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Abstract: Abstract In this article, we give a generalization of δ-twisted homology introduced by Jingyan Li, Vladimir Vershinin and Jie Wu, called Δ-twisted homology, which enriches the theory of δ-(co)homology introduced by Alexander Grigor’yan, Yuri Muranov and Shing-Tung Yau. We show that the Mayer—Vietoris sequence theorem holds for Δ-twisted homology. Applying the Δ-twisted ideas to Cartesian products, we introduce the notion of Δ-twisted Cartesian product on simplicial sets, which generalizes the classical work of Barratt, Gugenheim and Moore on twisted Cartesian products of simplicial sets. Under certain hypothesis, we show that the coordinate projection of Δ-twisted Cartesian product admits a fibre bundle structure. PubDate: 2022-10-01

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Abstract: Abstract We provide a construction of the moduli space of stable coherent sheaves in the world of non-archimedean geometry, where we use the notion of Berkovich non-archimedean analytic spaces. The motivation for our construction is Tony Yue Yu’s non-archimedean enumerative geometry in Gromov—Witten theory. The construction of the moduli space of stable sheaves using Berkovich analytic spaces will give rise to the non-archimedean version of Donaldson—Thomas invariants. In this paper we give the moduli construction over a non-archimedean field \({\mathbb{K}}\) . We use the machinery of formal schemes, that is, we define and construct the formal moduli stack of (semi)-stable coherent sheaves over a discrete valuation ring R, and taking generic fiber we get the non-archimedean analytic moduli space of semistable coherent sheaves over the fractional non-archimedean field \({\mathbb{K}}\) . We generalize Joyce’s d-critical scheme structure in [37] or Kiem—Li’s virtual critical manifolds in [38] to the world of formal schemes, and Berkovich non-archimedean analytic spaces. As an application, we provide a proof for the motivic localization formula for a d-critical non-archimedean \({\mathbb{K}}\) -analytic space using global motive of vanishing cycles and motivic integration on oriented formal d-critical schemes. This generalizes Maulik’s motivic localization formula for the motivic Donaldson—Thomas invariants. PubDate: 2022-10-01

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Abstract: Abstract We establish a blowing down criterion in the context of birational symplectic geometry in dimension 6. PubDate: 2022-10-01

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Abstract: Abstract It is already known that finitely-generated Fuchsian groups are profinitely rigid among all lattices of connected Lie groups by the result of Bridson, Conder and Reid. Hence the triangle groups are distinguished among themselves by their finite quotients. We focus on the question about quantifying the size of a quotient which separates two triangle groups and give an explicit upper bound. PubDate: 2022-10-01

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Abstract: Abstract We give an overview of five rationalization theories for spaces (Bousfield-Kan’s ℚ-completion; Sullivan’s rationalization; Bousfield’s homology rationalization; Casacuberta-Peschke’s Ω-rationalization; Gómez-Tato-Halperin-Tanré’s π1-fiberwise rationalization) that extend the classical rationalization of simply connected spaces. We also give an overview of the corresponding rationalization theories for groups (ℚ-completion; Hℚ-localization; Baumslag rationalization) that extend the classical Malcev completion. PubDate: 2022-10-01

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Abstract: Abstract Let Kn denote the complete graph consisting of n vertices, every pair of which forms an edge. We want to know the least possible number of the intersections, when we draw the graph Kn on the plane or on the sphere using continuous arcs for edges. In a paper published in 1960, Richard K. Guy conjectured that the least possible number of the intersections is \({1 \over {64}}{\left( {n - 1} \right)^2}{\left( {n - 3} \right)^2}\) if n is odd, or \({1 \over {64}}n{\left( {n - 2} \right)^2}\left( {n - 4} \right)\) if n is even. A virgin road Vn is a drawing of a Hamiltonian cycle in Kn consisting of n vertices and n edges such that no other edge-representing arcs cross Vn. A drawing of Kn is called virginal if it contains a virgin road. All drawings considered in this paper will be virginal with respect to a fixed virgin road Vn. We will present a certain drawing of a subgraph of Kn, for each n(≥ 5), which is “characteristic” in the sense that any minimal virginal drawing of Kn containing this subdrawing satisfies Guy’s conjecture. PubDate: 2022-10-01

