flow over a flat plate by looking at the pressure gradients implied by the Blasius solution for laminar fluid flow and adjusting the predicted flow, leading to a mathematically predictable flow separation in the boundary layer and the onset of turbulence (including both transition and fully turbulent regions - both with and without the presence of a flat plate). It then considers the implications for potential analytic solutions to the Navier-Stokes Equations of the fact that it is possible to predict turbulence and a singularity for many flows (at any velocity).

We begin with the comprehensible review of the basics of the Lorentz, (extended) Poincare Groups and O(3,2) and O(4,1). On the basis of the Gelfand-Tsetlin-Sokolik-Silagadze research~[1-3], we investigate the definitions of the discrete symmetry operators both on the classical level, and in the secondary-quantization scheme. We studied the physical content within several bases: light-front form formulation, helicity basis, angular momentum basis, on several practical examples. The conclusion is that we have ambiguities in the definitions of the the corresponding operators P, C; T, which lead to different physical consequences.

We present a generalization of the equations of hydrodynamics based on the noncommutative algebra of space-time sedeons. It is shown that for vortex-less flow the system of Euler and continuity equations is represented as a single non-linear sedeonic second-order wave equation for scalar and vector potentials, which is naturally generalized on viscous and vortex flows. As a result we obtained the closed system of four equations describing the diffusion damping of translational and vortex motions. The main peculiarities of the obtained equations are illustrated on the basis of the plane wave solutions describing the propagation of sound waves.

We discuss a generalization of the equations of hydrodynamics based on space-time algebra of sedeons. It is shown that the fluid dynamics can be described by sedeonic second-order wave equation for scalar and vector potentials. The generalized sedeonic Navier-Stokes equations for a viscous fluid and vortex flows are also discussed. The main peculiarities of the proposed approach are illustrated on the equations describing the propagation of sound waves.

We discuss the generalization of phenomenological equations for electromagnetic field in superconductor based on algebra of space-time sedeons. It is shown that the combined system of London and Maxwell equations can be reformulated as a single sedeonic wave equation for the field with nonzero mass of quantum, in which additional conditions are imposed on the scalar and vector potentials, relating them to the deviation of charge density and currents in the superconducting phase. Also we considered inhomogeneous equations including external sources in the form of charges and currents of the normal phase. In particular, a screening of the Coulomb interaction of external charges in a superconducting media is discussed.

An initial boundary value problem to a system of linear Schrodinger equations with nonlinear boundary conditions is considered. It is shown that attractor of the problem lies on circles in complex plane. Trajectories tend to xed points of hyperbolic type with unstable manifold which is formed by saddle points of codimension one. Each element of the attractor are periodic piecewise constant function on pase and amplitude of a wave function in WKB -approximation with nite or innite points of discontinuities on a period of the Julia type. More exactly, it has been obtained limit solutions of the problem which with accuracy O(h2) match the exact attractor of the boundary problem, which is independent on h > 0 in the zero WKB - approximation. The presented mathematical result are applied to the study of dynamics of two charged particles with opposite impulses, which are conned by two at walls with surface potentials of double-well type. It is shown that asymptotic behaviour of particles is similar to the behaviour of orbits that arise to well-known logistic map in complex plane. As example, there exist limit periodic nearly piecewise constant distributions of wave functions of Mandelbrot type with Julia type points of 'jumps' for amplitudes and phases of given free charged particles in a conned box with surface nonlinear double-well potential at walls in magnetic eld.

We consider the Schrodinger equation with nonlinear boundary conditions and ini- tial conditions. It is shown that attractor of the problem contains periodic piecewise constant function with nite, countable or uncountable points of discontinuities on a period. Solutions exists for a special class of initial data which are small perturbations of invariant solutions of dynamical system. The problem is considered with accuracy O(h2), where h is a small parameter of the problem. Applications to optical resonators with nonlinear feedback has been considered.

In this paper, we study the nonlinear waves in autonomous cellular neural networks (CNNs) having double layers of memristor coupling, by using the homotopy method. They can exhibit many interesting nonlinear waves, which are quite different from those in the single-layer autonomous CNNs. That is, the autonomous CNNs with double layers of memristor coupling can exhibit more complex nonlinear waves and more interesting bifurcation phenomena than those in the single layer autonomous CNNs. The above complex behaviors seem to be generated by the interaction with the two nonlinear waves, which are caused by the first layer and the second layer. The most remarkable point in this paper is that the autonomous CNNs with double layers can exhibit complex deformation behaviors of the nonlinear waves, due to the changes in the homotopy parameter. That is, we can generate many complex nonlinear waves by adjusting the homotopy parameter, and thereby we can control the complexity of the nonlinear waves. Furthermore, some autonomous CNNs exhibit the sensitive dependence on the homotopy parameter. That is, a small change in the homotopy parameter can result in large differences in a later state. Thus the homotopy method gives a new approach to the analysis of the complex nonlinear waves in the autonomous CNNs with double layers.

The problem of generation of plane and evanescent waves by elec- tric charge and current densities on a plane is considered. It is shown that, rst, both ordinary and evanescent waves can be emitted by such a source, second, that source of evanescent wave is perfectly static in an appropriate frame. The source is found as as explicit form of surface charge and current densities on a plane, which satisfy the continuity condition. One of components of retarded potential of the source is calculated. It is shown that the expression derived provides an erroneous representation of the eld.

No truth is truly true, the more we reveal the more we revere nature on our voyage of unprecedented discovery. We argue that the soul or anti-soul of Complex Multiscale Orbifold Spacetime (CMOSpacetime) in higher dimensional Non-Euclidean geometry, is the origin of intelligence, and the metric of metrizable intelligence is the sectional curvature's absolute value of CMOSpacetime's soul or anti-soul. We also argue that the intersecting souls and/or anti-souls , when their sectional curvatures approaching positive infinity and/or negative infinity as singularity, is the origin of quantum entanglement. We further argue that the sectional curvatures of CMOSpacetime's intersecting souls and/or anti-souls , is the origin of convergent evolution through conformal transformation. We derive CMOSpacetime, a N-dimensional orbifold $\mathbb{O}=\mathbb{M}/\mathbb{F_g}$ ($\mathbb {M}$ as manifold)/degree N projective algebraic variety $\mathbb X$ over $\mathbb{C}^{N}$ defined by degree N non-linear polynomial function $\mathbb{F_g}(X_1, ..., X_N) = \sum_{i,j=1}^N(w_iX_i^j+b_i)$ in hypercomplex number system with $X = x_1 + \sum_{m=2}^{N}(x_mi_m)$ on Non-Abelian quotient group \begin{math} SO(\frac{N}{2}, \frac{N}{2}) \end{math} (\begin{math} 8 \leq N \to \infty, N = 2^n \end{math}), neural networks by correlating general relativity and quantum mechanics based on mutual extensions from 3+1 dimensional spacetime $\mathbb{R}^{4}$ to N-dimensional CMOSpacetime $\mathbb{C}^{N}$. CMOSpacetime addresses both singularity and non-linearity as common issues faced by physics, AI and biology, and enables curvature-based second order optimization in orbifold-equivalent neural networks beyond gradient-based first order optimization in manifold-approximated a adopted in AI. We build CMOSpacetime theoretical framework based on General equivalence principle, a combination of Poincar\'{e} conjecture, Fermat's last theorem, Galois theory, Hodge conjecture, BSD conjecture, Riemann hypothesis, universal approximation theorem, and soul theorem. We also propose experiments on measuring intelligence of convolutional neural networks and transformers, as well as new ways of conducting Young's double-slit interference experiment. We believe that CMOSpacetime acting as a universal PDE, not only qualitatively and quantitatively tackles the black box puzzle in AI, quantum entanglement and convergent evolution, but also paves the way for CMOSpacetime synthesis to achieve true singularity.

In this paper we present a new symetric probability distribution with its properties and we show that is not a uniforme distribution using some standard proofs test like Kolmogorov-Smirnov test and also we may show that is derived from a new another special function by adjusting it using mean and deviation as two parameters , And in the second section we show that PDF present a wave function using rescaled plasma dispersion function such that we define it as a position of massive particle for such charged quantum system .

The Permanent Oscillatory Cosmology is confirmed by 75 formula giving the Hubble radius, with 7correlating to 10^−9. The computer shows the best formula, obtained using the Atiyah constant and the number 137, the Eddington’s integer part of the electric constant. This conforms with Atiyah’s testimony about the Physics-Mathematics unification and the central role of arithmetics in this unification process. The identification with the Eddington statistical formula gives G, compatible with the 10^−5 precise BIPM measurement and the 10^−6 precise sun-quasar non-Doppler Kotov period. The hypothesis of a computing Cosmos implies a π rationalization process which validates the Wyler’s theory and the Fermion Koide formula in the 10^−9 domain

We extend the Poincar´e group to the complex Minkowski spacetime. Special attention is paid to the corresponding algebra that we achieve through matrices as well as differential operators. We also point out the generalizations of the two Casimir operators.

The standard model is unified with gravity in an F4 gauge theory where the spacetime is a quasicrystalline compactication of an E9 Lorentzian lattice. The Higgs is played by a neutrino condensate. Using three languages where the first is a Hilbert loop, the second is a list of ver- tices in the E8 Lattice and the third is a two-dimensional quasicrystal exhibiting five-fold symmetry, the longtime sought generation formula for general five-fold symmetric quasicrystals are revealed.

The Betti-de Rham period isomorphism ("Abelian Geometry") is related to algebraic fundamental group (Anabelian Geometry), in analogy with the classical context of Hurewicz Theorem. To investigate this idea, the article considers an "Abstract Galois Theory", as a separated abstract structure from, yet compatible with, the Theory of Schemes, which has its historical origin in Commutative Algebra and motivation in the early stages of Algebraic Topology. The approach to Motives via Deformation Theory was suggested by Kontsevich as early as 1999, and suggests Formal Manifolds, with local models formal pointed manifolds, as the source of motives, and perhaps a substitute for a "universal Weil cohomology". The proposed research aims to gain additional understanding of periods via a concrete project, the discrete algebraic de Rham cohomology, a follow-up of author's previous work. The connection with Arithmetic Gauge Theory should provide additional intuition, by looking at covering maps as flat connection spaces, and considering branching covers of the Riemann sphere as the more general case. The research on Feynman/Veneziano Amplitudes and Gauss/Jacobi sums, allows to deepen the parallel between the continuum and discrete frameworks: an analog of Virasoro algebra in finite characteristic. A larger project is briefly considered, consisting in deriving Motives from the Theory of Deformations, as suggested by Kontsevich. Following Soibelman and Kontsevich, the idea of defining Formal Manifolds as groupoids of pointed formal manifolds (after Maurer-Cartan ``exponentiation''), with associated torsors as ``gluing data'' (transition functions) is suggested. This framework seems to be compatible with the ideas from Theory of Periods, sheaf theory / etale maps and Grothedieck's development of Galois Theory (Anabelian Geometry). The article is a preliminary evaluation of a research plan of the author. Further concrete problems are included, since they are related to the general ideas mentioned above, and especially relevant to understanding the applications to scattering amplitudes in quantum physics.

Feynman amplitudes are periods, and also coefficients of the QED partition function with a formal deformation parameter the fine structure constant $\alpha$. Moreover, this truly fundamental mathematical constant is the ratio of magnetic (fluxon) vs. electric charge, as well as the grading of the decay lifetimes telling apart weak from strong ``interactions''. On the other (Mathematical) hand $e$ is the ``inverse'' of $\pi$, another deformation parameter (no ordinary period), as Euler's famous identity $exp(2\pi i)=1$ suggests. In a recent work, Atiyah related $\alpha$ and the Todd function. But Todd classes are inverses of Chern classes, suggesting further ``clues'' to look for conceptual relationships between these mathematical constants, in an attempt to catch a a Platonic and Exceptional Universe by the TOE.

In this paper, we propose two-dimensional autonomous cellular neural networks (CNNs), which are formed by connecting single synaptic-input CNN cells to each node of an ideal memristor grid. Our computer simulations show that the proposed two-dimensional autonomous CNNs can exhibit interesting and complex nonlinear waves. In many two-dimensional autonomous CNNs, we have to use a locally active memristor grid, in order for the autonomous CNNs to exhibit the continuous evolution of nonlinear waves. Some other notable features of the two-dimensional autonomous CNNs are: The autonomous Van der Poll type CNN can exhibit various kinds of nonlinear waves by changing the characteristic curve of the nonlinear resistor in the CNN cell. Furthermore, if we choose a different step size in the numerical integration, it exhibits a different nonlinear wave. This property is similar to the sensitive dependence on initial conditions of chaos. The autonomous Lotka-Volterra CNN can also exhibit various kinds of nonlinear waves by changing the initial conditions. That is, it can exhibit different response for each initial condition. Furthermore, we have to choose a passive memristor grid to avoid an overflow in the numerical integration process. Our computer simulations show that the dynamics of the proposed autonomous CNNs are more complex than we expected.

This article traces a journey of discovery undertaken to search for wave phenomena in the incompressible Navier-Stokes equations. From the early days of my interest in Computational Fluid Dynamics (CFD) used for consulting purposes, and the use of various commercial solvers eventually leading to research programs, at a number of universities, spanning a number of years. It reviews research programs at Dortmund University (Dortmund, Germany), and this author’s post-graduate study at Chulalongkorn University (Bangkok, Thailand). During the latter, it was noticed that flow solutions became unstable when certain combinations of parameters were used - especially when real-life density and viscosity were used. Clarity was needed. This author had a perception that this research could lead to an understanding of the turbulence phenomenon, Tensor calculus was used to understand the macro nature of the NS Equations, and to place them firmly into the family of wave equations. My research continued in private for some 15 years, with the occasional presentation of findings at conferences, and an internet blog. In recent months major breakthroughs have been made, and now the evidence for wave phenomena is convincingly demonstrated.

There are many ad hoc expressions for the mass ratio of the proton to the electron. the models presented here are different from others in that they rely strictly on volumes and areas. One geometry is based on ellipsoids constructed with values taken from one of the two number sets: {(4pi), (4pi-1/pi), (4pi-2/pi)} or {(4pi+2), (4pi-2), (4pi-2/pi)}. The product of the three values of each number set approximates the value given by CODATA for the mass ratio of the proton to the electron. Another approximate is formed from a solid ball of radius, r = (4pi-1/pi), with a conical sector, wedge, or internal ellipsoid removed. Each extracted solid has curved surface area of (4pi-1/pi)/(pi^2). With the advent of the Higg’s Boson, its value can be approximated by H^0 = (4pi)(4pi-1/pi)(4pi-2/pi)(4pi-3/pi)(4pi-4/pi). Define the function F as follows: Let the initial set be the positive integers, the final set be the real numbers, and the rule assigning each member of the initial set to one member of the final set: F(m) =(4pi)...(4pi-(m-1)/pi). Conclusion: The function F(2)=1836.15... approximates the experimental value of the mass ratio of the proton to the electron and F(4) approximates the mass ratio of the Higg's Boson to the electron. The neutron-to-electron ratio is approximated with ln(4pi)+F(2). Email: harry.watson@att.net

In this thesis, I went through derivation of equations of motion for some free particles using symmetry, and also went through the math underneath spontaneous symmetry breaking and the Higgs mechanism in restoring missing mass in interaction between particles, which both showing the self-consistency of the Standard Model.

The present note adresses a paper by DiNunno & Matzner, in which the authors claim that 1) the volume of a J=Q=0 black hole as measured in "Schwarzschild coordinates" vanishes and 2) the volume itself is coordinate-dependent. We refute these statements as elementary conceptual mistakes, which originate from a basic misunderstanding of general covariance in the context of the gauge theory of General Relativity.

In this note we present a new approach to proof the Riemann hypothesis one of the most important open problem in pure mathematics using a new operator derived from unitary operator groups acts on Riemann-Siegal function and it uses partition function for Hamiltonian operator , The interest idea is to compute the compositional inverse of Riemann zeta function at $s=-\frac12$ such as we show that:$\zeta^{-1}(-\frac12) =\zeta(\frac12+i \beta)=0$ for some $\beta >0 $

Slow precession of the Earth rotation axis, and the Moon-Earth orbital resonance were accompanied during centuries by Newton's and Laplace's explanations. However, in the present paper, the author considers the possibility of additional factor: the small violation of the global energy-momentum conservation. Thus, the energy-momentum concept being not conserved, cannot be regarded as total (global) energy-momentum of system. The recently experimentally verified Lense-Thirring Effect and the Mercury's perihelion anomalous shift cannot be found in Newton Physics, and the latter demands the global energy-momentum conservation. Thus, the shift violates global energy-momentum conservation. Why? Because the energy-momentum is defined locally, not globally.

The present research contribution is devoted to solving the integrability problem of Liénard type differential equations. It is shown that such a problem may be solved by nonlocal transformation for some classes of equations. By doing so, it is observed that the integrability of a class of restricted Duffing type equations with integral power or fractional power nonlinearity may be secured by that of a general class of quadratic Liénard type differential equation, and vice versa. Such a restricted Duffing type equation is also shown to be closely related to a quadratic Li´enard type equation for which exact and explicit general solution may be computed. In this context it has been shown that exact and general periodic solutions may be computed for these two classes of restricted Duffing equations and quadratic Liénard type equations. The comparison of obtained solutions with some well-known results is carried out in some cases.

We suggest united models and specific predictions regarding elementary particles, dark matter, aspects of galaxy evolution, dark energy, and aspects of the cosmology timeline. Results include specific predictions for new elementary particles and specific descriptions of dark matter and dark energy. Some of our modeling matches known elementary particles and extrapolates to predict other elementary particles, including bases for dark matter. Some modeling explains observed ratios of effects of dark matter to effects of ordinary matter. Some models suggest aspects of galaxy formation and evolution. Some modeling correlates with eras of increases or decreases in the observed rate of expansion of the universe. Our modeling framework features mathematics for isotropic quantum harmonic oscillators and provides a framework for creating physics theories. Some aspects of our approach emphasize existence of elementary particles and de-emphasize motion. Some of our models complement traditional quantum field theory and, for example, traditional calculations of anomalous magnetic dipole moments.

The integrability of a general class of Liénard type equations is investigated through equation transformation theory. In this way it is shown that such a class of Liénard equations can generate a generalization of some interesting truly nonlinear oscillator equations like the cube and fifth root differential equations. It has then become possible to compute the exact and general solution to the generalized truly nonlinear oscillator equation. Under an appropriate choice of initial conditions, exact and explicit solution has been obtained in terms of Jacobi elliptic functions.

In this note, we study Einstein equations (EE) of general relativity considering a manifold M with a diagonal metric g_{ij}. We calculate the expression of the components of Ricci and Riemann tensors and the value of the scalar curvature R. Then we give the expression of the (EE) : -a- for the case where g_{ii}=g_i=g_i(x_i); -b- for the case where g_1=g_1(x_1=t) and g_i=g_i(t,x_i) for i=2,3,4$; -c- for the case (b) with x_4=z_0=constant.

Some ambiguities have recently been found in the definition of the partial derivative (in the case of presence of both explicit and implicit dependencies of the function subjected to differentiation). We investigate the possible influence of this subject on quantum mechanics and the classical/quantum field theory. Surprisingly, some commutators of operators of space-time 4-coordinates and those of 4-momenta are not equal to zero. We postulate the non-commutativity of 4-momenta and we derive mass splitting in the Dirac equation. Moreover, two iterated limits may not commute each other, in general. Thus, we present an example when the massless limit of the function of E, p, m does not exist in some calculations within quantum field theory

A C-loop algebra is assembled as the product 0f a Clifford algebra and a Cayley-Dickson algebra. Once the principle of spatial equivalence is invoked, a sub-algebra is identified with features that suggest it could provide an underlying basis for the standard model of fundamental particles.

In this paper, we show that the dynamics of a wide variety of nonlinear systems such as engineering, physical, chemical, biological, and ecological systems, can be simulated or modeled by the dynamics of memristor circuits. It has the advantage that we can apply nonlinear circuit theory to analyze the dynamics of memristor circuits. Applying an external source to these memristor circuits, they exhibit complex behavior, such as chaos and non-periodic oscillation. If the memristor circuits have an integral invariant, they can exhibit quasi-periodic or non-periodic behavior by the sinusoidal forcing. Their behavior greatly depends on the initial conditions, the parameters, and the maximum step size of the numerical integration. Furthermore, an overflow is likely to occur due to the numerical instability in long-time simulations. In order to generate a non-periodic oscillation, we have to choose the initial conditions, the parameters, and the maximum step size, carefully. We also show that we can reconstruct chaotic attractors by using the terminal voltage and current of the memristor. Furthermore, in many memristor circuits, the active memristor switches between passive and active modes of operation, depending on its terminal voltage. We can measure its complexity order by defining the binary coding for the operation modes. By using this coding, we show that the memristor's operation modes exhibit the higher complexity, in the forced memristor Toda lattice equations and the forced memristor Van der Pol equations. Furthermore, the memristor has the special operation modes in the memristor Chua circuit.

Given that the two-parameter $ p, q$ quantum-calculus deformations of the integers $ [ n ]_{ p, q} = (p^n - q^n)/ ( p - q) = F_n $ coincide precisely with the Fibonacci numbers (integers), as a result of Binet's formula when $ p = \tau = { 1 + \sqrt 5 \over 2}$, $ q = { \tilde \tau} = { 1 - \sqrt 5 \over 2 }$ (Galois-conjugate pairs), we extend this result to the $generalized$ Binet's formula (corresponding to generalized Fibonacci sequences) studied by Whitford. Consequently, the Galois-conjugate pairs $ (p, q = \tilde p ) = { 1\over 2} ( 1 \pm \sqrt m ) $, in the very special case when $ m = 4 k + 1$ and square-free, generalize Binet's formula $ [ n ]_{ p, q} = G_n$ generating integer-values for the generalized Fibonacci numbers $G_n$'s. For these reasons, we expect that the two-parameter $ (p, q = \tilde p)$ quantum calculus should play an important role in the physics of quasicrystals with $4k+1$-fold rotational symmetry.

