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This is the second installment in a series of papers applying descriptive set theoretic techniques to both analyze and enrich classical functors from homological algebra and algebraic topology. In it, we show that the Čech cohomology functors ˇH𝑛 on the category of locally compact separable metric spaces each factor into (i) what we term their definable version, a functor ˇH𝑛 def taking values in the category GPC of groups with a Polish cover (a category first introduced in this work's predecessor), followed by (ii) a forgetful functor from GPC to the category of groups. These definable cohomology functors powerfully refine their classical counterparts: we show that they are complete invariants, for example, of the homotopy types of mapping telescopes of d-spheres or d-tori for any 𝑑 ≥ 1, and, in contrast, that there exist uncountable families of pairwise homotopy inequivalent mapping telescopes of either sort on which the classical cohomology functors are constant. We then apply the functors ˇH𝑛 def to show that a seminal problem in the development of algebraic topology - namely, Borsuk and Eilenberg's 1936 problem of classifying, up to homotopy, the maps from a solenoid complement 𝑆3\Σ to the 2-sphere - is essentially hyperfinite but not smooth. Fundamental to our analysis is the fact that the Čech cohomology functors 𝑋 ↦→ ˇH𝑛(𝑋;𝐺) admit two main formulations: a more combinatorial one and a more homotopical formulation as the group [𝑋, 𝑃] of homotopy classes of maps from X to a polyhedral 𝐾(𝐺, 𝑛) space P. We describe the Borel-definable content of each of these formulations and prove a definable version of Huber's theorem reconciling the two. In the course of this work, we record definable versions of Urysohn's Lemma and the simplicial approximation and homotopy extension theorems, along with a definable Milnor-type short exact sequence decomposition of the space Map(𝑋, 𝑃) in terms of its subset of phantom maps; relatedly, we provide a topological characterization of this set for any locally compact Polish space X and polyhedron P. In aggregate, this work may be more broadly construed as laying foundations for the descriptive set theoretic study of the homotopy relation on such spaces Map(𝑋, 𝑃), a relation which, together with the more combinatorial incarnation of ˇH𝑛, embodies a substantial variety of classification problems arising throughout mathematics. We show, in particular, that if P is a polyhedral H-group, then this relation is both Borel and idealistic. In consequence, [𝑋, 𝑃] falls in the category of definable groups, an extension of the category GPC introduced herein for its regularity properties, which facilitate several of the aforementioned computations. [ABSTRACT FROM AUTHOR]

For an arbitrary field K and a K-variety V, we introduce the étale-open topology on the set V(K) of K-points of V. This topology agrees with the Zariski topology, Euclidean topology, or valuation topology when K is separably closed, real closed, or p-adically closed, respectively. Topological properties of the étale-open topology correspond to algebraic properties of K. For example, the étale-open topology on 픸¹ (K) is not discrete if and only if K is large. As an application, we show that a large stable field is separably closed. [ABSTRACT FROM AUTHOR]

Let A be the Steenrod algebra over the finite field k : = F 2 and G(q) be the general linear group of rank q over k. A well-known open problem in algebraic topology is the explicit determination of the cohomology groups of the Steenrod algebra, Ext A q , ∗ (k , k) , for all homological degrees q ⩾ 0. The Singer algebraic transfer of rank q, formulated by William Singer in 1989, serves as a valuable method for describing that Ext groups. This transfer maps from the coinvariants of a certain representation of G(q) to Ext A q , ∗ (k , k). Singer predicted that the algebraic transfer is always injective, but this has gone unanswered for all q ⩾ 4. This paper establishes Singer's conjecture for rank four in the generic degrees n = 2 s + t + 1 + 2 s + 1 - 3 whenever t ≠ 3 and s ⩾ 1 , and n = 2 s + t + 2 s - 2 whenever t ≠ 2 , 3 , 4 and s ⩾ 1. In conjunction with our previous results, this completes the proof of the Singer conjecture for rank four. [ABSTRACT FROM AUTHOR]

Persistent homology barcodes and diagrams are a cornerstone of topological data analysis that capture the "shape" of a wide range of complex data structures, such as point clouds, networks, and functions. However, their use in statistical settings is challenging due to their complex geometric structure. In this paper, we revisit the persistent homology rank function, which is mathematically equivalent to a barcode and persistence diagram, as a tool for statistics and machine learning. Rank functions, being functions, enable the direct application of the statistical theory of functional data analysis (FDA)—a domain of statistics adapted for data in the form of functions. A key challenge they present over barcodes in practice, however, is their lack of stability—a property that is crucial to validate their use as a faithful representation of the data and therefore a viable summary statistic. In this paper, we fill this gap by deriving two stability results for persistent homology rank functions under a suitable metric for FDA integration. We then study the performance of rank functions in functional inferential statistics and machine learning on real data applications, in both single and multiparameter persistent homology. We find that the use of persistent homology captured by rank functions offers a clear improvement over existing non‐persistence‐based approaches. [ABSTRACT FROM AUTHOR]

