Acta Mathematica Vietnamica最新文献

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 (kvarGamma )-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.

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+3leftlceil frac{s}{lfloor n/2 rfloor } rightrceil ) commutators of involutions if (mathrm {char,}Dne 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.

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.

We give a short survey of recent results on sparse-grid linear algorithms of approximate recovery and integration of functions possessing a unweighted or weighted Sobolev mixed smoothness based on their sampled values at a certain finite set. Some of them are extended to more general cases.

We study the question of which rings, and which families of ideals, have uniform symbolic topologies. In particular, we show that the uniform symbolic topology property holds for all dimension one primes in any normal complete local domain, provided dimension one primes in hypersurfaces have the uniform symbolic topology property. We also discuss bootstrapping techniques and provide a strong bootstrapping statement in positive characteristic. We apply these techniques to give families of primes in hypersurfaces of positive characteristic which have uniform symbolic topologies.

The notion of a nilpotent-invariant module was introduced and thoroughly investigated in Koşan and Quynh (Comm. Algebra 45, 2775–2782 2017) as a proper extension of an automorphism-invariant module. In this paper a ring is called a right (mathfrak {n})-ring if every right ideal is nilpotent-invariant. We show that a right (mathfrak {n})-ring is the direct sum of a square full semisimple artinian ring and a right square-free ring. Moreover, right (mathfrak {n})-rings are shown to be stably finite, and if the ring is also an exchange ring then it satisfies the substitution property, has stable range 1. These results are non-trivial extensions of similar ones on rings every right ideal is automorphism-invariant.

In this paper, we prove that the Watanabe-Yoshida conjecture holds up to dimension 7. Our primary new tool is a function, (varphi _J(R;z^t),) that interpolates between the Hilbert-Kunz multiplicities of a base ring, R, and various radical extensions, (R_n). We prove that this function is concave and show that its rate of growth is related to the size of (e_{textrm{HK}}(R)). We combine techniques from Celikbas et al. (Nagoya Math. J. 205, 149–165, 2012) and Aberbach and Enescu (Nagoya Math. J. 212, 59–85, 2013) to get effective lower bounds for (varphi ,) which translate to improved bounds on the size of Hilbert-Kunz multiplicities of singular rings. The improved inequalities are powerful enough to show that the conjecture of Watanabe and Yoshida holds in dimension 7.

We extend the asymptotic Samuel function of an ideal to a filtration and show that many of the good properties of this function for an ideal are true for filtrations. There are, however, interesting differences, which we explore. We study the notion of projective equivalence of filtrations and the relation between the asymptotic Samuel function and the multiplicity of a filtration. We further consider the case of discrete valued filtrations and show that they have particularly nice properties.

Let G be a finite simple connected graph on the vertex set (V(G)=[d]={1,dots ,d}) with edge set (E(G)={e_{1},dots , e_{n}}). Let (mathbb {K}[textbf{t}]=mathbb {K}[t_{1},dots ,t_{d}]) be the polynomial ring in d variables over a field (mathbb {K}). The edge ring of G is the affine semigroup ring (mathbb {K}[G]) generated by monomials (textbf{t}^{e}:=t_{i}t_{j}), for (e={i,j} in E(G)). In this paper, we will prove that, given integers d and n, where (dge 7) and (d+1le nle frac{d^{2}-7d+24}{2}), there exists a finite simple connected graph G with (|V(G)|=d) and (|E(G)|=n), such that (mathbb {K}[G]) is non-normal and satisfies ((S_{2}))-condition.

In this paper, we introduce a new inertial Tseng’s extragradient method with self-adaptive step sizes for approximating a common solution of split equalities of equilibrium problem (EP), non-Lipschitz pseudomonotone variational inequality problem (VIP) and fixed point problem (FPP) of nonexpansive semigroups in real Hilbert spaces. We prove that the sequence generated by our proposed method converges strongly to a common solution of the EP, pseudomonotone VIP and FPP of nonexpansive semigroups without any linesearch procedure nor the sequential weak continuity condition often assumed by authors when solving non-Lipschitz VIPs. Finally, we provide some numerical experiments for the proposed method in comparison with related methods in the literature. Our result improves, extends and generalizes several of the existing results in this direction.