Newton's mechanics is simple. His equivalence principle is simple, as is the inverse square law of gravitational force. A simple theory should have simple solutions to simple models. A system of n particles, given their initial speed and positions along with their masses, is such a simple model. Yet, solving for n>2 is not simple. This paper discusses what could be done to overcome that problem.

The Poincare and conformal groups are contenders for the most fundamental spacetime symmetry group. An 8-dimensional rep, putting two 4-spinors together, makes a suitable platform to install matrix representations of these two fundamental groups. But some of their generators do not commute, so new generators are introduced to keep the algebra closed. The combined algebra then has 37 basis generators, a dozen more than needed for the Poincare and conformal algebras. Interestingly, with two Lorentz subalgebras, one finds two distinct definitions of spin. For the adjoint representation, one set of Lorentz generators reduces to irreducible representations, all with integer spin. The other Lorentz group reduces to both integer and `half-integer' spin irreducible representations. Also, one finds that the various representations confirm the spin rules for matrix translation generators with the spins of both Lorentz subgroups.

The transport of material through the atmosphere is an issue with wide ranging implications for fields as diverse as agriculture, aviation, and human health. Due to the unsteady nature of the atmosphere, predicting how material will be transported via Earth's wind field is challenging. Lagrangian diagnostics, such as Lagrangian coherent structures (LCSs), have been used to discover the most significant regions of material collection or dispersion. However, Lagrangian diagnostics can be time consuming to calculate and often rely on weather forecasts that may not be completely accurate. Recently, Eulerian diagnostics have been developed which can provide indications of LCS and have computational advantages over their Lagrangian counterparts. In this paper, a methodology is developed for estimating local Eulerian diagnostics from wind velocity data measured by a fixed wing unmanned aircraft system (UAS) flying in circular arcs. Using a simulation environment, it is shown that the Eulerian diagnostic estimates from UAS measurements approximate the true local Eulerian diagnostics, therefore also predicting the passage of LCSs. This methodology requires only a single flying UAS, making it more easy to implement in the field than existing alternatives.

In the present article we investigate the spin-1/2 and spin-1 cases in different bases. Next, we look for relations with the Majorana-like field operator. We show explicitly incompatibility of the Majorana anzatzen with the Dirac-like field operators in both the original Majorana theory and its generalizations. Several explicit examples are presented for higher spins too. It seems that the calculations in the helicity basis give mathematically and physically reasonable results only.

Is math in harmony with existence? Is it possible to calculate any property of existence over math? Is exact proof of something possible without pre-acceptance of some physical properties? This work is realized to analysis these arguments somehow as simple as possible over short cuts, and it came up with some compatible results finally. It seems that both free space and moving bodies in this space are dependent on the same rule as there is no alternative, and the rule is determined by mathematics.

We continue the discussion of several explicit examples of generalizations in relativistic quantum mechanics. We discussed the generalized spin-1/2 equations for neutrinos and the spin-1 equations for photon. The equations obtained by means of the Gersten-Sakurai method and those of Weinberg for spin-1 particles have been mentioned. Thus, we generalized the Maxwell and Weyl equations. Particularly, we found connections of the well-known solutions and the dark 4-spinors in the Ahluwalia-Grumiller elko model. They are also not the eigenstates of the chirality and helicity. The equations may lead to the dynamics which are different from those accepted at the present time. For instance, the photon may have non-transverse components and the neutrino may be {\it not} in the energy states and in the chirality states. The second-order equations have been considered too. They have been obtained by the Ryder method.

In a non-equilibrium thermodynamical physics, there has been al- most no universal theory for representing the far from equilibrium sys- tems. In this work, I formulated the thermodynamical path integral from macroscopic view, using the analogy of optimal transport and large deviations to calculate the non-equilibrium indicators quantita- tively. As a result, I derived Jarzynski equality, fluctuation theorem, and second law of thermodynamics as its corollaries of this formula. In addition, the latter result implies the connection between non- equilibrium thermodynamics and Riemannian geometry via entropic flow.

In this paper, by using Uhlenbeck-Yau's continuity method, we prove that the existence of approximation $\alpha$-Hermitian-Einstein strusture and the $\alpha$-semi-stability on $I_{\pm}$-holomorphic bundles over compact bi-Hermitian manifolds are equivalent.

In this paper, we solve the Dirichlet problem for $\alpha$-Hermitian-Einstein equations on $I_{\pm}$-holomorphic bundles over bi-Hermitian manifolds. As a corollary, we obtain an analogue result about generalized holomorphic bundles on generalized K\"{a}hler manifolds.

This unique mathematical method for understanding the flow of gas through each individual objects shape will show us how we can produce physical functions for each object based on the dissemination of gas particles in accordance to its shape. We analyze its continuum per shape of the object and the forces acting on the gas which in return produces its own unique function for the given object due to the rate at which forces were applied to the gas. We also get to examine the different changes in the working rate due to the effect of its volume and mass from the given objects shape with our working equation discovered through Green’s and Gaussian functions.

We formulate and solve the problem of boundary values in non-relativistic quantum mechanics in non-commutative boundary spaces-times. The formalism developed can be useful to the formulation of the boundary value problem in in Noncommutative Quantum Mechanics

Scientists have discovered a mysterious pattern that somehow connects a bus system in Mexico and chicken eyes to quantum physics and number theory. The observed universality reveals properties for a large class of systems that are independent of the dynamical details, revealing underlying mathematical connections described by the classical Poisson distribution. This note suggests that their origin can be found in the wavefunction as modeled by the geometric interpreation of Clifford algebra.

In a day from days, when the famous x is lengthened to x_2 and lost its virginity... Hey-o! Here comes the danger up in this club again. Listen up! Here's the story about a little guy, that lives in a dark world and uses power of wisdom as a torch to find way in darkness; and all day and all night and everything he sees is just illusion. I have been working about the laws of existence for a time. I developed new formulas which were based on a strong mechanism over philosophical hypotheses. Nobody can answer easily; but I thought many times better mathematical infrastructure. Actually at the beginning, I noticed, that a fixed observer does observation of moving bodies being the bodies do a circular motion because of emerging and changing angles over time even if the objects move parallel manner relatively to the observer at that time . This would not happen accidentally even if abstract math says, nothing is going to change. Eureka! Finally while I was in a cafe today, I remembered and developed in a few hours a new method on some note papers which I demanded from cafe to explain existence, and thereupon I asked to my friends for leave, and I am writing towards morning in the name of giving a shoulder to the tired giants. The ancients smile on me!

It is widely known that the E8 polytope can be folded into two Golden Ratio (Phi) scaled copies of the 4 dimensional (4D) 120 vertex 720 edge H4 600-cell. While folding an 8D object into a 4D one is done by applying the dot product of each vertex to a 4x8 folding matrix, we use an 8x8 rotation matrix to produce four 4D copies of H4 600-cells, with the original two left side scaled 4D copies related to the two right side 4D copies in a very specific way. This paper will describe and visualize in detail the specific symmetry relationships which emerge from that rotation of E8 and the emergent fourfold copies of H4. It will also introduce a projection basis using the Icosahedron found within the 8x8 rotation matrix. It will complete the detail for constructing E8 from the 3D Platonic solids, Icosians, and the 4D H4 600-cell. Eight pairs of Phi scaled concentric Platonic solids are identified directly using the sorted and grouped 3D projected vertex norms present within E8.

It was shown in [1] that gravitational interaction can be expressed as an algebraic quadratic invariant form of energies. This allows the decomposition of the entire gravitational system into the sum of squares of energies of its composing particles. Still then, we ran into serious problems, when it came to figure out the Hamiltonian and calculate the total energy of the system from that. (Equivalently put, the algebraic invariant above is not a Hamiltonian one.) The problem is: What goes wrong? This is what this article is about, and the answer is very simple.

This paper shows that the solution of some classes of Schrödinger equations may be performed in terms of the solution of equations of constant coefficients. In this context, it has been possible to generate new exactly solvable potentials and to show that the Schrödinger equation for some well known potentials may also be solved in terms of elementary functions.

In this paper, a general modeling principle is introduced that was found useful for modeling complex physical systems for engineering applications. The technique is a nonlinear asymptotic method (NLAM), constructed from simplified physical theories, i.e., physical theories that were developed from particular points of view, that can be used to construct a more global theory. Originally, the technique was envisioned primarily for engineering applications, but its success has led to a more general principle. Four examples are presented to discuss and illustrate this method.

This paper is an appendix to the article "From Bernoulli to Laplace and Beyond" (refenced below), and discusses different aspects of it: electromagnetism, field tensors, general relativity, and probability.

Reviewing Laplace's equation of gravitation from the perspective of D. Bernoulli, known as Poisson-equation, it will be shown that Laplace's equation tacitly assumes the temperature T of the mass system to be approximately 0 degrees of Kelvin. For temperatures greater zero, the gravitational field will have to be given an additive correctional field. Now, temperature is intimately related to the heat, and heat is known to be radiated as an electromagnetic field. It is shown to take two things in order to get at the gravitational field in the low temperature limit: the total square energy density of the source in space-time and a (massless) field, which expresses the equivalence of inert and gravitational mass/energy in a quadratic, Lorentz-invariant form. This field not only necessarily must include electromagnetic interaction, it also will be seen to behave like it.

This work shows that the Bratu equation belongs to a general class of Liénard-type equations for which the general solution may be exactly and explicitly computed within the framework of the generalized Sundman transformation. In this perspective the exact solution of the Bratu nonlinear two-point boundary value problem as well as of some well-known Bratu-type problems have been determined.

I sketch roughly how an Alcubierre drive could work, by examining exotic geometries consisting of soliton solutions to the dynamics of space filling curves. I also briefly consider how remote sensing might work for obstacle avoidance concerning a craft travelling through space via a 'wormhole wave'. Finally I look into how one might adopt remote sensing ideas to build intrasolar wormhole networks, as well as extrasolar jump gates.

The geometric significance of complex numbers is well known, such as the meaning of imaginary unit i is to rotate a vector with pi/2, etc. In this article, we will try to find some intuitive geometric significances of Pauli matrices, split-complex numbers, SU(2), SO(3), and their relations, and some other operators often used in quantum physics, including a new method to lead to the spinor-space and Dirac equation.

Authors introduce a generalized singular differential equation of quadratic Liénard type for study of exact classical and quantum mechanical solutions. The equation is shown to exhibit periodic solutions and to include the linear harmonic oscillator equation and the Painlevé-Gambier XVII equation as special cases. It is also shown that the equation may exhibit discrete eigenstates as quantum behavior under Nikiforov-Uvarov approach after several point transformations.

Usually, physics students don't like thermodynamics: it is incomprehensible. They commonly get told to get used to it. Later on, as an expert, they'll find that the thermodynamic calculations come with surprises: sometimes evil, sometimes good. That can mean only one thing: The theory is inconsistent. In here, it will be shown where that is.

From the equations of the problem of $n$ body, we consider that $t$ is a function of the variables $(x_k,y_k,z_k)_{k=1,n}$. We write a new formulation of the equations of the $n$ body problem.

The G(4,8) Double Conformal Space-Time Algebra (DCSTA) is a high-dimensional 12D Geometric Algebra that extends the concepts introduced with the G(8,2) Double Conformal / Darboux Cyclide Geometric Algebra (DCGA) with entities for Darboux cyclides (incl. parabolic and Dupin cyclides, general quadrics, and ring torus) in spacetime with a new boost operator. The base algebra in which spacetime geometry is modeled is the G(1,3) Space-Time Algebra (STA). Two G(2,4) Conformal Space-Time subalgebras (CSTA) provide spacetime entities for points, hypercones, hyperplanes, hyperpseudospheres (and their intersections) and a complete set of versors for their spacetime transformations that includes rotation, translation, isotropic dilation, hyperbolic rotation (boost), planar reflection, and (pseudo)spherical inversion. G(4,8) DCSTA is a doubling product of two orthogonal G(2,4) CSTA subalgebras that inherits doubled CSTA entities and versors from CSTA and adds new 2-vector entities for general (pseudo)quadrics and Darboux (pseudo)cyclides in spacetime that are also transformed by the doubled versors. The "pseudo" surface entities are spacetime surface entities that use the time axis as a pseudospatial dimension. The (pseudo)cyclides are the inversions of (pseudo)quadrics in hyperpseudospheres. An operation for the directed non-uniform scaling (anisotropic dilation) of the 2-vector general quadric entities is defined using the boost operator and a spatial projection. Quadric surface entities can be boosted into moving surfaces with constant velocities that display the Thomas-Wigner rotation and length contraction of special relativity. DCSTA is an algebra for computing with general quadrics and their inversive geometry in spacetime. For applications or testing, G(4,8) DCSTA can be computed using various software packages, such as the symbolic computer algebra system SymPy with the GAlgebra module.

In this paper, using proposed three new transformation methods we have solved general solutions and exact solutions of the problems of definite solutions of the Laplace equation, Poisson equation, Schrödinger equation, the homogeneous and non-homogeneous wave equations, Helmholtz equation and heat equation. In the process of solving, we find that in the more universal case, general solutions of partial differential equations have various forms such as basic general solution, series general solution, transformational general solution, generalized series general solution and so on.

This book proposes a review and, on important points, a new formulation of the main concepts of Theoretical Physics. Rather than offering an interpretation based on exotic physical assumptions (additional dimension, new particle, cosmological phenomenon,...) or a brand new abstract mathematical formalism, it proceeds to a systematic review of the main concepts of Physics, as Physicists have always understood them : space, time, material body, force fields, momentum, energy... and proposes the right mathematical objects to deal with them, chosen among well grounded mathematical theories. Proceeding this way, the reader will have a comprehensive, consistent and rigorous understanding of the main topics of the Physics of the XXI° century, together with many tools to do practical computations. After a short introduction about the meaning of Theories in Physics, a new interpretation of the main axioms of Quantum Mechanics is proposed. It is proven that these axioms come actually from the way mathematical models are expressed, and this leads to theorems which validate most of the usual computations and provide safe and clear conditions for their use, as it is shown in the rest of the book. Relativity is introduced through the construct of the Geometry of General Relativity, from 5 propositions and the use of tetrads and fiber bundles, which provide tools to deal with practical problems, such as deformable solids. A review of the concept of motion leads to associate a frame to all material bodies, whatever their scale, and to the representation of motion in Clifford Algebras. Momenta, translational and rotational, are then represented by spinors, which provide a clear explanation for the spin and the existence of anti-particles. The force fields are introduced through connections, in the framework of gauge theories, which is here extended to the gravitational field. It shows that this field has actually a rotational and a transversal component, which are masked under the usual treatment by the metric and the Levy-Civita connection. A thorough attention is given to the topic of the propagation of fields with new and important results. The general theory of lagrangians in the application of the Principle of Least Action is reviewed, and two general models, incorporating all particles and fields are explored, and used for the introduction of the concepts of currents and energy-momentum tensor. Precise guidelines are given to find solutions for the equations representing a system in the most general case. The topic of the last chapter is discontinuous processes. The phenomenon of collision is studied, and we show that bosons can be understood as discontinuities in the fields.

In this article, we give a general exact mathematical framework that all the fundamental relations and conservation equations of continuum mechanics can be derived based on it. We consider a general integral equation contains the parameters that act on the volume and the surface of the integral's domain. The idea is to determine how many local relations can be derived from this general integral equation and what these local relations are. After obtaining the general Cauchy lemma, we derive two other local relations by a new general exact tetrahedron argument. So, there are three local relations that can be derived from the general integral equation. Then we show that all the fundamental laws of continuum mechanics, including the conservation of mass, linear momentum, angular momentum, energy, and the entropy law, can be considered in this general framework. Applying the general three local relations to the integral form of the fundamental laws of continuum mechanics in this new framework leads to exact derivation of the mass flow, continuity equation, Cauchy lemma for traction vectors, existence of stress tensor, general equation of motion, symmetry of stress tensor, existence of heat flux vector, differential energy equation, and differential form of the Clausius-Duhem inequality for entropy law. The general exact tetrahedron argument is an exact proof that removes all the challenges on derivation of the fundamental relations of continuum mechanics. In this proof, there is no approximate or limited process and all the parameters are exact point-based functions. Also, it gives a new understanding and a deep insight into the origins and the physics and mathematics of the fundamental relations and conservation equations of continuum mechanics. This general mathematical framework can be used in many branches of continuum physics and the other sciences.

In 1822, Cauchy presented the idea of traction vector that contains both the normal and tangential components of the internal surface forces per unit area and gave the tetrahedron argument to prove the existence of stress tensor. These great achievements form the main part of the foundation of continuum mechanics. For about two centuries, some versions of tetrahedron argument and a few other proofs of the existence of stress tensor are presented in every text on continuum mechanics, fluid mechanics, and the relevant subjects. In this article, we show the birth, importance, and location of these Cauchy's achievements, then by presenting the formal tetrahedron argument in detail, for the first time, we extract some fundamental challenges. These conceptual challenges are related to the result of applying the conservation of linear momentum to any mass element, the order of magnitude of the surface and volume terms, the definition of traction vectors on the surfaces that pass through the same point, the approximate processes in the derivation of stress tensor, and some others. In a comprehensive review, we present the different tetrahedron arguments and the proofs of the existence of stress tensor, discuss the challenges in each one, and classify them in two general approaches. In the first approach that is followed in most texts, the traction vectors do not exactly define on the surfaces that pass through the same point, so most of the challenges hold. But in the second approach, the traction vectors are defined on the surfaces that pass exactly through the same point, therefore some of the relevant challenges are removed. We also study the improved works of Hamel and Backus, and indicate that the original work of Backus removes most of the challenges. This article shows that the foundation of continuum mechanics is not a finished subject and there are still some fundamental challenges.

The birth of modern continuum mechanics is the Cauchy's idea for traction vectors and his achievements of the existence of stress tensor and derivation of the general equation of motion. He gave a proof of the existence of stress tensor that is called Cauchy tetrahedron argument. But there are some challenges on the different versions of tetrahedron argument and the proofs of the existence of stress tensor. We give a new proof of the existence of stress tensor and derivation of the general equation of motion. The exact tetrahedron argument gives us, for the first time, a clear and deep insight into the origins and the nature of these fundamental concepts and equations of continuum mechanics. This new approach leads to the exact definition and derivation of these fundamental parameters and relations of continuum mechanics. By the exact tetrahedron argument we derived the relation for the existence of stress tensor and the general equation of motion, simultaneously. In this new proof, there is no limited, average, or approximate process and all of the effective parameters are exact values. Also in this proof, we show that all the challenges on the previous tetrahedron arguments and the proofs of the existence of stress tensor are removed.

Physicists are proposing different mechanics to describe the nature, physical body is measured by intrinsic properties like electric charge, and extrinsic properties being related to space like generalized coordinates or velocities etc., with these properties we can predict what event will happen. We can naturally define the fact of the event and the cause of the event as information, the information grasped by physicist must be originated from something objective, information must have its object container. Intrinsic property information is contained by object itself, but container of extrinsic property information like position is ambiguous, position is a relation based on multiple objects, it's hard to define which one is the information container. With such ambiguity, no mechanics is a complete theory, errors hidden in assumptions are hard to find. Here we show a new theoretical framework with strict information container restriction, on which we can build complete determinism theories to approach grand unification.

We investigate relationships between two forms of Hilbert-Schmidt two-re[al]bit and two-qubit "separability functions''--those recently advanced by Lovas and Andai (arXiv:1610.01410), and those earlier presented by Slater ({\it J. Phys. A} {\bf{40}} [2007] 14279). In the Lovas-Andai framework, the independent variable $\varepsilon \in [0,1]$ is the ratio $\sigma(V)$ of the singular values of the $2 \times 2$ matrix $V=D_2^{1/2} D_1^{-1/2}$ formed from the two $2 \times 2$ diagonal blocks ($D_1, D_2$) of a randomly generated $4 \times 4$ density matrix $D$. In the Slater setting, the independent variable $\mu$ is the diagonal-entry ratio $\sqrt{\frac{d_ {11} d_ {44}}{d_ {22} d_ {33}}}$--with, importantly, $\mu=\varepsilon$ or $\mu=\frac{1}{\varepsilon}$ when both $D_1$ and $D_2$ are themselves diagonal. Lovas and Andai established that their two-rebit function $\tilde{\chi}_1 (\varepsilon )$ ($\approx \varepsilon$) yields the previously conjectured Hilbert-Schmidt separability probability of $\frac{29}{64}$. We are able, in the Slater framework (using cylindrical algebraic decompositions [CAD] to enforce positivity constraints), to reproduce this result. Further, we similarly obtain its new (much simpler) two-qubit counterpart, $\tilde{\chi}_2(\varepsilon) =\frac{1}{3} \varepsilon ^2 \left(4-\varepsilon ^2\right)$. Verification of the companion conjecture of a Hilbert-Schmidt separability probability of $\frac{8}{33}$ immediately follows in the Lovas-Andai framework. We obtain the formulas for $\tilde{\chi}_1(\varepsilon)$ and $\tilde{\chi}_2(\varepsilon)$ by taking $D_1$ and $D_2$ to be diagonal, allowing us to proceed in lower (7 and 11), rather than the full (9 and 15) dimensions occupied by the convex sets of two-rebit and two-qubit states. The CAD's themselves involve 4 and 8 variables, in addition to $\mu=\varepsilon$. We also investigate extensions of these analyses to rebit-retrit and qubit-qutrit ($6 \times 6$) settings.