Nucleic acids are considered as promising materials for developing exquisite nanostructures from one to three dimensions. The advances of DNA nanotechnology facilitate ingenious design of DNA nanostructures with diverse shapes and sizes. Especially, the algebraic topological framework nucleic acids (ATFNAs) are functional DNA nanostructures that engineer guest molecules (e. g., nucleic acids, proteins, small molecules, and nanoparticles) stoichiometrically and spatially. The intrinsic precise properties and tailorable functionalities of ATFNAs hold great promise for biological applications, such as cell recognition and immunotherapy. This Perspective highlights the concept and development of precisely assembled ATFNAs, and outlines the new frontiers and opportunities for exploiting the structural advantages of ATFNAs for biological applications. [ABSTRACT FROM AUTHOR]

Quantum walks have emerged as a transformative paradigm in quantum information processing and can be applied to various graph problems. This study explores discrete-time quantum walks on simplicial complexes, a higher-order generalization of graph structures. Simplicial complexes, encoding higher-order interactions through simplices, offer a richer topological representation of complex systems. Since the conventional classical random walk cannot directly detect community structures, we present a quantum walk algorithm to detect higher-order community structures called simplicial communities. We utilize the Fourier coin to produce entangled translation states among adjacent simplices in a simplicial complex. The potential of our quantum algorithm is tested on Zachary's karate club network. This study may contribute to understanding complex systems at the intersection of algebraic topology and quantum walk algorithms. [ABSTRACT FROM AUTHOR]

A d$d$‐dimensional invertible topological field theory (TFT) is a functor from the symmetric monoidal (∞,n)$(\infty,n)$‐category of d$d$‐bordisms (embedded into R∞$\mathbb {R}^\infty$ and equipped with a tangential (X,ξ)$(X,\xi)$‐structure) that lands in the Picard subcategory of the target symmetric monoidal (∞,n)$(\infty,n)$‐category. We classify these field theories in terms of the cohomology of the (n−d)$(n-d)$‐connective cover of the Madsen–Tillmann spectrum. This is accomplished by identifying the classifying space of the (∞,n)$(\infty,n)$‐category of bordisms with Ω∞−nMTξ$\Omega ^{\infty -n}MT\xi$ as an E∞$E_\infty$‐space. This generalizes the celebrated result of Galatius–Madsen–Tillmann–Weiss (Acta Math. 202 (2009), no. 2, 195–239) in the case n=1$n=1$, and of Bökstedt–Madsen (An alpine expedition through algebraic topology, vol. 617, Contemp. Math., Amer. Math. Soc., Providence, RI, 2014, pp. 39–80) in the n$n$‐uple case. We also obtain results for the (∞,n)$(\infty,n)$‐category of d$d$‐bordisms embedding into a fixed ambient manifold M$M$, generalizing results of Randal–Williams (Int. Math. Res. Not. IMRN 2011 (2011), no. 3, 572–608) in the case n=1$n=1$. We give two applications: (1) we completely compute all extended and partially extended invertible TFTs with target a certain category of n$n$‐vector spaces (for n⩽4$n \leqslant 4$), and (2) we use this to give a negative answer to a question raised by Gilmer and Masbaum (Forum Math. 25 (2013), no. 5, 1067–1106. arXiv:0912.4706). [ABSTRACT FROM AUTHOR]

Soft set theory, initially introduced through the seminal article "Soft set theory—First results" in 1999, has gained considerable attention in the field of mathematical modeling and decision-making. Despite its growing prominence, a comprehensive survey of soft set theory, encompassing its foundational concepts, developments, and applications, is notably absent in the existing literature. We aim to bridge this gap. This survey delves into the basic elements of the theory, including the notion of a soft set, the operations on soft sets, and their semantic interpretations. It describes various generalizations and modifications of soft set theory, such as N-soft sets, fuzzy soft sets, and bipolar soft sets, highlighting their specific characteristics. Furthermore, this work outlines the fundamentals of various extensions of mathematical structures from the perspective of soft set theory. Particularly, we present basic results of soft topology and other algebraic structures such as soft algebras and σ -algebras. This article examines a selection of notable applications of soft set theory in different fields, including medicine and economics, underscoring its versatile nature. The survey concludes with a discussion on the challenges and future directions in soft set theory, emphasizing the need for further research to enhance its theoretical foundations and broaden its practical applications. Overall, this survey of soft set theory serves as a valuable resource for practitioners, researchers, and students interested in understanding and utilizing this flexible mathematical framework for tackling uncertainty in decision-making processes. [ABSTRACT FROM AUTHOR]

Algebraic topology has proved to be a useful tool in the study of distributed computing. In this paper, we take a geometric realization problem that arises in distributed computing and formulate it as a more general problem in algebraic topology. We go on to give solutions in some initial cases, show that a modified problem has solutions in a wider range of cases, and relate the solutions back to distributed computing. [ABSTRACT FROM AUTHOR]

We describe the homotopy classes of loops in the space of 2 × 2 simple (=non-degenerate) matrices with various symmetries. This turns out to be an elementary exercise in the homotopy of closed curves in R 3 / { 0 }. Since closed curves in R 3 / { 0 } can be readily visualized, no advanced tools of algebraic topology are needed. The matrices represent gapped Bloch Hamiltonians in 1D with a two dimensional Hilbert space per unit cell. [ABSTRACT FROM AUTHOR]

