Viewpoint Of Linear Algebra Research Articles

Linear Coded Caching Scheme for Centralized Networks

Coded caching systems have been widely studied to reduce the data transmission during the peak traffic time. In practice, two important parameters of a coded caching system should be considered, i.e., the transmission rate which is the maximum amount of the data transmission during the peak traffic time, and the subpacketization level, the number of divided packets of each file when we implement a coded caching scheme. Although there exists a tradeoff between transmission rate and subpacketization, we prefer to design a scheme with transmission rate and subpacketization as small as possible since they reflect the transmission efficiency and complexity of the caching scheme, respectively. In this paper, we first characterize a coded caching scheme from the viewpoint of linear algebra and show that designing a linear coded caching scheme is equivalent to constructing three classes of matrices satisfying some rank conditions. Then based on the invariant subspaces in linear algebra and combinatorial design theory, a new class of coded caching schemes over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> is obtained by constructing these three classes of matrices. It turns out that the transmission rate of our new scheme is the same as the scheme construct by Yan et al. (IEEE Trans. Inf. Theory 63, 5821-5833, 2017), but the subpacketization is significantly reduced. Finally by means of these matrices, we show that the minimum storage regenerating codes can also be used to construct coded caching schemes.

Combinatorics in Pearson residuals

This paper focuses on residual analysis of statistical independence of multiple variables from the viewpoint of linear algebra and combinatorics. The results show that multidimensional residuals are represented as linear sum of determinants of 2 × 2 submatrices, which can be viewed as information granules measuring the degree of statistical dependence. Furthermore, all the elements of information granules have combinatorial characteristics of index sets.

Nonsingular plane filling curves of minimum degree over a finite field and their automorphism groups: Supplements to a work of Tallini

Our concern is a nonsingular plane curve defined over a finite field Fq which includes all the Fq-rational points of the projective plane. The possible degree of such a curve is at least q+2. We prove that nonsingular plane curves of degree q+2 having the property actually exist. More precisely, we write down explicitly all of those curves. Actually, Giuseppe Tallini studied such curves in his old paper in 1961. We explain the connection between his work and ours. Also we give another proof of his result on the automorphism group of such a curve, from the viewpoint of linear algebra.

Contingency matrix theory: Statistical dependence in a contingency table

Chance discovery aims at understanding the meaning of functional dependency from the viewpoint of unexpected relations. One of the most important observations is that such a chance is hidden under a huge number of coocurrencies extracted from a given data. On the other hand, conventional data-mining methods are strongly dependent on frequencies and statistics rather than interestingness or unexpectedness. This paper discusses some limitations of ideas of statistical dependence, especially focusing on the formal characteristics of Simpson’s paradox from the viewpoint of linear algebra. Theoretical results show that such a Simpson’s paradox can be observed when a given contingency table as a matrix is not regular, in other words, the rank of a contingency matrix is not full. Thus, data-ordered evidence gives some limitations, which should be compensated by human-oriented reasoning.

A Modeling of Wrinkled Membranes Using a Projection Matrix (Derivation of Modified Elasticity Matrix Using a Projection Technique)

A modeling of wrinkled membranes based on projection operator is presented within the classical membrane theory. The total strains in wrinkled membranes are decomposed into elastic and zero-strain energy parts. A projection matrix that extracts the elastic parts from the total strains is derived from the view point of linear algebra. The resulting modified elasticity matrix that represents the stress-strain relations in wrinkled membranes is thus obtained as product of the classical elasticity matrix and projection matrix. A qualitative comparison is conducted between the presented modified elasticity matrix and the existing modified elasticity matrices.

On inertia and schur complement in optimization

Partitioned symmetric matrices, in particular the Hessian of the Lagrangian, play a fundamental role in nonlinear optimization. For this type of matrices S.-P. Han and O. Fujiwara recently presented an inertia theorem under a certain regularity assumption. We prove that this theorem is true without any regularity assumption. Then we consider matrix extensions preserving the sign of the determinant. Such extensions are shown to be related with the positive definiteness of some Schur complement. Under a regularity assumption this shows, from the viewpoint of linear algebra, the equivalence of strong stability in the sense of M. Kojima and strong regularity in the sense of S. M. Robinson. Finally, we discuss the inertia of a typical one-parameter family of symmetric matrices, occurring in various places in optimization (augmented Lagrangians, focal-point theory, etc.).

Fault diagnosis of compartmental systems

AbstractThis paper investigates the fault diagnosis problems of compartmental systems from the viewpoint of linear algebra and graph theory. A system is considered faulty if a part of the system parameters (rate constants) takes an abnormal value. The problems of whether or not one can detect faults (fault detectability) and whether or not one can distinguish different faults (distinguishability) through observation of steady states are discussed. First, it is shown that the difference between the value of an observation vector for a faulty system and that for the normal system lies in a linear subspace corresponding to the fault, and a specific subspace is given. According to which linear subspace the difference in the observation vector belongs, a fault is detected and distinguished. Next, it is shown that conditions for fault detectability and distinguishability can be represented by using the dimension of the linear subspace corresponding to the fault and that an equivalent representation can be given in terms of the rank condition of a certain matrix. Finally, as the main result of this paper a representation is given for the distinguishability conditions on a compartment graph. In particular, when the number of the rate constants taking an abnormal value is at most one, these conditions can be replaced by directed path conditions which can be tested easily.