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Abstract: Abstract We give estimates of approximation by Riesz–Zygmund, Euler and Abel–Poisson means in Stepanets spaces of functions f with finite norm equal to the \(l^p\) -norm of the sequence of Vilenkin–Fourier coefficients ( \(1\leqslant p<\infty \) ) in terms of appropriate K-functional. The Vilenkin systems considered in the paper are of bounded type. For this K-functional instead of modulus of continuity we obtain direct and converse approximation theorems. Also we characterize Lipschitz classes connected with Stepanets spaces and standard modulus of continuity in terms of approximation by cited above means. PubDate: 2024-08-01

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Abstract: Abstract We study moduli space of holomorphic triples \(E_{1}\xrightarrow {\scriptscriptstyle \phi } E_{2}\) , composed of torsion-free sheaves \(E_{i}\) , \(i=1,2\) , and a holomorphic mophism between them, over a smooth complex projective surface S. The triples are equipped with Schmitt stability condition (Schmitt in Algebras Represent Theory 6(1):1–32, 2000). We observe that when Schmitt stability parameter q(m) becomes sufficiently large, the moduli space of triples benefits from having a perfect relative and absolute deformation-obstruction theory in some cases. We further generalize our construction by gluing triple moduli spaces, and extend the earlier work (Gholampour et al. in Nested Hilbert schemes on surfaces: virtual fundamental class, preprint, arXiv:1701.08899) where the obstruction theory of nested Hilbert schemes over the surface was studied. Here we extend the earlier results to the moduli space of chains $$\begin{aligned} E_{1}\xrightarrow { \ \phi _{1} \ } E_{2}\xrightarrow { \ \phi _{2} \ } \cdots \xrightarrow { \ \phi _{n-1} \ } E_{n}, \end{aligned}$$ where \(\phi _{i}\) are injective morphisms and \(\textrm{rk}\hspace{0.55542pt}(E_{i})\geqslant 1\) for all i. There is a connection, by wallcrossing in the master space, between the theory of such higher rank flags, and the theory of Higgs pairs on the surface, which provides the means to relate the flag invariants to the local DT invariants of threefold given by a line bundle on the surface, . PubDate: 2024-07-16

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Abstract: Abstract In 2010, Turaev introduced knotoids as a variation on knots that replaces the embedding of a circle with the embedding of a closed interval with two endpoints. A variety of knot invariants have been extended to knotoids. Here we provide definitions of hyperbolicity for both spherical and planar knotoids. We prove that the product of hyperbolic spherical knotoids is hyperbolic and the volumes add. We also determine the least volume of a rational spherical knotoid and provide various classes of hyperbolic knotoids. We also include tables of hyperbolic volumes for both spherical and planar knotoids. PubDate: 2024-07-15

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Abstract: Abstract In 2000, W.A. Kirk introduced the concept of directionally nonexpansive mappings. Here, we present a more comprehensive study of this class of mappings, including a fixed-point result for them. We further present a class of mappings, properly containing the directionally nonexpansive ones, for which Kirk’s theorem still holds. PubDate: 2024-07-12

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Abstract: Abstract The goal of this paper is to characterize generalized Alexander quandles of finite groups in the language of the underlying groups. Firstly, we prove that if finite groups G are simple, then the quandle isomorphic classes of generalized Alexander quandles of G one-to-one correspond to the conjugacy classes of the automorphism groups of G. This correspondence can be also claimed for the case of symmetric groups. Secondly, we give a characterization of generalized Alexander quandles of finite groups G under some assumptions in terms of G. As corollaries of this characterization, we obtain several characterizations to some particular groups, e.g., abelian groups and dihedral groups. Finally, we perform a characterization of generalized Alexander quandles arising from groups with their order up to 15. PubDate: 2024-07-09

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Abstract: Abstract Given a complete atomic Boolean algebra, we show there is a commutative BCK-algebra whose ideal lattice is that Boolean algebra. This result is shown to exist within a larger framework involving BCK-algebras of functions, whose ideals and prime ideals are analyzed by way of a specific Galois connection. As a corollary of the main theorem, we show that every discrete topological space is the prime spectrum of a cBCK-algebra. PubDate: 2024-06-28

