Algebraic Topology

En esta investigación se aborda una novedosa herramienta para el análisis de datos, conocida como Análisis Topológico de Datos (TDA por sus siglas en inglés). Subyace de un área de las Matemáticas conocida como Álgebra Combinatoria o más recientemente Topología Algebraica, la cual a través de hacer un fuerte uso de Computación, Estadística, Probabilidad y Topología, entre otros conceptos, extrae características matemáticas de un conjunto de datos que nos permiten asociar, crear e inferir información general y de calidad sobre estos. Palabras clave: Topología Algebraica, Topología Combinatoria, TDA, Análisis topológico de datos. Con el advenimiento y uso de nuevas tecnologías de información, entre ellos dispositivos móviles, documentos o datos creados en redes sociales y su comentarios adicionales, han generado una cantidad enorme de datos. Estos datos sin un propio uso, carecen de valor, sin embargo, imagine que es un banco y desea disminuir el riesgo de atraer un nuevo cliente, ya sea por evitar un posible lavado de dinero, o una aseguradora que busca mitigar el riesgo utilizando un análisis de riesgo no estructurado y de fuentes informales.

Addressing the fundamental limitations of the standard cosmological model (ΛCDM) in deriving initial conditions and core parameters from first principles, this paper proposes the “Fantasia Forest” meta-framework. This framework constructs a maternal structure beyond conventional spacetime from a top-down perspective. The core innovation lies in identifying the observable universe as a specific dynamical path (geodesic) within the “Fantasia Forest”—a pre-geometric, pre-physical configu ration phase space of information. The laws of classical physics and spacetime itself emerge as low-energy effective phenomena. This research builds a complete theoretical system through four key break throughs:

1. Utilizing symmetric monoidal higher groupoids as the ground state of the “Fantasia Forest” and constructing the configuration phase space via fac torization homology;

2. Introducing the information geometric action principle to elucidate the dynamical mechanism of cosmic evolution, validated for logical self consistency through a toy model;

3. Proposing the concept of “spectral dimension” to explain the spacetime emergence process, unifying dark matter and dark energy as tidal effects (geodesic convergence and divergence, respectively) induced by the curvature of the configuration space;

4. Formulating three falsifiable observational predictions: an annual weak drift of the gravitational constant, anisotropic features in dark matter distribution, and non-standard correlations among cosmological pa rameters. This framework provides a novel ontological perspective for interdisciplinary re search in quantum gravity and cosmology, breaking through the inherent boundaries of traditional bottom-up approaches. It offers a new direction with both theoretical depth and experimental verifiability for solving core cosmological puzzles like dark matter and dark energy

We present a base-parametric framework in which a theory is defined relative to a chosen evaluation base (a small quantaloid), a cover structure, and an aggregation rule. Building on right-written convolution and Kleene-type path closure, Čech local

We derive the complete structure of fundamental physics from a single principle: quantum coherence maximization under self-consistency constraints. This uniquely determines the golden ratio ϕ = (1 + √ 5)/2 as the fundamental scaling parameter through the functional equation ϕ 2 = ϕ + 1. The theory employs a holographic E8 boundary architecture (2+1D) that projects to 3+1D bulk spacetime, providing forward causality and resolving circular logic problems. From this foundation, we derive both General Relativity and the Standard Model with zero free parameters. All coecients emerge from E8/SO(10)/SU(5) representation theory: the ne structure constant α-1 = [(4 + 3ϕ)/(7-3ϕ)] × π 3 (0.017% error), the Weinberg angle sin 2 θ W = ϕ/7 (0.03% error), and all fermion mass ratios with sub-percent precision. The theory makes ten independent Tier-1 predictions conrmed experimentally with combined statistical signicance p < 10-40. Key results include: mutual information ratios I(A : B)/I(B : C) = ϕ in quantum systems (0.18% error), decoherence optimization at coupling ratio g 2 /g 1 = ϕ (0.4% error), and Fibonacci anyon quantum dimension d τ = ϕ (machine precision). The framework resolves long-standing problems including the hierarchy problem, strong CP problem, and cosmological constant problem through ϕ-scaling. Physics structure emerges as mathematically necessary rather than contingent, with immediate experimental tests possible on quantum computers and topological quantum computing platforms.

