This document is devoted to understanding and implementing the energy numbers, which were recently explicated very clearly by Emmerson in his recent paper. Through this line of reasoning, it becomes apparent that the algebra defined by the energy numbers are indeed the natural algebra for categorifying quantization. We also develop a notion of symmetric topological vector spaces, and forcing on said spaces motivated by homological mirror symmetry.

We consider strings from the perspective of stable motivic, homotopical QFT. Some predictions for the behavior of gauginos in both a Minkowski light cone and $5$-dimensional $mathcal{A}dmathcal{S}_5$-space are given. We show that there is a duality between working locking in a system of dendrites, and threshold edging at the periphery of a manifold.This work extends the work of [4] and [7] by providing a more mathematical interpretation of the realization of quasi-quanta in open topological dynamical systems. This interpretation incidentally involves the category of pure motives over $mathfrak{C}$, and projections of fiber spectra to the category of stable homotopies.

One of the possible explanations for entanglement is a sort of perverse holonomy which acts on sheaves whose germs are eigenvectors for a tuple of local variables. We take baby steps towards realizing this model by introducing an equivariant form of holonomy. As a test category, we take U(1)-bundles whose outbound fibrations are Koszul nerves of degree (p+q)=n.

Let $mathfrak{X}$ be a topological stack, and $LocSys(mathfrak{X})$ a local system taking varieties $v in mathfrak{X}$ to their projective resolutions over an affine coordinate system. Let $alpha$ and $beta$ be smooth charts encompassing non-degenerate loci of the upper-half plane, and let $varphi$ be the map $beta circ alpha^{-1}$. Our goal is to describe a class of vector bundles, called $emph{geometric sub-bundles}$, which provide holonomic transport for n-cells (for small values of n) over a $G_delta$-space which models the passage $mathfrak{X} ightrightarrows LocSys(mathfrak{X})$. We will first establish the preliminary definitions before advancing our core idea, which succinctly states that for a pointed, stratified space $Strat_M^ast$, there is a canonical selection of transition maps $[varphi]$ which preserves the intersection of a countable number of fibers in some sub-bundle of the bundle $Bun_V$ over $LocSys(mathfrak{X})$

The Four Colour Theorem is one of the mathematical problems with a fairly short history. This problem originated from coloring areas on a map, but has been dealt with graph and topological theory. Since the discovery of the problem, there have been many proofs by people interested in this mathematical problem, but in 1976 it was recognized as a proof by computer. The method of proof was to show that many graphs or many patterns can be colored with four colors. This proposed algorithm aims to show that all graphs are satisfied with the four color theorem regardless of the topology and the four color problem has no more non-deterministic polynomial time complexity.

Safety is critical in robotic tasks. Energy function based methods have been introduced to address the problem. To ensure safety in the presence of control limits, we need to design an energy function that results in persistently feasible safe control at all system states.However, designing such an energy function for high-dimensional nonlinear systems remains challenging.Considering the fact that there are redundant dynamics in high dimensional systems with respect to the safety specifications, this paper proposes a novel approach called abstract safe control.We propose a system abstraction method that enables the design of energy functions on a low-dimensional model.Then we can synthesize the energy function with respect to the low-dimensional model to ensure persistent feasibility.The resulting safe controller can be directly transferred to other systems with the same abstraction, e.g., when a robot arm holds different tools. The proposed approach is demonstrated on a 7-DoF robot arm (14 states) both in simulation and real-world. Our method always finds feasible control and achieves zero safety violations in 500 trials on 5 different systems.

In this paper, we developed a set of linear constraints to test whether 4 points form a square. Traditionally people use Euclidean distance to test whether the 4 points form a square. It forms a square if the four sides are of equal length and the diagonals are of equal length. My test function using a set of linear constraints is much simpler without the use of quadratic operations in Euclidean distance test function. This is needed in the future to prove Square Peg Problem for any arbitary closed curve.