This paper shows that explicit and exact general periodic solutions for various types of Lienard equations can be computed by applying the generalized Sundman transformation. As an il- lustration of the efficiency of the proposed theory, the cubic Duffing equation and Painleve- Gambier equations were considered. As a major result, it has been found, for the first time, that equation XII of the Painleve-Gambier classication can exhibit, according to an appropriate parametric choice, trigonometric solutions, but with a shift factor.

I have mainly analyzed the mathematical meaning of non-classical mathematical theory for three fundamental physics equations - Maxwell’s equations, Dirac’s equations, Einstein’s equations from the quantized core theory of ancient China’s Taoism, and found they have some structures described in the core of the theory of ancient China’s Taoism, especially they all obviously own the yin-yang induction structure. This reveals the relations between the ancient China’s Taoism and modern mathematics and physics in a way, which may help us to understand some problems of the fundamental theory of physics.

Calculating certain aspects of geometry has been difficult. They have defied analytics. Here I propose a method of analysing shape and space in terms of two variables (n,m).

In the year 2017 it was formally conjectured that if the Bender-Brody-M\"uller (BBM) Hamiltonian can be shown to be self-adjoint, then the Riemann hypothesis holds true. Herein we discuss the domain and eigenvalues of the Bender-Brody-M\"uller conjecture. Moreover, a second quantization of the BBM Schr\"odinger equation is performed, and a closed-form solution for the nontrivial zeros of the Riemann zeta function is obtained. Finally, it is shown that all of the nontrivial zeros are located at $\Re(z)=1/2$.

Given an adiabatic system of particles as defined in [4], the problem is whether and to what degree one can break it into its constituents and describe their mutual interaction.

The quantum mechanics of inverse square potentials in one dimension is usually studied through renormalization, self-adjoint extension and WKB approximation. This paper shows that such potentials may be investigated within the framework of the position-dependent mass quantum mechanics formalism under the usual boundary conditions. As a result, exact discrete bound state solutions are expressed in terms of associated Laguerre polynomials with negative energy spectrum using the Nikiforov-Uvarov method for the repulsive inverse square potential.

This article proved two theorems and presented one conjecture about the real-zeros of Jones Polynomial of Torus. Topological quantum computer is related to knots/braids theory where Jones polynomials are characters of the quantum computing. Since the real-zeros of Jones polynomials of torus are observable physical quantities, except the real-zero at 1.0 there exists another distinguished real-zero in 1 < r < 2 for every Jones polynomial of Torus, these unique real zeros can be IDs of torus knots in topological quantum computing.

The Double Conformal Space-Time Algebra (DCSTA) is a high-dimensional 12D Geometric Algebra G(4,8) that extends the concepts introduced with the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA) G(8,2) with entities for Darboux cyclides (incl. parabolic and Dupin cyclides, general quadrics, and ring torus) in spacetime with a new boost operator. The base algebra in which spacetime geometry is modeled is the Space-Time Algebra (STA) G(1,3). Two Conformal Space-Time subalgebras (CSTA) G(2,4) provide spacetime entities for points, flats (incl. worldlines), and hyperbolics, and a complete set of versors for their spacetime transformations that includes rotation, translation, isotropic dilation, hyperbolic rotation (boost), planar reflection, and (pseudo)spherical inversion in rounds or hyperbolics. The DCSTA G(4,8) is a doubling product of two G(2,4) CSTA subalgebras that inherits doubled CSTA entities and versors from CSTA and adds new bivector entities for (pseudo)quadrics and Darboux (pseudo)cyclides in spacetime that are also transformed by the doubled versors. The "pseudo" surface entities are spacetime hyperbolics or other surface entities using the time axis as a pseudospatial dimension. The (pseudo)cyclides are the inversions of (pseudo)quadrics in rounds or hyperbolics. An operation for the directed non-uniform scaling (anisotropic dilation) of the bivector general quadric entities is defined using the boost operator and a spatial projection. DCSTA allows general quadric surfaces to be transformed in spacetime by the same complete set of doubled CSTA versor (i.e., DCSTA versor) operations that are also valid on the doubled CSTA point entity (i.e., DCSTA point) and the other doubled CSTA entities. The new DCSTA bivector entities are formed by extracting values from the DCSTA point entity using specifically defined inner product extraction operators. Quadric surface entities can be boosted into moving surfaces with constant velocities that display the length contraction effect of special relativity. DCSTA is an algebra for computing with quadrics and their cyclide inversions in spacetime. For applications or testing, DCSTA G(4,8) can be computed using various software packages, such as Gaalop, the Clifford Multivector Toolbox (for MATLAB), or the symbolic computer algebra system SymPy with the GAlgebra module.

This work lays the foundations of the theory of kinematic changeable sets ("abstract kinematics"). Theory of kinematic changeable sets is based on the theory of changeable sets. From an intuitive point of view, changeable sets are sets of objects which, unlike elements of ordinary (static) sets, may be in the process of continuous transformations, and which may change properties depending on the point of view on them (that is depending on the reference frame). From the philosophical and imaginative point of view the changeable sets may look like as "worlds" in which evolution obeys arbitrary laws. Kinematic changeable sets are the mathematical objects, consisting of changeable sets, equipped by different geometrical or topological structures (namely metric, topological, linear, Banach, Hilbert and other spaces). In author opinion, theories of changeable and kinematic changeable sets (in the process of their development and improvement), may become some tools of solving the sixth Hilbert problem at least for physics of macrocosm. Investigations in this direction may be interesting for astrophysics, because there exists the hypothesis, that in the large scale of Universe, physical laws (in particular, the laws of kinematics) may be different from the laws, acting in the neighborhood of our solar System. Also these investigations may be applied for the construction of mathematical foundations of tachyon kinematics. We believe, that theories of changeable and kinematic changeable sets may be interesting not only for theoretical physics but also for other fields of science as some, new, mathematical apparatus for description of evolution of complex systems.

This paper is on the mathematical structure of space, time, and gravity. It is shown that electrodynamics is neither charge inversion invariant, nor is it time inversion invariant.

It was proposed gravity propulsion method by using asymmetric conical capacitor charged by high voltage. It was used linear approximation of general relativity equations for derivation of gravity field potential of charged conical capacitor and was shown that negative gravity capabilities of conical capacitor depends only on ratio of electric energy and capacitor mass density, where electric energy density depends on applied voltage and geometric parameters of conical capacitor.

Riemann's hypothesis (1859) is the conjecture stating that: The real part of every non trivial zero of Riemann's zeta function is 1/2. The main contribution of this paper is to achieve the proof of Riemann's hypothesis. The key idea is to provide an Hamiltonian operator whose real eigenvalues correspond to the imaginary part of the non trivial zeros of Riemann's zeta function and whose existence, according to Hilbert and Polya, proves Riemann's hypothesis.

In this article we present an analytical method for deriving the relationship between the pressure drop and flow rate in laminar flow regimes, and apply it to the flow of power-law fluids through axially-symmetric corrugated tubes. The method, which is general with regards to fluid and tube shape within certain restrictions, can also be used as a foundation for numerical integration where analytical expressions are hard to obtain due to mathematical or practical complexities. Five converging-diverging geometries are used as examples to illustrate the application of this method.

The one-dimensional Navier-Stokes equations are used to derive analytical expressions for the relation between pressure and volumetric flow rate in capillaries of five different converging-diverging axisymmetric geometries for Newtonian fluids. The results are compared to previously-derived expressions for the same geometries using the lubrication approximation. The results of the one-dimensional Navier-Stokes are identical to those obtained from the lubrication approximation within a non-dimensional numerical factor. The derived flow expressions have also been validated by comparison to numerical solutions obtained from discretization with numerical integration. Moreover, they have been certified by testing the convergence of solutions as the converging-diverging geometries approach the limiting straight geometry.

Euler-Lagrange variational principle is used to obtain analytical and numerical flow relations in cylindrical tubes. The method is based on minimizing the total stress in the flow duct using the fluid constitutive relation between stress and rate of strain. Newtonian and non-Newtonian fluid models; which include power law, Bingham, Herschel-Bulkley, Carreau and Cross; are used for demonstration.

We use a method based on the lubrication approximation in conjunction with a residual-based mass-continuity iterative solution scheme to compute the flow rate and pressure field in distensible converging–diverging tubes for Navier–Stokes fluids. We employ an analytical formula derived from a one-dimensional version of the Navier–Stokes equations to describe the underlying flow model that provides the residual function. This formula correlates the flow rate to the boundary pressures in straight cylindrical elastic tubes with constant-radius. We validate our findings by the convergence toward a final solution with fine discretization as well as by comparison to the Poiseuille-type flow in its convergence toward analytic solutions found earlier in rigid converging–diverging tubes. We also tested the method on limiting special cases of cylindrical elastic tubes with constant-radius where the numerical solutions converged to the expected analytical solutions. The distensible model has also been endorsed by its convergence toward the rigid Poiseuille-type model with increasing the tube wall stiffness. Lubrication-based one-dimensional finite element method was also used for verification. In this investigation five converging–diverging geometries are used for demonstration, validation and as prototypes for modeling converging–diverging geometries in general.

In this paper, we use a generic and general variational method to obtain solutions to the flow of generalized Newtonian fluids through circular pipes and plane slits. The new method is not based on the use of the Euler-Lagrange variational principle and hence it is totally independent of our previous approach which is based on this principle. Instead, the method applies a very generic and general optimization approach which can be justified by the Dirichlet principle although this is not the only possible theoretical justification. The results that were obtained from the new method using nine types of fluid are in total agreement, within certain restrictions, with the results obtained from the traditional methods of fluid mechanics as well as the results obtained from the previous variational approach. In addition to being a useful method in its own for resolving the flow field in circular pipes and plane slits, the new variational method lends more support to the old variational method as well as for the use of variational principles in general to resolve the flow of generalized Newtonian fluids and obtain all the quantities of the flow field which include shear stress, local viscosity, rate of strain, speed profile and volumetric flow rate. The theoretical basis of the new variational method, which rests on the use of the Dirichlet principle, also provides theoretical support to the former variational method.

We continue our investigation to the use of the variational method to derive flow relations for generalized Newtonian fluids in confined geometries. While in the previous investigations we used the straight circular tube geometry with eight fluid rheological models to demonstrate and establish the variational method, the focus here is on the plane long thin slit geometry using those eight rheological models, namely: Newtonian, power law, Ree-Eyring, Carreau, Cross, Casson, Bingham and Herschel-Bulkley. We demonstrate how the variational principle based on minimizing the total stress in the flow conduit can be used to derive analytical expressions, which are previously derived by other methods, or used in conjunction with numerical procedures to obtain numerical solutions which are virtually identical to the solutions obtained previously from well established methods of fluid dynamics. In this regard, we use the method of Weissenberg-Rabinowitsch-Mooney-Schofield (WRMS), with our adaptation from the circular pipe geometry to the long thin slit geometry, to derive analytical formulae for the eight types of fluid where these derived formulae are used for comparison and validation of the variational formulae and numerical solutions. Although some examples may be of little value, the optimization principle which the variational method is based upon has a significant theoretical value as it reveals the tendency of the flow system to assume a configuration that minimizes the total stress. Our proposal also offers a new methodology to tackle common problems in fluid dynamics and rheology.

This article deals with the flow of Newtonian fluids through axially-symmetric corrugated tubes. An analytical method to derive the relation between volumetric flow rate and pressure drop in laminar flow regimes is presented and applied to a number of simple tube geometries of converging-diverging nature. The method is general in terms of fluid and tube shape within the previous restrictions. Moreover, it can be used as a basis for numerical integration where analytical relations cannot be obtained due to mathematical difficulties.

We derive analytical expressions for the flow of Newtonian and power law fluids in elastic circularly-symmetric tubes based on a lubrication approximation where the flow velocity profile at each cross section is assumed to have its axially-dependent characteristic shape for the given rheology and cross sectional size. Two pressure-area constitutive elastic relations for the tube elastic response are used in these derivations. We demonstrate the validity of the derived equations by observing qualitatively correct trends in general and quantitatively valid asymptotic convergence to limiting cases. The Newtonian formulae are compared to similar formulae derived previously from a one-dimensional version of the Navier-Stokes equations.

In this paper, analytical expressions correlating the volumetric flow rate to the pressure drop are derived for the flow of Carreau and Cross fluids through straight rigid circular uniform pipes and long thin slits. The derivation is based on the application of Weissenberg-Rabinowitsch-Mooney-Schofield method to obtain flow solutions for generalized Newtonian fluids through pipes and our adaptation of this method to the flow through slits. The derived expressions are validated by comparing their solutions to the solutions obtained from direct numerical integration. They are also validated by comparison to the solutions obtained from the variational method which we proposed previously. In all the investigated cases, the three methods agree very well. The agreement with the variational method also lends more support to this method and to the variational principle which the method is based upon.

In this paper we investigate the yield condition in the mobilization of yield-stress materials in distensible tubes. We discuss the two possibilities for modeling the yield-stress materials prior to yield: solid-like materials and highly-viscous fluids and identify the logical consequences of these two approaches on the yield condition. As part of this investigation we derive an analytical expression for the pressure field inside a distensible tube with a Newtonian flow using a one-dimensional Navier-Stokes flow model in conjunction with a pressure-area constitutive relation based on elastic tube wall characteristics.

A residual-based lubrication method is used in this paper to find the flow rate and pressure field in converging-diverging rigid tubes for the flow of time-independent category of non-Newtonian fluids. Five converging-diverging prototype geometries were used in this investigation in conjunction with two fluid models: Ellis and Herschel-Bulkley. The method was validated by convergence behavior sensibility tests, convergence to analytical solutions for the straight tubes as special cases for the converging-diverging tubes, convergence to analytical solutions found earlier for the flow in converging-diverging tubes of Newtonian fluids as special cases for non-Newtonian, and convergence to analytical solutions found earlier for the flow of power-law fluids in converging-diverging tubes. A brief investigation was also conducted on a sample of diverging-converging geometries. The method can in principle be extended to the flow of viscoelastic and thixotropic/rheopectic fluid categories. The method can also be extended to geometries varying in size and shape in the flow direction, other than the perfect cylindrically-symmetric converging-diverging ones, as long as characteristic flow relations correlating the flow rate to the pressure drop on the discretized elements of the lubrication approximation can be found. These relations can be analytical, empirical and even numerical and hence the method has a wide applicability range.

A Geometric Model A family of models in Euclidean space is developed from the following approximation. m_p/m_e = 4pi(4pi- 1\pi)(4pi-2/pi) = 1836.15 (1) where (m_p) and (m_e) are the numeric values for the mass of the proton and the mass of electron, respectively. In particular, we will develop models (1) that agree with the recommended value of the mass ratio of the proton to the electron to six significant figures, (2) that explain the “shape-shifting” behavior of the proton, and (3) that are formed concisely from the sole transcendental number pi. This model is solely geometric, relying on volume as the measure of mass. Claim that inclusion of quantum/relativistic properties enhance the accuracy of the model. The goal is to express the ratio of the proton mass to the electron mass in terms of (1) pure mathematical constants and (2) a quantum corrective factor. harry.watson@att.net

n Riemannian geometry there is a unique combination of the Riemann-Christoffel curvature tensor, Ricci tensor and Ricci scalar that defines a fourth-order Lagrangian for conformal gravity theory. This Lagrangian can be greatly simplified by eliminating the curvature tensor term, leaving a unique combination of just the Ricci tensor and scalar. The resulting formalism and the associated equations of motion provide a tantalizing alternative to Einstein-Hilbert gravity that may have application to the problems of dark matter and dark energy without the imposition of the cosmological constant or extraneous scalar, vector and spinor terms typically employed in attempts to generalize the Einstein-Hilbert formalism. Gauss-Bonnet gravity specifies that the full Lagrangian hides an ordinary divergence (or surface term) that can be used to eliminate the curvature tensor term. In this paper we show that the overall formalism, outside of surface terms necessary for integration by parts, does not involve any such divergence. Instead, it is the Bianchi identities that are hidden in the formalism, and it is this fact that allows for the simplification of the conformal Lagrangian.

We use a generic and general numerical method to obtain solutions for the flow of generalized Newtonian fluids through circular pipes and plane slits. The method, which is simple and robust can produce highly accurate solutions which virtually match any analytical solutions. The method is based on employing the stress, as a function of the pipe radius or slit thickness dimension, combined with the rate of strain function as represented by the fluid rheological constitutive relation that correlates the rate of strain to stress. Nine types of generalized Newtonian fluids are tested in this investigation and the solutions obtained from the generic method are compared to the analytical solutions which are obtained from the Weissenberg-Rabinowitsch-Mooney-Schofield method. Very good agreement was obtained in all the investigated cases. All the required quantities of the flow which include local viscosity, rate of strain, flow velocity profile and volumetric flow rate, as well as shear stress, can be obtained from the generic method. This is an advantage as compared to some traditional methods which only produce some of these quantities. The method is also superior to the numerical meshing techniques which may be used for resolving the flow in these systems. The method is particularly useful when analytical solutions are not available or when the available analytical solutions do not yield all the flow parameters.

We investigate the possibility that the spatial dependency of stress in generalized Newtonian flow systems is a function of the applied pressure field and the conduit geometry but not of the fluid rheology. This possibility is well established for the case of a one-dimensional flow through simply connected regions, specifically tubes of circular uniform cross sections and plane thin slits. If it can also be established for the more general case of generalized Newtonian flow through non-circular or multiply connected geometries, such as the two-dimensional flow through conduits of rectangular or elliptical cross sections or the flow through annular circular pipes, then analytical or semi-analytical or highly accurate numerical solutions; regarding stress, rate of strain, velocity profile and volumetric flow rate; for these geometries can be obtained from the stress function, which can be easily obtained from the Newtonian case, in combination with the constitutive rheological relation for the particular non-Newtonian fluid, as done previously for the case of the one-dimensional flow through simply connected regions.

In this article we challenge the claim that the previously proposed variational method to obtain flow solutions for generalized Newtonian fluids in circular tubes and plane slits is exact only for power law fluids. We also defend the theoretical foundation and formalism of the method which is based on minimizing the total stress through the application of the Euler-Lagrange principle.

In this article, the extensional flow and viscosity and the converging-diverging geometry were examined as the basis of the peculiar viscoelastic behavior in porous media. The modified Bautista-Manero model, which successfully describes shearthinning, elasticity and thixotropic time-dependency, was used for modeling the flow of viscoelastic materials which also show thixotropic attributes. An algorithm, originally proposed by Philippe Tardy, that employs this model to simulate steadystate time-dependent flow was implemented in a non-Newtonian flow simulation code using pore-scale modeling and the initial results were analyzed. The findings are encouraging for further future development.

Yield-stress is a problematic and controversial non-Newtonian flow phenomenon. In this article, we investigate the flow of yield-stress substances through porous media within the framework of pore-scale network modeling. We also investigate the validity of the Minimum Threshold Path (MTP) algorithms to predict the pressure yield point of a network depicting random or regular porous media. Percolation theory as a basis for predicting the yield point of a network is briefly presented and assessed. In the course of this study, a yield-stress flow simulation model alongside several numerical algorithms related to yield-stress in porous media were developed, implemented and assessed. The general conclusion is that modeling the flow of yield-stress fluids in porous media is too difficult and problematic. More fundamental modeling strategies are required to tackle this problem in the future.

An algebra for unit multivector components for a manifold of five poly-complex dimensions is presented. The algebra has many properties that suggest it may provide a basis for a grand unification theory.

Effects of L{\'{e}}vy noise on self-propelled particles in a two-dimensional potential is investigated. The current reversal phenomenon appear in the system. $V$($x$ direction average velocity) changes from negative to positive with increasing asymmetry parameter $\beta$, and changes from positive to negative with increasing self-propelled velocity $v_0$. The $x$ direction average velocity $V$ has a maximum with increasing modulation constant $\lambda$.

$N$-scales are a generalization of time-scales that has been put forward to unify continuous and discrete analyses to higher dimensions. In this paper we investigate massive scalar field theory on $n$-scales. In a specific case of a regular 2-scale, we find that the IR energy spectrum is almost unmodified when there are enough spatial points. This is regarded as a good sign because the model reproduces the known results in the continuum approximation. Then we give field equation on a general $n$-scale. It has been seen that the field equation can only be solved via computer simulations. Lastly, we propose that $n$-scales might be a good way to model singularities encountered in the general theory of relativity.

Generalization of fractal density on a fractals for spaces with positive and negative fractal dimensions. Fractal-fractional generalized physics (i.e. classical or quantum physics). Generalized Hausdorff measures. Numbers and generalized functions as generalized logical values. Beyond logics and numbers. Generalized concept of physical field, i.e generalized universes and multiverses. Fractional generalization of path integrals.

This note represents an attempt to give a solution of Navier-Stokes equations under the assumptions $(A)$ of the problem as described by the Clay Institute \cite{bib1}. We give a proof of the condition of bounded energy when the velocity vector $u$ and vorticity vector $\Omega=curl(u)$ are collinear.

The second-order approach to the entropy gradient maximization for systems with many degrees of freedom provides the dynamic equations of first order and light-like second order without additional ergodicity conditions like conservation laws. The first order dynamics lead to the definition of the conserved kinetic energy and potential energy. In terms of proper degrees of freedom the total energy conservation reproduces the Einstein's mass-energy relation. The newtonian interpretation of the second order dynamic equations suggests the definition for general inertial mass and for the interaction potential.