The interactions between topology and algebraic geometry expose various interesting properties. This paper proposes the deformations of topological n-manifolds over the automorphic polynomial ring maps and associated isomorphic imbedding of locally flat submanifolds within the n-manifolds. The manifold deformations include topologically homeomorphic bending of submanifolds at multiple directions under algebraic operations. This paper introduces the concept of a topological equivalence class of manifolds and the associated equivalent class of polynomials in a real ring. The concepts of algebraic compositions in a real polynomial ring and the resulting topological properties (homeomorphism, isomorphism and deformation) of manifolds under algebraic compositions are introduced. It is shown that a set of ideals in a polynomial ring generates manifolds retaining topological isomorphism under algebraic compositions. The numerical simulations are presented in this paper to illustrate the interplay of topological properties and the respective real algebraic sets generated by polynomials in a ring within affine 3-spaces. It is shown that the coefficients of polynomials generated by a periodic smooth function can induce mirror symmetry in manifolds. The proposed formulations do not consider the simplectic class of manifolds and associated quantizable deformations. However, the proposed formulations preserve the properties of Nash representations of real algebraic manifolds including Nash isomorphism. [ABSTRACT FROM AUTHOR]

We prove that (under appropriate orientation conditions, depending on R) a Hamiltonian isotopy ψ 1 of a symplectic manifold (M , ω) fixing a relatively exact Lagrangian L setwise must act trivially on R ∗ (L) , where R ∗ is some generalised homology theory. We use a strategy inspired by that of Hu et al. (Geom Topol 15:1617–1650, 2011), who proved an analogous result over Z / 2 and over Z under stronger orientation assumptions. However the differences in our approaches let us deduce that if L is a homotopy sphere, ψ 1 | L is homotopic to the identity. Our technical set-up differs from both theirs and that of Cohen et al. (in: Algebraic topology, Springer, Berlin, 2019) and Cohen (in: The Floer memorial volume, Birkhäuser, Basel). We also prove (under similar conditions) that ψ 1 | L acts trivially on R ∗ (L L) , where L L is the free loop space of L. From this we deduce that when L is a surface or a K (π , 1) , ψ 1 | L is homotopic to the identity. Using methods of Lalonde and McDuff (Topology 42:309–347, 2003), we also show that given a family of Lagrangians all of which are Hamiltonian isotopic to L over a sphere or a torus, the associated fibre bundle cohomologically splits over Z / 2 . [ABSTRACT FROM AUTHOR]

We define the k-cut complex of a graph G with vertex set V(G) to be the simplicial complex whose facets are the complements of sets of size k in V(G) inducing disconnected subgraphs of G. This generalizes the Alexander dual of a graph complex studied by Fröberg [Topics in Algebra, Part 2, PWN, Warsaw, 1990, pp. 57-70] and Eagon and Reiner [J. Pure Appl. Algebra, 130 (1998), pp. 265-275]. We describe the effect of various graph operations on the cut complex and study its shellability, homotopy type, and homology for various families of graphs, including trees, cycles, complete multipartite graphs, and the prism Kn X K2, using techniques from algebraic topology, discrete Morse theory, and equivariant poset topology. [ABSTRACT FROM AUTHOR]

Based on a constructive generalization of the results of works on the topology of phase diagrams of multi-component condensed systems, the contribution of these studies to the general theory of phase diagrams is presented. General fundamental specific features characteristic of certain classes of diagrams were identified as a result of the systematic study of this topology. The topology of subsolidus isobaric-isothermal phase diagrams, as well as melting diagrams, is considered. This review can be useful for the formulation and solution of new problems in the study of phase diagrams of multi-component systems. [ABSTRACT FROM AUTHOR]

Each complex hyperplane arrangement gives rise to a Milnor fibration of its complement. Although the Betti numbers of the Milnor fiber F can be expressed in terms of the jump loci for rank 1 local systems on the complement, explicit formulas are still lacking in full generality, even for b1(F). We study here the "generic" case (in which b1(F) is as small as possible), and look deeper into the algebraic topology of such Milnor fibrations with trivial algebraic monodromy. Our main focus is on the cohomology jump loci and the lower central series quotients of π1(F). In the process, we produce a pair of arrangements for which the respective Milnor fibers have the same Betti numbers, yet non-isomorphic fundamental groups: the difference is picked by the higher-depth characteristic varieties and by the Schur multipliers of the second nilpotent quotients. [ABSTRACT FROM AUTHOR]

Let $ G $ be a graph that can be embedded in a surface $ \Sigma $ of Euler characteristic $ c'(\Sigma) $. In this paper, we proved that there exists an integer $ \Delta_0 = 4\cdot (5+\sqrt{49-24c'})\cdot [2-c'+\frac{1}{2} (5+\sqrt{49-24c'})] $ such that the total chromatic number of $ G $ is $ \Delta(G) +1 $ if $ \Delta(G) \geq \Delta_0 $.

