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Abstract: Abstract Let \((R,\mathfrak {m})\) be a Noetherian local ring such that \(\widehat{R}\) is reduced. We prove that, when \(\widehat{R}\) is \(S_2\) , if there exists a parameter ideal \(Q\subseteq R\) such that \(\bar{e}_1(Q)=0\) , then R is regular and \(\nu (\mathfrak {m}/Q)\le 1\) . This leads to an affirmative answer to a problem raised by Goto-Hong-Mandal [Goto, S., Hong, J., Mandal, M.: The positivity of the first coefficients of normal Hilbert polynomials. Proc. Amer. Math. Soc. 139(7), 2399–2406 (2011)]. We also give an alternative proof (in fact a strengthening) of their main result. In particular, we show that if \(\widehat{R}\) is equidimensional, then \(\bar{e}_1(Q)\ge 0\) for all parameter ideals \(Q\subseteq R\) , and in characteristic \(p>0\) , we actually have \(e_1^*(Q)\ge 0\) . Our proofs rely on the existence of big Cohen-Macaulay algebras. PubDate: 2024-08-06
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Abstract: Abstract In this paper, we establish a series of new results on strong duality and solution existence for conic linear programs in locally convex Hausdorff topological vector spaces and finite-dimensional Euclidean spaces. Namely, under certain regularity conditions based on quasi-relative interiors of convex sets, we prove that if one problem in the dual pair consisting of a primal program and its dual has a solution, then the other problem also has a solution, and the optimal values of the problems are equal. In addition, we show that if the cones are generalized polyhedral convex, then the regularity conditions can be omitted. Moreover, if the spaces are finite-dimensional and the ordering cones are closed convex, then instead of the solution existence condition, it suffices to require the finiteness of the optimal value. The present paper complements our recent research work [Luan, N.N., Yen, N.D.: Strong duality and solution existence under minimal assumptions in conic linear programming. J. Optim. Theory Appl. (https://doi.org/10.1007/s10957-023-02318-w)]. PubDate: 2024-08-06
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Abstract: Abstract Let \(H = \langle n_1, n_2,n_3\rangle \) be a numerical semigroup. Let \(\widetilde{H}\) be the interval completion of H, namely the semigroup generated by the interval \(\langle n_1,n_1+1, \ldots , n_3\rangle \) . Let K be a field and K[H] the semigroup ring generated by H. Let \(I_H^{*}\) be the defining ideal of the tangent cone of K[H]. In this paper, we describe the defining equations of \(I_H^{*}\) . From that, we prove the Herzog-Stamate conjecture for monomial space curves stating that \(\beta _i(I_H^{*}) \le \beta _i(I_{\widetilde{H}}^{*})\) for all i, where \(\beta _i(I_H^{*})\) and \(\beta _i(I_{\widetilde{H}}^{*})\) are the ith Betti numbers of \(I_H^{*}\) and \(I_{\widetilde{H}}^{*}\) respectively. PubDate: 2024-08-06
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Abstract: Abstract In this paper we introduce minimal balanced neighborly polynomials and show some methods to construct such polynomials. In particular, using this notion, we prove the existence of balanced neighborly polynomials of the following types: (i) type \((p,\dots ,p)\) for most prime numbers p, (ii) types \((d-1,d,d,d)\) , \((d-1,d-1,d,d)\) and \((d-1,d-1,d-1,d)\) when d is odd or is divisible by 4. We also construct balanced neighborly simplicial spheres of type \((2,4k-1,4k-1,4k-1)\) . PubDate: 2024-08-06
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Abstract: Abstract Let K be a division ring with center Z(K), and n a positive integer. Let \(\textrm{SL}(n,K)\) be the special linear group of degree n over K and \(\textrm{SD}(n,K)\) its subgroup consisting of all diagonal matrices whose Dieudonne’s determinant is \(\overline{1}\) . We prove that \(\textrm{SD}(n,K)\) is weakly pronormal, but not pronormal in \(\textrm{SL}(n,K)\) provided either Z(K) is an infinite field in case \(n\ge 3\) or Z(K) is a finite field containing at least seven elements in case \(n\ge 5\) . PubDate: 2024-08-02
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Abstract: Abstract Let G be a graph with n vertices and let \(S=\mathbb {K}[x_1,\dots ,x_n]\) be the polynomial ring in n variables over a field \(\mathbb {K}\) . Assume that I(G) and J(G) denote the edge ideal and the cover ideal of G, respectively. We provide a combinatorial upper bound for the index of depth stability of symbolic powers of J(G). As a consequence, we compute the depth of symbolic powers of cover ideals of fully clique-whiskered graphs. Meanwhile, we determine a class of graphs G with the property that the Castelnuovo–Mumford regularity of S/I(G) is equal to the induced matching number of G. PubDate: 2024-07-31
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Abstract: Abstract We prove Bertini type theorems and give some applications of them. The applications are in the context of Lefschetz theorem for Nori fundamental group for normal varieties as well as for geometric formal orbifolds. In another application, it is shown that a certain class of a smooth quasi-projective variety contains a smooth curve such that irreducible lisse \(\ell \) –adic sheaves on the variety with “ramification bounded by a branch data” remains irreducible when restricted to the curve. PubDate: 2024-07-17