Wick rotation produces numbers that agree with experiment and yet the method is mathematically wrong and not allowed by any self-consistent rule. We explore a small slice of wiggle room in complex analysis and show that it may be possible to use QFT without reliance on Wick rotations.

The bio-heat transfer equation for homogeneous material model can be easily calculated by using second order finite difference approximation to discretize the spatial derivatives and explicit finite-difference time-domain (FDTD) scheme for time domain discretization. Mr. Gandhi and colleagues solved the bio-heat equation for inhomogeneous models utilizing implicit finite-difference method. Whereas we appreciate their research, we would like to address a few issues that may help further clarify or confirm the research.

A C-loop algebra, designated U is assembled as the product: M4(C)x T. When M4(C) is assigned to represent Cl{1,3}(R) x C and the principle of spatial equivalence is invoked, a sub-algebra designated W is found to have features that suggest it could provide an underlying basis for the standard model of fundamental particles.U is of the same order as Cl{0,10}(R), but has a ``natural" partition into Cl{1,3}(R) x C x W, suggesting that its use in string/M theories in the place of Cl{0,10}(R) may generate a description of reality.

In this paper, an arbitrary high-order compact method is developed for compressible multi-component flows with a stiffened gas equations of state(EOS). The main contribution is combining the high-order, conservative, compact spectral volume scheme(SV) with the non-oscillatory kinetic scheme(NOK) to solve the quasi-conservative extended Euler equations of compressible multi-component flows. The new scheme consists of two parts: the conservative part and the non-conservative part. The original high order compact SV scheme is used to discretize the conservative part directly. In order to treat the equation of state of the stiffened gas, the NOK scheme is utilized to compute the numerical flux. Then, careful analysis is made to satisfy the necessary condition to avoid unphysical oscillation near the material interfaces. After that, a high-order compact scheme for the non-conservative part is obtained. This new scheme has the following advantages for numerical simulations of compressible multi-component stiffened gas: high order accuracy with compact stencil and oscillation-free near the material interfaces. Numerical tests demonstrate the good performance and the efficiency of the new scheme for multi-component flow simulations.

The papier presents an essay of the resolution of Navier-Stokes equations under the hypothesis (A) of the open problem cited by Clay Institute (C.L. Fefferman, 2006).

Kaluza's 1921 theory of gravity and electromagnetism using a fifth wrapped-up spatial dimension is inspiration for many modern attempts to develop new physical theories. The original theory has problems which may well be overcome, and thus Kaluza theory should be looked at again: it is a natural, if not necessary, geometric unification of gravity and electromagnetism. Here a general demonstration that the Lorentz force law can be derived from a range of Kaluza theories is presented. This is investigated via non-Maxwellian kinetic definitions of charge that are divergence-free and relate Maxwellian charge to 5D components of momentum. The possible role of torsion is considered as an extension. It is shown, however, that symmetric torsion components are likely not admissible in any prospective theory. As a result Kaluza's original theory is rehabilitated and a call for deeper analysis made.

This paper introduces the G(4,8) Double Conformal Space-Time Algebra (DCSTA). G(4,8) DCSTA is a straightforward extension of the G(2,8) Double Conformal Space Algebra (DCSA), which is a different form of the G(8,2) Double Conformal / Darboux Cyclide Geometric Algebra (DCGA). G(4,8) DCSTA extends G(2,8) DCSA with spacetime boost operations and differential operators for differentiation with respect to the pseudospatial time w=ct direction and time t. The spacetime boost operation can implement anisotropic dilation (directed non-uniform scaling) of quadric surface entities. DCSTA is a high-dimensional 12D embedding of the G(1,3) Space-Time Algebra (STA) and is a doubling of the G(2,4) Conformal Space-Time Algebra (CSTA). The 2-vector quadric surface entities of the DCSA subalgebra appear in DCSTA as quadric surfaces at zero velocity that can be boosted into moving surfaces with constant velocities that display the length contraction effect of special relativity. DCSTA inherits doubled forms of all CSTA entities and versors. The doubled CSTA entities (standard DCSTA entities) include points, hypercones, hyperplanes, hyperpseudospheres, and other entities formed as their intersections, such as planes, lines, spatial spheres and circles, and spacetime hyperboloids (pseudospheres) and hyperbolas (pseudocircles). The doubled CSTA versors (DCSTA versors) include rotor, hyperbolic rotor (boost), translator, dilator, and their compositions such as the translated-rotor, translated-boost, and translated-dilator. The DCSTA versors provide a complete set of spacetime transformation operators on all DCSTA entities. DCSTA inherits the DCSA 2-vector spatial entities for Darboux cyclides (incl. parabolic and Dupin cyclides, general quadrics, and ring torus) and gains Darboux pseudocyclides formed in spacetime with the pseudospatial time dimension. All DCSTA entities can be reflected in, and intersected with, the standard DCSTA entities. To demonstrate G(4,8) DCSTA as concrete mathematics with possible applications, this paper includes sample code and example calculations using the symbolic computer algebra system SymPy.

This book proposes a review and, on some important points, a new interpretation of the main concepts of Theoretical Physics. Rather than offering an interpretation based on exotic physical assumptions (additional dimension, new particle, cosmological phenomenon,…) or a brand new abstract mathematical formalism, it proceeds to a systematic review of the main concepts of Physics, as Physicists have always understood them : space, time, material body, force fields, momentum, energy… and propose the right mathematical tools to deal with them, chosen among well known mathematical theories. After a short introduction about the place of Mathematics in Physics, a new interpretation of the main axioms of Quantum Mechanics is proposed. It is proven that these axioms come actually from the way mathematical models are expressed, and this leads to theorems which validate most of the usual computations and provide safe and clear conditions for their use, as it is shown in the rest of the book. Relativity is introduced through the construct of the Geometry of General Relativity, based on 5 propositions and the use of tetrads and fiber bundles, which provide tools to deal with practical problems, such as deformable solids. A review of the concept of momenta leads to the introduction of spinors in the framework of Clifford algebras. It gives a clear understanding of spin and antiparticles. The force fields are introduced through connections, in the, now well known, framework of gauge theories, which is here extended to the gravitational field. It shows that this field has actually a rotational and a transversal component, which are masked under the usual treatment by the metric and the Levy-Civita connection. A thorough attention is given to the topic of the propagation of fields with interesting results, notably to explore gravitation. The general theory of lagrangians in the application of the Principle of Least Action is reviewed, and two general models, incorporating all particles and fields are explored, and used for the introduction of the concepts of currents and energy-momentum tensor. Precise guidelines are given to find operational solutions of the equations of the gravitational field in the most general case. The last chapter shows that bosons can be understood as discontinuities in the fields. In this 4th version of this book, changes have been made : - in Relativist Geometry : the ideas are the same, but the chapter has been rewritten, notably to introduce the causal structure and explain the link with the practical measures of time and space; - in Spinors : the relation with momenta has been introduced explicitly - in Force fields : the section dedicated to the propagation of fields is new, and is an important addition. - in Continuous Models : the section about currents and energy-momentum tensor are new. - in Discontinuous Processes : the section about bosons has been rewritten and the model improved.

I open a dusty old drawer, and I found this article, rejected by every journal, with a complete, total loss of time; a thing that I'll never make; and I change only the bibliography, that is to initially conceived . The old idea sounds interesting, and here, on vixra, there is not rejection. I don't remember the whole theory, and the whole programs, but it can be useful to others; so I share it with you. It seem that without the complication of the least common divisor, the calculation is more simple, and elegant.

The theory of idealiscience is an accurate theoretical model, by the model we can deduce most important laws of Physics, explain a lot of physical mysteries, even a lot of basic and important philosophical questions. we can also get the theoretical values of a lot of physical constants, even some of the constants can not be deduced by traditional physical theories, such as neutron mass and magnetic moment,Avogadro constant and so on.

In this paper we use the steerable special relativistic (space-time) Fourier transform (SFT), and relate the classical convolution of the algebra for space-time $Cl(3,1)$-valued signals over the space-time vector space $\R^{3,1}$, with the (equally steerable) Mustard convolution. A Mustard convolution can be expressed in the spectral domain as the point wise product of the SFTs of the factor functions. In full generality do we express the classical convolution of space-time signals in terms of finite linear combinations of Mustard convolutions, and vice versa the Mustard convolution of space-time signals in terms of finite linear combinations of classical convolutions.

Die Einführung 'quaternionischer Differentialformen' auf dem Tangentialraum einer vierdimensionalen Mannigfaltigkeit ergibt ein vielversprechendes mathematisches System zur Beschreibung unserer physikalischen (3+1)-Raumzeit schon mit einem Minimum von Grundannahmen. Hier wird die Grundlage dieses Modells einer 'Quaternionischen Raumzeit' dargestellt.

We present a new decomposition of unitary matrices particularly useful for mixing matrices. The decomposition separates the complex phase information from the mixing angle information of the matrices and leads to a new type of parameterization. We show that the mixing angle part of U(n) is equivalent to U(n-1). We give closed form parameterizations for 3x3 unitary mixing matrices (such as the CKM and MNS matrices) that treat the mixing angles equally. We show the relationship between Berry-Pancharatnam or quantum phase and the Jarlskog invariant Jcp that gives the CP-violation in the standard model. We established the likely existence of the new decomposition by computer simulation in 2008. Philip Gibbs proved the n=3 case in 2009 and in 2011, Samuel Lisi proved the general case using Floer theory in symplectic geometry. We give an accessible version of Lisi's proof.

Merriam-Webster's Collegiate Dictionary, Eleventh Edition, gives a technical definition of curvature, "the rate of change of the angle through which the tangent to a curve turns in moving along the curve and which for a circle is equal to the reciprocal of the radius". That precisely describes a curve's intuitive curvature, but the Riemann "curvature" tensor is zero for all curves! We work out the natural extension of intuitive curvature to hypersurfaces, based on the rates that their tangents develop components which are orthogonal to the local tangent hyperplane. Intuitive curvature is seen to have the form of a second-rank symmetric tensor which cannot be algebraically expressed in terms of the metric tensor and a finite number of its partial derivatives. The Riemann "curvature" tensor contrariwise is a fourth-rank tensor with both antisymmetric and symmetric properties that famously is algebraically expressed in terms of the metric tensor and its first and second partial derivatives. Thus use of the word "curvature" with regard to the Riemann tensor is misleading, and since it can't encompass intuitive curvature, Gauss-Riemann "geometry" oughtn't be termed differential geometry either. That "geometry" is no more than the class of the algebraic functions of the metric and any finite number of the metric's partial derivatives, which it is convenient to organize into generally covariant entities such as the Riemann tensor because those potentially play a role in generally-covariant metric-based field theories.

In this paper we extend the so-called dual or mirror image formalism and Caldirola's- Kanai's formalism for damped harmonic oscillator to the case that both frictional coefficient and time-dependent frequency depend on time explicitly. As an solvable example, we consider the case that frictional coefficient $ \ga (t) = \frac{ \ga_0}{1 + q t} , (q > 0 )$ and angular frequency function $ w(t) = \frac{ w_0}{ 1 + q t } $. For this choice, we construct the quantum harmonic Hamiltonian and express it in terms of $su(2)$ algebra generators. Using the exact invariant for the Hamiltonian and its unitary transform, we solve the time-dependent Schro\"dinger equation with time-dependent frictional coefficient and time-dependent frequency.

This article begins by defining two sets of related constants: The first set is used to define a nonstandard cubic equation. The second set is used to define equations that constrain a pair of rotation matrices. Curiously, these sets each contain an angle, each defined much like the other, where these angles separately play a central role in identifying solutions to their respective, very different equations. In this way these similarly-defined angles provide good evidence of a non-coincidental relationship between the nonstandard cubic equation and the pair of matrices. Moreover, a special case of this nonstandard cubic neatly combines with these matrices to produce values that map over to multiple physical constants (the fine structure constant reciprocal, the Weinberg angle, and the quark and lepton mixing angles), providing further evidence of a non-coincidental relationship.

The fact that a red shift of galaxies is observed to increase with the galaxies approaching the cosmic horizon is taken as plain evidence that the universe is acceleratedly expanding, hopefully perpetually. That implication does however not take into account the gravitational effects of the cosmic horizon on the universe; the cosmic horizon is the center of mass of our universe, and taking this into account, a very different picture of the state of our universe results: Based on [1] it is shown that the observed red shift, implies a huge deceleration of the universe, already. Whether the cosmos is currently before, at, or after the turn from expansion to collapse, this could only be answered by repeated red shift measurements over the next decades.

There is ongoing interest in designing data fusion systems that make use of human opinions (i.e.,\soft" data) alongside readings from various sensors that use mechanical, electromagnetic, optical,and acoustic transducers (i.e., \hard" data). One of the major challenges in the development of these hard/soft fusion systems is to determine accurately and flexibly the impact of human responses on the performance of the fusion operator.

In this paper, a new scheme of arbitrary high order accuracy in both space and time is proposed to solve hyperbolic conservative laws. Based on the idea of °ux vector splitting(FVS) scheme, we split all the space and time derivatives in the Taylor expansion of the numerical °ux into two parts: one part with positive eigenvalues, another part with negative eigenvalues. According to a Lax-Wendro® procedure, all the time derivatives are then replaced by space derivatives. And the space derivatives is calculated by WENO reconstruction polynomial. One of the most important advantages of this new scheme is easy to implement.In addition, it should be pointed out, the procedure of calculating the space and time derivatives in numerical °ux can be used as a building block to extend the current ¯rst order schemes to very high order accuracy in both space and time. Numerous numerical tests for linear and nonlinear hyperbolic conservative laws demonstrate that new scheme is robust and can be high order accuracy in both space and time.

As far as the writer is aware, the Bianchi identities associated with a Weyl space have never been presented. That space was discovered by the noted German mathematical physicist Hermann Weyl in 1918, and represented the geometry underlying a tantalizing theory that appeared to successfully unify the gravitational and electromagnetic fields. One of theory’s problems involved one form of the Bianchi identities, which in Riemannian space are used to derive the divergenceless Einstein tensor. Such a derivation is generally not applicable in a non-Riemannian geometry like Weyl’s, in which the covariant derivative of the metric tensor is non-zero. But it turns out that such a derivation is not only possible but straightforward, with a result that hints at a fundamental relationship between Weyl’s geometry and electromagnetism.

These introductory lectures on affine and quantum algebras are motivated by the idea that triality and exceptional structures are crucial to gravity and color symmetry. We start with the ADE classification.

Misconceptions have recently been found in the definition of a partial derivative (in the case of the presence of both explicit and implicit dependencies of the function subjected to differentiation) in the classical analysis. We investigate the possible influence of this discovery on quantum mechanics and the classical/quantum field theory. Surprisingly, some commutators of operators of space-time 4-coordinates do not equal to zero. Thus, we provide the bases for new-fashioned noncommutative field theory.

This book provides an introduction to quaternions and Clifford geometric algebras. Quaternion rotations are covered extensively. A reference manual on the entities and operations of conformal (CGA) and quadric geometric algebras (QGA) is given. The Space-Time Algebra (STA) is introduced and offers an example of its use for special relativity velocity addition. Advanced algebraic techniques for the symbolic expansion of the geometric product of blades is explained with numerous examples.

It is shown that the electromagnetic field is not a U(1) gauge theory, and it is shown that the Wightman axioms are inconsistent with the principle of conservation of energy and momentum.

We develop some ideas that can be used to show relationships between quantum state tensors and gravitational metric tensors. After firmly grasping the math by $\alpha$ and Einstein's equation, this is another attempt to shake it and see what goes and what stays. We introduce slightly more rigorous definitions for some familiar objects and find an unexpected connection between the chirological phase $\Phi^n$ and the quaternions $\bm{q}\in\mathbb{H}$. Torsion, the only field in string theory not already present in the theory of infinite complexity, is integrated. We propose a solution to the Ehrenfest paradox and a way to prove the twin primes conjecture. The theory's apparent connections to negative frequency resonant radiation and time reversal symmetry violation are briefly treated.

This work presents a formalism of the notions of space and of time which contains that of the special relativity, which is compatible with the quantum theories, and which distinguishes itself from the general relativity by the fact that it allows us to define the possible states of motion between two observers arbitrarily chosen in the nature. Before calculating the advance of the perihelion of an orbit, it is necessary to define the existence of a perihelion and its possible movement. In other words, it is necessary to express the use of a physical space which is a set of spatial positions, a set of world lines constantly at rest according to a unique observer. This document defines all the physical spaces of the nature (some compared with the others) by noting that to choose a temporal variable in one of these spaces, it is enough to choose a particular parametrization along each of its points. If the world lines of a family of observers are not elements of a unique physical space, then even in classical physics, how can they manage to put end to end their rulers to determine the measure of a segment of curve of their reference frame (each will have to ask to his neighbor: a little seriousness please, do not move until the measurement is ended) ? This question is the basis of the solution which will be proposed to paradox of Ehrenfest. A notion of expansion of the universe is established as being a structural reality and a rigorous theoretical formulation of the Hubble's experimental law is proposed. We shall highlight the fact that a relative motion occurs only along specific trajectories and this notion of authorized trajectories is not a novelty in physics as it is stated in the Bohr atomic model. We shall also highlight the fact that a non-uniform rectilinear motion possesses a horizon having the structure of a plan.

A gravitational ellipse is the mathematical result of Newton's law of gravitation. [Ref.1] The equation describing such an ellipse, is obtained by differentiating space-by-time twice. Le Verrier [Ref.2] stated: 'rotating gravitational ellipses are observed in the solar system'. One could be asked, to adjust the existing gravitational equation in such a way, that a rotating gravitational ellipse is obtained. The additional rotation is an extra variable, so the equation will be a three times space-by-time differentiated equation. In order to obtain a three times space-by-time differentiated equation we need to differentiate space-by-time for the third time. Differentiating space-by-time twice gives the following result.[Ref.3] \begin{equation} \centerline{ $(\ddot{X})^2 + (\ddot{Y})^2 = (\ddot{R} - R \dot{a}^2 )^2 + (R\ddot{a} + 2 \dot{R} \dot{a} )^2 $} \end{equation} A third time differentiation of space-by-time gives the result: \begin{equation} \centerline{ $(\dddot X )^2 + (\dddot{Y})^2 = (\dddot{R} - 3\dot{R} \dot{a}^2 - 3 R \dot{a} \ddot{a} )^2 + (R\dddot{a} + 3 \dot{R} \ddot{a} + 3 \ddot{R} \dot{a} - R\dot{a}^3 )^2 $} \end{equation} We are now simply performing the necessary mathematical exercise to produce the new equation, which describes rotating gravitational ellipses. \newline \centerline{\includegraphics{20150202_RotatingEllipse.png} }\newline I assume that the reader accepts the mathematical differential equation, which defines a rotating gravitational motion as observed. But we now have two equations defining rotating gravitational ellipses as observed in nature: the EIH equations (Ref.4) and the above equation 2, which obeys the Euclidean space premises.

Symmetry information beneath wave mechanics is re-examined. Homogeneity of space is the symmetry, fundamental to the quantum free particle. The unitary information of the Canonical Commutation Relation is shown not to be implied by that symmetry. Keywords:quantum mechanics, wave mechanics, Canonical Commutation Relation, symmetry, homogeneity of space unitary, non-unitary.

This short note presents the structures of lattices and continuous geometries in the energy spectrum of a quantum bound state. Quantum logic, in von Neumann's original sense, is used to construct these structures. Finally, a quantum logical understanding of the emergence of discreteness is suggested.

The problem of finding a better immunization strategy for controlling the spreading of the epidemic with limited resources has attracted much attention since its great theoretical significance and wide application. In this paper, we propose a novel and successful targeted immunization strategy based on percolation transition. Our strategy immunizes the fraction of critical nodes which lead to the emergence of giant connected component. To test the effectiveness of the proposed method, we conduct the experiments on several artificial networks and real-world networks. The results show that the proposed method outperforms the existing well-known methods with 18% to 50% fewer immunized nodes for same degree of immunization.

The number of independent components in the Riemann-Christoffel curvature tensor, being composed of the metric tensor and its first and second derivatives, varies considerably with the dimension of space. Since few texts provide an explicit derivation of component number, we present here a simplified method using only the curvature tensor’s antisymmetry property and the cyclicity condition. For generality and comparison, the method for computing component number in both Riemannian and non-Riemannian space is presented.

Another ... friendly and creative ... author-editor interaction is presented in which several basic conundrums in physics are mentioned, conundrums no physicist seems to care about ...

I propose to ask mathematics itself for the possible behaviour of nature, with the focus on starting with a most simple realistic model, employing a philosophy of investigation rather than invention when looking for a unified theory of physics. Doing a 'mathematical experiment' of putting a least set of conditions on a general time-dependent manifold results in mathematics itself inducing a not too complex 4-dimensional object similar to our physical spacetime, with candidates for gravitational and electromagnetic fields emerging on the tangent bundle. This suggests that the same physics might govern spacetime not only on a macroscopic scale, but also on the microscopic scale of elementary particles, with possible junctions to quantum mechanics.