Higher-order network models are becoming increasingly relevant for their ability to explicitly capture interactions between three or more entities in a complex system at once. In this paper, we study homophily, the tendency for alike individuals to form connections, as it pertains to higher-order interactions. We find that straightforward extensions of classical homophily measures to interactions of size 3 and larger are often inflated by homophily present in pairwise interactions. This inflation can even hide the presence of anti-homophily in higher-order interactions. Hence, we develop a structural measure of homophily, simplicial homophily, which decouples homophily in pairwise interactions from that of higher-order interactions. The definition applies when the network can be modeled as a simplicial complex, a mathematical abstraction which makes a closure assumption that for any higher-order relationship in the network, all corresponding subsets of that relationship occur in the data. Whereas previous work has used this closure assumption to develop a rich theory in algebraic topology, here we use the assumption to make empirical comparisons between interactions of different sizes. The simplicial homophily measure is validated theoretically using an extension of a stochastic block model for simplicial complexes and empirically in large-scale experiments across 16 datasets. We further find that simplicial homophily can be used to identify when node features are valuable for higher-order link prediction. Ultimately, this highlights a subtlety in studying node features in higher-order networks, as measures defined on groups of size k can inherit features described by interactions of size ℓ< k. [ABSTRACT FROM AUTHOR]

Coverage is one of the fundamental problems in directional sensor networks (DSNs). This problem is more complicated when we deal with heterogeneous DSNs (HDSNs). Prolonging the network lifetime is another important problem in this area. The problem of finding k disjoint cover sets known as the Set k-Cover problem can solve both the coverage and lifetime issues. In this paper a distributed algorithm is proposed for the Set k-Cover problem in HDSNs and then the method is applied for target k-tracking problem. In the Set k-Cover problem, directional sensors are partitioned into k disjoint sets where each set covers the entire area, and in object k-tracking problem, the object must be tracked by at least k sensors. The proposed algorithms are based on the notion of homology in algebraic topology. We consider the nerve complex corresponding to the HDSN and demonstrate how topological properties of the nerve complex of the network can be used to formulate the Set k-Cover problem as an integer linear programming problem. Then, we propose a distributed algorithm based on the subgradient method for this problem. After that, we propose a distributed algorithm for object k-tracking based on the solution of the Set k-Cover problem. Finally, we evaluate the performance of the proposed algorithms by conducting simulation experiments. [ABSTRACT FROM AUTHOR]

We present a solution of Exercise 1.2.1 in [1] which yields a short new proof of a key step in one of proofs of Brouwer's fixed point theorem (1910). A few people asked the author about the details of the solution, and this note might be interesting to a broader audience. Our approach is absolutely different from the ones using algebraic or differential topology or differential calculus and is based on a simple observation which somehow escaped many authors treating this theorem in the past. [ABSTRACT FROM AUTHOR]

In this study, we focus on training recurrent spiking neural networks to generate spatiotemporal patterns in the form of closed two-dimensional trajectories. Spike trains in the trained networks are examined in terms of their dissimilarity using the Victor-Purpura distance. We apply algebraic topology methods to the matrices obtained by rank-ordering the entries of the distance matrices, specifically calculating the persistence barcodes and Betti curves. By comparing the features of different types of output patterns, we uncover the complex relations between low-dimensional target signals and the underlying multidimensional spike trains. [ABSTRACT FROM AUTHOR]

Using tools from algebraic topology and algebraic geometry (fundamental polygons and blowups, respectively), we prove that cutting along the center of a Möbius strip yields a cylinder. [ABSTRACT FROM AUTHOR]

A toric origami manifold, introduced by Cannas da Silva, Guillemin and Pires, is a generalization of a toric symplectic manifold. For a toric symplectic manifold, its equivariant Chern classes can be described in terms of the corresponding Delzant polytope and the stabilization of its tangent bundle splits as a direct sum of complex line bundles. But in general a toric origami manifold is not simply connected, so the algebraic topology of a toric origami manifold is more difficult than a toric symplectic manifold. In this paper they give an explicit formula of the equivariant Chern classes of an oriented toric origami manifold in terms of the corresponding origami template. Furthermore, they prove the stabilization of the tangent bundle of an oriented toric origami manifold also splits as a direct sum of complex line bundles. [ABSTRACT FROM AUTHOR]

We study how the topology of feature embedding space changes as it passes through the layers of a well-trained deep neural network (DNN) through Betti numbers. Motivated by existing studies using simplicial complexes on shallow fully connected networks (FCN), we present an extended analysis using Cubical homology instead, with a variety of popular deep architectures and real image datasets. We demonstrate that as depth increases, a topologically complicated dataset is transformed into a simple one, resulting in Betti numbers attaining their lowest possible value. The rate of decay in topological complexity (as a metric) helps quantify the impact of architectural choices on the generalization ability. Interestingly from a representation learning perspective, we highlight several invariances such as topological invariance of (1) an architecture on similar datasets; (2) embedding space of a dataset for architectures of variable depth; (3) embedding space to input resolution/size, and (4) data sub-sampling. In order to further demonstrate the link between expressivity & the generalization capability of a network, we consider the task of ranking pre-trained models for downstream classification task (transfer learning). Compared to existing approaches, the proposed metric has a better correlation to the actually achievable accuracy via fine-tuning the pre-trained model. • This paper categorizes the change in topological complexity through Betti-Numbers. • Attempts to characterize the learning ability of the DNNs through topological transformation of feature space. • Attempts to use the topological complexity of feature space for the task of ranking pre-trained models. [ABSTRACT FROM AUTHOR]