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Abstract: Abstract The invariant \(\textrm{v}\) -number was introduced very recently in the study of Reed-Muller-type codes. Jaramillo and Villarreal (J. Combin. Theory Ser. A 177:105310, 2021) initiated the study of the \(\textrm{v}\) -number of edge ideals. Inspired by their work, we take the initiation to study the \(\textrm{v}\) -number of binomial edge ideals in this paper. We discuss some properties and bounds of the \(\textrm{v}\) -number of binomial edge ideals. We explicitly find the \(\textrm{v}\) -number of binomial edge ideals locally at the associated prime corresponding to the cutset \(\emptyset \) . We show that the \(\textrm{v}\) -number of Knutson binomial edge ideals is less than or equal to the \(\textrm{v}\) -number of their initial ideals. Also, we classify all binomial edge ideals whose \(\textrm{v}\) -number is 1. Moreover, we try to relate the \(\textrm{v}\) -number with the Castelnuvo-Mumford regularity of binomial edge ideals and give a conjecture in this direction. PubDate: 2024-07-04
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Abstract: Abstract Convergence analysis of Mann’s iteration method using Kantorovich’s theorem in the context of connected and complete Riemannian manifolds has been examined in this paper. We also provide an algorithm for Mann’s method to find a singularity in a two dimensional sphere \(S^2\) . Finally, we provide an example that shows the better convergence result of Mann’s method in comparison to that of Newton’s method. PubDate: 2024-06-29
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Abstract: Abstract We propose a primal-dual backward reflected forward splitting method for solving structured primal-dual monotone inclusions in real Hilbert spaces. The algorithm allows to use the inexact computations of Lipschitzian and cocoercive operators. The strong convergence of the generated iterative sequence is proved under the strong monotonicity condition, whilst the weak convergence is formally proved under several conditions used in the literature. An application to a structured minimization problem is provided. PubDate: 2024-06-28
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Abstract: Abstract In this paper, we present the complete convergence for weighted sums of coordinatewise pairwise negative quadrant dependent random variables taking values in Hilbert spaces. As an application of the results, the complete convergence of degenerate von Mises statistics is investigated. PubDate: 2024-06-15 DOI: 10.1007/s40306-024-00537-5
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Abstract: Abstract In this paper, we propose an LMI-based approach to study stability and \(H_\infty \) filtering for linear singular continuous equations with time-varying delay. Particularly, the delay pattern is quite general and includes non-differentiable time-varying delay. First, new delay-dependent sufficient conditions for the admissibility of the equation are extended to the time-varying delay case. Then, we propose a design of \(H_\infty \) filters via feasibility problem involving linear matrix inequalities, which can be solved by the standard numerical algorithm. The proposed result is demonstrated through an example and simulations. PubDate: 2024-06-14 DOI: 10.1007/s40306-024-00534-8
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Abstract: Abstract We study the problem of reconstructing an unknown source term in parabolic equations from integral observations. It is reformulated into a variational problem in combination with Tikhonov regularization and then a formula for the gradient of the objective functional to be minimized is computed via a solution of an adjoint problem. The variational problem is discretized by the splitting method based on finite difference schemes and solved by the conjugate gradient method. A numerical scheme for numerically estimating singular values of the solution operator in the inverse problem is suggested. Some numerical examples are presented to show the efficiency of the method. PubDate: 2024-06-11 DOI: 10.1007/s40306-024-00536-6
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Abstract: Abstract We obtain some weighted \(L^{p}\) -Sobolev estimates with gain on p and the weight for solutions of the \(\overline{\partial }\) -equation in linearly convex domains of finite type in \(\mathbb {C}^{n}\) and apply them to obtain weighted \(L^{p}\) -Sobolev estimates for weighted Bergman projections of convex domains of finite type for quite general weights equivalent to a power of the distance to the boundary. PubDate: 2024-06-11 DOI: 10.1007/s40306-024-00530-y
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Abstract: Abstract We define a Real version of smooth Deligne cohomology for manifolds with involution which interpolates between equivariant sheaf cohomology and smooth imaginary-valued forms. Our main result is a classification of Real line bundles with Real connection on manifolds with involution. PubDate: 2024-06-05 DOI: 10.1007/s40306-024-00538-4
