Sharif Digital Repository

In this paper, we complement some recent results of L. Márki, J. Meyer, J. Szigeti and L. van Wyk, by investigating the constant-trace representations of a Clifford algebra C(V) of an arbitrary quadratic form q: V→ F (possibly degenerate) and we present some relevant applications. In particular, the existence of the polynomial identities of C(V) of particular form when the characteristic of the base field is zero is looked at. Furthermore, a lower bound is found on the minimal number t, such that C(V) can be embedded in a matrix ring of degree t, over some commutative F-algebra. Also, some results on the dimension of commutative subalgebras of C(V) are obtained. © 2021, The Author(s), under... 

Let F be a field, char (F) ≠ 2, and S ⊆ GLn (F), where n is a positive integer. In this paper we show that if for every distinct elements x, y ∈ S, x + y is singular, then S is finite. We conjecture that this result is true if one replaces field with a division ring. © 2009 Elsevier Inc. All rights reserved  

Let be a ring and be the set of all non-trivial left ideals of. The intersection graph of ideals of, denoted by, is a graph with the vertex set and two distinct vertices and are adjacent if and only if. In this paper, we classify all rings (not necessarily commutative) whose domination number of the intersection graph of ideals is at least 2. Moreover, some results on the intersection graphs of ideals of matrix algebras over a finite field are given. For instance, we determine the domination number, the clique number and the independence number of. We prove that if is a positive integer and, then the domination number of is. Among other results, we show that if, where is a positive integer... 

In this paper, we investigate the recovery of a sparse weight vector (parameters vector) from a set of noisy linear combinations. However, only partial information about the matrix representing the linear combinations is available. Assuming a low-rank structure for the matrix, one natural solution would be to first apply a matrix completion to the data, and then to solve the resulting compressed sensing problem. In big data applications such as massive MIMO and medical data, the matrix completion step imposes a huge computational burden. Here, we propose to reduce the computational cost of the completion task by ignoring the columns corresponding to zero elements in the sparse vector. To... 

Given a metric graph G, we are concerned with finding a spanning tree of G where the maximum weighted degree of its vertices is minimum. In a metric graph (or its spanning tree), the weighted degree of a vertex is defined as the sum of the weights of its incident edges. In this paper, we propose a 4.5-approximation algorithm for this problem. We also prove it is NP-hard to approximate this problem within a 2 - ε factor. © 2007 Elsevier B.V. All rights reserved  

Let G be a graph of order n and rank(G) denotes the rank of its adjacency matrix. Clearly, n ≤ rank (G) + rank (over(G, -)) ≤ 2 n. In this paper we characterize all graphs G such that rank (G) + rank (over(G, -)) = n, n + 1 or n + 2. Also for every integer n ≥ 5 and any k, 0 ≤ k ≤ n, we construct a graph G of order n, such that rank (G) + rank (over(G, -)) = n + k. © 2006 Elsevier Inc. All rights reserved  

We study graphs whose adjacency matrices have determinant equal to 1 or -1, and characterize certain subclasses of these graphs. Graphs whose adjacency matrices are totally unimodular are also characterized. For bipartite graphs having a unique perfect matching, we provide a formula for the inverse of the corresponding adjacency matrix, and address the problem of when that inverse is diagonally similar to a nonnegative matrix. Special attention is paid to the case that such a graph is unicyclic. © 2006 Elsevier Inc. All rights reserved  

Let D be a division ring with center F and n {greater than or slanted equal to} 1 a natural number. For S ⊆ Mn(D) the commuting graph of S, denoted by Γ(S), is the graph with vertex set S{minus 45 degree rule}Z(S) such that distinct vertices a and b are adjacent if and only if ab = ba. In this paper we prove that if n > 2 and A, N, I, T are the sets of all non-invertible, nilpotent, idempotent matrices, and involutions, respectively, then for any division ring D, Γ ( A ), Γ ( N ), Γ ( I ), and Γ ( T ) are connected graphs. We show that if n > 2 and U is the set of all upper triangular matrices, then for every algebraic division ring D, Γ ( U ) is a connected graph. Also it is shown that if R... 

In this article we show that for any forest there exists a labelling of the vertices for which the row-reduced echelon form of its adjacency matrix is a {-1, 0, 1}-matrix. This result clearly provides an affirmative answer to the conjecture: The null space of the adjacency matrix of every forest has a {-1, 0, 1}-basis. © 2005 Elsevier Inc. All rights reserved  

In 2011, Haemers asked the following question: If S is the Seidel matrix of a graph of order n and S is singular, does there exist an eigenvector of S corresponding to 0 which has only ±1 elements? In this paper, we construct infinite families of graphs which give a negative answer to this question. One of our constructions implies that for every natural number N, there exists a graph whose Seidel matrix S is singular such that for any integer vector in the nullspace of S, the absolute value of any entry in this vector is more than N. We also derive some characteristics of vectors in the nullspace of Seidel matrices, which lead to some necessary conditions for the singularity of Seidel... 