A tools and techniques of neutrosophic graph have found many applications in different areas such as topology, networks, computer of science, etc. In addition, neutrosophic graph is a generalization of intuitionistic fuzzy graph. Therefore, in this paper we study some characteristics of neutrosopheic graphs (NTCG) and some basic definitions. Moreover we investigate several kinds of arcs, 𝜂 𝜇-strong, 𝜎 𝜇-strong, 𝜌 𝜇-arc, and 𝜂 𝜉-strong, 𝜎 𝜉-strong, 𝜌 𝜉-arc in neutrosopheic graphs (NTCG) , Finally we give neutrosophic μ-bridge and neutrosophic 𝜉-bridge (NTC 𝜉-bridge) and some interesting properties of neutrosophic bridge (NTCB), which is being taught for the first time, and obtain several important properties.

The main purpose of this paper is to define the notion of neutrosophic based normal and regular spaces. This study investigates and open new class and conception of generalization of classical regular and normal spaces. The hereditary and topological properties of neutrosophic based normal and regular spaces have been analyzed and investigated. It also examines neutrosophic topological subspaces, providing insights into their characteristics. Furthermore, the paper investigates neutrosophic regular spaces and demonstrates their hereditary nature, specifically focusing on 𝑅 1 , 𝑅 2 , 𝑅 3 and 𝑅 4. Additionally, we explain some example of a neutrosophic-regular based space X which is a neutrosophic based normal-space but it is not necessary to neutrosophic-𝑇 1 spaces. Eventually, it is shown that under certain conditions that the images are preserved in neutrosophic based normal and regular spaces.

Exponential Fuzzy Graphs (EFGs) are a new family of fuzzy graphs in which the vertices and edges' membership functions exhibit exponential decay. An EFG, which is characterised as a pair G = (V, E), captures the uncertainty and degradation of influence in complex systems by giving each edge (r, w) ∈ E a membership value ϑ E (r, w) = α E (r, w) • e-λα E (r,w) and each vertex v ∈ V a membership value ϑ V (w) = α V (w) • e-λα V (w) , where λ > 0 is a decay parameter. To maintain consistency inside the fuzzy structure, the edge membership values are limited by ϑ E (r, w) ≤ min{ϑ V (r), ϑ V (w)}. Some fundamental aspects such as vertex degree, order and size are explored in relation to the idea of exponential fuzzy graphs (EFGs). EFGs are characterised as complete and complement. Some basic operations like semi-strong product, union, join, composition and cartesian product are defined with graphical representing examples. The vertex degree of the generated vertices is examined for each operation and associated theorems are demonstrated. The theoretical findings are shown using examples. The use of EFGs in modelling real-life imprecise and uncertain data is explained in an application related to environmental contamination.

In this paper, I_g^*- closed sets, and I_g^*- open are used to investigate and define a new class of functions is said to be I_g^*-Continues functions, I_g^*-irresolute functions in ideal topological space topological spaces. Morover, I introduce I_g^*- compact spaces and I_g^*-connected spaces, and maximal I_g^*-closed sets. I obtain their characterizations and study their basic properties.

In this paper we show if R is a filtered ring and M a filtered R module then we can define a valuation on a module for M. Then we show that we can find an skeleton of valuation on M, and we prove some properties such that derived form it for a filtered module.

I establish UCET-7, extending the Gauss-Bonnet-Chern theorem to discrete and stochastic curvature estimators. Building on UCET-6, which proved local equivalence, I show that global invariants-Euler characteristic, Pontryagin numbers, and normal Euler classes-are conserved independently of estimator family or stochastic perturbation. Deterministic convergence holds with O(h 2) bias; stochastic convergence admits explicit high-probability bounds balancing bias and variance at the optimal scale h ≍ N-1/(6+n) , yielding minimax rate O(N-2/(6+n)). Numerical experiments on spheres, tori, hyperbolic surfaces, and K3 manifolds confirm theory within 10-3 relative error. Conceptually, UCET-7 formulates a Topological Conservation Principle: local estimator noise redistributes curvature but integrated invariants remain fixed, paralleling a Noether-type law for topology.

As is well-known, the homology groups of the complement of a complex hyperplane arrangement are torsion-free. Nevertheless, as we showed in a recent paper [2], the homology groups of the Milnor fiber of such an arrangement can have non-trivial integer torsion. We give here a brief account of the techniques that go into proving this result, outline some of its applications, and indicate some further questions that it brings to light.