We give an elementary calculation of the internal hom from Kähler differentials to Kähler differentials, and from Milnor K-theory to Kähler differentials in the category of presheaves with transfers. This answers a question of Merici and Saito. [ABSTRACT FROM AUTHOR]

We develop the motivic integration theory over formal Deligne-Mumford stacks over a power series ring of arbitrary characteristic. This is a generalization of the corresponding theory for tame and smooth Deligne-Mumford stacks constructed in earlier papers of the author. As an application, we obtain the wild motivic McKay correspondence for linear actions of arbitrary finite groups, which has been known only for cyclic groups of prime order. In particular, this implies the motivic version of Bhargava's mass formula as a special case. In fact, we prove a more general result, the invariance of stringy motives of (stacky) log pairs under crepant morphisms. [ABSTRACT FROM AUTHOR]

The representation theory of groups is one of the most interesting examples of the interaction between physics and pure mathematics, where group rings play the main role. The group ring RΓ is actually an associative ring that inherits the properties of the group Γ and the ring of coefficients R. In addition to the fact that the theory of group rings is clearly the meeting point of group theory and ring theory, it also has applications in algebraic topology, homological algebra, algebraic K-theory and algebraic coding theory. In this article, we provide a complete description of Gorenstein flat-cotorsion modules over the group ring RΓ, where Γ is a group and R is a commutative ring. It will be shown that if Γ' ≥ Γ is a finite-index subgroup, then the restriction of scalars along the ring homomorphism RΓ' - RΓ as well as its right adjoint RΓ RΓ RΓ', preserve the class of Gorenstein flat-cotorsion modules. Then, as a result, Serre's Theorem is proved for the invariant GhdRΓ, which refines the Gorenstein homological dimension of Γ over R, GhdRΓ, and is defined using flat-cotorsion modules. Moreover, we show that the inequality GFCD(RΓ) ≤ GFCD(R)+cdRΓ holds for the group ring RΓ, where GFCD(R) denotes the supremum of Gorenstein flat-cotorsion dimensions of all R-modules and cdRΓ is the cohomological dimension of Γ over R. [ABSTRACT FROM AUTHOR]

Developing reliable methods to discriminate different transient brain states that change over time is a key neuroscientific challenge in brain imaging studies. Topological data analysis (TDA), a novel framework based on algebraic topology, can handle such a challenge. However, existing TDA has been somewhat limited to capturing the static summary of dynamically changing brain networks. We propose a novel dynamic-TDA framework that builds persistent homology over a time series of brain networks. We construct a Wasserstein distance based inference procedure to discriminate between time series of networks. The method is applied to the resting-state functional magnetic resonance images of the human brain. We demonstrate that our proposed dynamic-TDA approach can distinctly discriminate between the topological patterns of male and female brain networks. [ABSTRACT FROM AUTHOR]

The prevalence of teamwork in contemporary science has raised new questions about collaboration networks and the potential impact on research outcomes. Previous studies primarily focused on pairwise interactions between scientists when constructing collaboration networks, potentially overlooking group interactions among scientists. In this study, we introduce a higher-order network representation using algebraic topology to capture multi-agent interactions, i.e., simplicial complexes. Our main objective is to investigate the influence of higher-order structures in local collaboration networks on the productivity of the focal scientist. Leveraging a dataset comprising more than 3.7 million scientists from the Microsoft Academic Graph, we uncover several intriguing findings. Firstly, we observe an inverted U-shaped relationship between the number of disconnected components in the local collaboration network and scientific productivity. Secondly, there is a positive association between the presence of higher-order loops and individual scientific productivity, indicating the intriguing role of higher-order structures in advancing science. Thirdly, these effects hold across various scientific domains and scientists with different impacts, suggesting strong generalizability of our findings. The findings highlight the role of higher-order loops in shaping the development of individual scientists, thus may have implications for nurturing scientific talent and promoting innovative breakthroughs. [ABSTRACT FROM AUTHOR]