Recent advances have begun to blur the lines between theoretical mathematics and applied mathematics. Oftentimes, in a variety of fields, concepts from not only applied mathematics but theoretical mathematics have been employed to great effect. As more and more researchers come to utilize, deploy, and develop both abstract and concrete mathematical models (both theoretical and applied), the demand for highly generalizable, accessible, and versatile mathematical models has increased drastically (Rosen, 2011). Specifically in the case of Complex Systems and the accompanying field of Complex Systems Analysis, this phenomenon has had profound effects. As researchers, academics, and scholars from these fields turn to mathematical models to assist in their scientific inquiries (specifically, concepts and ideas taken from various subsets of graph theory), the limitations of our current mathematical frameworks becomes increasingly apparent. To remedy this, we present the Chang Graph, a simple graph defined by an n-sided regular polygon surrounding a 2n-sided regular polygon. Various properties and applications of this graph are discussed, and further research is proposed for the study of this mathematical model.

That every Euclidean subset homeomorphic to the ambient Euclidean space is open, a version of invariance of domain, is a relatively deep result whose typical proof is far from elementary. When it comes to the real line, the version of invariance of domain admits a simple proof that depends precisely on some elementary results of ``common sense''. It seems a pity that an elementary proof of the version of invariance of domain for the real line is not well-documented in the related literature even as an exercise, and it certainly deserves a space. Apart from the main purpose, as we develop the ideas we also make present some pedagogically enlightening remarks, which may or may not be well-documented.

Besides the proof of the mathematical conjecture, a new form for the three-dimensional euclidean sphere is given. This sphere can be embedded into pseudo-euclidean metric, making the new description for the Universe.

In this paper we give a comprehensive presentation of the notions of filter base, filter and ultrafilter on single valued neutrosophic set and we investigate some of their properties and relationships. More precisely, we discuss properties related to filter completion, the image of neutrosophic filter base by a neutrosophic induced mapping and the infimum and supremum of two neutrosophic filter bases.

In 1999, Molodtsov initiated the concept of Soft Sets Theory as a new mathematical tool and a completely different approach for dealing with uncertainties in many fields of applied sciences. In 2011, Shabir and Naz introduced and studied the theory of soft topological spaces, also defining and investigating many new soft properties as generalization of the classical ones. In this paper, we introduce the notions of soft separation between soft points and soft closed sets in order to obtain a generalization of the well-known Embedding Lemma for soft topological spaces.

We introduce the idea of an Essential Spaces for 2-dimensional compact manifolds. We raise the question whether essential spaces do exist also for 3-dimensional manifolds.

The aim of this paper is to introduce the concepts of IFS α -open sets. Also we discussed the relationship between this type of Open set and other existing Open sets in Intuitionistic fuzzy topological spaces. Also we introduce new class of closed sets namely IFS α -closed sets and its properties are studied.

In this paper we introduce two new classes of topological spaces called pre-R0 and pre-R1 spaces in terms of the concept of preopen sets and investigate some of their fundamental properties.

In 1996, Dontchev [14] introduced and investigated a new notion of non-continuity called contra-continuity. Recently, Baker et al. [6] of- fered a new generalization of contra-continuous functions via $\lambda$-closed sets, called almost contra $\lambda$-continuous functions. It is the objective of this paper to further study some more properties of such functions.

The purpose of this paper is to introduce some new classes of topological spaces by utilizing b-open sets and study some of their fundamental properties

In the present paper, we offer a new form of firm continuity, called firm contra-continuity, by which we characterize strongly S-closed spaces. Moreover, we investigate the basic properties of firmly contra-continuous functions. We also introduce and investigate the notion of locally contra-closed graphs.

It is the object of this paper to study further the notion of $\Lambda s$-semi- $\theta$-closed sets which is defined as the intersection of a $\theta$- $\Lambda s$-set and a semi--closed set. Moreover, we introduce some low separation axioms using the above notions. Also we present and study the notions of $\Lambda s$- continuous functions, $\Lambda s$-compact spaces and $\Lambda s$-connected spaces.

M. Ganster and I.L. Reilly [2] introduced a new decomposition of continuity called LC-continuity. In this paper, we introduce and investigate a generalization LC- continuity called weakly LC-continuity.