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Abstract: Abstract This is a set of lecture notes for the first author’s lectures on the difference equations in 2019 at the Institute of Advanced Study for Mathematics at Zhejiang University. We focus on explicit computations and examples. The convergence of local solutions is discussed. PubDate: 2022-10-01

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Abstract: Abstract In this paper we study the cubical stuctures and fundamental groups of real toric spaces. We give an explicit presentation of the fundamental group of the real toric space over a simple polytope. Then using this presentation, we give a description of the existence of non-degenerate colourings on a simple polytope from a homotopy point of view. PubDate: 2022-10-01

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Abstract: Abstract Let M and N be topological spaces, let G be a group, and let τ: G × M → M be a proper free action of G. In this paper, we define a Borsuk—Ulam-type property for homotopy classes of maps from M to N with respect to the pair (G, τ) that generalises the classical antipodal Borsuk—Ulam theorem of maps from the n-sphere \({\mathbb{S}^n}\) to ℝn. In the cases where M is a finite pathwise-connected CW-complex, G is a finite, non-trivial Abelian group, τ is a proper free cellular action, and N is either ℝ2 or a compact surface without boundary different from \({\mathbb{S}^2}\) and ℝℙ2, we give an algebraic criterion involving braid groups to decide whether a free homotopy class β ∈ [M, N] has the Borsuk—Ulam property. As an application of this criterion, we consider the case where M is a compact surface without boundary equipped with a free action τ of the finite cyclic group ℤn. In terms of the orientability of the orbit space Mτ of M by the action τ, the value of n modulo 4 and a certain algebraic condition involving the first homology group of Mτ, we are able to determine if the single homotopy class of maps from M to ℝ2 possesses the Borsuk—Ulam property with respect to (ℤn, τ). Finally, we give some examples of surfaces on which the symmetric group acts, and for these cases, we obtain some partial results regarding the Borsuk—Ulam property for maps whose target is ℝ2. PubDate: 2022-10-01

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Abstract: Abstract With the great advancement of experimental tools, a tremendous amount of biomolecular data has been generated and accumulated in various databases. The high dimensionality, structural complexity, the nonlinearity, and entanglements of biomolecular data, ranging from DNA knots, RNA secondary structures, protein folding configurations, chromosomes, DNA origami, molecular assembly, to others at the macromolecular level, pose a severe challenge in their analysis and characterization. In the past few decades, mathematical concepts, models, algorithms, and tools from algebraic topology, combinatorial topology, computational topology, and topological data analysis, have demonstrated great power and begun to play an essential role in tackling the biomolecular data challenge. In this work, we introduce biomolecular topology, which concerns the topological problems and models originated from the biomolecular systems. More specifically, the biomolecular topology encompasses topological structures, properties and relations that are emerged from biomolecular structures, dynamics, interactions, and functions. We discuss the various types of biomolecular topology from structures (of proteins, DNAs, and RNAs), protein folding, and protein assembly. A brief discussion of databanks (and databases), theoretical models, and computational algorithms, is presented. Further, we systematically review related topological models, including graphs, simplicial complexes, persistent homology, persistent Laplacians, de Rham—Hodge theory, Yau—Hausdorff distance, and the topology-based machine learning models. PubDate: 2022-10-01

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Abstract: Abstract In this article we estimate mutual distances of simple choreographic solutions for the Newtonian three-body problem. Explicit formulas will be proved and our applications include the famous figure-8 orbit. PubDate: 2022-10-01

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Abstract: Abstract We study the homology of the dual de Rham complex as functors on the category of abelian groups. We give a description of homology of the dual de Rham complex up to degree 7 for free abelian groups and present a corrected version of the proof of Jean’s computations of the zeroth homology group. PubDate: 2022-10-01