The Orlik--Solomon algebra of a simple matroid M is the  quotient A=E/I, where E is the exterior algebra over the ground set of M and some field k and I is a homogeneous ideal defined in terms of the circuits in M. From this graded algebra, one may construct several other algebraic invariants, such as the holonomy Lie algebra h=h(A), the holonomy Chen Lie algebra h'/h'', and the Koszul modules W_i(A) for i\ge 1. One then may seek to compute the Hilbert series of these graded objects and determine the support loci and the resonance varieties (or schemes) associated with  the modules W_i(A). In this talk, I will explain some of the known---or conjectured---relationships between these various objects, and how to interpret them topologically in the case when the lattice of flats of M can be realized as the intersection lattice of a complex hyperplane arrangement. I will also present some structural results and intriguing examples for "decomposable" matroids.

The resonance varieties, the holonomy Lie algebra, and the holonomy Chen Lie algebra associated with the Orlik-Solomon algebra of a matroid provide an algebraic lens through which to examine the rich combinatorial structure of matroids and their geometric realizations as complex hyperplane arrangements. In this survey paper, we emphasize the commonalities between these objects but also the possible ways in which realizable and non-realizable matroids differ, leading to some open questions.

This book offers a rigorous and comprehensive introduction to the fundamental principles of Real Analysis,designed for advanced undergraduate and beginning graduate students. The journey begins with a detailed construction of the real number system, establishing a solid axiomatic foundation. It then proceeds to a thorough exploration of the core concepts of limits, continuity, differentiation, and integration, with a strong emphasis on formal proof and ε-δ techniques. A significant portion of the text is dedicated to the theory of infinite sequences and series, including uniform convergence and power series. The narrative culminates in a modern treatment of the Lebesgue integral, providing a powerful and elegant extension of the Riemann integral. With its clear exposition, extensive collection of challenging exercises, and careful progression from concrete intuition to abstract reasoning, this text equips students with the deep understanding and technical proficiency necessary for further study in pure and applied mathematics.

Batanin and Leinster's work on globular operads has provided one of many potential definitions of a weak ω-category. Through the language of globular operads they construct a monad whose algebras encode weak ω-categories. The purpose of this work is to show how to construct a similar monad which will allow us to formulate weak ω-categorifications of any equational algebraic theory. We first review the classical theory of operads and PROs. We then present how Leinster's globular operads can be extended to a theory of globular PROs via categorical enrichment over the category of collections. It is then shown how a process called globularization allows us to construct from a classical PRO P a globular PRO whose algebras are those algebras for P which are internal to the category of strict ω-categories and strict ω-functors. Leinster's notion of a contraction structure on a globular operad is then extended to this setting of globular PROs in order to build a monad whose algebras are globular PROs with contraction over the globularization of the classical PRO P . Among these PROs with contraction over P is the globular PRO whose algebras are by construction the fully weakened ω-categorifications of the algebraic theory encoded by P . Globular PROs and the weak ω-categorification of algebraic theories by

By means of the spinorial representation of matrices, two constructed determinant-induced metrics of conformal-Euclidean Riemannian and of Finsler types, respectively, are shown to produce a (h, v)-structure, whose properties are investigated from the point of view of naturally developed Einstein and Maxwell-like equations. MSC2010: 54B40, 53C60.

Studying distributed computing through the lens of algebraic topology has been the source of many significant breakthroughs during the last two decades, especially in the design of lower bounds or impossibility results for deterministic algorithms. In a nutshell, this approach consists of capturing all the possible states of a distributed system at a certain time as a simplicial complex P(t), called protocol complex, and viewing computation as a simplicial map from that complex to the so-called output complex, O, that captures all possible legal output states of the system. The topological properties (e.g., homotopy) of the protocol complex depends on the

We provide a machinery for transferring some properties of metrizable AN R-spaces to metrizable LC n -spaces. As a result, we show that for complete metrizable spaces the properties ALC n , LC n and W LC n coincide to each other. We also provide the following spectral characterizations of ALC n and cell-like compacta: A compactum X is ALC n if and only if X is the limit space of a σ-complete inverse system S = {X α , p β α , α < β < τ } consisting of compact metrizable LC n -spaces X α such that all bonding projections p β α , as a well all limit projections p α , are U V n -maps. A compactum X is a cell-like (resp., U V n ) space if and only if X is the limit space of a σ-complete inverse system consisting of cell-like (resp., U V n ) metric compacta.