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Abstract: Abstract The quandle coloring quiver was introduced by Cho and Nelson as a categorification of the quandle coloring number. In some cases, it has been shown that the quiver invariant offers more information than other quandle enhancements. In this paper, we compute the quandle coloring quivers of 2-bridge links with respect to the dihedral quandles. PubDate: 2024-06-12

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Abstract: Abstract We show that any pseudo-effective divisor on a normal surface decomposes uniquely into its “integral positive” part and “integral negative” part, which is an integral analog of Zariski decompositions. By using this decomposition, we give three applications: a vanishing theorem of divisors on surfaces (a generalization of Kawamata–Viehweg and Miyaoka vanishing theorems), Reider-type theorems of adjoint linear systems on surfaces (including a log version and a relative version of the original one) and extension theorems of morphisms defined on curves on surfaces (generalizations of Serrano and Paoletti’s results). PubDate: 2024-06-03

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Abstract: Abstract We deal with topological spaces homeomorphic to their respective squares. Primarily, we investigate the existence of large families of such spaces in some subclasses of compact metrizable spaces. As our main result we show that there is a family of size continuum of pairwise non-homeomorphic compact metrizable zero-dimensional spaces homeomorphic to their respective squares. This answers a question of Włodzimierz J. Charatonik. We also discuss the situation in the classes of continua, Peano continua and absolute retracts. PubDate: 2024-06-03

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Abstract: Abstract Let X be a complex projective K3 surface and let \(T_X\) be its transcendental lattice; the characteristic polynomials of isometries of \(T_X\) induced by automorphisms of X are powers of cyclotomic polynomials. Which powers of cyclotomic polynomials occur' The aim of this note is to answer this question, as well as related ones, and give an alternative approach to some results of Kondō, Machida, Oguiso, Vorontsov, Xiao and Zhang; this leads to questions and results concerning orthogonal groups of lattices. PubDate: 2024-05-30 DOI: 10.1007/s40879-024-00745-1

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Abstract: Abstract For a projective 2n-dimensional irreducible holomorphic symplectic manifold Y of generalized Kummer deformation type and j the smallest prime number dividing \(n+1\) , we prove the Lefschetz standard conjectures in degrees \(<2(n+1)(j-1)/j\) . We show that the restriction homomorphism from the cohomology of a projective deformation of a moduli space of Gieseker-stable sheaves on an Abelian surface to the cohomology of Y is surjective in these degrees. An immediate corollary is that the Lefschetz standard conjectures hold for Y when \(n+1\) is prime. The proofs rely on Markman’s description of the monodromy of generalized Kummer varieties and construction of a universal family of moduli spaces of sheaves, Verbitsky’s theory of hyperholomorphic sheaves, and the cohomological decomposition theorem. PubDate: 2024-05-28 DOI: 10.1007/s40879-024-00744-2

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Abstract: Abstract We propose two possible generalizations of the A-integral within the Riemann measurable functions class and study their basic properties, relation to one another, and to the Birkhoff integral. PubDate: 2024-05-28 DOI: 10.1007/s40879-024-00751-3

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Abstract: Abstract We classify G-solid rational surfaces over the field of complex numbers for finite group actions. PubDate: 2024-05-17 DOI: 10.1007/s40879-024-00747-z

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Abstract: Abstract We extend the notion of proper elements to all finite Coxeter groups. For all infinite families of finite Coxeter groups we prove that the probability a random element is proper goes to zero in the limit. This proves a conjecture of the third author and Alexander Yong regarding the proportion of Schubert varieties that are Levi spherical for all infinite families of Weyl groups. We also enumerate the proper elements in the exceptional Coxeter groups. PubDate: 2024-05-16 DOI: 10.1007/s40879-024-00746-0

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Abstract: Abstract We prove that the finite -algebra \(U(\mathfrak {osp}_{1 2n}, f_\textrm{prin})\) associated to \(\mathfrak {osp}_{1 2n}\) and its principal nilpotent element \(f_\textrm{prin}\) is isomorphic to Gorelik’s ghost center of \(\mathfrak {osp}_{1 2n}\) . It is an analogue for \(\mathfrak {osp}_{1 2n}\) of a theorem of Kostant (Invent Math 48(2):101–184, 1978). PubDate: 2024-05-15 DOI: 10.1007/s40879-024-00743-3