In this paper, we introduce a new class of continuous functions as an application of $\Lambda$-generalized closed sets (namely $\Lambda_g$-closed set, $\Lambda$-g-closed set and $g \Lambda$-closed set) namely $\Lambda$-generalized continuous functions (namely $\Lambda g$-continuous, $\Lambda$-g-continuous and $g \Lambda$-continuous) and study their properties in topological space.

Recently the notion of rarely $\alpha$-continuous functions has been introduced and investigated by Jafari [1]. This paper is devoted to the study of upper (and lower) rarely $\alpha$-continuous multifunctions.

In this paper, we first introduce a new class of closed map called $\rho$- closed map. Moreover, we introduce a new class of homeomorphism called a $\rho$-homeomorphism.We also introduce another new class of closed map called $\rho*$-closed map and introduce a new class of homeomorphism called a $\rho*$-homeomorphism and prove that the set of all $\rho*$-homeomorphisms forms a group under the operation of composition of maps.

In this paper, we introduce kT½ -spaces, k*T½ -spaces, kT_b -spaces, kT_c -spaces, kT_d -spaces, kT_f -spaces, kT_^g* -spaces and T^k_b-spaces and investigate their characterizations.

In this paper we define the concept of $\Lambda_b$-sets (resp. $V_b$-sets) of a topological space, i.e., the intersection of b-open (resp. the union of b-closed) sets. We study the fundamental property of $\Lambda_b$-sets (resp. $V_b$-sets) and investigate the topologies defined by these families of sets.

The aim of this paper is to consider compactness notions by utilizing $\lambda$-sets, V - sets, locally closed sets, locally open sets, $\lambda$-closed sets and $\lambda$-open sets. We are able to completely characterize these variations of compactness, and also provide various interesting examples that support our results.

In this paper we consider a new class of topological spaces, called pc-compact spaces. This class of spaces lies strictly between the classes of strongly compact spaces and C- compact spaces. Also, every pc-compact space is p-closed in the sense of Abo-Khadra. We will investigate the fundamental properties of pc-compact spaces, and consider their behaviour under certain mappings.

Balachandran [1] introduced the notion of GO-compactness by involving g-open sets. Quite recently, Caldas et al. in [8] and [9] investigated this class of compactness and characterized several of its properties. In this paper, we further investigate this class of compactness and obtain several more new properties. Moreover, we introduce and study the new class of GO-(m, n)-compact spaces.

We generalize the very well known boundary operator of the ordinary singular homology theory. We describe a variant of this ordinary simplicial boundary operator, where the usual boundary (n-1)-simplices of each n-simplex, i.e. the `faces´, are replaced by combinations of internal (n-1)-simplices parallel to the faces. This construction may lead to an infinte class of extraordinary non-isomorphic homology theories. Further, we show some interesting constructions on the standard simplex.

The main purpose of this paper is to introduce the notions of D-sets, DS-sets, D-continuity and DS-continuity and to obtain decompositions of continuous functions, A-continuous functions and AB-continuous functions. Also, properties of the classes of D-sets and DS-sets are discussed.

In this paper, the notions of DS*-sets and DS*-continuous functions are introduced and their properties and their relationships with some other types of sets are investigated. Moreover, some new decompositions of continuous functions are obtained by using DS*-continuous functions, DS-continuous functions and D-continuous functions.

In this paper, we consider the class of $\alpha_{[\gamma, \gamma']}generalized closed set in topological spaces and investigate some of their properties. We also present and study new separation axioms by using the notions of $\alpha$-open and $\alpha$-bioperations. Also, we analyze the relations with some well known separation axioms.

In this paper we introduce new types of functions called g*bp-continuous function, almost g*bp-continuous function, and weakly g*bp-continuous function in topological spaces and study some of their basic properties and relations among them.

The note studies further properties and results of analysis in the setting of hypermetric spaces. Among others, we present some results concerning the hyper uniform limit of a sequence of continuous functions, the hypermetric identication theorem and the metrization problem for hypermetric space.