For any countable CW -complex K and a cardinal number τ ≥ ω we construct a completely metrizable space X(K, τ ) of weight τ with the following properties: e-dim X(K, τ ) ≤ K, X(K, τ ) is an absolute extensor for all normal spaces Y with e-dim Y ≤ K, and for any completely metrizable space Z of weight ≤ τ and e-dim Z ≤ K the set of closed embeddings Z → X(K, τ ) is dense in the space C(Z, X(K, τ )) of all continuous maps from Z into X(K, τ ) endowed with the limitation topology. This result is applied to prove the existence of universal spaces for all metrizable spaces of given weight and with a given cohomological dimension.

Cubic power graph of dihedral group D n having order 2n and identity element e, Γ cpg (D n) is a finite, simple, undirected graph for which two different vertices d 1 , d 2 ∈ D n are adjacent if and only if d 1 d 2 = d 3 or d 2 d 1 = d 3 for some d ∈ D n with d 3 ̸ = e. In this research, conditions on n under which Γ cpg (D n) is planar, are calculated. Also conditions on n under which Γ cpg (D n) is Hamiltonian, are calculated. It is also shown that Γ cpg (D n) is not an Eulerian graph. A polynomial that counts how many different ways there are to color a graph with a specific number of colors is called a chromatic polynomial. We have calculated the chromatic polynomial of Γ cpg (D n) when gcd(n, 3) = 1. Also chromatic polynomial of almost complete graph obtained by deleting k number of non-adjacent edges is obtained. Spectral graph theory is a branch of mathematics that examines a graph's characteristics in relation to its characteristic polynomial, eigenvalues, and eigenvectors of related matrices, such as its adjacency matrix, Laplacian matrix or degree based matrices. Characteristic polynomials of degree based matrices such as Laplacian matrix, Signless Laplacian matrix, Maximum matrix, Minimum matrix and Greatest common divisor degree matrix of Γ cpg (D n) when gcd(n, 3) = 1.

What if the universe isn't made of objects and particles first-but from simple lines, threads, or even nothing more than dimensions layered on top of each other? This hypothesis, called the Dimensional Weave Hypothesis, is an imaginative explanation of how higher dimensions might emerge from lower ones like how a cloth is made from threads. Here, I explore how 1D threads (lines) could be woven into a 2D surface, how this might happen inside a static 4D space, and what role mass, time, and energy play in making this possible-or impossible-in the real world.

This paper was initially motivated by efforts to develop video performance in the time of COVID that incorporates split screen clips of dancers virtually interacting with each other. This led to play with several topological surface patterns in which connections between side by side video clips explores connection in a time when people are necessarily separated. The attention to the paths of objects in these videos led to an examination of paths of various types in dance composition, which touch on several other connections between dance and topological structures, including floor paths, winding and turning numbers, and especially knots and links. This is an initial effort to examine several topological properties found in many dance forms.

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $p$ and size $q$. For $k\ge 1$, a bijection $f: V(G)\cup E(G) \to \{k, k+1, k+2, \ldots, k+p+q-1\}$ such that $f(uv)= |f(u) - f(v)|$ for every edge $uv\in E(G)$ is said to be a $k$-super graceful labeling of $G$. We say $G$ is $k$-super graceful if it admits a $k$-super graceful labeling. In this paper, we study the $k$-super gracefulness of some regular and bi-regular graphs.

Après avoir exposé l&#39;importance des dualités géométrico-algébriques dans l&#39;histoire des mathématiques, la thèse propose de rassembler bon nombre d&#39;entre elle sous une approche unifiée abstraite, la dualité d&#39;Isbell. La dualité d&#39;Isbell est formellement définie comme une adjonction entre un préfaisceau et un copréfaisceau, et permet de définir un nouveau paradigme de constructivité baptisé P3. En mathématique, nous montrons que cette dualité est présente en géométrie algébrique, en géométrie algébrique dérivée, en topologie algébrique et en analyse fonctionnelle. En logique contemporaine, nous montrons qu&#39;elle peut être rendue explicite dans la géométrie de l&#39;interaction de Girard. Nous montrons ensuite comment les sciences appliquées peuvent faire usage de la dualité d&#39;Isbell, en permettant de renouveler significativement les théories. En sciences physiques, nous montrons qu&#39;elle ouvre une perspective dans la théorie quantique des champs, vers la ...

