arXiv subjects

Farhad Rahmati

Publications and source records attributed to Farhad Rahmati.

18 recordsLinked to original sources

arXiv2026

Linked sheaves of modules

We introduce a notion of linkage for sheaves of modules on connected Noetherian schemes, extending classical linkage of modules. Linkage is defined for stable sheaves admitting finite free resolutions via the transpose and syzygy functors. We show that linkedness is a local property and that, on affine schemes, a coherent sheaf is linked if and only if its module of global sections is linked. We further show that linkage is preserved under restriction and, under suitable rank conditions, under gluing over connected schemes. We also obtain criteria for the existence of linked subsheaves when the structure sheaf is not a domain. In the projective setting, we compare invariants of linked sheaves, including graded cohomology modules, Castelnuovo-Mumford regularity, and Hilbert polynomials.

math.AC↗

arXiv2025

Linkage of sheaves of modules

Inspired by the works in linkage theory of modules, we define the concept of linkage of sheaves of modules as a generalization of linkage of modules. Thus, we expressed it in geometry algebraic language. We show that the linkedness of sheaves is a locally property. As an important result, we have shown that the sheaf of modules made of Glueing schemes and Glueing linked sheaves of modules is a linked sheaf. Also, it has been shown that for every sheaf of modules on non-domain, it is possible to obtain a maximal linked subsheaf of modules.

math.AG↗

arXiv2024

Symbolic Powers and Symbolic Rees Algebras of Binomial Edge Ideals of Some Classes of Block Graphs

In this paper, we investigate some properties of symbolic powers and symbolic Rees algebras of binomial edge ideals associated with some classes of block graphs. First, it is shown that symbolic powers of binomial edge ideals of pendant cliques graphs coincide with the ordinary powers. Furthermore, we see that binomial edge ideals of a generalization of these graphs are symbolic $F$-split. Consequently, net-free generalized caterpillar graphs are also a class of block graphs with symbolic $F$-split binomial edge ideals. Finally, it turns out that symbolic Rees algebras of binomial edge ideals associated with these two classes, namely pendant cliques graphs and net-free generalized caterpillar graphs, are strongly $F$-regular.

math.AC↗

arXiv2024

Gapset Extensions, Theory and Computations

In this paper we extend some set theoretic concepts of numerical semigroups for arbitrary sub-semigroups of natural numbers. Then we characterized gapsets which leads to a more efficient computational approach towards numerical semigroups and finally we introduce the extension of gapsets and prove that the sequence of the number of gapsets of size $g$ is non-decreasing as a weak version of Bras-Amorós's conjecture.

math.CO↗

arXiv2024

The Limit Space of Self-similar Groups and Schreier graphs

The present paper investigates the limit $G$-space $\mathcal{J}_{G}$ generated by the self-similar action of automatic groups on a regular rooted tree. The limit space $\mathcal{J}_{G}$ is the Gromov-Hausdorff limit of the family of Schreier graphs $Γ_{n}$; therefore, $\mathcal{J}_{G}$ can be approximated by Schreier graphs on level $n$-th when $n$ tends to infinity. We propose a computer program whose code is written in Wolfram language computes the adjacency matrix of Schreier graph $Γ_{n}$ at each specified level of the regular rooted tree. In this paper, the Schreier graphs corresponding to each automatic group is computed by applying the program to some collection of automata groups, including classic automatic groups.

math.GR↗

arXiv2024

Quantum resistant multi-signature scheme with optimal communication round: A Blockchain-based approach

Blockchain is a decentralized network to increase trust, integrity, and transparency of transactions. With the exponential growth of transactions in the realm of Blockchain, especially in Bitcoin, Blockchain size increases as all transactions must be stored and verified. In Bitcoin, validating M of N transactions involves the necessity of M authentic signatures out of the total N transactions. This procedure is so time-consuming and needs a significant storage capacity. To address these issues, several multi signature schemes have been proposed, enabling users to interactively generate a common signature on a single message. Recently, some lattice based multi signature schemes have been presented to deal with the threats of quantum computers. However, none of them have met all desirable features of multi signature schemes like aggregate public key, low numbers of communication rounds, or resistant to quantum computers. Within this paper, we present a new multi signature scheme based on lattices, known as Razhims, that has aggregate public key, necessitates solely a single round of communication, and is resistant to quantum computers. In Razhims, the aggregate public key size and the final signature size are equal to the public key size and the final signature size of a standard signature respectively, and are independent of the number of signers.

cs.CR↗

arXiv2024

The Rank of the Cartier operator on Picard Curves

For an algebraic curve $\mathcal{X}$ defined over an algebraically closed field of characteristic $p > 0$, the $a$-number $a(\mathcal{X})$ is the dimension of the space of exact holomorphic differentials on $\mathcal{X}$. We compute the $a$-number for a family of certain Picard curves, using the action of the Cartier operator on $H^0(\mathcal{X},Ω^1)$.

math.AG↗

arXiv2024

A remark on a result of Huber and Kahn

A. Huber and B. Kahn construct a relative slice filtration on the motive M(X) associated to a principal T-bundle X over a smooth scheme Y. As a consequence of their result, one can observe that the mixed Tateness of the motive M(Y) implies that the motive M(X) is mixed Tate. In this note we prove the inverse implication for a principal G-bundle, for a split reductive group G.

math.AG↗

arXiv2023

A fully decentralized auditing approach for edge computing: A Game-Theoretic Perspective