We analyze cross-correlation between runs scored over a time interval in cricket matches of different teams using methods of random matrix theory (RMT). We obtain an ensemble of cross-correlation matrices $C$ from runs scored by eight cricket playing nations for (i) test cricket from 1877 -2014 (ii)one-day internationals from 1971 -2014 and (iii) seven teams participating in the Indian Premier league T20 format (2008-2014) respectively. We find that a majority of the eigenvalues of C fall within the bounds of random matrices having joint probability distribution $P(x_1\ldots,x_n)=C_{N \beta} \, \prod_{j<k}w(x_j)\left | x_j-x_k \right |^\beta$ where $w(x)=x^{N\beta a}\exp\left(-N\beta b x\right)$ and $\beta$ is the Dyson parameter. The corresponding level density gives Marchenko-Pastur (MP) distribution while fluctuations of every participating team agrees with the universal behavior of Gaussian Unitary Ensemble (GUE). We analyze the components of the deviating eigenvalues and find that the largest eigenvalue corresponds to an influence common to all matches played during these periods.

In this paper I invite you to take a step aside current quantum field theory (QFT): QFT has been said to be "well-established" since the 80's of the last century by its foremost theorists), and the majority of physicists consider it to be essentially complete since the discovery of the Higgs particle. It will be interesting to see, what that really means: What are the problems left over to the younger generations? I'll show you that a.o. it fails in its Lagrangian formalism, its postulate of positivity of energy, I'll show the uselessness of the uncertainty principle as to electromagnetic fields, and we'll see that there are serious doubts as to its conception of the photonic nature of electromagnetic fields, which a simple experiment could test against.

Some Lie-algebraic structures of three-dimensional quantum Nambu mechanics are studied. From our result, we argue that the three-dimensional quantum Nambu mechanics is a natural extension of the ordinary Heisenberg quantum theory, and we give our insight that we can construct several candidates "beyond the Heisenberg quantum theory".

Various geometric aspects of the Nambu-Goldstone ( NG ) type symmetry breakings ( normal, generalized, and anomalous NG theorems ) are summarized, and their relations are discussed. By the viewpoint of Riemannian geometry, Laplacian, curvature and geodesics are examined. Theory of Ricci ow is investigated in complex geometry of the NG-type theorems, and its diusion and stochastic forms are derived. In our anomalous NG theorems, the structure of symplectic geometry is emphasized, Lagrangian submanifolds and mirror duality are noticed. Possible relations between the Langlands correspondence, the Riemann hypothesis and the geometric nature of NG-type theorems are given.

On the basis of the Silagadze research[1], we investigate the question of the definitions of the discrete symmetry operators both on the classical level, and in the secondary-quantization scheme [2,3]. We studied the physical content within several bases: light-front form formulation [4], helicity basis, angular momentum basis, and so on, on several practical examples. The conclusion is that we have ambiguities in the definitions of the corresponding operators P, C; T, which lead to different physical consequences [5,6].

Introducing a special quaternionic vector calculus on the tangent bundle of a 4-dimensional space, and by forcing a condition of holomorphism, a Minkowski-type spacetime emerges, from which special relativity, gravitation and also the whole Maxwell theory of electromagnetic fields arises.

Recently, several discussions on the possible observability of 4-vector fields have been published in literature. Furthermore, several authors recently claimed existence of the helicity=0 fundamental field. We re-examine the theory of antisymmetric tensor fields and 4-vector potentials. We study the massless limits. In fact, a theoretical motivation for this venture is the old papers of Ogievetskii and Polubarinov, Hayashi, and Kalb and Ramond. They proposed the concept of the notoph, whose helicity properties are complementary to those of the photon. We analyze the quantum field theory with taking into account mass dimensions of the notoph and the photon. We also proceed to derive equations for the symmetric tensor of the second rank on the basis of the Bargmann-Wigner formalism They are consistent with the general relativity. Particular attention has been paid to the correct definitions of the energy-momentum tensor and other Noether currents. We estimate possible interactions, fermion-notoph, graviton-notoph, photon-notoph. PACS number: 03.65.Pm , 04.50.-h , 11.30.Cp

It has for long been been overlooked that, quite easily, infinitely many {\it ultrapower} field extensions $\mathbb{F}_{\cal U}$ can be constructed for the usual field $\mathbb{R}$ of real numbers, by using only elementary algebra. This allows a simple and direct access to the benefit of both infinitely small and infinitely large scalars, {\it without} the considerable usual technical difficulties involved in setting up and then using the Transfer Principle in Nonstandard Analysis. A natural Differential and Integral Calculus - which extends the usual one on the field $\mathbb{R}$ - is set up in these fields $\mathbb{F}_{\cal U}$ without any use of the Transfer Principle in Nonstandard Analysis, or of any topological type structure. Instead, in the case of the Riemann type integrals introduced, three simple and natural axioms in Set Theory are assumed. The case when these three axioms may be inconsistent with the Zermelo-Fraenkel Set Theory is discussed in section 5.

Infinitely many {\it ultrapower} field extensions $\mathbb{F}_{\cal U}$ are constructed for the usual field $\mathbb{R}$ of real numbers by using only elementary algebra, thus allowing for the benefit of both infinitely small and infinitely large scalars, and doing so {\it without} the considerable usual technical difficulties involved in setting up the Transfer Principle in Nonstandard Analysis. A natural Integral Calculus - which extends the usual one on the field $\mathbb{R}$ - is set up in these fields $\mathbb{F}_{\cal U}$. A separate paper presents the same for the Differential Calculus.

Arguments in favor of existence of Green's functions for all linear equations are analyzed. In case of equation for electromagnetic field, these arguments have been widely used through formal considerations according to which electromagnetic field equations are nothing but some non-covariant scalar equations. We criticize these considerations and show that justification of applying the method of Green's functions to equations of classical electrodynamics are invalid. Straightforward calculations are presented which show that in case of dipole radiation the method gives incorrect results.

The fluid equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term - hence describing viscous flow. Due to specific of NS equations they could be transformed into full/partial inhomogeneous parabolic differential equations: differential equations in respect of space variables and the full differential equation in respect of time variable and time dependent inhomogeneous part. Velocity and outer forces densities components were expressed in form of curl for obtaining solutions satisfying continuity condition specifying divergence of velocities equality to zero. Finally, solution in 3D space for any shaped boundary was expressed in terms of 3D Green function and inverse Laplace transform accordantly.

The phenomenon of reflection from conducting surface is considered in terms of exact solutions of Maxwell equations. Matching of waves and current density at the plane is completed. Amplitudes of reflected and transmitted waves are found as functions of incident wave and conductivity of the plane. This work is completed also for conducting plane lying between two distinct media. It is shown that in case of conducting interface waves with some certain parameters (polarization, incidence angle and frequency) and transform completely into waves of current density whereas amplitude of the reflected wave is equal to zero that is equivalent to total absorption.

This paper established a physical axiom system. By the axiom system we can derive important physical laws such as momentum conservation law, Newton's second law, Newton's law of gravity, Schrodinger equation and Maxwell's equations, simplified existing physical theories and explain some physical phenomenons those unresolved by traditional physical theories. We can also derive Schwarzschild solution of external spherically symmetric gravitational field, gravitational red shift equation, proved that if in large-scale distance, Newton's law of gravity and red shift equation must be corrected, the data by corrected formulas meet the astronomical observed results well.

The phenomenon of total internal reflection is considered in terms of exact solutions of Maxwell equations. Matching of plane and evanescent waves at the interface is completed. It is shown that amplitude of the reflected wave cannot be obtained from the matching alone. Since it can differ from that of incident wave due to possible energy loss which may occur in the evanescent wave zone if there is another layer of optically dense medium, this loss is to be specified via amplitude of the reflected wave. Besides, reflected wave potential has phase shift which also depends on this specification.

It is shown that the action function of a macroscopic adiabatic system of particles described as continuously differentable functions of energy-momentum in space-time, exists, that this is a plane wave, and that this function can in turn be integrated to a 4-vector field, which satisfies the Maxwell equations in the Lorentz gauge. Also, it is shown, how to formulate these results in terms of Functional Analysis of Hilbert spaces. With it, we show a.o. that PCT = -CPT = ±1 holds, which is a strong form of the PCT-theorem; we show that - in order to capture the concept of mass - the standard model gauge group has to be augmented by a factor group U(2), such that the complete gauge group becomes U(4). It is shown that the sourceless action field in itself suffices to describe the long ranged interaction of matter, both, electromagnetic and gravitational. This turns Einstein’s conception of photons as real particles and subsequently the concept of gravitons into physically unproven assumptions, which complicate, but not simplify the theory. The results appear to imply that the fields themselves do not interact with their sources. Though, this has never been checked by an experiment. As shown, a simple experiment could be carried out to answer this question.

We check that the relation between the angle and the radius of the movement of an object following a logarithm spiral of p p SO(2) is constant.

We calculate the torsion free spin connection on the quantum group Bq[SU2] at the fourth root of unity. From this we deduce the covariant derivative and the Riemann curvature. Next we compute the Dirac operator of this quantum group and we give numerical approximations of its eigenvalues.

We study the noncommutative geometry of the dihedral group D6 using the tools of quantum group theory. We explicit the torsion free regular spin connection and the corresponding 'Levi-Civita' connection. Next, we nd the Riemann curvature and its Ricci tensor. The main result is the Dirac operator of a representation of the group which we nd the eigenvalues and the eigenmodes

We study the Borel algebra dene by [xa; xb] = 2a;1xb as a noncommutative manifold R 3 . We calculate its noncommutative dierential form relations. We deduce its partial derivative relations and the derivative of a plane wave. After calculating its de Rham cohomology, we deduce the wave operator and its corresponding magnetic solution

In a series of papers written over the period 1944-1948, the great Austrian physicist Erwin Schrödinger presented his ideas on symmetric and non-symmetric affine connections and their possible application to general relativity. Several of these ideas were subsequently presented in his notable 1950 book "Space-Time Structure," in which Schrödinger outlined the case for both metric and general connections, symmetric and otherwise. In the following discussion we focus on one particular connection presented by Schrödinger in that book and its relationship with the non-metricity tensor. We also discuss how this connection overcomes a problem that Hermann Weyl experienced with the connection he proposed in his failed 1918 theory of the combined gravitational-electromagnetic field. A simple physical argument is then presented demonstrating that Schrödingers’s formalism accommodates electromagnetism in a more natural way than Weyl’s theory.

A special case of the cubic equation, distinguished by having an unusually economical solution, is shown to relate to both the fine structure constant inverse (approximately 137.036) and the sines squared of the quark and lepton mixing angles.

The fluid equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term - hence describing viscous flow. Due to specific of NS equations they could be transformed to full/partial inhomogeneous parabolic differential equations: differential equations in respect of space variables and the full differential equation in respect of time variable and time dependent inhomogeneous part. Finally, orthogonal polynomials as the partial solutions of obtained Helmholtz equations were used for derivation of analytical solution of incompressible fluid equations in 1D, 2D and 3D space for rectangular boundary. Solution in 2D and 3D space for any shaped boundary was expressed in term of 2D and 3D global solution of Helmholtz equation accordantly.

A framework for calculations in a semi-Riemannian space with the typical metric connection and curvature expressions is developed, with an emphasis on deriving them from an embedding function as a more fundamental object than the metric tensor. The scale-invariant and 'linearizing' logarithmic nature of an 'infinitesimal embedding' of a tangent space into its neighbourhood is observed, and a composition scheme of spacetime scenarios from 'outer' non-invariant and 'inner' scale-invariant embeddings is briefly outlined.

Mathematical provability , then classification, of Saint-Venant's Principle are discussed. Beginning with the simplest case of Saint-Venant's Principle, four problems of elasticity are discussed mathematically. It is concluded that there exist two categories of elastic problems concerning Saint-Venant's Principle: Experimental Problems, whose Saint-Venant's Principle is established in virtue of supporting experiment, and Analytical Problems, whose Saint-Venant's decay is proved or disproved mathematically, based on fundamental equations of linear elasticity. The boundary-value problems whose stress boundary condition consists of Dirac measure, a "singular distribution ", can not be dealt with by the mathematics of elasticity for " proof " or "disproof " of their Saint-Venant's decay, in terms of mathematical coverage.

The Simulation Hypothesis proposes that all of reality is an artificial simulation, analogous to a computer simulation. Outlined here is a low computational cost method for programming cosmic microwave background parameters in Planck time Simulation Hypothesis Universe. The model initializes `micro Planck-size black-holes' as entities that embed the Planck units. For each incremental unit of Planck time, the universe expands by adding 1 micro black-hole, a dark energy is not required. The mass-space parameters increment linearly, the electric parameters in a sqrt-progression, thus for electric parameters the early black-hole transforms most rapidly. The velocity of expansion is constant and is the origin of the speed of light, the Hubble constant becomes a measure of the black-hole radius and the CMB radiation energy density correlates to the Casimir force. A peak frequency of 160.2GHz correlates to a 14.624 billion year old black-hole. The cosmological constant, being the age when the simulation reaches the limit, approximates $t$ = 10$^{123} t_p$.

It was proposed new gauge invariant Lagrangian, where the gauge field interact with the charged electromagnetic fields. Gauge invariance was archived by replacing of particle mass with new one invariant of the field $F_{\mu\nu}F^{\mu\nu}$ multiplied with calibration constant $\alpha_g$. It was shown that new proposed Lagrangian generates similar Dirac and electromagnetic field equations. Solution of Dirac equations for a free no massless particle answers to the 'question of the age' why free particle deal in experiments like a de Broil wave. Resulting wave functions of the new proposed Lagrangian will describe quantized list of bespinor particles of different masses. Finally, it was shown that renormalization of the new proposed Lagrangian is similar to QED in case similarity of new proposed Lagrangian to classic QED.

The Simulation Hypothesis proposes that all of reality, including the earth and the universe, is in fact an artificial simulation, analogous to a computer simulation, and as such our reality is an illusion. In this essay I describe a method for programming mass, length, time and charge (MLTA) as geometrical objects derived from the formula for a virtual electron; $f_e = 4\pi^2r^3$ ($r = 2^6 3 \pi^2 \alpha \Omega^5$) where the fine structure constant $\alpha$ = 137.03599... and $\Omega$ = 2.00713494... are mathematical constants and the MLTA geometries are; M = (1), T = ($2\pi$), L = ($2\pi^2\Omega^2$), A = ($4\pi \Omega)^3/\alpha$. As objects they are independent of any set of units and also of any numbering system, terrestrial or alien. As the geometries are interrelated according to $f_e$, we can replace designations such as ($kg, m, s, A$) with a rule set; mass = $u^{15}$, length = $u^{-13}$, time = $u^{-30}$, ampere = $u^{3}$. The formula $f_e$ is unit-less ($u^0$) and combines these geometries in the following ratio M$^9$T$^{11}$/L$^{15}$ and (AL)$^3$/T, as such these ratio are unit-less. To translate MLTA to their respective SI Planck units requires an additional 2 unit-dependent scalars. We may thereby derive the CODATA 2014 physical constants via the 2 (fixed) mathematical constants ($\alpha, \Omega$), 2 dimensioned scalars and the rule set $u$. As all constants can be defined geometrically, the least precise constants ($G, h, e, m_e, k_B$...) can also be solved via the most precise ($c, \mu_0, R_\infty, \alpha$), numerical precision then limited by the precision of the fine structure constant $\alpha$.

The Statement of Modified Saint-Venant’s Principle is suggested. The axisymmetrical deformation of the infinite circular cylinder loaded by an equilibrium system of forces on its near end is discussed and its formulation of Modified Saint-Venant’s Principle is established. It is evident that finding solutions of boundary-value problems is a precise and pertinent approach to establish Saint-Venant type decay of elastic problems.

The formulation of quantum mechanics with a complex Hilbert space is equivalent to a formulation with a real Hilbert space and particular density matrix and observables. We study the real representations of the Poincare group, motivated by the fact that the localization of complex unitary representations of the Poincare group is incompatible with causality, Poincare covariance and energy positivity. We review the map from the complex to the real irreducible representations—finite- dimensional or unitary—of a Lie group on a Hilbert space. Then we show that all the finite-dimensional real representations of the identity component of the Lorentz group are also representations of the parity, in contrast with many complex representations. We show that any localizable unitary representation of the Poincare group, compatible with Poincare covariance, verifies: 1) it is self-conjugate (regardless it is real or complex); 2) it is a direct sum of irreducible representations which are massive or massless with discrete helicity. 3) it respects causality; 4) it is an irreducible representation of the Poincare group (including parity) if and only if it is: a)real and b)massive with spin 1/2 or massless with helicity 1/2. Finally, the energy positivity problem is discussed in a many-particles context.

We construct self/anti-self charge conjugate (Majorana-like) states for the (1/2,0)+(0,1/2)$ representation of the Lorentz group, and their analogs for higher spins within the quantum field theory. The problem of the basis rotations and that of the selection of phases in the Dirac-like and Majorana-like field operators are considered. The discrete symmetries properties (P, C, T) are studied. Particular attention has been paid to the question of (anti)commutation of the Charge conjugation operator and the Parity in the helicity basis. Dynamical equations have also been presented. In the (1/2,0)+(0,1/2) representation they obey the Dirac-like equation with eight components, which has been first introduced by Markov. Thus, the Fock space for corresponding quantum fields is doubled (as shown by Ziino). The chirality and the helicity (two concepts which are frequently confused in the literature) for Dirac and Majorana states have been discussed.

We propose a preliminary algorithm which is designed to reduce aspects of the n-body problem to a 2-body problem for holographic principle compliance. The objective is to share an alternative view-point on the n-body problem to try and generate a simpler solution in the future. The algorithm operates 2D and 3D data structures to initiate the encoding of the chaotic dynamical system equipped with modified superfluid order parameter fields in both 3D and 4D versions of the Inopin holographic ring (IHR) topology. For the algorithm, we arbitrarily select one point-mass to be the origin and, from that reference frame, we subsequently engage a series of instructions to consolidate the residual (n-1)-bodies to the IHR. Through a step-by-step example, we demonstrate that the algorithm yields "IHR effective" (IHRE) net quantities that enable us to hypothetically define an IHRE potential, kinetic, and Lagrangian.

There are Poincare group representations on complex Hilbert spaces, like the Dirac spinor field, or real Hilbert spaces, like the electromagnetic field tensor. The Majorana spinor is an element of a 4 dimensional real vector space. The Majorana spinor field is a space-time dependent Majorana spinor, solution of the free Dirac equation. The Majorana-Fourier and Majorana-Hankel transforms of Majorana spinor fields are defined and related to the linear and angular momenta of a spin one-half representation of the Poincare group. We show that the Majorana spinor field with finite mass is an unitary irreducible projective representation of the Poincare group on a real Hilbert space. Since the Bargmann-Wigner equations are valid for all spins and are based on the free Dirac equation, these results open the possibility to study Poincare group representations with arbitrary spins on real Hilbert spaces.

Zanaboni Theory is mathematically analyzed in this paper. The conclusion is that Zanaboni Theorem is invalid and not a proof of Saint-Venant's Principle; Discrete Zanaboni Theorem and Zanaboni's energy decay are inconsistent with Saint-Venant's decay; the inconsistency, discussed here, between Zanaboni Theory and Saint-Venant's Principle provides more proofs that Saint-Venant's Principle is not generally true.

The problem of statement of Saint-Venant's Principle is concerned. Statement of Boussinesq or Love is ambiguous so that its interpretations are in contradiction with each other. Rationalized Statement of Saint-Venant’s Principle of elasticity is suggested to rule out the ambiguity of Statements of Boussinesq and Love. Rational Saint-Venant's Principle is suggested to fit and guide applications of the principle to fields of continuum physics and cover the analogical case as well as the non-analogical case discovered and discussed in this paper . `` Constraint-free " problems are suggested and `` Constraint-free " Rational Saint-Venant's Principle or Rational Saint-Venant's Principle with Relaxed Boundary Condition is developed to generalize the principle and promote its applications to fields of continuum physics . Applications of Analogical Rational Saint-Venant's Principle and `` Constraint-free " Rational Saint-Venant's Principle are exemplified, emphasizing `` properness " of the boundary-value problems. Three kinds of properly posed boundary-value problems, i.e., the boundary-value problem with the undetermined boundary function, the boundary-value problem with the implicit boundary condition and the boundary-value problem with the explicit boundary condition, are suggested for both `` constrained " and `` constraint-free " problems.

Orbital averages are employed to compute the secular variation of the elliptical planetary elements in the orbital plane in presence of perturbing forces of various kinds. They are also useful as an aid in the computation of certain complex integrals. An extensive list of computed integrals is given.

In this work, we forge a powerful, easy-to-visualize, flexible, consistent, and disciplined abstract vector framework for particle and astro physics that is compliant with the holographic principle. We demonstrate that the structural properties of the complex number and the sphere enable us to introduce and define the triplex number---an influential information structure that is similar to the 3D hyper-complex number by D. White and P. Nylander---which identifies a 3D analogue of (2D) complex space. Consequently, we engage the complex and triplex numbers as abstract vectors to systematically encode the state space of the Riemannian dual 3D and 4D space-time topologies, where space and time are dual and interconnected; we use the triplex numbers (with triplex multiplication) to extend 1D and 2D algebraic systems to 3D and 4D configurations. In doing so, we equip space-time with order parameter fields for topological deformations. Finally, to exemplify our motivation, we provide three example applications for this framework.

It is demonstrated that, for the recently introduced classical magnetized Kepler problems in dimension $2k+1$, the non-colliding orbits in the ``external configuration space" $\mathbb R^{2k+1}\setminus\{\mathbf 0\}$ are all conics, moreover, a conic orbit is an ellipse, a parabola, and a branch of a hyperbola according as the total energy is negative, zero, and positive. It is also demonstrated that the Lie group ${\mr {SO}}^+(1,2k+1)\times {\bb R}_+$ acts transitively on both the set of oriented elliptic orbits and the set of oriented parabolic orbits.