Urban renewal planning and development are vital for enhancing the living quality of city residents. However, such improvement activities are often expensive, time-consuming, and in need of standardization. The convergence of remote sensing technologies, social big data, and artificial intelligence solutions has created unprecedented opportunities for comprehensive digital planning and analysis in urban renewal development and management. However, fast interdisciplinary development imposes some challenges because the data collected and the solutions built are defined piece by piece and require further fusion and integration of knowledge, evaluation standards, systematic analyses, and new methodologies. To address these challenges, we propose a municipal and urban renewal development index (MUDI) system with data modeling and mathematical analysis models. The MUDI system is applied and studied in three circumstances: (1) at regional level, 337 cities are selected in China to demonstrate the MUDI system's comparable analysis capabilities on a large scale across cities; (2) at city level, 285 residential communities are selected in Xiamen to demonstrate the use of remote sensing data as key MUDIs for a temporal urban land change analysis; and (3) at the level of residential neighborhoods' urban renewal practices, Xiamen's Yingping District is selected to demonstrate the MUDI system's project management capabilities. We find that the MUDI system is highly effective in municipal and urban data model building through the abstraction and summation of grid-based satellite and social big data. Secondly, the MUDI system enables comprehension of the high dimensionality and complexity of multisource datasets for municipal and urban renewal development. Thirdly, the system is applied to enable the use of the newly developed UMAP algorithm, a model based on Riemannian geometry and algebraic topology, and the carrying out of a principal component analysis for the key dimensions and an index correlation analysis. Fourthly, various artificial intelligence-driven algorithms can be developed for urban renewal analyses based on the MUDIs. The MUDI system is a new and effective method for urban renewal planning and management that can be flexibly extended and applied to various cities and urban districts. [ABSTRACT FROM AUTHOR]

Since the 1974 paper by Peskine and Szpiro, liaison theory via complete intersections, and more generally via Gorenstein varieties, has become a standard tool kit in commutative algebra and algebraic geometry, allowing to compare algebraic features of linked varieties. In this paper we develop a liaison theory via quasi-Gorenstein varieties, a much broader class than Gorenstein varieties. As applications, we derive a connectedness property of quasi-Gorenstein subspace arrangements generalizing previous results by Benedetti and the first author, and we deduce the classical topological Lefschetz duality via the Stanley-Reisner correspondence. [ABSTRACT FROM AUTHOR]

The formulations of polynomials over a topological simplex combine the elements of topology and algebraic geometry. This paper proposes the formulation of simplicial polynomials and the properties of resulting topological manifolds in two classes, non-degenerate forms and degenerate forms, without imposing the conditions of affine topological spaces. The non-degenerate class maintains the degree preservation principle of the atoms of the polynomials of a topological simplex, which is relaxed in the degenerate class. The concept of hybrid decomposition of a simplicial polynomial in the non-degenerate class is introduced. The decompositions of simplicial polynomial for a large set of simplex vertices generate ideal components from the radical, and the components preserve the topologically isolated origin in all cases within the topological manifolds. Interestingly, the topological manifolds generated by a non-degenerate class of simplicial polynomials do not retain the homeomorphism property under polynomial extension by atom addition if the simplicial condition is violated. However, the topological manifolds generated by the degenerate class always preserve isomorphism with varying rotational orientations. The hybrid decompositions of the non-degenerate class of simplicial polynomials give rise to the formation of simplicial chains. The proposed formulations do not impose strict positivity on simplicial polynomials as a precondition. [ABSTRACT FROM AUTHOR]

Topological data analysis (TDA) is a recent approach for analyzing and interpreting complex data sets based on ideas a branch of mathematics called algebraic topology. TDA has proven useful to disentangle non-trivial data structures in a broad range of data analytics problems including the study of cardiovascular signals. Here, we aim to provide an overview of the application of TDA to cardiovascular signals and its potential to enhance the understanding of cardiovascular diseases and their treatment in the form of a literature or narrative review. We first introduce the concept of TDA and its key techniques, including persistent homology, Mapper, and multidimensional scaling. We then discuss the use of TDA in analyzing various cardiovascular signals, including electrocardiography, photoplethysmography, and arterial stiffness. We also discuss the potential of TDA to improve the diagnosis and prognosis of cardiovascular diseases, as well as its limitations and challenges. Finally, we outline future directions for the use of TDA in cardiovascular signal analysis and its potential impact on clinical practice. Overall, TDA shows great promise as a powerful tool for the analysis of complex cardiovascular signals and may offer significant insights into the understanding and management of cardiovascular diseases. [ABSTRACT FROM AUTHOR]

Given a smooth, projective curve Y, a finite group G and a positive integer n we study smooth, proper families X → Y x S → S of Galois covers of Y with Galois group isomorphic to G branched in n points, parameterized by algebraic varieties S. When G is with trivial center we prove that the Hurwitz space HnG (Y) is a fine moduli variety for this moduli problem and construct explicitly the universal family. For arbitrary G we prove that HnG (Y) is a coarse moduli variety. For families of pointed Galois covers of (Y, y0) we prove that the Hurwitz space HnG (Y, y0) is a fine moduli variety, and construct explicitly the universal family, for arbitrary group G. We use classical tools of algebraic topology and of complex algebraic geometry. [ABSTRACT FROM AUTHOR]

In this paper first we construct the Eilenberg-MacLane space through Moore space then we study this space through an infinite symmetric product functor. Finally, we study some applications. [ABSTRACT FROM AUTHOR]

Let G be a graph that can be embedded in a surface Σ of Euler characteristic ć(Σ). In this paper, we proved that there exists an integer Δ0 = 4 ·... such that the total chromatic number of G is Δ(G) + 1 if Δ(G) ≥ Δ0. [ABSTRACT FROM AUTHOR]