Much of the fascination with arrangements of complex hyperplanes comes from the rich interplay between the combinatorics of the intersection lattice and the algebraic topology of the complement and related spaces. These lectures will emphasize the topological and geometric aspects of the theory, including some interactions with low-dimensional topology, singularity theory, and group theory. Topics will include: Milnor fibration, boundary manifold, and branched covers, cohomology jump loci, and Lie algebras associated to the fundamental group. Several classes of arrangements will be discussed, with emphasis on concrete examples and computer-aided computations.

Much of the fascination with arrangements of complex hyperplanes comes from the rich interplay between the combinatorics of the intersection lattice and the algebraic topology of the complement and related spaces. These lectures will emphasize the topological and geometric aspects of the theory, including some interactions with low-dimensional topology, singularity theory, and group theory. Topics will include: Milnor fibration, boundary manifold, and branched covers, cohomology jump loci, and Lie algebras associated to the fundamental group. Several classes of arrangements will be discussed, with emphasis on concrete examples and computer-aided computations.

Much of the fascination with arrangements of complex hyperplanes comes from the rich interplay between the combinatorics of the intersection lattice and the algebraic topology of the complement and related spaces. These lectures will emphasize the topological and geometric aspects of the theory, including some interactions with low-dimensional topology, singularity theory, and group theory. Topics will include: Milnor fibration, boundary manifold, and branched covers, cohomology jump loci, and Lie algebras associated to the fundamental group. Several classes of arrangements will be discussed, with emphasis on concrete examples and computer-aided computations.

Much of the fascination with arrangements of complex hyperplanes comes from the rich interplay between the combinatorics of the intersection lattice and the algebraic topology of the complement and related spaces. These lectures will emphasize the topological and geometric aspects of the theory, including some interactions with low-dimensional topology, singularity theory, and group theory. Topics will include: Milnor fibration, boundary manifold, and branched covers, cohomology jump loci, and Lie algebras associated to the fundamental group. Several classes of arrangements will be discussed, with emphasis on concrete examples and computer-aided computations.

Given a compact symplectic manifold M with the Hamiltonian action of a torus T , let zero be a regular value of the moment map, and M 0 the symplectic reduction at zero. Denote by κ 0 the Kirwan map H * T (M ) → H * (M 0 ). For an equivariant cohomology class η ∈ H * T (M ) we present new localization formulas which express M 0 κ 0 (η) as sums of certain integrals over the connected components of the fixed point set M T . To produce such a formula we apply a residue operation to the Atiyah-Bott-Berline-Vergne localization formula for an equivariant form on the symplectic cut of M with respect to a certain cone, and then, if necessary, iterate this process using other cones. When all cones used to produce the formula are one-dimensional we recover, as a special case, the localization formula of Guillemin and Kalkman [GK]. Using similar ideas, for a special choice of the cone (whose dimension is equal to that of T ) we give a new proof of the Jeffrey-Kirwan localization formula [JK1]. This paper is dedicated to Alan Weinstein on the occasion of his 60th birthday.

We compute the symplectic volume of the symplectic reduced space of the product of N coadjoint orbits of a compact connected Lie group G. We compare our result with the result of Suzuki and Takakura , who study this in the case G = SU(3) starting from geometric quantization.

8 Equivariant Cohomology 33 8.1 Homotopy quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8.2 The Cartan model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 8.3 Characteristic classes of bundles over BU(1) and BT . . . . . . . . . 35 8.4 Characteristic classes in terms of the Cartan model . . . . . . . . . . 37 8.5 Equivariant first Chern class of a prequantum line bundle . . . . . . . 38 8.6 Euler classes and equivariant Euler classes . . . . . . . . . . . . . . . 38 8.7 Localization formula for torus actions . . . . . . . . . . . . . . . . . . 39 8.8 Proof of localization theorem when M consists of isolated fixed points 40 8.9 Equivariant characteristic classes . . . . . . . . . . . . . . . . . . . . 43 8.10 The abelian localization theorem . . . . . . . . . . . . . . . . . . . . 45