In this paper, the concept of generalized neutrosophic pre-closed sets and generalized neutrosophic pre-open sets are introduced. We also study relations and various properties between the other existing neutrosophic open and closed sets. In addition, we discuss some applications of generalized neutrosophic pre-closed sets, namely neutrosophic pT_{1/2} space and neutrosophic gpT_{1/2} space. The concepts of generalized neutrosophic connected spaces, generalized neutrosophic compact spaces and generalized neutrosophic extremally disconnected spaces are established. Some interesting properties are investigated in addition to giving some examples.

In this paper, we initiate the concept of intuitionistic fuzzy ideals on rough sets. Using a new relation we discuss some of the algebraic nature of intuitionistic fuzzy ideals of a ring.

In this paper, we obtain new separation axioms by using the notion of $(\delta; p)$-open sets introduced by Jafari [3] via the notion of pre-regular $p$-open sets [2].

In this note, we are intended to offer some theoretical consideration concerning the introduction of point-free topological monoids on the locales and frames. Moreover, we define a quantum group on locales by utilizing the Drinfeld-Jimbo group.

The purpose of this paper is to introduce the notion of nano ideal topological spaces and investigate the relation between nano topological space and nano ideal topological space. Moreover, we offer some new open and closed sets in the context of nano ideal topological spaces and present some of their basic properties and characterizations.

In this paper, we introduce and study upper and lower slightly $\delta$-$\beta$- continuous multifunctions in topological spaces and obtain some characterizations of these new continuous multifunctions.

In this paper, we introduce and study the concept of qI-open set. Based on this new concept, we dene new classes of functions, namely qI-continuous functions, qI-open functions and qI- closed functions, for which we prove characterization theorems.

In this paper we apply the method of compression to construct a dense set of points in the plane at rational distance from each other. We provide a positive solution to the Erd˝os-Ulam problem.

The Khalimsky topology plays a significant role in the digital image processing. In this paper we define a topology $\kappa_1$ on the set of integers generated by the triplets of the form $\{2n, 2n+1, 2n+3\}$. We show that in this space $(\mathbb{Z}, \kappa_1)$, every point has a smallest neighborhood and hence this is an Alexandroff space. This topology is homeomorphic to Khalimskt topology. We prove, among others, that this space is connected and $T_{3/4}$. Moreover, we introduce the concept of $^*g\hat{\alpha}$-closed sets in a topological space and characterize it using $^*g\alpha o$-kernel and closure. We investigate the properties of $^*g\hat{\alpha}$-closed sets in digital plane. The family of all $^*g\hat{\alpha}$-open sets of $(\mathbb{Z}^2, \kappa^2)$, forms an alternative topology of $\mathbb{Z}^2$. We prove that this plane $(\mathbb{Z}^2, ^*g\hat{\alpha}O)$ is $T_{1/2}$. It is well known that the digital plane $(\mathbb{Z}^2, \kappa^2)$ is not $T_{1/2}$, even if $(\mathbb{Z}, \kappa)$ is $T_{1/2}$.

On December 22, 2006, the journal Science honored Perelman's proof of the Poincare Conjecture as the scientific ``Breakthrough of the Year", the first time this honor was bestowed in the area of mathematics. However, I have critical questions about Perelman's proof of Poincare Conjecture. The conjecture states, that ``Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.'' The ``homeomorphic" means that by non-singular deformation one produces perfect sphere - the equivalent of initial space. However, pasting in foreign caps will not make such deformation. My short proofs are given.

We introduce the idea of Solid Strip Configurations which is a way of construction 3-dimensional compact manifolds alternative to $\Delta$-complexes and CW complexes. The proposed method is just an idea which we believe deserve further formal mathematical investigation.

To a topological space X we attach in two equivalent ways a profinite space X' and a continuous map F: X --> X' such that, for any continuous map f: X --> Y, where Y is a profinite space, there is a unique continuous map f': X' --> Y such that f'oF = f.

