Issue 4, Volume 49, December 2024

2024

1. On Topological Representation Theory from Quivers Fang Li , Zhihao Wang , Jie Wu , Bin Yu PDF

   ▸ 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.

   563-594
2. Linear Singular Continuous Time-varying Delay Equations: Stability and Filtering via LMI Approach Vu Ngoc Phat , Nguyen Truong Thanh PDF

   ▸ 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.

   595-609
3. The v-Number of Binomial Edge Ideals Siddhi Balu Ambhore , Kamalesh Saha , Indranath Sengupta PDF

   ▸ 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 -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.

   611-628
4. Kantorovich’s Theorem on Mann’s Iteration Method in Riemannian Manifold Babita Mehta , P. K. Parida , Sapan Kumar Nayak PDF

   ▸ 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.

   629-648
5. Bertini Type Results and Their Applications Indranil Biswas , Manish Kumar , A. J. Parameswaran PDF

   ▸ 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.

   649-665
6. New Algorithms for Solving the Split Equality Problems in Hilbert Spaces Nguyen Song Ha , Truong Minh Tuyen PDF

   ▸ Abstract

   We introduce a new approach by using unconstrained optimization to find a solution to the system of the split equality problems in real Hilbert spaces. Our new algorithms do not depend on the norm of the transfer mappings. We also give the relaxed iterative algorithms corresponding to the proposed algorithms. Finally, we present some numerical experiments to demonstrate the performance of the main results.

   667-689
7. A Degenerate Forward-backward Problem Involving the Spectral Dirichlet Laplacian Nguyen Ngoc Trong , Bui Le Trong Thanh , Tan Duc Do PDF

   ▸ Abstract

   Let $\varOmega$ be an open bounded subset of ${\mathbb {R}}$, $s \in (\frac{1}{2},1)$ and $\epsilon > 0$. We investigate the problem \begin{equation*}\begin{aligned} (P_\epsilon ) \quad \left\{ \begin{array}{ll} {\partial }_t u = -(-\Delta )^s \big ( \varphi (u) + \epsilon \, {\partial }_t(\psi (u)) \big )\text { in } \varOmega \times (0,T],\\ \varphi (u) + \epsilon \, {\partial }_t(\psi (u)) = 0 \text { on } {\partial }\varOmega \times (0,T], \\ u = u_0 \text { in } \varOmega \times \{0\}, \end{array}\right. \end{aligned}\end{equation*} where $\varphi , \psi \in C^\infty ({\mathbb {R}})$ and $u_0 \in {\mathcal {M}}^+(\varOmega )$ satisfy certain assumptions. Here $(-\Delta )^s$ denotes the spectral Dirichlet Laplacian and ${\mathcal {M}}^+(\varOmega )$ is the set of positive Radon measures on $\varOmega$. We show that $(P_\epsilon )$ has a unique weak solution.

   691-718