I have been reading two highly interesting articles by Robert Kiehn. There are very many contacts on TGD inspired vision and its open interpretational problems. The notion of Falaco soliton has surprisingly close resemblance with K\"ahler magnetic flux tubes defining fundamental structures in TGD Universe. Fermionic strings are also fundamental structures of TGD accompanying magnetic flux tubes and this supports the vision that these string like objects could allow reduction of various condensed matter phenomena such as sound waves -usually regarded as emergent phenomena allowing only highly phenomenological description - to the fundamental microscopic level in TGD framework. This can be seen as the basic outcome of this article. Kiehn proposed a new description for the generation of various instability patterns of hydrodynamics flows (Kelvin-Helmholtz and Rayleigh-Taylor instabilities) in terms of hyperbolic dynamics so that a connection with wave phenomena like interference and diffraction would emerge. The role of characteristic surfaces as surfaces of tangential and also normal discontinuities is central for the approach. In TGD framework the characteristic surfaces have as analogs light-like wormhole throats at which the signature of the induced 4-metric changes and these surfaces indeed define boundaries of two phases and of material objects in general. This inspires a more detailed comparison of Kiehn's approach with TGD.

The original focus of this article was p-adic icosahedron. The discussion of attempt to define this notion however leads to the challenge of defining the concept of p-adic sphere, and more generally, that of p-adic manifold, and this problem soon became the main target of attention since it is one of the key challenges of also TGD. There exists two basic philosophies concerning the construction of both real and p-adic manifolds: algebraic and topological approach. Also in TGD these approaches have been competing: algebraic approach relates real and p-adic space-time points by identifying the common rationals. Finite pinary cutoff is however required to achieve continuity and has interpretation in terms of finite measurement resolution. Canonical identification maps p-adics to reals and vice versa in a continuous manner but is not consistent with p-adic analyticity nor field equations unless one poses a pinary cutoff. It seems that pinary cutoff reflecting the notion of finite measurement resolution is necessary in both approaches. This represents a new notion from the point of view of mathematics. a) One can try to generalize the theory of real manifolds to p-adic context. The basic problem is that p-adic balls are either disjoint or nested so that the usual construction by gluing partially overlapping spheres fails. This leads to the notion of Berkovich disk obtained as a completion of p-adic disk having path connected topology (non-ultrametric) and containing p-adic disk as a dense subset. This plus the complexity of the construction is heavy price to be paid for path-connectedness. A related notion is Bruhat-Tits tree defining kind of skeleton making p-adic manifold path connected. The notion makes sense for the p-adic counterparts of projective spaces, which suggests that p-adic projective spaces (S² and CP₂ in TGD framework) are physically very special. b) Second approach is algebraic and restricts the consideration to algebraic varieties for which also topological invariants have algebraic counterparts. This approach looks very natural in TGD framework - at least for imbedding space. Preferred extremals of Kähler action can be characterized purely algebraically - even in a manner independent of the action principle - so that they might make sense also p-adically. Number theoretical universality is central element of TGD. Physical considerations force to generalize the number concept by gluing reals and various p-adic number fields along rationals and possible common algebraic numbers. This idea makes sense also at the level of space-time and of "world of classical worlds" (WCW). Algebraic continuation between different number fields is the key notion. Algebraic continuation between real and p-adic sectors takes place along their intersection which at the level of WCW correspond to surfaces allowing interpretation both as real and p-adic surfaces for some value(s) of prime p. The algebraic continuation from the intersection of real and p-adic WCWs is not possible for all p-adic number fields. For instance, real integrals as functions of parameters need not make sense for all p-adic number fields. This apparent mathematical weakness can be however turned to physical strength: real space-time surfaces assignable to elementary particles can correspond only some particular p-adic primes. This would explain why elementary particles are characterized by preferred p-adic primes. The p-adic prime determining the mass scale of the elementary particle could be fixed number theoretically rather than by some dynamical principle formulated in real context (number theoretic anatomy of rational number does not depend smoothly on its real magnitude!). Although Berkovich construction of p-adic disk does not look promising in TGD framework, it suggests that the difficulty posed by the total disconnectedness of p-adic topology is real. TGD in turn suggests that the difficulty could be overcome without the completion to a non-ultrametric topology. Two approaches emerge, which ought to be equivalent. a) The TGD inspired solution to the construction of path connected effective p-adic topology is based on the notion of canonical identification mapping reals to p-adics and vice versa in a continuous manner. The trivial but striking observation was that canonical identification satisfies triangle inequality and thus defines an Archimedean norm allowing to induce real topology to p-adic context. Canonical identification with finite measurement resolution defines chart maps from p-adics to reals and vice versa and preferred extremal property allows to complete the discrete image to hopefully space-time surface unique within finite measurement resolution so that topological and algebraic approach are combined. Finite resolution would become part of the manifold theory. p-Adic manifold theory would also have interpretation in terms of cognitive representations as maps between realities and p-adicities. b) One can ask whether the physical content of path connectedness could be also formulated as a quantum physical rather than primarily topological notion, and could boil down to the non-triviality of correlation functions for second quantized induced spinor fields essential for the formulation of WCW spinor structure. Fermion fields and their n-point functions could become part of a number theoretically universal definition of manifold in accordance with the TGD inspired vision that WCW geometry - and perhaps even space-time geometry - allow a formulation in terms of fermions. This option is a mere conjecture whereas the first one is on rigorous basis.

We study the relation between the Guzwiller Trace for a dynamical system and the Riemann-Weil trace formula for the Riemann zeros, using the Bohr-Sommerfeld quantization condition and the fractional calculus we obtain a method to define implicitly a potential , we apply this method to define a Hamiltonian whose energies are the square of the Riemann zeros (imaginary part) , also we show that for big ‘x’ the potential is very close to an exponential function. In this paper and for simplicity we use units so • Keywords: = Riemann Hypothesis, WKB semiclassical approximation, Gutzwiller trace formula, Bohr-Sommerfeld quantization,exponential potential.

In this paper, I introduce a particular discrete spacetime that should be seriously considered as part of physics because it allows to explain the characteristics of the motion properly, contrary to what happens with the continuous spacetime of the common conception.

The study of magnetic fields produced by steady currents is a full-valued physical theory which like any other physical theory employs a certain mathematics. This theory has two limiting cases in which source of the field is confined on a surface or a curve. It turns out that mathematical methods to be used in these cases are completely different and differ from from that of the main of the main part of this theory, so, magnetostatics actually consists of three distinct theories. In this work, these three theories are discussed with special attention to the case current carried by a curve. In this case the source serves as a model of thin wire carrying direct current, therefore this theory can be termed magnetostatics of thin wires. The only mathematical method used in this theory till now, is the method of Green's functions. Critical analysis of this method completed in this work, shows that application of this method to the equation for vector potential of a given current density has no foundation and application of this method yields erroneous results

Let ${\mathbb R}^{2k+1}_*={\mathbb R}^{2k+1}\setminus\{\vec 0\}$ ($k\ge 1$) and $\pi$: ${\mathbb R}^{2k+1}_*\to \mathrm{S}^{2k}$ be the map sending $\vec r\in {\mathbb R}^{2k+1}_*$ to ${\vec r\over |\vec r|}\in \mathrm{S}^{2k}$. Denote by $P\to {\mathbb R}^{2k+1}_*$ the pullback by $\pi$ of the canonical principal $\mathrm{SO}(2k)$-bundle $\mathrm{SO}(2k+1)\to \mathrm{S}^{2k} $. Let $E_\sharp\to {\mathbb R}^{2k+1}_*$ be the associated co-adjoint bundle and $E^\sharp\to T^*{\mathbb R}^{2k+1}_*$ be the pullback bundle under projection map $T^*{\mathbb R}^{2k+1}_*\to {\mathbb R}^{2k+1}_*$. The canonical connection on $\mathrm{SO}(2k+1)\to \mathrm{S}^{2k} $ turns $E^\sharp$ into a Poisson manifold. The main result here is that the real Lie algebra $\mathfrak{so}(2, 2k+2)$ can be realized as a Lie subalgebra of the Poisson algebra $(C^\infty(\mathcal O^\sharp), \{, \})$, where $\mathcal O^\sharp$ is a symplectic leave of $E^\sharp$ of special kind. Consequently, in view of the earlier result of the author, an extension of the classical MICZ Kepler problems to dimension $2k+1$ is obtained. The hamiltonian, the angular momentum, the Lenz vector and the equation of motion for this extension are all explicitly worked out.

Violating the law of energy conservation, Zanaboni Theorem is invalid and Zanaboni's proof is wrong. Zanaboni's mistake of " proof " is analyzed. Energy Theorem for Zanaboni Problem is suggested and proved. Equations and conditions are established in this paper for Zanaboni Problem, which are consistent with , equivalent or identical to each other. Zanaboni Theorem is, for its invalidity , not a mathematical formulation or proof of Saint-Venant's Principle. AMS Subject Classifications: 74-02, 74G50

Constraints imposed directly on accelerations of the system leading to the relation of constants of motion with appropriate local projectors occurring in the derived equations are considered. In this way a generalization of the Noether's theorem is obtained and constraints are also considered in the phase space.

How to relate the physical \emph{real} reality with the logical \emph{true} abstract mathematics concepts is nothing but pure postulate. The most basic postulates of physics are by using what kind of mathematics to describe the most fundamental concepts of physics. Main point of relativity theories is to remove incorrect and simplify the assumptions about the nature of space-time. There are plentiful bonus of doing so, for example gravity emerges as natural consequence of curvature of spacetime. We argue that the Einstein version of general relativity is not complete, since it can't explain quantum phenomenon. If we want to reconcile quantum, we should give up one implicit assumption we tend to forget: the differentiability. What would be the benefits of these changes? It has many surprising consequences. We show that the weird uncertainty principle and non-commutativity become straightforward in the circumstances of non-differentiable functions. It's just the result of the divergence of usual definition of \emph{velocity}. All weirdness of quantum mechanics are due to we are trying to making sense of nonsense. Finally, we proposed a complete relativity theory in which the spacetime are non-differentiable manifold, and physical law takes the same mathematical form in all coordinate systems, under arbitrary differentiable or non-differentiable coordinate transformations. Quantum phenomenon emerges as natural consequence of non-differentiability of spacetime.

Memristor was postulated by Chua in 1971 by analyzing mathematical relations between pairs of fundamental circuit variables and realized by HP laboratory in 2008. This relation can be generalized to include any class of two-terminal devices whose properties depend on the state and history of the system. These are called memristive systems, including current-voltage for the memristor, charge-voltage for the memcapacitor, and current-flux for the meminductor. This paper further enlarge the family of elementary circuit elements, in order to model many irregular and exotic nondifferentiable phenomena which are common and dominant to the nonlinear dynamics of many biological, molecular and nanodevices.

• ABSTRACT: We give and interpretation of the Riemann Xi-function as the quotient of two functional determinants of an Hermitian Hamiltonian . To get the potential of this Hamiltonian we use the WKB method to approximate and evaluate the spectral Theta function over the Riemann zeros on the critical strip . Using the WKB method we manage to get the potential inside the Hamiltonian , also we evaluate the functional determinant by means of Zeta regularization, we discuss the similarity of our method to the method applied to get the Zeros of the Selberg Zeta function. In this paper and for simplicity we use units so • Keywords: = Riemann Hypothesis, Functional determinant, WKB semiclassical Approximation , Trace formula ,Bolte’s law, Quantum chaos.

A new interpretation of the basic vector of the free Fock space (FFS) and the FFS is proposed. The approximations to various equations with additional parameters, for n-point information (n-pi), are also considered in the case of non-polynomial nonlinearities. Key words: basic, generating and state vectors, local and global, Cuntz relations, perturbation and closure principles, homotopy analysis method, Axiom of Choice, consilience.

The problem of a cylinder with a small spherical cavity loaded by an equilibrium system of forces is suggested and discussed and its formulation of Saint-Venant's Principle is established. It is evident that finding solutions of boundary-value problems is a precise and pertinent approach to establish Saint-Venant type decay of elastic problems. Keywords : Saint-Venant’s Principe, proof, provability, solution, decay, formulation, cavity AMS Subject Classifications: 74-02, 74G50

The Statement of Modified Saint-Venant's Principle is suggested. The axisymmetrical deformation of the infinite circular cylinder loaded by an equilibrium system of forces on its near end is discussed and its formulation of Modified Saint-Venant's Principle is established. It is evident that finding solutions of boundary-value problems is a precise and pertinent approach to establish Saint-Venant type decay of elastic problems. AMS Subject Classifications: 74-02, 74G50

It appears not to be known that subjecting the axioms to certain conditions, such as for instance to be physically meaningful, may interfere with the logical essence of axiomatic systems, and do so in unforeseen ways, ways that should be carefully considered and accounted for. Consequently, the use of "physical intuition" in building up axiomatic systems for various theories of Physics may lead to situations which have so far not been carefully considered.

This paper presents the phenomenon of disconnect in the axiomatic approach to theories of Physics, a phenomenon which appears due to the insistence on axioms which have a physical meaning. This insistence introduces a restriction which is foreign to the abstract nature of axiomatic systems as such. Consequently, it turns out to introduce as well the mentioned disconnect. The axiomatic approach in Physics has a longer tradition. It is there already in Newton's Principia. Recently for instance, a number of axiomatic approaches have been proposed in the literature related to Quantum Mechanics. Special Relativity, [2], had from its beginning in 1905 been built upon two axioms, namely, the Galilean Relativity and the Constancy of the Speed of Light in inertial reference frames. Hardly noticed in wider circles, the independence of these two axioms had quite early been subjected to scrutiny, [5,3], and that issue has on occasion been addressed ever since, see [8,4,24] and the literature cited there. Recently, [24], related to these two axioms in Special Relativity, the following phenomenon of wider importance in Physics was noted. As the example of axiomatization of Special Relativity shows it, it is possible to face a disconnect between a system of physically meaningful axioms, and on the other hand, one or another of the mathematical models used in the study of the axiomatized physical theory. The consequence is that, seemingly unknown so far, one faces in Physics the possibility that the axiomatic method has deeper, less obvious, and in fact not considered, or simply overlooked limitations. As there is no reason to believe that the system of the usual two axioms of Special Relativity is the only one subjected to such a disconnect, the various foundational ventures in modern Physics, related for instance to gravitation, quanta, or their bringing together in an overarching theory, may benet from the study of the possible sources and reasons for such a disconnect. An attempt of such study is presented in this paper.

A generalization of number concept is proposed. One can replace integer n with n-dimensional Hilbert space and sum + and product × with direct sum ⊕ and tensor product ⊗ and introduce their co-operations, the definition of which is highly non-trivial. </p><p> This procedure yields also Hilbert space variants of rationals, algebraic numbers, p-adic number fields, and even complex, quaternionic and octonionic algebraics. Also adeles can be replaced with their Hilbert space counterparts. Even more, one can replace the points of Hilbert spaces with Hilbert spaces and repeat this process, which is very similar to the construction of infinite primes having interpretation in terms of repeated second quantization. This process could be the counterpart for construction of n^th order logics and one might speak of Hilbert or quantum mathematics. The construction would also generalize the notion of algebraic holography and provide self-referential cognitive representation of mathematics. </p><p> This vision emerged from the connections with generalized Feynman diagrams, braids, and with the hierarchy of Planck constants realized in terms of coverings of the imbedding space. Hilbert space generalization of number concept seems to be extremely well suited for the purposes of TGD. For instance, generalized Feynman diagrams could be identifiable as arithmetic Feynman diagrams describing sequences of arithmetic operations and their co-operations. One could interpret ×_q and +_q and their co-algebra operations as 3-vertices for number theoretical Feynman diagrams describing algebraic identities X=Y having natural interpretation in zero energy ontology. The two vertices have direct counterparts as two kinds of basic topological vertices in quantum TGD (stringy vertices and vertices of Feynman diagrams). The definition of co-operations would characterize quantum dynamics. Physical states would correspond to the Hilbert space states assignable to numbers. One prediction is that all loops can be eliminated from generalized Feynman diagrams and diagrams are in projective sense invariant under permutations of incoming (outgoing legs).

Absolute Galois Group defined as Galois group of algebraic numbers regarded as extension of rationals is very difficult concept to define. The goal of classical Langlands program is to understand the Galois group of algebraic numbers as algebraic extension of rationals - Absolute Galois Group (AGG) - through its representations. Invertible adeles -ideles - define Gl₁ which can be shown to be isomorphic with the Galois group of maximal Abelian extension of rationals (MAGG) and the Langlands conjecture is that the representations for algebraic groups with matrix elements replaced with adeles provide information about AGG and algebraic geometry. </p><p> I have asked already earlier whether AGG could act is symmetries of quantum TGD. The basis idea was that AGG could be identified as a permutation group for a braid having infinite number of strands. The notion of quantum adele leads to the interpretation of the analog of Galois group for quantum adeles in terms of permutation groups assignable to finite l braids. One can also assign to infinite primes braid structures and Galois groups have lift to braid groups. </p><p> Objects known as dessins d'enfant provide a geometric representation for AGG in terms of action on algebraic Riemann surfaces allowing interpretation also as algebraic surfaces in finite fields. This representation would make sense for algebraic partonic 2-surfaces, and could be important in the intersection of real and p-adic worlds assigned with living matter in TGD inspired quantum biology, and would allow to regard the quantum states of living matter as representations of AGG. Adeles would make these representations very concrete by bringing in cognition represented in terms of p-adics and there is also a generalization to Hilbert adeles.

Information being a relatively new concept in science, the likelihood is pointed out that we do not yet have a good enough grasp of its nature and relevance. This likelihood is further enhanced by the ubiquitous use of information which creates the perception of a manifest, yet in fact, rather superficial familiarity. The paper suggests several aspects which may be essential features of information, or on the contrary, may not be so. In this regard, further studies are obviously needed, studies which may have to avoid with care various temptations to reductionism, like for instance the one claiming that ``information is physical".

<p> The arguments of this article support the view that in TGD Universe number theoretic and geometric Langlands conjectures could be understood very naturally. The basic notions are following. </p><p> <OL> <LI>Zero energy ontology (ZEO) and the related notion of causal diamond CD (CD is short hand for the cartesian product of causal diamond of M⁴ and of CP₂). ZEO leads to the notion of partonic 2-surfaces at the light-like boundaries of CD and to the notion of string world sheet. These notions are central in the recent view about TGD. One can assign to the partonic 2-surfaces a conformal moduli space having as additional coordinates the positions of braid strand ends (punctures). By electric-magnetic duality this moduli space must correspond closely to the moduli space of string world sheets. </p><p> <LI>Electric-magnetic duality realized in terms of string world sheets and partonic 2-surfaces. The group G and its Langlands dual ^LG would correspond to the time-like and space-like braidings. Duality predicts that the moduli space of string world sheets is very closely related to that for the partonic 2-surfaces. The strong form of 4-D general coordinate invariance implying electric-magnetic duality and S-duality as well as strong form of holography indeed predicts that the collection of string world sheets is fixed once the collection of partonic 2-surfaces at light-like boundaries of CD and its sub-CDs is known. </p><p> <LI> The proposal is that finite measurement resolution is realized in terms of inclusions of hyperfinite factors of type II₁ at quantum level and represented in terms of confining effective gauge group. This effective gauge group could be some associate of G: gauge group, Kac-Moody group or its quantum counterpart, or so called twisted quantum Yangian strongly suggested by twistor considerations. At space-time level the finite measurement resolution would be represented in terms of braids at space-time level which come in two varieties correspond to braids assignable to space-like surfaces at the two light-like boundaries of CD and with light-like 3-surfaces at which the signature of the induced metric changes and which are identified as orbits of partonic 2-surfaces connecting the future and past boundaries of CDs. </p><p> There are several steps leading from G to its twisted quantum Yangian. The first step replaces point like particles with partonic 2-surfaces: this brings in Kac-Moody character. The second step brings in finite measurement resolution meaning that Kac-Moody type algebra is replaced with its quantum version. The third step brings in zero energy ontology: one cannot treat single partonic surface or string world sheet as independent unit: always the collection of partonic 2-surfaces and corresponding string worlds sheets defines the geometric structure so that multilocality and therefore quantum Yangian algebra with multilocal generators is unavoidable. </p><p> In finite measurement resolution geometric Langlands duality and number theoretic Langlands duality are very closely related since partonic 2-surface is effectively replaced with the punctures representing the ends of braid strands and the orbit of this set under a discrete subgroup of G defines effectively a collection of "rational" 2-surfaces. The number of the "rational" surfaces in geometric Langlands conjecture replaces the number of rational points of partonic 2-surface in its number theoretic variant. The ability to compute both these numbers is very relevant for quantum TGD. </p><p> <LI>The natural identification of the associate of G is as quantum Yangian of Kac-Moody type group associated with Minkowskian open string model assignable to string world sheet representing a string moving in the moduli space of partonic 2-surface. The dual group corresponds to Euclidian string model with partonic 2-surface representing string orbit in the moduli space of the string world sheets. The Kac-Moody algebra assigned with simply laced G is obtained using the standard tachyonic free field representation obtained as ordered exponentials of Cartan algebra generators identified as transversal parts of M⁴ coordinates for the braid strands. The importance of the free field representation generalizing to the case of non-simply laced groups in the realization of finite measurement resolution in terms of Kac-Moody algebra cannot be over-emphasized. </p><p> <LI>Langlands duality involves besides harmonic analysis side also the number theoretic side. Galois groups (collections of them) defined by infinite primes and integers having representation as symplectic flows defining braidings. I have earlier proposed that the hierarchy of these Galois groups define what might be regarded as a non-commutative homology and cohomology. Also G has this kind of representation which explains why the representations of these two kinds of groups are so intimately related. This relationship could be seen as a generalization of the MacKay correspondence between finite subgroups of SU(2) and simply laced Lie groups. </p><p> <LI>Symplectic group of the light-cone boundary acting as isometries of the WCW geometry kenociteallb/compl1 allowing to represent projectively both Galois groups and symmetry groups as symplectic flows so that the non-commutative cohomology would have braided representation. This leads to braided counterparts for both Galois group and effective symmetry group. </p><p> <LI>The moduli space for Higgs bundle playing central role in the approach of Witten and Kapustin to geometric Landlands program is in TGD framework replaced with the conformal moduli space for partonic 2-surfaces. It is not however possible to speak about Higgs field although moduli defined the analog of Higgs vacuum expectation value. Note that in TGD Universe the most natural assumption is that all Higgs like states are "eaten" by gauge bosons so that also photon and gluons become massive. This mechanism would be very general and mean that massless representations of Poincare group organize to massive ones via the formation of bound states. It might be however possible to see the contribution of p-adic thermodynamics depending on genus as analogous to Higgs contribution since the conformal moduli are analogous to vacuum expectation of Higgs field. </OL></p>