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Abstract: Abstract In this paper, using Nevanlinna theory and linear algebra, we characterize transcendental meromorphic solutions of nonlinear differential equation of the form $$\begin{aligned} f^n+Q_d(z,f)=\sum _{i=1}^{l}p_{i}(z)e^{\alpha _{i}(z)}, \end{aligned}$$ where \(l\ge 2\) , \(n\ge l+2\) are integers, f(z) is a meromorphic function, \(Q_d(z,f)\) is a differential polynomial in f(z) of degree \(d\le n-(l+1)\) with rational functions as its coefficients, \(p_{1}(z)\) , \(p_{2}(z)\) , \(\dots \) , \(p_{l}(z)\) are non-vanishing rational functions and \(\alpha _{1}(z)\) , \(\alpha _{2}(z)\) , \(\dots \) , \(\alpha _{l}(z)\) are nonconstant polynomials such that \(\alpha _{1}^\prime (z)\) , \(\alpha _{2}^\prime (z)\) , \(\dots \) , \(\alpha _{l}^\prime (z)\) are distinct. Further, we give the necessary conditions for the existence of meromorphic solutions of the above equation, and supply the example to demonstrate the sharpness of the condition of the obtained theorem. PubDate: 2024-05-31 DOI: 10.1007/s40306-024-00539-3
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Abstract: Abstract In this note, a condition (open persistence) is presented under which a (pre)closure operation on submodules (resp. ideals) over rings of global sections over a scheme X can be extended to a (pre)closure operation on sheaves of submodules of a coherent \(\mathcal {O}_X\) -module (resp. sheaves of ideals in \(\mathcal {O}_X\) ). A second condition (glueability) is given for such an operation to behave nicely. It is shown that for an operation that satisfies both conditions, the question of whether the operation commutes with localization at single elements is equivalent to the question of whether the new operation preserves quasi-coherence. It is shown that both conditions hold for tight closure and some of its important variants, thus yielding a geometric reframing of the open question of whether tight closure localizes at single elements. A new singularity type (semi F-regularity) arises, which sits between F-regularity and weak F-regularity. The paper ends with (1) a case where semi F-regularity and weak F-regularity coincide, and (2) a case where they cannot coincide without implying a solution to a major conjecture. PubDate: 2024-05-24 DOI: 10.1007/s40306-024-00533-9
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Abstract: Abstract In this work, we introduce topological representations of a quiver as a system consisting of topological spaces and its relationships determined by the quiver. Such a setting gives a natural connection between topological representations of a quiver and diagrams of topological spaces. Firstly, we investigate the relation between the category of topological representations and that of linear representations of a quiver via \(P(\varGamma )\) - \(\mathcal {TOP}^o\) and \(k\varGamma \) -Mod, concerning (positively) graded or vertex (positively) graded modules. Secondly, we discuss the homological theory of topological representations of quivers via the \(\varGamma \) -limit functor \(lim ^{\varGamma }\) , and use it to define the homology groups of topological representations of quivers via \(H _n\) . It is found that some properties of a quiver can be read from homology groups. Thirdly, we investigate the homotopy theory of topological representations of quivers. We define the homotopy equivalence between two morphisms in \({\textbf {Top}}\text{- }{} {\textbf {Rep}}\varGamma \) and show that the parallel Homotopy Axiom also holds for top-representations based on the homotopy equivalence. Last, we obtain the functor \(At^{\varGamma }\) from \({\textbf {Top}}\text{- }{} {\textbf {Rep}}\varGamma \) to \({\textbf {Top}}\) and show that \(At^{\varGamma }\) preserves homotopy equivalence between morphisms. The relationship between the homotopy groups of a top-representation (T, f) and the homotopy groups of \(At^{\varGamma }(T,f)\) is also established. PubDate: 2024-05-21 DOI: 10.1007/s40306-024-00531-x
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Abstract: Abstract In connection with [Theorem 4.6, Linear Algebra Appl. 646, 119–131, (2022)], we show that each matrix in the commutator subgroup of the general linear group over a centrally-finite division ring D, in which each element in the commutator subgroup of D is a product of at most s commutators, can be written as a product of at most \(3+3\left\lceil \frac{s}{\lfloor n/2 \rfloor } \right\rceil \) commutators of involutions if \(\mathrm {char\,}D\ne 2\) , where \({\displaystyle \lceil x \rceil }\) , \({\displaystyle \lfloor x \rfloor }\) denote the ceiling and floor functions of x, respectively. Moreover, we also present the special case when \(D= \mathbb {H}\) , the division ring of quaternions, and an application in real group algebras. PubDate: 2024-05-09 DOI: 10.1007/s40306-024-00532-w
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Abstract: Abstract The Schmidt subspace theorem has been studied extensively for both cases of fixed and moving targets in projective spaces over number fields and the case of fixed targets in projective spaces over function fields. This paper studies the case of function fields with moving targets; in particular, we extend the result of Min Ru and Paul Vojta in the Inventiones Mathematicae (1997) to the case of moving hyperplane targets in projective spaces over function fields. PubDate: 2024-05-07 DOI: 10.1007/s40306-024-00529-5
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