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Abstract: Abstract In this paper we consider the Monge–Ampère type equations on compact almost Hermitian manifolds. We derive C∞ a priori estimates under the existence of an admissible \(\cal{C}\) -subsolution. Finally, we obtain an existence result if there exists an admissible supersolution. PubDate: 2022-10-01

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Abstract: Abstract For a finite group G, the power graph \(\cal{P}(G)\) is a simple connected graph whose vertex set is the set of elements of G and two distinct vertices x and y are adjacent if and only if xi = y or yj = x, for 2 ≤ i, j ≤ n. In this paper, we obtain the distance Laplacian spectrum of power graphs of finite groups such as cyclic groups, dihedral groups, dicyclic groups, abelian groups and elementary abelian p groups. Moreover, we find the largest and second smallest distance Laplacian eigenvalue of power graphs of such groups. PubDate: 2022-10-01

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Abstract: Abstract A Steinberg-type conjecture on circular coloring is recently proposed that for any prime p ≥ 5, every planar graph of girth p without cycles of length from p + 1 to p(p − 2) is Cp-colorable (that is, it admits a homomorphism to the odd cycle Cp). The assumption of p ≥ 5 being prime number is necessary, and this conjecture implies a special case of Jaeger’s Conjecture that every planar graph of girth 2p — 2 is Cp-colorable for prime p ≥ 5. In this paper, combining our previous results, we show the fractional coloring version of this conjecture is true. Particularly, the p = 5 case of our fractional coloring result shows that every planar graph of girth 5 without cycles of length from 6 to 15 admits a homomorphism to the Petersen graph. PubDate: 2022-10-01

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Abstract: Abstract In this paper we compute the cohomology of moduli space of cubic fourfolds with ADE type singularities relying on Kirwan’s blowup and Laza’s GIT construction. More precisely, we obtain the Betti numbers of Kirwan’s resolution of the moduli space. Furthermore, by applying decomposition theorem we obtain the Betti numbers of the intersection cohomology of Baily-Borel compactification of the moduli space. PubDate: 2022-10-01

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Abstract: Abstract Let A be a basic connected finite dimensional associative algebra over an algebraically closed field k and G be a cyclic group. There is a quiver QG with relations ρG such that the skew group algebras A[G] is Morita equivalent to the quotient algebra of path algebra kQG modulo ideal (ρG). Generally, the quiver QG is not connected. In this paper we develop a method to determine the number of connect components of QG. Meanwhile, we introduce the notion of weight for underlying quiver of A such that A is G-graded and determine the connect components of smash product A#kG*. PubDate: 2022-10-01

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Abstract: Abstract In this paper, it is proved that the weak convergence of the Lp Gaussian surface area measures implies the convergence of the corresponding convex bodies in the Hausdorff metric for p ≥ 1. Moreover, continuity of the solution to the Lp Gaussian Minkowski problem with respect to p is obtained. PubDate: 2022-09-15

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Abstract: Abstract This paper focuses on the higher order fractional differentiability of weak solution pairs to the following nonlinear stationary Stokes system $$\left\{ {\matrix{{{\rm{div}}\,{\cal A}\left( {x,D{\bf{u}}} \right) - \nabla \pi = {\rm{div}}\,\,{\bf{F}},} \hfill & {{\rm{in}}\,\,\Omega ,} \hfill \cr {{\rm{div}}\,{\bf{u}} = 0,} \hfill & {{\rm{in}}\,\,\Omega .} \hfill \cr } } \right.$$ In terms of the difference quotient method, our first result reveals that if F ∈ B p,q,loc β (Ω,ℝn)for p = 2 and \(1 \le q \le {{2n} \over {n - 2\beta }}\) , then such extra Besov regularity can transfer to the symmetric gradient Du and its pressure π with no losses under a suitable fractional differentiability assumption on \(x \mapsto {\cal A}\left( {x,\xi } \right)\) . Furthermore, when the vector field \({\cal A}\left( {x,D{\bf{u}}} \right)\) is simplified to the full gradient ∇u, we improve the aforementioned Besov regularity for all integrability exponents p and q by establishing a new Campanato-type decay estimates for (∇u, π). PubDate: 2022-09-15