<p> Infinite primes is a purely TGD inspired notion. The notion of infinity is number theoretical and infinite primes have well defined divisibility properties. One can partially order them by the real norm. p-Adic norms of infinite primes are well defined and finite. The construction of infinite primes is a hierarchical procedure structurally equivalent to a repeated second quantization of a supersymmetric arithmetic quantum field theory. At the lowest level bosons and fermions are labelled by ordinary primes. At the next level one obtains free Fock states plus states having interpretation as bound many particle states. The many particle states of a given level become the single particle states of the next level and one can repeat the construction ad infinitum. The analogy with quantum theory is intriguing and I have proposed that the quantum states in TGD Universe correspond to octonionic generalizations of infinite primes. It is interesting to compare infinite primes (and integers) to the Cantorian view about infinite ordinals and cardinals. The basic problems of Cantor's approach which relate to the axiom of choice, continuum hypothesis, and Russell's antinomy: all these problems relate to the definition of ordinals as sets. In TGD framework infinite primes, integers, and rationals are defined purely algebraically so that these problems are avoided. It is not surprising that these approaches are not equivalent. For instance, sum and product for Cantorian ordinals are not commutative unlike for infinite integers defined in terms of infinite primes. </p><p> Set theory defines the foundations of modern mathematics. Set theory relies strongly on classical physics, and the obvious question is whether one should reconsider the foundations of mathematics in light of quantum physics. Is set theory really the correct approach to axiomatization? </p><p> <OL> <LI> Quantum view about consciousness and cognition leads to a proposal that p-adic physics serves as a correlate for cognition. Together with the notion of infinite primes this suggests that number theory should play a key role in the axiomatics. <LI> Algebraic geometry allows algebraization of the set theory and this kind of approach suggests itself strongly in physics inspired approach to the foundations of mathematics. This means powerful limitations on the notion of set. <LI> Finite measurement resolution and finite resolution of cognition could have implications also for the foundations of mathematics and relate directly to the fact that all numerical approaches reduce to an approximation using rationals with a cutoff on the number of binary digits. <LI> The TGD inspired vision about consciousness implies evolution by quantum jumps meaning that also evolution of mathematics so that no fixed system of axioms can ever catch all the mathematical truths for the simple reason that mathematicians themselves evolve with mathematics. </OL> I will discuss possible impact of these observations on the foundations of physical mathematics assuming that one accepts the TGD inspired view about infinity, about the notion of number, and the restrictions on the notion of set suggested by classical TGD. </p>

<p> In this article the goal is to find whether the general mathematical structures associated with twistor approach, superstring models and M-theory could have a generalization or a modification in TGD framework. The contents of the chapter is an outcome of a rather spontaneous process, and represents rather unexpected new insights about TGD resulting as outcome of the comparisons. </p><p> <I>1. Infinite primes, Galois groups, algebraic geometry, and TGD</I> </p><p> In algebraic geometry the notion of variety defined by algebraic equation is very general: all number fields are allowed. One of the challenges is to define the counterparts of homology and cohomology groups for them. The notion of cohomology giving rise also to homology if Poincare duality holds true is central. The number of various cohomology theories has inflated and one of the basic challenges to find a sufficiently general approach allowing to interpret various cohomology theories as variations of the same motive as Grothendieck, who is the pioneer of the field responsible for many of the basic notions and visions, expressed it. </p><p> Cohomology requires a definition of integral for forms for all number fields. In p-adic context the lack of well-ordering of p-adic numbers implies difficulties both in homology and cohomology since the notion of boundary does not exist in topological sense. The notion of definite integral is problematic for the same reason. This has led to a proposal of reducing integration to Fourier analysis working for symmetric spaces but requiring algebraic extensions of p-adic numbers and an appropriate definition of the p-adic symmetric space. The definition is not unique and the interpretation is in terms of the varying measurement resolution. </p><p> The notion of infinite has gradually turned out to be more and more important for quantum TGD. Infinite primes, integers, and rationals form a hierarchy completely analogous to a hierarchy of second quantization for a super-symmetric arithmetic quantum field theory. The simplest infinite primes representing elementary particles at given level are in one-one correspondence with many-particle states of the previous level. More complex infinite primes have interpretation in terms of bound states. </p><p> <OL> </p><p> <LI>What makes infinite primes interesting from the point of view of algebraic geometry is that infinite primes, integers and rationals at the n:th level of the hierarchy are in 1-1 correspondence with rational functions of n arguments. One can solve the roots of associated polynomials and perform a root decomposition of infinite primes at various levels of the hierarchy and assign to them Galois groups acting as automorphisms of the field extensions of polynomials defined by the roots coming as restrictions of the basic polynomial to planes x_n=0, x_n=x_n-1=0, etc... </p><p> <LI>These Galois groups are suggested to define non-commutative generalization of homotopy and homology theories and non-linear boundary operation for which a geometric interpretation in terms of the restriction to lower-dimensional plane is proposed. The Galois group G_k would be analogous to the relative homology group relative to the plane x_k-1=0 representing boundary and makes sense for all number fields also geometrically. One can ask whether the invariance of the complex of groups under the permutations of the orders of variables in the reduction process is necessary. Physical interpretation suggests that this is not the case and that all the groups obtained by the permutations are needed for a full description. </p><p> <LI>The algebraic counterpart of boundary map would map the elements of G_k identified as analog of homotopy group to the commutator group [G_k-2,G_k-2] and therefore to the unit element of the abelianized group defining cohomology group. In order to obtains something analogous to the ordinary homology and cohomology groups one must however replaces Galois groups by their group algebras with values in some field or ring. This allows to define the analogs of homotopy and homology groups as their abelianizations. Cohomotopy, and cohomology would emerge as duals of homotopy and homology in the dual of the group algebra. </p><p> <LI>That the algebraic representation of the boundary operation is not expected to be unique turns into blessing when on keeps the TGD as almost topological QFT vision as the guide line. One can include all boundary homomorphisms subject to the condition that the anticommutator δⁱ_kδ^j_k-1+δ^j_kδⁱ_k-1 maps to the group algebra of the commutator group [G_k-2,G_k-2]. By adding dual generators one obtains what looks like a generalization of anticommutative fermionic algebra and what comes in mind is the spectrum of quantum states of a SUSY algebra spanned by bosonic states realized as group algebra elements and fermionic states realized in terms of homotopy and cohomotopy and in abelianized version in terms of homology and cohomology. Galois group action allows to organize quantum states into multiplets of Galois groups acting as symmetry groups of physics. Poincare duality would map the analogs of fermionic creation operators to annihilation operators and vice versa and the counterpart of pairing of k:th and n-k:th homology groups would be inner product analogous to that given by Grassmann integration. The interpretation in terms of fermions turns however to be wrong and the more appropriate interpretation is in terms of Dolbeault cohomology applying to forms with homomorphic and antiholomorphic indices. </p><p> <LI> The intuitive idea that the Galois group is analogous to 1-D homotopy group which is the only non-commutative homotopy group, the structure of infinite primes analogous to the braids of braids of braids of ... structure, the fact that Galois group is a subgroup of permutation group, and the possibility to lift permutation group to a braid group suggests a representation as flows of 2-D plane with punctures giving a direct connection with topological quantum field theories for braids, knots and links. The natural assumption is that the flows are induced from transformations of the symplectic group acting on δ M²_+/-× CP₂ representing quantum fluctuating degrees of freedom associated with WCW ("world of classical worlds"). Discretization of WCW and cutoff in the number of modes would be due to the finite measurement resolution. The outcome would be rather far reaching: finite measurement resolution would allow to construct WCW spinor fields explicitly using the machinery of number theory and algebraic geometry. </p><p> <LI>A connection with operads is highly suggestive. What is nice from TGD perspective is that the non-commutative generalization homology and homotopy has direct connection to the basic structure of quantum TGD almost topological quantum theory where braids are basic objects and also to hyper-finite factors of type II₁. This notion of Galois group makes sense only for the algebraic varieties for which coefficient field is algebraic extension of some number field. Braid group approach however allows to generalize the approach to completely general polynomials since the braid group make sense also when the ends points for the braid are not algebraic points (roots of the polynomial). </p><p> </OL> </p><p> This construction would realize the number theoretical, algebraic geometrical, and topological content in the construction of quantum states in TGD framework in accordance with TGD as almost TQFT philosophy, TGD as infinite-D geometry, and TGD as generalized number theory visions. </p><p> <I>2. p-Adic integration and cohomology</I> </p><p> This picture leads also to a proposal how p-adic integrals could be defined in TGD framework. </p><p> <OL> <LI> The calculation of twistorial amplitudes reduces to multi-dimensional residue calculus. Motivic integration gives excellent hopes for the p-adic existence of this calculus and braid representation would give space-time representation for the residue integrals in terms of the braid points representing poles of the integrand: this would conform with quantum classical correspondence. The power of 2π appearing in multiple residue integral is problematic unless it disappears from scattering amplitudes. Otherwise one must allow an extension of p-adic numbers to a ring containing powers of 2π. </p><p> <LI> Weak form of electric-magnetic duality and the general solution ansatz for preferred extremals reduce the Kähler action defining the Kähler function for WCW to the integral of Chern-Simons 3-form. Hence the reduction to cohomology takes places at space-time level and since p-adic cohomology exists there are excellent hopes about the existence of p-adic variant of Kähler action. The existence of the exponent of Kähler gives additional powerful constraints on the value of the Kähler fuction in the intersection of real and p-adic worlds consisting of algebraic partonic 2-surfaces and allows to guess the general form of the Kähler action in p-adic context. </p><p> <LI>One also should define p-adic integration for vacuum functional at the level of WCW. p-Adic thermodynamics serves as a guideline leading to the condition that in p-adic sector exponent of Kähler action is of form (m/n)^r, where m/n is divisible by a positive power of p-adic prime p. This implies that one has sum over contributions coming as powers of p and the challenge is to calculate the integral for K= constant surfaces using the integration measure defined by an infinite power of Kähler form of WCW reducing the integral to cohomology which should make sense also p-adically. The p-adicization of the WCW integrals has been discussed already earlier using an approach based on harmonic analysis in symmetric spaces and these two approaches should be equivalent. One could also consider a more general quantization of Kähler action as sum K=K₁+K₂ where K₁=rlog(m/n) and K₂=n, with n divisible by p since exp(n) exists in this case and one has exp(K)= (m/n)^r × exp(n). Also transcendental extensions of p-adic numbers involving n+p-2 powers of e^1/n can be considered. </p><p> <LI>If the Galois group algebras indeed define a representation for WCW spinor fields in finite measurement resolution, also WCW integration would reduce to summations over the Galois groups involved so that integrals would be well-defined in all number fields. </OL> </p><p> <I>3. Floer homology, Gromov-Witten invariants, and TGD</I> </p><p> Floer homology defines a generalization of Morse theory allowing to deduce symplectic homology groups by studying Morse theory in loop space of the symplectic manifold. Since the symplectic transformations of the boundary of δ M⁴_+/-× CP₂ define isometry group of WCW, it is very natural to expect that Kähler action defines a generalization of the Floer homology allowing to understand the symplectic aspects of quantum TGD. The hierarchy of Planck constants implied by the one-to-many correspondence between canonical momentum densities and time derivatives of the imbedding space coordinates leads naturally to singular coverings of the imbedding space and the resulting symplectic Morse theory could characterize the homology of these coverings. </p><p> One ends up to a more precise definition of vacuum functional: Kähler action reduces Chern-Simons terms (imaginary in Minkowskian regions and real in Euclidian regions) so that it has both phase and real exponent which makes the functional integral well-defined. Both the phase factor and its conjugate must be allowed and the resulting degeneracy of ground state could allow to understand qualitatively the delicacies of CP breaking and its sensitivity to the parameters of the system. The critical points with respect to zero modes correspond to those for Kähler function. The critical points with respect to complex coordinates associated with quantum fluctuating degrees of freedom are not allowed by the positive definiteness of Kähler metric of WCW. One can say that Kähler and Morse functions define the real and imaginary parts of the exponent of vacuum functional. </p><p> The generalization of Floer homology inspires several new insights. In particular, space-time surface as hyper-quaternionic surface could define the 4-D counterpart for pseudo-holomorphic 2-surfaces in Floer homology. Holomorphic partonic 2-surfaces could in turn correspond to the extrema of Kähler function with respect to zero modes and holomorphy would be accompanied by super-symmetry. </p><p> Gromov-Witten invariants appear in Floer homology and topological string theories and this inspires the attempt to build an overall view about their role in TGD. Generalization of topological string theories of type A and B to TGD framework is proposed. The TGD counterpart of the mirror symmetry would be the equivalence of formulations of TGD in H=M⁴× CP₂ and in CP₃× CP₃ with space-time surfaces replaced with 6-D sphere bundles. </p><p> <I>4. K-theory, branes, and TGD </I> </p><p> K-theory and its generalizations play a fundamental role in super-string models and M-theory since they allow a topological classification of branes. After representing some physical objections against the notion of brane more technical problems of this approach are discussed briefly and it is proposed how TGD allows to overcome these problems. A more precise formulation of the weak form of electric-magnetic duality emerges: the original formulation was not quite correct for space-time regions with Euclidian signature of the induced metric. The question about possible TGD counterparts of R-R and NS-NS fields and S, T, and U dualities is discussed. </p><p> <I>5. p-Adic space-time sheets as correlates for Boolean cognition</I> </p><p> p-Adic physics is interpreted as physical correlate for cognition. The so called Stone spaces are in one-one correspondence with Boolean algebras and have typically 2-adic topologies. A generalization to p-adic case with the interpretation of p pinary digits as physically representable Boolean statements of a Boolean algebra with 2ⁿ>p>p^n-1 statements is encouraged by p-adic length scale hypothesis. Stone spaces are synonymous with profinite spaces about which both finite and infinite Galois groups represent basic examples. This provides a strong support for the connection between Boolean cognition and p-adic space-time physics. The Stone space character of Galois groups suggests also a deep connection between number theory and cognition and some arguments providing support for this vision are discussed. </p>

<p> Witten was awarded by Fields medal from a construction recipe of Jones polynomial based on topological QFT assigned with braids and based on Chern-Simons action. Recently Witten has been working with an attempt to understand in terms of quantum theory the so called Khovanov polynomial associated with a much more abstract link invariant whose interpretation and real understanding remains still open. </p><p> The attempts to understand Witten's thoughts lead to a series of questions unavoidably culminating to the frustrating "Why I do not have the brain of Witten making perhaps possible to answer these questions?". This one must just accept. In this article I summarize some thoughts inspired by the associations of the talk of Witten with quantum TGD and with the model of DNA as topological quantum computer. In my own childish manner I dare believe that these associations are interesting and dare also hope that some more brainy individual might take them seriously. </p><p> An idea inspired by TGD approach which also main streamer might find interesting is that the Jones invariant defined as vacuum expectation for a Wilson loop in 2+1-D space-time generalizes to a vacuum expectation for a collection of Wilson loops in 2+2-D space-time and could define an invariant for 2-D knots and for cobordisms of braids analogous to Jones polynomial. As a matter fact, it turns out that a generalization of gauge field known as gerbe is needed and that in TGD framework classical color gauge fields defined the gauge potentials of this field. Also topological string theory in 4-D space-time could define this kind of invariants. Of course, it might well be that this kind of ideas have been already discussed in literature. </p>

<p> The vanishing of ordinary determinant tells that a group of linear equations possesses non-trivial solutions. Hyperdeterminant generalizes this notion to a situation in which one has homogenous multilinear equations. The notion has applications to the description of quantum entanglement and has stimulated interest in physics blogs. Hyperdeterminant applies to hyper-matrices with n matrix indices defined for an n-fold tensor power of vector space - or more generally - for a tensor product of vector spaces with varying dimensions. Hyper determinant is an n-linear function of the arguments in the tensor factors with the property that all partial derivatives of the hyper determinant vanish at the point, which corresponds to a non-trivial solution of the equation. A simple example is potential function of n arguments linear in each argument. </p><p> Why the notion of hyperdeterminant- or rather its infinite-dimensional generalization- might be interesting in TGD framework relates to the quantum criticality of TGD stating that TGD Universe involves a fractal hierarchy of criticalities: phase transitions inside phase transitions inside... At classical level the lowest order criticality means that the extremal of Kähler action possesses non-trivial second variations for which the action is not affected. The system is critical. In QFT context one speaks about zero modes. The vanishing of the so called Gaussian (of functional) determinant associated with second variations is the condition for the existence of critical deformations. In QFT context this situation corresponds to the presence of zero modes. </p><p> The simplest physical model for a critical system is cusp catastrophe defined by a potential function V(x) which is fourth order polynomial. At the edges of cusp two extrema of potential function stable and unstable extrema co-incide and the rank of the matrix defined by the potential function vanishes. This means vanishing of its determinant. At the tip of the cusp the also the third derivative vanishes of potential function vanishes. This situation is however not describable in terms of hyperdeterminant since it is genuinely non-linear rather than only multilinear. </p><p> In a complete analogy, one can consider also the vanishing of n:th variations in TGD framework as higher order criticality so that the vanishing of hyperdeterminant might serve as a criterion for the higher order critical point and occurrence of phase transition. Why multilinearity might replace non-linearity in TGD framework could be due to the non-locality. Multilinearty with respect to imbedding space-coordinates at different space-time points would imply also the vanishing of the standard local divergences of quantum field theory known to be absent in TGD framework on basis of very general arguments. In this article an attempt to concretize this idea is made. The challenge is highly non-trivial since in finite measurement resolution one must work with infinite-dimensional system. </p>

<p> There have been impressive steps in the understanding of N=4 maximally sypersymmetric YM theory possessing 4-D super-conformal symmetry. This theory is related by AdS/CFT duality to certain string theory in AdS₅× S⁵ background. Second stringy representation was discovered by Witten and is based on 6-D Calabi-Yau manifold defined by twistors. The unifying proposal is that so called Yangian symmetry is behind the mathematical miracles involved. </p><p> In the following I will discuss briefly the notion of Yangian symmetry and suggest its generalization in TGD framework by replacing conformal algebra with appropriate super-conformal algebras. Also a possible realization of twistor approach and the construction of scattering amplitudes in terms of Yangian invariants defined by Grassmannian integrals is considered in TGD framework and based on the idea that in zero energy ontology one can represent massive states as bound states of massless particles. There is also a proposal for a physical interpretation of the Cartan algebra of Yangian algebra allowing to understand at the fundamental level how the mass spectrum of n-particle bound states could be understood in terms of the n-local charges of the Yangian algebra. </p><p> Twistors were originally introduced by Penrose to characterize the solutions of Maxwell's equations. Kähler action is Maxwell action for the induced Kähler form of CP₂. The preferred extremals allow a very concrete interpretation in terms of modes of massless non-linear field. Both conformally compactified Minkowski space identifiable as so called causal diamond and CP₂ allow a description in terms of twistors. These observations inspire the proposal that a generalization of Witten's twistor string theory relying on the identification of twistor string world sheets with certain holomorphic surfaces assigned with Feynman diagrams could allow a formulation of quantum TGD in terms of 3-dimensional holomorphic surfaces of CP₃× CP₃ mapped to 6-surfaces dual CP₃× CP₃, which are sphere bundles so that they are projected in a natural manner to 4-D space-time surfaces. Very general physical and mathematical arguments lead to a highly unique proposal for the holomorphic differential equations defining the complex 3-surfaces conjectured to correspond to the preferred extremals of Kähler action. </p>

Outlined here is a programming approach for use in Planck level simulation hypothesis models. It is based around an expanding (the simulation clock-rate measured in units of Planck time) 4-axis hyper-sphere and mathematical particles that oscillate between an electric wave-state and a mass (unit of Planck mass per unit of Planck time) point-state. Particles are assigned a spin axis which determines the direction in which they are pulled by this (hyper-sphere pilot wave) expansion, thus all particles travel at, and only at, the velocity of expansion (the origin of $c$), however only the particle point-state has definable co-ordinates within the hyper-sphere. Photons are the mechanism of information exchange, as they lack a mass state they can only travel laterally (in hypersphere co-ordinate terms) between particles and so this hypersphere expansion cannot be directly observed, relativity then becomes the mathematics of perspective translating between the absolute (hypersphere) and the relative motion (3D space) co-ordinate systems. A discrete `pixel' lattice geometry is assigned as the gravitational space. Units of $\hbar c$ `physically' link particles into orbital pairs. As these are direct particle to particle links, a gravitational force between macro objects is not required, the gravitational orbit as the sum of these individual orbiting pairs. A 14.6 billion year old hyper-sphere (the sum of Planck black-hole units) has similar parameters to the cosmic microwave background. The Casimir force is a measure of the background radiation density.