This short article follows an earlier document, wherein I indicated how the foundations of game theory could be reformulated within the lens of a more information theoretic and topological approach. Building on said work, herein I intend to generalise this to meta games, where one game (the meta-game) is built on top of a game, and then to meta-meta-games. Finally I indicate how one might take these ideas further, in terms of constructing frameworks to study policies, which relate to the solution of various algebraic invariants.

Within this paper, I combine ideas from information theory, topology and game theory, to develop a framework for the determination of optimal strategies within iterated cooperative games of incomplete information.

We show that if one has ever loved reading Prasolov’s books, then one can move on reading our recent article [3] and several words following to deduce that partitioning a graph into triangles is not an easy problem.

It is proved that the irreducible map according to Franklin consists of 5 regions and, as a consequence, 4 colors are sufficient for colouring any map on the sphere

Using Artin Presentation Theory, we mathematically augment a remark of Atiyah on physics and Donaldson's 4D theory which, conversely, explicitly introduces the theoretical physical relevance of AP Theory into Modern Physics. AP Theory is a purely discrete group-theoretic, in fact, a framed pure braid theory, which, in the sharpest possible holographic manner, encodes all closed, orientable 3-manifolds and their knot and linking theories, and a large class of compact, connected, simply-connected, smooth 4-manifolds with a connected boundary, whose physical relevance for Atiyah's remark we explain.

There is Prize committee (claymath.org), which requires publication in worldwide reputable mathematics journal and at least two years of following scientific admiration. Why then the God-less Grisha Perelman has published only in a God-less forum (arXiv), publication was unclear as the crazy sketch; but mummy child "Grisha" has being forced to accept the Millennium Prize? Am I simply ugly or poor? Please respect my copyrights!

By means of a computer, all the possible homogeneous compact generalised triangulations made up of a small number of 3-simplexes (from 1 to 3) have been classified in homology classes. The analysis shows that, with a small number of simplexes, it is already possible to build quite a large number of separate topological spaces.

In this preliminary work, we focus on a particular iso-geometrical, iso-topological facet of iso-mathematics by suggesting a developing, generalized approach for encoding the states and transitions of spherically-symmetric structures that vary in size. In particular, we introduce the notion of "effective iso-radius" to facilitate a heightened characterization of dynamic iso-sphere Inopin holographic rings (IHR) as they undergo "iso-transitions" between "iso-states". In essence, we propose the existence of "effective dynamic iso-sphere IHRs". In turn, this emergence drives the construction of a new "effective iso-state" platform to encode the generalized dynamics of such iso-complex, non-linear systems in a relatively straightforward approach of spherical-based iso-topic liftings. The initial results of this analysis are significant because they lead to alternative modes of research and application, and thereby pose the question: do these effective dynamic iso-sphere IHRs have application in physics and chemistry? Our hypothesis is: yes. To answer this inquiry and assess this conjecture, this developing work should be subjected to further scrutiny, collaboration, improvement, and hard work via the scientific method in order to advance it as such.

In this preliminary work, we use a dynamic iso-unit function to iso-topically lift the "static" Inopin holographic ring (IHR) of the unit sphere to an interconnected pair of "dynamic iso-sphere IHRs" (iso-DIHR), where the IHR is simultaneously iso-dual to both a magnified "exterior iso-DIHR" and de-magnified ``interior iso-DIHR". For both the continuously-varying and discretely-varying cases, we define the dynamic iso-amplitude-radius of one iso-DIHR as being equivalent to the dynamic iso-amplitude-curvature of its counterpart, and conversely. These initial results support the hypothesis that a new IHR-based mode of iso-geometry and iso-topology may be in order, which is significant because the interior and exterior zones delineated by the IHR are fundamentally "iso-dual inverses" and may be inferred from one another.