Given a compact symplectic manifold M with the Hamiltonian action of a torus T , let zero be a regular value of the moment map, and M 0 the symplectic reduction at zero. Denote by κ 0 the Kirwan map H * T (M ) → H * (M 0 ). For an equivariant cohomology class η ∈ H * T (M ) we present new localization formulas which express M 0 κ 0 (η) as sums of certain integrals over the connected components of the fixed point set M T . To produce such a formula we apply a residue operation to the Atiyah-Bott-Berline-Vergne localization formula for an equivariant form on the symplectic cut of M with respect to a certain cone, and then, if necessary, iterate this process using other cones. When all cones used to produce the formula are one-dimensional we recover, as a special case, the localization formula of Guillemin and Kalkman [GK]. Using similar ideas, for a special choice of the cone (whose dimension is equal to that of T ) we give a new proof of the Jeffrey-Kirwan localization formula [JK1]. This paper is dedicated to Alan Weinstein on the occasion of his 60th birthday.

Equivariant cohomology was designed to allow the study of spaces which are the quotient of a manifold M by the action of a compact group G. This can be accomplished by studying the fixed point sets of subgroups of G, notably the maximal torus T . The Cartan model replaces the study of infinite-dimensional manifolds by families of differential forms on finite-dimensional G-manifolds parametrized by an element X in the Lie algebra of G with polynomial dependence on X . A version of de Rham cohomology can be developed for the Cartan model. The localization theorem of Atiyah-Bott and Berline-Vergne describes the evaluation of such an equivariantly closed differential form on the fundamental class of the manifold. In this chapter, we first define homotopy quotients in Sect. 9.2. We introduce the Cartan model in Sect. 9.3. We treat characteristic classes of bundles over classifying spaces in Sect. 9.4. We consider these characteristic classes in terms of the Cartan model in Sect. 9.5. We treat the equivariant first Chern class of a prequantum line bundle in Sect. 9.6. We treat Euler classes and equivariant Euler classes in Sect. 9.7 We treat the localization formula for torus actions in Sect. 9.8. Finally, we treat the abelian localization theorem of Atiyah-Bott and Berline-Vergne in Sect. 9.10. References for this chapter are Audin [1], Sect. 5 and Berline-Getzler-Vergne [2], Sect. 7. In this section, we first define classifying spaces, and then use them to define homotopy quotients.

This document explores an arithmetic interpretation of knot group theory, focusing on the relationship between knot relators, their corresponding Alexander polynomials, and a newly defined measure termed global arithmetic torsion. By mapping knot group generators to specific arithmetic operations involving a parameter t, we demonstrate how the closure condition of a relator path can yield the Alexander polynomial equation ∆(t) = 0. Furthermore, the global arithmetic torsion of such paths, τ (γR)(t), exhibits a remarkable structure, consistently factoring into the Alexander polynomial ∆(t) and a cyclotomic component of the form t K-1. This suggests a deep connection between knot topology, the non-commutative nature of arithmetic operation sequences, and algebraic properties related to roots of unity and cyclotomic fields. Examples, including the figure-eight knot (41) and knots 62 and 63, are used to illustrate these findings.

Homosexuality has become a global trend both in developing and developed countries. While it has been legalized in some countries, it's still illegal in others. In this paper, a comprehensive mathematical model of sexual orientations with recovery centers was developed using a set of ordinary differential equations and solved using Wolfram Mathematica and the fourth-order Runge-Kutta method. The population was divided into ten compartments; Males (M), Females (F), gays (G), lesbians (L), heterosexual males (Hm), heterosexual females (Hf), bisexual males (Bm), bisexual females (Bf), recovery centers for males (Rm) and recovery centers for females (Rf). Positivity, boundedness, the homosexuality epidemic threshold, and equilibria for the model were determined, and stabilities were investigated. Moreover, control reproduction number and bifurcation analysis were also studied. The sensitivity of the parameters was investigated using the partial rank correlation coefficient (PRCC) method and Latin hypercube sampling (LHS). Numerical simulation was done using MATLAB 45-ODE Solver and showed that increasing homosexuality recruitment rates increased homosexuality in the population, and vice versa. It was also established numerically that increasing transfer rates from the bisexual classes to recovery centers increased the heterosexual populations. In conclusion, the effective transfer to recovery centers of the homosexual populations and low contact rates are most significant in reducing the spread of homosexuality in the community.