We introduce the Majorana spinors in the momentum representation. They obey the Dirac-like equation with eight components, which has been first introduced by Markov. Thus, the Fock space for corresponding quantum fields is doubled (as shown by Ziino). Particular attention has been paid to the questions of chirality and helicity (two concepts which frequently are confused in the literature) for Dirac and Majorana states.

In this paper we applied a finite element method to finding the effects on the reverberation times of common irregularities like curved surfaces, non-parallel walls and large open-walled ante-rooms, found in auditoria. The number of modes having a reverberation time in a specified time interval is expressed as a function of the total allowed degrees of freedom and it is shown that even when the number of degrees of freedom of the model is large there is, in general, no one dominant group. Curved surfaces in particular lead to a situation where some modes have very long reverberation times, leading to bad acoustics. In such situations a knowledge of the offending mode shapes give an indication on where to position absorptive material for optimum effect.

The Simulation Hypothesis proposes that all of reality is in fact an artificial simulation, analogous to a computer simulation, and as such our reality is an illusion. Outlined here is a method for programming relativistic mass, space and time at the Planck level as applicable for use in (Planck-level) Universe-as-a-Simulation Hypothesis. For the virtual universe the model uses a 4-axis hyper-sphere that expands in incremental steps (the simulation clock-rate). Virtual particles that oscillate between an electric wave-state and a mass point-state are mapped within this hyper-sphere, the oscillation driven by this expansion. Particles are assigned an N-S axis which determines the direction in which they are pulled along by the expansion, thus an independent particle momentum may be dispensed with. Only in the mass point-state do particles have fixed hyper-sphere co-ordinates. The rate of expansion translates to the speed of light and so in terms of the hyper-sphere co-ordinates all particles (and objects) travel at the speed of light, time (as the clock-rate) and velocity (as the rate of expansion) are therefore constant, however photons, as the means of information exchange, are restricted to lateral movement across the hyper-sphere thus giving the appearance of a 3-D space. Lorentz formulas are used to translate between this `3-D space' and the hyper-sphere co-ordinates, relativity resembling the mathematics of perspective.

The fine structure constant α = e²/hc ~ 1/137.036 and the blackbody radiation constant α_R = e²(a_R/k⁴_B)^1/3 ~ 1/157.555 are linked by prime numbers. The blackbody radiation constant is a new method to measure the fine structure constant. It also links the fine structure constant to the Boltzmann constant.

The following open problem is presented and motivated : Are there physical systems whose state spaces do not compose according to either the Cartesian product, as classical systems do, or the usual tensor product, as quantum systems do ?

Much of Mathematics, and therefore Physics as well, have been limited by four rather consequential restrictions. Two of them are ancient taboos, one is an ancient and no longer felt as such bondage, and the fourth is a surprising omission in Algebra. The paper brings to the attention of those interested these four restrictions, as well as the fact that each of them has by now ways, even if hardly yet known ones, to overcome them.

We study in some detail the structure of the projective quadric Q' obtained by taking the quotient of the isotropic cone in a standard pseudohermitian space H_p,q with respect to the positive real numbers R⁺ and, further, by taking the quotient ~Q = Q'/U(1). The case of signature (1. 1) serves as an illustration. ~Q is studied as a compactication of RxH_p-1,q-1

Our obtaining the analytical equations for the gravitation of a particular type of mathematical annulus, which we called a 'Special Gravitating Annulus' (SGA), greatly facilitates studying its orbital properties by computer programming. This includes isomorphism, periodic and chaotic polar orbits, and orbits in three dimensions. We provide further insights into the gravitational properties of this annulus and describe our computer algorithms and programs. We study a number of periodic orbits, giving them names to aid identification. 'Ellipses extraordinaires' which are bisected by the annulus, have no gravitating matter at either focus and represent a fundamental departure from the normal association of elliptical orbits with Keplerian motion. We describe how we came across this type of orbit and the analysis we performed. We present the simultaneous differential equations of motion of 'ellipses extraordinaires' and other orbits as a mathematical challenge. The 'St.Louis Gateway Arch' orbit contains two 'instantaneous static points' (ISP). Polar elliptical orbits can wander considerably without tending to form other kinds of orbit. If this type of orbit is favoured then this gives a similarity to spiral galaxies containing polar orbiting material. Annular oscillatory orbits and rotating polar elliptical orbits are computed in isometric projection. A 'daisy' orbit is computed in stereo-isometric projection. The singularity at the centre of the SGA is discussed in relation to mechanics and computing, and it appears mathematically different from a black hole. In the Appendix, we prove by a mathematical method that a thin plane self-gravitating Newtonian annulus, free from external influence, exhibiting radial gravitation that varies inversely with the radius in the annular plane, must have an area mass density which also varies inversely with the radius and this exact solution is the only exact solution.

This appendix contains basic facts about CP₂ as a symmetric space and Kähler manifold. The coding of the standard model symmetries to the geometry of CP₂, the physical interpretation of the induced spinor connection in terms of electro-weak gauge potentials, and basic facts about induced gauge fields are discussed

The original justification for the hierarchy of Planck constants came from the indications that Planck constant could have large values in both astrophysical systems involving dark matter and also in biology. The realization of the hierarchy in terms of the singular coverings and possibly also factor spaces of CD and CP₂ emerged from consistency conditions. It however seems that TGD actually predicts this hierarchy of covering spaces. The extreme non-linearity of the field equations defined by Kähler action means that the correspondence between canonical momentum densities and time derivatives of the imbedding space coordinates is 1-to-many. This leads naturally to the introduction of the covering space of CD x CP₂, where CD denotes causal diamond defined as intersection of future and past directed light-cones.

The notion of electric magnetic duality emerged already two decades ago in the attempts to formulate the Kähler geometry of the "world of classical worlds". Quite recently a considerable step of progress took place in the understanding of this notion. This concept leads to the identification of the physical particles as string like objects defined by magnetic charged wormhole throats connected by magnetic ux tubes. The second end of the string contains particle having electroweak isospin neutralizing that of elementary fermion and the size scale of the string is electro-weak scale would be in question. Hence the screening of electro-weak force takes place via weak confinement. This picture generalizes to magnetic color confinement. Electric-magnetic duality leads also to a detailed understanding of how TGD reduces to almost topological quantum field theory. A surprising outcome is the necessity to replace CP₂ Kähler form in Kähler action with its sum with S² Kähler form.

Generalized Feynman diagrams have become the central notion of quantum TGD and one might even say that space-time surfaces can be identified as generalized Feynman diagrams. The challenge is to assign a precise mathematical content for this notion, show their mathematical existence, and develop a machinery for calculating them. Zero energy ontology has led to a dramatic progress in the understanding of generalized Feynman diagrams at the level of fermionic degrees of freedom. In particular, manifest finiteness in these degrees of freedom follows trivially from the basic identifications as does also unitarity and non-trivial coupling constant evolution. There are however several formidable looking challenges left. <ol> <li>One should perform the functional integral over WCW degrees of freedom for fixed values of on mass shell momenta appearing in the internal lines. After this one must perform integral or summation over loop momenta.</li> <li>One must define the functional integral also in the p-adic context. p-Adic Fourier analysis relying on algebraic continuation raises hopes in this respect. p-Adicity suggests strongly that the loop momenta are discretized and ZEO predicts this kind of discretization naturally.</li> </ol> In this article a proposal giving excellent hopes for achieving these challenges is discussed.

<p> The focus of this book is the number theoretical vision about physics. This vision involves three loosely related parts. </p><p> <OL><LI> The fusion of real physic and various p-adic physics to a single coherent whole by generalizing the number concept by fusing real numbers and various p-adic number fields along common rationals. Extensions of p-adic number fields can be introduced by gluing them along common algebraic numbers to reals. Algebraic continuation of the physics from rationals and their their extensions to various number fields (generalization of completion process for rationals) is the key idea, and the challenge is to understand whether how one could achieve this dream. A profound implication is that purely local p-adic physics would code for the p-adic fractality of long length length scale real physics and vice versa, and one could understand the origins of p-adic length scale hypothesis. <LI> Second part of the vision involves hyper counterparts of the classical number fields defined as subspaces of their complexifications with Minkowskian signature of metric. Allowed space-time surfaces would correspond to what might be called hyper-quaternionic sub-manifolds of a hyper-octonionic space and mappable to M⁴× CP₂ in natural manner. One could assign to each point of space-time surface a hyper-quaternionic 4-plane which is the plane defined by the modified gamma matrices but not tangent plane in general. Hence the basic variational principle of TGD would have deep number theoretic content. <LI> The third part of the vision involves infinite primes identifiable in terms of an infinite hierarchy of second quantized arithmetic quantum fields theories on one hand, and as having representations as space-time surfaces analogous to zero loci of polynomials on the other hand. Single space-time point would have an infinitely complex structure since real unity can be represented as a ratio of infinite numbers in infinitely many manners each having its own number theoretic anatomy. Single space-time point would be in principle able to represent in its structure the quantum state of the entire universe. This number theoretic variant of Brahman=Atman identity would make Universe an algebraic hologram. </p><p> Number theoretical vision suggests that infinite hyper-octonionic or -quaternionic primes could could correspond directly to the quantum numbers of elementary particles and a detailed proposal for this correspondence is made. Furthermore, the generalized eigenvalue spectrum of the Chern-Simons Dirac operator could be expressed in terms of hyper-complex primes in turn defining basic building bricks of infinite hyper-complex primes from which hyper-octonionic primes are obtained by dicrete SU(3) rotations performed for finite hyper-complex primes. </OL> </p><p> Besides this holy trinity I will discuss loosely related topics. Included are possible applications of category theory in TGD framework; TGD inspired considerations related to Riemann hypothesis; topological quantum computation in TGD Universe; and TGD inspired approach to Langlands program. <p>

Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields discussed in this article, and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article the connection between standard model symmetries and classical number fields is discussed. The basis vision is that the geometry of the infinite-dimensional WCW ("world of classical worlds") is unique from its mere existence. This leads to its identification as union of symmetric spaces whose Kähler geometries are fixed by generalized conformal symmetries. This fixes space-time dimension and the decomposition M⁴ x S and the idea is that the symmetries of the Kähler manifold S make it somehow unique. The motivating observations are that the dimensions of classical number fields are the dimensions of partonic 2-surfaces, space-time surfaces, and imbedding space and M⁸ can be identified as hyper-octonions- a sub-space of complexified octonions obtained by adding a commuting imaginary unit. This stimulates some questions. Could one understand S = CP₂ number theoretically in the sense that M⁸ and H = M⁴ x CP₂ be in some deep sense equivalent ("number theoretical compactification" or M⁸ - H duality)? Could associativity define the fundamental dynamical principle so that space-time surfaces could be regarded as associative or co-associative (defined properly) sub-manifolds of M⁸ or equivalently of H. One can indeed define the associativite (co-associative) 4-surfaces using octonionic representation of gamma matrices of 8-D spaces as surfaces for which the modified gamma matrices span an associate (co-associative) sub-space at each point of space-time surface. Also M⁸ - H duality holds true if one assumes that this associative sub-space at each point contains preferred plane of M⁸ identifiable as a preferred commutative or co-commutative plane (this condition generalizes to an integral distribution of commutative planes in M⁸). These planes are parametrized by CP₂ and this leads to M⁸ - H duality. WCW itself can be identified as the space of 4-D local sub-algebras of the local Clifford algebra of M⁸ or H which are associative or co-associative. An open conjecture is that this characterization of the space-time surfaces is equivalent with the preferred extremal property of Kähler action with preferred extremal identified as a critical extremal allowing infinite-dimensional algebra of vanishing second variations.

Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article p-adic physics and the technical problems relates to the fusion of p-adic physics and real physics to a larger structure are discussed. The basic technical problems relate to the notion of definite integral both at space-time level, imbedding space level and the level of WCW (the "world of classical worlds"). The expressibility of WCW as a union of symmetric spacesleads to a proposal that harmonic analysis of symmetric spaces can be used to define various integrals as sums over Fourier components. This leads to the proposal the p-adic variant of symmetric space is obtained by a algebraic continuation through a common intersection of these spaces, which basically reduces to an algebraic variant of coset space involving algebraic extension of rationals by roots of unity. This brings in the notion of angle measurement resolution coming as Δφ = 2π/pⁿ for given p-adic prime p. Also a proposal how one can complete the discrete version of symmetric space to a continuous p-adic versions emerges and means that each point is effectively replaced with the p-adic variant of the symmetric space identifiable as a p-adic counterpart of the real discretization volume so that a fractal p-adic variant of symmetric space results. If the Kähler geometry of WCW is expressible in terms of rational or algebraic functions, it can in principle be continued the p-adic context. One can however consider the possibility that that the integrals over partonic 2-surfaces defining ux Hamiltonians exist p-adically as Riemann sums. This requires that the geometries of the partonic 2-surfaces effectively reduce to finite sub-manifold geometries in the discretized version of δM₊⁴. If Kähler action is required to exist p-adically same kind of condition applies to the space-time surfaces themselves. These strong conditions might make sense in the intersection of the real and p-adic worlds assumed to characterized living matter.

There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first approach relies on a direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach discussed in this article relies on the construction of spinor structure based on the hypothesis that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for the second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of spinor structure. This implies a geometrization of fermionic statistics. The basic philosophy is that at fundamental level the construction of WCW geometry reduces to the second quantization of the induced spinor fields using Dirac action. This assumption is parallel with the bosonic emergence stating that all gauge bosons are pairs of fermion and antifermion at opposite throats of wormhole contact. Vacuum function is identified as Dirac determinant and the conjecture is that it reduces to the exponent of Kähler function. In order to achieve internal consistency induced gamma matrices appearing in Dirac operator must be replaced by the modified gamma matrices defined uniquely by Kähler action and one must also assume that extremals of Kähler action are in question so that the classical space-time dynamics reduces to a consistency condition. This implies also super-symmetries and the fermionic oscillator algebra at partonic 2-surfaces has intepretation as N = 1 generalization of space-time supersymmetry algebra different however from standard SUSY algebra in that Majorana spinors are not needed. This algebra serves as a building brick of various super-conformal algebras involved. The requirement that there exist deformations giving rise to conserved Noether charges requires that the preferred extremals are critical in the sense that the second variation of the Kähler action vanishes for these deformations. Thus Bohr orbit property could correspond to criticality or at least involve it. Quantum classical correspondence demands that quantum numbers are coded to the properties of the preferred extremals given by the Dirac determinant and this requires a linear coupling to the conserved quantum charges in Cartan algebra. Effective 2-dimensionality allows a measurement interaction term only in 3-D Chern-Simons Dirac action assignable to the wormhole throats and the ends of the space-time surfaces at the boundaries of CD. This allows also to have physical propagators reducing to Dirac propagator not possible without the measurement interaction term. An essential point is that the measurement interaction corresponds formally to a gauge transformation for the induced Kähler gauge potential. If one accepts the weak form of electric-magnetic duality Kähler function reduces to a generalized Chern-Simons term and the effect of measurement interaction term to Kähler function reduces effectively to the same gauge transformation. The basic vision is that WCW gamma matrices are expressible as super-symplectic charges at the boundaries of CD. The basic building brick of WCW is the product of infinite-D symmetric spaces assignable to the ends of the propagator line of the generalized Feynman diagram. WCW Kähler metric has in this case "kinetic" parts associated with the ends and "interaction" part between the ends. General expressions for the super-counterparts of WCW ux Hamiltoniansand for the matrix elements of WCW metric in terms of their anticommutators are proposed on basis of this picture.

There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first one relies on a direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach relies on the construction of spinor structure assuming that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for the second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of spinor structure. In this article the construction of Kähler form and metric based on symmetries is discussed. The basic vision is that WCW can be regarded as the space of generalized Feynman diagrams with lines thickned to light-like 3-surfaces and vertices identified as partonic 2-surfaces. In zero energy ontology the strong form of General Coordinate Invariance (GCI) implies effective 2-dimensionality and the basic objects are pairs partonic 2-surfaces X² at opposite light-like boundaries of causal diamonds (CDs). The hypothesis is that WCW can be regarded as a union of infinite-dimensional symmetric spaces G/H labeled by zero modes having an interpretation as classical, non-quantum uctuating variables. A crucial role is played by the metric 2-dimensionality of the light-cone boundary δM₊⁴ + and of light-like 3-surfaces implying a generalization of conformal invariance. The group G acting as isometries of WCW is tentatively identified as the symplectic group of δM₊⁴ x CP₂ localized with respect to X². H is identified as Kac-Moody type group associated with isometries of H = M₊⁴ x CP₂ acting on light-like 3-surfaces and thus on X². An explicit construction for the Hamiltonians of WCW isometry algebra as so called ux Hamiltonians is proposed and also the elements of Kähler form can be constructed in terms of these. Explicit expressions for WCW ux Hamiltonians as functionals of complex coordinates of the Cartesisian product of the infinite-dimensional symmetric spaces having as points the partonic 2-surfaces defining the ends of the the light 3-surface (line of generalized Feynman diagram) are proposed.

There are two basic approaches to quantum TGD. The first approach, which is discussed in this article, is a generalization of Einstein's geometrization program of physics to an infinitedimensional context. Second approach is based on the identification of physics as a generalized number theory. The first approach relies on the vision of quantum physics as infinite-dimensional Kähler geometry for the "world of classical worlds" (WCW) identified as the space of 3-surfaces in in certain 8-dimensional space. There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first approach relies on direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach relies on the construction of spinor structure based on the hypothesis that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of WCW spinor structure. In this article the proposal for Kähler function based on the requirement of 4-dimensional General Coordinate Invariance implying that its definition must assign to a given 3-surface a unique space-time surface. Quantum classical correspondence requires that this surface is a preferred extremal of some some general coordinate invariant action, and so called Kähler action is a unique candidate in this respect. The preferred extremal has intepretation as an analog of Bohr orbit so that classical physics becomes and exact part of WCW geometry and therefore also quantum physics. The basic challenge is the explicit identification of WCW Kähler function K. Two assumptions lead to the identification of K as a sum of Chern-Simons type terms associated with the ends of causal diamond and with the light-like wormhole throats at which the signature of the induced metric changes. The first assumption is the weak form of electric magnetic duality. Second assumption is that the Kähler current for preferred extremals satisfies the condition jK ^ djK = 0 implying that the ow parameter of the ow lines of jK defines a global space-time coordinate. This would mean that the vision about reduction to almost topological QFT would be realized. Second challenge is the understanding of the space-time correlates of quantum criticality. Electric-magnetic duality helps considerably here. The realization that the hierarchy of Planck constant realized in terms of coverings of the imbedding space follows from basic quantum TGD leads to a further understanding. The extreme non-linearity of canonical momentum densities as functions of time derivatives of the imbedding space coordinates implies that the correspondence between these two variables is not 1-1 so that it is natural to introduce coverings of CD x CP₂. This leads also to a precise geometric characterization of the criticality of the preferred extremals.

There are two basic approaches to the construction of quantum TGD. The first approach relies on the vision of quantum physics as infinite-dimensional Kähler geometry for the "world of classical worlds" identified as the space of 3-surfaces in in certain 8-dimensional space. Essentially a generalization of the Einstein's geometrization of physics program is in question. The second vision is the identification of physics as a generalized number theory. This program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article brief summaries of physics as infinite-dimensional geometry and generalized number theory are given to be followed by more detailed articles.

This work is sequel to the book "A Treatise in Information Geometry", submitted to vixra in late 2009. The aim of this dissertation is to continue the development of fractal geometry initiated in the former volume. This culminates in the construction of first order self-referential geometry, which is a special form of 8-tensor construction on a differential manifold with nice properties. The associated information theory has many powerful and interesting consequences. Additionally within this treatise, various themes in modern mathematics are surveyed- Galois theory, Category theory, K-theory, and Sieve theory, and various connections between these structures and information theory investigated. In particular it is demonstrated that the exotic geometric analogues of these constructions - save for Category theory, which is foundational - form special cases of the self referential calculus.

This paper examines the difference between infinite and finite domains of a Stefan Problem. It is pointed out that attributes of solutions to the Diffusion Equation suggest assumptions of an infinite domain are invalid during initial times for finite domain Stefan Problems. The paper provides a solution for initial and early times from an analytical approach using a perturbation. This solution can then easily be applied to numerical models for later times. The differences of the two domains are examined and discussed.

In early 1999, Professor Frieden of the University of Arizona published a book through Cambridge University Press titled "Physics from Fisher Information". It is the purpose of this dissertation to further develop some of his ideas, as well as explore various exotic differentiable structures and their relationship to physics. In addition to the original component of this work, a series of survey chapters are provided, in the interest of keeping the treatise self-contained. The first summarises the main preliminary results on the existence of non-standard structures on manifolds from the Milnor-Steenrod school. The second is a standard introduction to semi-riemannian geometry. The third introduces the language of geometric measure theory, which is important in justifying the existence of smooth solutions to variational problems with smooth structures and smooth integrands. The fourth is a short remark on PDE existence theory, which is needed for the fifth, which is essentially a typeset version of a series of lectures given by Ben Andrews and Gerhard Huisken on the Hamilton-Perelman program for proving the Geometrisation Conjecture of Bill Thurston.

It is shown that the wave equation cannot be solved for the general spreading of the cylindrical wave using the method of separation of variables. But an equation is presented in case of its solving the above act will have occurred. Also using this equation the above-mentioned general spreading of the cylindrical wave for large distances is obtained which contrary to what is believed consists of arbitrary functions.