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Abstract: Abstract We investigate the behaviour of the geometric directional bundles, associated to arbitrary subsets in \(\mathbb {R}^n\) , under bi-Lipschitz homeomorphisms, and give conditions under which their bi-Lipschitz type is preserved. The most general sets we consider satisfy the sequence selection property (SSP) and, consequently, we investigate the behaviour of such sets under bi-Lipschitz homeomorphisms as well. In particular, we show that the bi-Lipschitz images of subanalytic sets generically satisfy (SSP). PubDate: 2024-05-03 DOI: 10.1007/s40879-024-00742-4

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Abstract: Abstract In Bierstone and Parusiński (Duke Math J 167(16):3115–3128, 2018) proved the existence of global smoothings for closed subanalytic sets, both in an embedded and a non-embedded sense. In particular, in the non-embedded desingularization procedure the authors constructed smoothings of (generically) even degree, indeed it is well-known the existence of subanalytic sets which do not admit non-embedded smoothings of (generically) odd degree. In this paper we introduce a natural topological notion of nonbounding equator for subanalytic sets and prove a criterion to determine whether a closed subanalytic set X only admits global smoothings of even degree along the nonbounding equator. More in detail, we prove that if X has a nonbounding equator Y then every smoothing of X which is a covering on a connected neighborhood W of Y has even degree over W. PubDate: 2024-04-29 DOI: 10.1007/s40879-024-00740-6

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Abstract: Abstract We compute the CM volume, that is the degree of the descended CM line bundle on the coarse moduli space in two cases: on the Fano K-moduli space of quartic del Pezzo in any dimension, and on the K-moduli space of the log Fano hyperplane arrangements of dimension one and two. Furthermore, we relate these volumes to the Weil–Petersson volumes by extending the notion of Weil–Petersson metric in the log case. PubDate: 2024-04-23 DOI: 10.1007/s40879-024-00741-5

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Abstract: Abstract We consider the time-dependent 3-D Navier–Stokes equations (NSE) on a multi-connected bounded domain \(\Omega \subset \mathbb {R^{\textrm{3}}}\) with inhomogeneous boundary data \(\beta \in H^{1/2}(\Gamma )\) on \(\partial \Omega =\Gamma \) , where \(\Gamma \) is a union of Lipschitz continuous surfaces \(\Gamma _{0},\Gamma _{1},\dots ,\Gamma _{l}\) . This assumption includes the particular case when the \(\Gamma _{i}\) are disjoint, the stationary version of which is classically known as Leray’s problem. Existence results for Leray’s problem have either assumed flux conditions beyond the general flux condition necessitated by compatibility constraints, or required size restrictions on the data. Here we incorporate a spectral hyperviscosity term in the time-dependent case and obtain existence and foundational results, assuming only the general flux condition and without imposing size restrictions on the boundary data \(\beta \) . For any such \(\beta \in H^{1/2}(\Gamma )\) we establish global existence and uniqueness of mild solutions. Then on any interval [0, T] on which these solutions and the corresponding NSE solution share a common \(H^{1}\) -bound (as is present on local intervals of existence of strong solutions, in certain special cases, or as is commonly assumed in achieving strong convergence results in numerical studies) we show for slightly more regular \(\beta \) that the spectrally-hyperviscous solutions converge strongly and uniformly in \(H^{1}(\Omega )\) to the NSE solution as the spectral hyperviscosity term vanishes in the limit of key parameters. To achieve this robust sense of approximation of the NSE system, an involved setup and specially-adapted semigroup techniques assume essential roles, and the exposition is new for the case \(\beta =0\) as well. Our final results adapt the NSE reformulation in Liu et al. (J Comput Phys 229(9):3428–3453, 2010) to recast our approximating system in a form potentially more adaptable to computation. PubDate: 2024-04-22 DOI: 10.1007/s40879-024-00739-z

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Abstract: Abstract We develop the Darboux theory of integrability for polynomial vector fields in the n-dimensional torus \(\mathbb {T}^n\) . We also provide the maximum number of invariant parallels and meridians that a polynomial vector field X on \(\mathbb {T}^n\) can have in function of its degree. PubDate: 2024-04-05 DOI: 10.1007/s40879-024-00737-1