Edge storage presents a viable data storage alternative for application vendors (AV), offering benefits such as reduced bandwidth overhead and latency compared to cloud storage. However, data cached in edge computing systems is susceptible to intentional or accidental disturbances. This paper proposes a decentralized integrity auditing scheme to safeguard data integrity and counter the traditional reliance on centralized third-party auditors (TPA), which are unfit for distributed systems. Our novel approach employs edge servers (ES) as mutual auditors, eliminating the need for a centralized entity. This decentralization minimizes potential collusion with malicious auditors and biases in audit outcomes. Using a strategic game model, we demonstrate that ESs are more motivated to audit each other than TPAs. The auditing process is addressed as a Nash Equilibrium problem, assuring accurate integrity proof through incentives for ESs. Our scheme's security and performance are rigorously assessed, showing it is secure within the random oracle model, offers improved speed, and is cost-effective compared to existing methods.

cs.CR↗

arXiv2023

On the Equality of Symbolic and Ordinary Powers of Binomial Edge Ideals

In this paper, we investigate whether the symbolic and ordinary powers of a binomial edge ideal $J_{G}$ are equal. We show that the equality $J_{G}^{t}=J_{G}^{(t)}$ holds for every $t \geq 1$ when $|Ass(J_{G})|=2$. Moreover, if $G$ is a caterpillar tree, then one has the same equality. Finally, we characterize the generalized caterpillar graphs which the equality of symbolic and ordinary powers of $J_{G}$ occurs.

math.AC↗

arXiv2023

Comparison of symbolic and ordinary powers of parity binomial edge ideals

In this paper, we investigate when symbolic and ordinary powers of the parity binomial edge ideal of a graph fail to be equal. It turns out that if $\mathcal{I}_{G}$ is the parity binomial edge ideal of a graph $G$, then in each of the following cases the symbolic power $\mathcal{I}_{G}^{(t)}$ and the ordinary power $\mathcal{I}_{G}^t$ are not equal for some $t$: (i) the clique number of $G$ is greater than 3; (ii) $G$ has a net; or (iii) $G$ has a PT as an induced subgraph.

math.AC↗

arXiv2020

Approximation Algorithms for the Load Balanced Capacitated Vehicle Routing Problem

We study the load balanced capacitated vehicle routing problem (LBCVRP): the problem is to design a collection of tours for a fixed fleet of vehicles with capacity Q to distribute a supply from a single depot between a number of predefined clients, in a way that the total traveling cost is a minimum, and the vehicle loads are balanced. The unbalanced loads cause the decrease of distribution quality especially in business environments and exibility in the logistics activities. The problem being NP-hard, we propose two approximation algorithms. When the demands are equal, we present a (1-1/Q)p+3/2approximation algorithm that finds balanced loads. Here, p is the approximation ratio for the known metric traveling salesman problem (TSP). This result leads to a 2.5-1/Q approximation ratio for the tree metrics since an optimal solution can be found for the TSP on a tree. We present an improved 2 approximation algorithm. When the demands are unequal, we focus on obtaining approximate solutions since finding balanced loads is NP-complete. We propose an algorithm that provides a 4 approximation for the balance of the loads. We assume a second approach to get around the difficulties of the feasibility. In this approach, we redefine and convert the problem into a multi-objective problem. The algorithm we propose has a 4 factor of approximation.

cs.DS↗

arXiv2019

Lyubeznik Tables of Ideals of Cycle Graphs

Let R = K[x_1; : : : ; x_n] be a polynomial ring over a field K, and I := I_C_n be an edge ideal of n-cycle graph C_n. In the present paper, we compute the last column of the Lyubeznik table of R/I.

math.AC↗

arXiv2019

The $a$-number of Certain Hyperelliptic Curves

In this paper, we compute a formula for the $a$-number of certain hyperelliptic curves given by the equation $y^2= x^m+1$ for infinitely many values of $m$. The same question is studied for the curve corresponding to $y^2= x^m+x$.

math.AC↗

arXiv2012

Shellable and Cohen-Macaulay complete t-partite graphs

Let G be a simple undirected graph. We find the number of maximal independent sets in complete t-partite graphs. We will show that vertex decomposability and shellability are equivalent in this graphs. Also, we obtain an equivalent condition for Cohen-Macaulay in complete t-partite graphs.

math.AC↗

arXiv2012

Toricness of Binomial Edge Ideals

Let G be a finite simple graph. In this paper we will show that the binomial edge ideal of G, JG is toric if and only if each connected component of G is complete and in this case it is the sum of toric ideal associated to bipartite complete graphs.

math.AC↗

arXiv2012

Modification of the Elite Ant System in Order to Avoid Local Optimum Points in the Traveling Salesman Problem

This article presents a new algorithm which is a modified version of the elite ant system (EAS) algorithm. The new version utilizes an effective criterion for escaping from the local optimum points. In contrast to the classical EAC algorithms, the proposed algorithm uses only a global updating, which will increase pheromone on the edges of the best (i.e. the shortest) route and will at the same time decrease the amount of pheromone on the edges of the worst (i.e. the longest) route. In order to assess the efficiency of the new algorithm, some standard traveling salesman problems (TSPs) were studied and their results were compared with classical EAC and other well-known meta-heuristic algorithms. The results indicate that the proposed algorithm has been able to improve the efficiency of the algorithms in all instances and it is competitive with other algorithms.

cs.AI↗

arXiv2011

Squarefree vertex cover algebras

In this paper we introduce squarefree vertex cover algebras. We study the question when these algebras coincide with the ordinary vertex cover algebras and when these algebras are standard graded. In this context we exhibit a duality theorem for squarefree vertex cover algebras.

math.AC↗