This is the first in a series of survey papers featuring the mathematical contributions of Themba Dube to pointfree topology and ordered algebraic structures. We cover Dube's distinguished career and benefactions to the discipline with the early beginnings in nearness frames. We envelope the essential aspects of Dube's work in structured frames. The paper radars across the initial themes of nearness, metrization, and uniform structures that Dube conceives and presents in his independent and joint published papers. Pertinent subcategories of these structured frames are discussed. We also feature Dube's imprints on certain categorical aspects of his work on βL, λL, νL and ßL. [ABSTRACT FROM AUTHOR]

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In order to study the equivariant cobordism classification of involutions with fixed point set Dold manifolds, for a special Dold manifold F=P(2,15), the equivariant cobordism classification of all involutions (M,T) with fixed point set was determined. Firstly, the Stiefel-Whitney classes of tangent bundle and normal bundle over P(2,15) were given. Secondly, according to Kosniowski-Stong Theorem, either the contradiction was obtained by constructing a suitable symmetric polynomial function to prove that the hypothesis was wrong and the involution did not exist, or that arbitrary symmetric polynomial functions satisfy Kosniowski-Stong Theorem was proved, which illustrate the existence of involution. Finally, every involution (M,T) with fixed point set P(2,15) bounds was obtained. The results show that there exists an involution with fixed point set P(2,15), and the equivariant cobordism classification of all involutions can be determined. The research results popularize the conclusion of studying involutions fixing F=P(2,n)(n=1,3,5), enrich the equivariant cobordism classification of involutions with fixed point set Dold manifolds, and provide some reference for researching involutions of fixed point set with other special manifold.

We present an algorithm for computing A-annihilated elements of the form QI [1] in H*QS0 where I runs through admissible sequences of positive excess. This is an algorithm with polynomial time complexity to address a sub-problem of an unsolved problem in algebraic topology known as the hit problem of Peterson which is likely to be NP-hard. [ABSTRACT FROM AUTHOR]

In the quest for mathematical foundations of M-theory, the Hypothesis H that fluxes are quantized in Cohomotopy theory, implies, on flat but possibly singular spacetimes, that M-brane charges locally organize into equivariant homotopy groups of spheres. Here, we show how this leads to a correspondence between phenomena conjectured in M-theory and fundamental mathematical concepts/results in stable homotopy, generalized cohomology and Cobordism theory M f : — stems of homotopy groups correspond to charges of probe p -branes near black b -branes; — stabilization within a stem is the boundary-bulk transition; — the Adams d-invariant measures G 4 -flux; — trivialization of the d-invariant corresponds to H 3 -flux; — refined Toda brackets measure H 3 -flux; — the refined Adams e-invariant sees the H 3 -charge lattice; — vanishing Adams e-invariant implies consistent global C 3 -fields; — Conner–Floyd's e-invariant is the H 3 -flux seen in the Green–Schwarz mechanism; — the Hopf invariant is the M2-brane Page charge ( G ̃ 7 -flux); — the Pontrjagin–Thom theorem associates the polarized brane worldvolumes sourcing all these charges. In particular, spontaneous K3-reductions with 24 branes are singled out from first principles : — Cobordism in the third stable stem witnesses spontaneous KK-compactification on K3-surfaces; — the order of the third stable stem implies the 24 NS5/D7-branes in M/F-theory on K3. Finally, complex-oriented cohomology emerges from Hypothesis H, connecting it to all previous proposals for brane charge quantization in the chromatic tower: K-theory, elliptic cohomology, etc. : — quaternionic orientations correspond to unit H 3 -fluxes near M2-branes; — complex orientations lift these unit H 3 -fluxes to heterotic M-theory with heterotic line bundles. In fact, we find quaternionic/complex Ravenel-orientations bounded in dimension; and we find the bound to be 10, as befits spacetime dimension 1 0 + 1. [ABSTRACT FROM AUTHOR]

This paper tackles the problem of coefficient field choice in persistent homology. When we compute a persistence diagram, we need to select a coefficient field before computation. We should understand the dependence of the diagram on the coefficient field to facilitate computation and interpretation of the diagram. We clarify that the dependence is strongly related to the torsion part of Z relative homology in the filtration. We show the sufficient and necessary conditions of the independence of coefficient field choice. An efficient algorithm is proposed to verify the independence. A slight modification of the standard persistence algorithm gives the verification algorithm. In a numerical experiment with the algorithm, a persistence diagram rarely changes even when the coefficient field changes if we consider a filtration in R 3 . The experiment suggests that, in practical terms, changes in the field coefficient will not change persistence diagrams when the data are in R 3 . [ABSTRACT FROM AUTHOR]