In this introductory work, we use Santilli's iso-topic lifting as a cutting-edge platform to explore Mandelbrot's set. The objective is to upgrade Mandelbrot's complex quadratic polynomial with iso-multiplication and then computationally probe the effects on this revolutionary fractal. For this, we define the "iso-complex quadratic polynomial" and engage it to generate a locally iso-morphic array of "Mandelbrot iso-sets" by varying the iso-unit, where the connectedness property is topologically preserved in each case. The iso-unit broadens and strengthens the chaotic analysis, and authorizes an enhanced classification and demystification such complex systems because it equips us with an additional degree of freedom: the new Mandelbrot iso-set array is an improvement over the traditional Mandelbrot set because it is significantly more general. In total, the experimental results exemplify dynamic iso-spaces and indicate two modes of topological effects: scale-deformation and boundary-deformation. Ultimately, these new and preliminary developments spark further insight into the emerging realm of iso-fractals.

We propose a preliminary framework that engages iso-triplex numbers and deformation order parameters to encode the spatial states of Iso Open Topological Strings (Iso-OTS) for fermions and the temporal states of Iso Closed Topological Strings (Iso-CTS) for bosons, where space and time are iso-dual. The objective is to introduce an elementary Topological Iso-String Theory (TIST) that complies with the holographic principle and fundamentally represents the twisting, winding, and deforming of helical, spiral, and vortical information structures---by default---for attacking superfluidic motion patterns and energy states with iso-topic lifting. In general, these preliminary results indicate a cutting-edge, flexible, consistent, and powerful iso-mathematical framework with considerable representational capability that warrants further examination, collaboration, construction, and discipline.

In this exploration, we introduce and define "dynamic iso-spaces", which are cutting-edge iso-mathematical constructions that are built with "dynamic iso-topic liftings" for "dynamic iso-unit functions". For this, we consider both the continuous and discrete cases. Subsequently, we engineer two simple examples that engage Fibonacci's sequence and Mandelbrot's set to define a "Fibonacci dynamic iso-space" and a "Mandelbrot dynamic iso-space", respectively. In total, this array of resulting iso-structures indicates that a new branch of iso-mathematics may be in order.

Recently in 1997, Sarker in [8] introduced the concept of fuzzy ideal and fuzzy local function between fuzzy topological spaces. In the present paper, we introduce some new fuzzy notions via fuzzy ideals. Also, we generalize the notion of L-open sets due to Jankovic and Homlett [6]. In addition to, we generalize the concept of L-closed sets, L- continuity due to Abd El-Monsef et al. [2]. Relationships between the above new fuzzy notions and other relevant classes are investigated.1

In a preliminary assessment, we begin to apply Santilli's iso-mathematics to triplex numbers, Euclidean triplex space, triplex fractals, and Inopin's 2-sphere holographic ring (HR) topology. In doing so, we successfully identify and define iso-triplex numbers for iso-fractal geometry in a Euclidean iso-triplex space that is iso-metrically equipped with an iso-2-sphere HR topology. As a result, we state a series of lemmas that aim to characterize these emerging iso-mathematical structures. These initial outcomes indicate that it may be feasible to engage this encoding framework to systematically attack a broad range of problems in the disciplines of science and mathematics, but a thorough, rigorous, and collaborative investigation should be in order to challenge, refine, upgrade, and implement these ideas.

A homology theory based on both near-standard and non-near-standard microsimplices is constructed. Its basic properties, including Eilenberg-Steenrod axioms for homology and continuity with respect to resolutions of spaces, are proved.

It is a part of my Algebraic General Topology research. In this article, I introduce the concepts of funcoids, which generalize proximity spaces and reloids, which generalize uniform spaces. The concept of funcoid is generalized concept of proximity, the concept of reloid is cleared from superfluous details (generalized) concept of uniformity. Also funcoids generalize pretopologies and preclosures. Also funcoids and reloids are generalizations of binary relations whose domains and ranges are filters (instead of sets). Also funcoids and reloids can be considered as a generalization of (oriented) graphs, this provides us with a common generalization of analysis and discrete mathematics. The concept of continuity is defined by an algebraic formula (instead of old messy epsilondelta notation) for arbitrarymorphisms (including funcoids and reloids) of a partially ordered category. In one formula are generalized continuity, proximity continuity, and uniform continuity.