In 2011, Haemers asked the following question: If S is the Seidel matrix of a graph of order n and S is singular, does there exist an eigenvector of S corresponding to 0 which has only ±1 elements? In this paper, we construct infinite families of graphs which give a negative answer to this question. One of our constructions implies that for every natural number N, there exists a graph whose Seidel matrix S is singular such that for any integer vector in the nullspace of S, the absolute value of any entry in this vector is more than N. We also derive some characteristics of vectors in the nullspace of Seidel matrices, which lead to some necessary conditions for the singularity of Seidel... 

The existence of periodic solutions for a forced Hill's equation is proved. The proof is then extended to the case of a non-homogeneous matrix valued Hill's equation. Under the stated conditions, using Lyapunov's criteria [Proc. AMS 13 (1962) 601; Hill's Equation, Interscience Publishers, New York, 1966] some results on the stability oh Hill's equation are obtained. © 2004 Elsevier Inc. All rights reserved  

In this investigation we have taken the viewpoint of the Landau theory for the normal state and have suggested a covariant equation for current density. This suggestion is based on the exact analysis of electrodynamic relations in the intermediate state of superconductors. Our analysis is restricted to the simplest electrodynamic relations. At first we have employed Maxwell's equations with the general Ohm-London equation and have found the general covariant current which takes a covariant form after Lorentz transformation. Finally we have checked it by considering two inertial frames in a simple state  

In this work, parallel solution of the Navier-Stokes equations for a mixed convection heat problem is achieved using a finite-element-based finite-volume method in fully coupled and semi coupled algorithms. A major drawback with the implicit methods is the need for solving the huge set of linear algebraic equations in large scale problems. The current parallel computation is developed on distributed memory machines. The matrix decomposition and solution are carried out using PETSc library. In the fully coupled algorithm, there is a 36-diagonal global matrix for the two-dimensional governing equations. In order to reduce the computational time, the matrix is suitably broken in several... 

A nonlinear comprehensive model has been developed in this paper to study the train derailment and hunting in severe braking conditions. The train consists of cars each having 40 dof, connected to each other by couplers and buffers. The car model is nonlinear and three-dimensional and includes nonlinear springs and dampers of primary and secondary suspensions, dry friction between different parts such as car body and side bearers, center-plate parts, wheelset bearings and bogie frames, and also clearances and mechanical stops. Nonlinearities of wheel and rail profiles, pressure build-up delay in brake circuit, and nonlinearities of connecting parts have also been included in the model. A... 

We present an analytical method for solution of one-dimensional optical systems, based on the differential transfer matrices. This approach can be used for exact calculation of various functions including reflection and transmission coefficients, band structures, and bound states. We show the consistency of the WKB method with our approach and discuss improvements for even symmetry and infinite periodic structures. Moreover, a general variational representation of bound states is introduced. As application examples, we consider the reflection from a sinusoidal grating and the band structure of an infinite exponential grating. An excellent agreement between the results from our differential... 

Kernel learning is one of the most important and recent approaches to constrained clustering. Until now many kernel learning methods have been introduced for clustering when side information in the form of pairwise constraints is available. However, almost all of the existing methods either learn a whole kernel matrix or learn a limited number of parameters. Although the non-parametric methods that learn whole kernel matrix can provide capability of finding clusters of arbitrary structures, they are very computationally expensive and these methods are feasible only on small data sets. In this paper, we propose a kernel learning method that shows flexibility in the number of variables between... 

Let G be a graph and L(G) be the Laplacian matrix of G. In this paper, we explicitly determine the multiplicity of Laplacian eigenvalue 2 for any unicyclic graph containing a perfect matching  

In this paper we analyse two variants of SIMON family of light-weight block ciphers against variants of linear cryptanalysis and present the best linear cryptanalytic results on these variants of reducedround SIMON to date. We propose a time-memory trade-off method that finds differential/ linear trails for any permutation allowing low Hamming weight differential/ linear trails. Our method combines low Hamming weight trails found by the correlation matrix representing the target permutation with heavy Hamming weight trails found using a Mixed Integer Programming model representing the target differential/linear trail. Our method enables us to find a 17-round linear approximation for SIMON-48...