An orthogonal representation of a graph G $G$ over a field F ${\mathbb{F}}$ is an assignment of a vector uv∈Ft ${u}_{v}\in {{\mathbb{F}}}^{t}$ to every vertex v $v$ of G $G$, such that 〈uv,uv〉≠0 $\langle {u}_{v},{u}_{v}\rangle \ne 0$ for every vertex v $v$ and 〈uv,uv′〉=0 $\langle {u}_{v},{u}_{{v}^{^{\prime} }}\rangle =0$ whenever v $v$ and v′ $v^{\prime} $ are adjacent in G $G$. The locality of the orthogonal representation is the largest dimension of a subspace spanned by the vectors associated with a closed neighborhood in the graph. We introduce a novel graph parameter, called the local orthogonality dimension, defined for a given graph G $G$ and a given field F ${\mathbb{F}}$, as the smallest possible locality of an orthogonal representation of G $G$ over F ${\mathbb{F}}$. This is a local variant of the well‐studied orthogonality dimension of graphs, introduced by Lovász, analogously to the local variant of the chromatic number introduced by Erdös et al. We investigate the usefulness of topological methods for proving lower bounds on the local orthogonality dimension. Such methods are known to imply tight lower bounds on the chromatic number of several graph families of interest, such as Kneser graphs and Schrijver graphs. We prove that graphs for which topological methods imply a lower bound of t $t$ on their chromatic number have local orthogonality dimension at least ⌈t∕2⌉+1 $\lceil t\unicode{x02215}2\rceil +1$ over every field, strengthening a result of Simonyi and Tardos on the local chromatic number. We show that for certain graphs this lower bound is tight, whereas for others, the local orthogonality dimension over the reals is equal to the chromatic number. More generally, we prove that for every complement of a line graph, the local orthogonality dimension over R ${\mathbb{R}}$ coincides with the chromatic number. This strengthens a recent result by Daneshpajouh, Meunier, and Mizrahi, who proved that the local and standard chromatic numbers of these graphs are equal. As another extension of their result, we prove that the local and standard chromatic numbers are equal for some additional graphs, from the family of Kneser graphs. We also study the computational aspects of the local orthogonality dimension and show that for every integer k≥3 $k\ge 3$ and a field F ${\mathbb{F}}$, it is NP‐hard to decide whether the local orthogonality dimension of an input graph over F ${\mathbb{F}}$ is at most k $k$. We finally present an application of the local orthogonality dimension to the index coding problem from information theory, extending a result of Shanmugam, Dimakis, and Langberg. [ABSTRACT FROM AUTHOR]

During the last decades, the structure of mod-2 cohomology of the Steenrod ring $\mathscr {A}$ became a major subject in Algebraic topology. One of the most direct attempt in studying this cohomology by means of modular representations of the general linear groups was the surprising work [ Math. Z. 202 (1989), 493–523] by William Singer, which introduced a homomorphism, the so-called algebraic transfer , mapping from the coinvariants of certain representation of the general linear group to mod-2 cohomology group of the ring $\mathscr A.$ He conjectured that this transfer is a monomorphism. In this work, we prove Singer's conjecture for homological degree $4.$ [ABSTRACT FROM AUTHOR]

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The definition of climate itself cannot be given without a proper understanding of the key ideas of long-term behavior of a system, as provided by dynamical systems theory. Hence, it is not surprising that concepts and methods of this theory have percolated into the climate sciences as early as the 1960s. The major increase in public awareness of the socio-economic threats and opportunities of climate change has led more recently to two major developments in the climate sciences: (i) the Intergovernmental Panel on Climate Change's successive Assessment Reports and (ii) an increasing understanding of the interplay between natural climate variability and anthropogenically driven climate change. Both of these developments have benefited from remarkable technological advances in computing resources, relating throughput as well as storage, and in observational capabilities, regarding both platforms and instruments. Starting with the early contributions of nonlinear dynamics to the climate sciences, we review here the more recent contributions of (a) the theory of non-autonomous and random dynamical systems to an understanding of the interplay between natural variability and anthropogenic climate change and (b) the role of algebraic topology in shedding additional light on this interplay. The review is thus a trip leading from the applications of classical bifurcation theory to multiple possible climates to the tipping points associated with transitions from one type of climatic behavior to another in the presence of time-dependent forcing, deterministic as well as stochastic. [ABSTRACT FROM AUTHOR]

Materials made of graphyne, graphyne oxide, and graphyne quantum dots have drawn a lot of interest due to their potential uses in medicinal nanotechnology. Their remarkable physical, chemical, and mechanical qualities, which make them very desirable for a variety of prospective purposes in this area, are mostly to blame for this. In the subject of mathematical chemistry, molecular topology deals with the algebraic characterization of molecules. Molecular descriptors can examine a compound's properties and describe its molecular topology. By evaluating these indices, researchers can predict a molecule's behavior including its reactivity, solubility, and toxicity. Amidst the captivating realm of carbon allotropes, γ -graphyne has emerged as a mesmerizing tool, with exquisite attention due to its extraordinary electronic, optical, and mechanical attributes. Research into its possible applications across numerous scientific and technological fields has increased due to this motivated attention. The exploration of molecular descriptors for characterizing γ -graphyne is very attractive. As a result, it is crucial to investigate and predict γ -graphyne's molecular topology in order to comprehend its physicochemical characteristics fully. In this regard, various characterizations of γ -graphyne and zigzag γ -graphyne nanoribbons, by computing and comparing distance-degree-based topological indices, leap Zagreb indices, hyper leap Zagreb indices, leap gourava indices, and hyper leap gourava indices, are investigated. [ABSTRACT FROM AUTHOR]