In the first half of this thesis the algebraic properties of a class of minimal, polynomial systems on IRn are considered. Of particular interest in the sequel are the results that (i) a tensor algebra generated by the observation space and strong accessibility algebra is equal to the Lie algebra of polynomial vector fields on IRn and (ii) the observation algebra of such a system is equal to the ring of polynomial functions on IRn. The former result is proved directly, but to establish the second we construct a canonical form for which the claim is trivial, the general case then following from the properties of the diffeomorphism relating the two realisations. It is also shown that, as a consequence of the structure of the observation space, any system in the class considered has a finite Volterra series solution, thereby showing that the canonical form developed is dual to that of Crouch. The second part of the work is devoted to the algebraic aspects of nonlinear filtering. The fu...

We give a structural description of the finite subsets A of an arbitrary group G which obey the polynomial growth condition |A n | ď n d |A| for some bounded d and sufficiently large n, showing that such sets are controlled by (a bounded number of translates of) a coset nilprogression in a certain precise sense. This description recovers some previous results of Breuillard-Green-Tao and Breuillard-Tointon concerning sets of polynomial growth; we are also able to describe the subsequent growth of |A m | fairly explicitly for m ě n, at least when A is a symmetric neighbourhood of the identity. We also obtain an analogous description of symmetric probability measures µ whose nfold convolutions µ ˚n obey the condition }µ ˚n} ´2 ℓ 2 ď n d }µ} ´2 ℓ 2 . In the abelian case, this description recovers the inverse Littlewood-Offord theorem of Nguyen-Vu, and gives a "symmetrised" variant of a recent nonabelian inverse Littlewood-Offord theorem of Tiep-Vu. Our main tool to establish these results is the inverse theorem of Breuillard, Green, and the author that describes the structure of approximate groups.

The notions of compactness and Hausdorff separation for generalized enriched categories allow us, as classically done for the category Top of topological spaces and continuous functions, to study compactly generated spaces and quasi-spaces in this setting. Moreover, for a class C of objects we generalize the notion of C-generated spaces, from which we derive, for instance, a general concept of Alexandroff spaces. Furthermore, as done for Top, we also study, in our level of generality, the relationship between compactly generated spaces and quasi-spaces.

We introduce and comprehensively develop the Finite-Derivational Involution Class (FDIC), a novel class of real-valued functions characterized by three fundamental properties: involution (f (f (x)) = x), co-domain differentiability (D f = D f ′), and finite non-empty intersection of function and derivative values. This framework establishes deep connections between analysis, algebra, and differential geometry, with applications spanning cryptography, biological modeling, quantum mechanics, and computational mathematics. We provide rigorous definitions, complete classification theorems with full proofs, existence results, and explore extensions to complex domains, stochastic settings, and geometric manifolds.

🇫🇷 Résumé
Ce travail développe un cadre mathématique rigoureux pour la propagation quantique sur les groupes localement compacts stratifiés, à l’aide des C*-algèbres croisées. Un théorème de non-propagation est démontré, et la dualité holographique est formalisée via un morphisme
Φ
Φ, établissant un isomorphisme en K-théorie entre le bulk et le boundary. Des validations numériques confirment les résultats théoriques.

🇬🇧 Abstract
This work develops a rigorous mathematical framework for quantum propagation on stratified locally compact groups using crossed product C*-algebras. A non-propagation theorem is proved, and holographic duality is formalized via a morphism
Φ
Φ, establishing a K-theoretic isomorphism between the bulk and the boundary. Numerical validations support the theoretical results.

This paper constructs an h-model structure for diagrams of streams, locally preordered spaces. Along the way, the paper extends some classical characterizations of Hurewicz fibrations and closed Hurewicz cofibrations. The usual characterization of classical closed Hurewicz cofibrations as inclusions of neighborhood deformation retracts extends. A characterization of classical Hurewicz fibrations as algebras over a pointed Moore cocylinder endofunctor also extends. An immediate consequence is a long exact sequence for directed homotopy monoids, with applications to safety verifications for database protocols.

This is a review paper about symmetric products of spaces $SP^n(X):= X^n/S_n$. We focus our attention on the symmetric products of 2-manifolds and make a journey through selected topics of algebraic topology, algebraic geometry, mathematical physics, theoretical mechanics etc. where these objects play an important role, demonstrating along the way the fundamental unity of diverse fields of physics and mathematics.