Matrix In SL Research Articles

Utilizing disease transmission and response capacities to optimize covid-19 control in Malaysia

ObjectivesPublic Health Social Measures (PHSM) such as movement restriction movement needed to be adjusted accordingly during the COVID-19 pandemic to ensure low disease transmission alongside adequate health system capacities based on the COVID-19 situational matrix proposed by the World Health Organization (WHO). This paper aims to develop a mechanism to determine the COVID-19 situational matrix to adjust movement restriction intensity for the control of COVID-19 in Malaysia.MethodsSeveral epidemiological indicators were selected based on the WHO PHSM interim guidance report and validated individually and in several combinations to estimate the community transmission level (CT) and health system response capacity (RC) variables. Correlation analysis between CT and RC with COVID-19 cases was performed to determine the most appropriate CT and RC variables. Subsequently, the CT and RC variables were combined to form a composite COVID-19 situational matrix (SL). The SL matrix was validated using correlation analysis with COVID-19 case trends. Subsequently, an automated web-based system that generated daily CT, RC, and SL was developed.ResultsCT and RC variables were estimated using case incidence and hospitalization rate; Hospital bed capacity and COVID-19 ICU occupancy respectively. The estimated CT and RC were strongly correlated [ρ = 0.806 (95% CI 0.752, 0.848); and ρ = 0.814 (95% CI 0.778, 0.839), p < 0.001] with the COVID-19 cases. The estimated SL was strongly correlated with COVID-19 cases (ρ = 0.845, p < 0.001) and responded well to the various COVID-19 case trends during the pandemic. SL changes occurred earlier during the increase of cases but slower during the decrease, indicating a conservative response. The automated web-based system developed produced daily real-time CT, RC, and SL for the COVID-19 pandemic.ConclusionsThe indicators selected and combinations formed were able to generate validated daily CT and RC levels for Malaysia. Subsequently, the CT and RC levels were able to provide accurate and sensitive information for the estimation of SL which provided valuable evidence on the progression of the pandemic and movement restriction adjustment for the control of Malaysia.

On the parametrized modular group

Generalizing the modular and Hecke groups, we consider the subgroup Π of SL(2, ℤ[ξ]) generated by the parabolic element \(\left( {_{01}^{1\xi } } \right)\) and the elliptic element \(\left( {_{10}^{0 - 1} } \right)\), where ℤ[ξ] is the ring of polynomials in the variable ξ. For ζ ∈ ℂ and W ∈ Π, we denote by W (ζ) the matrix in SL(2, ℂ) obtained when evaluating the parameter ξ at ζ. We enumerate the elements of Π and study the relators, defined as those W ∈ ℂ for which there exists ζ ∈ ℂ with W(ζ) = ±I. Then, for W ∈ Π, we investigate the sets of ζ for which W(ζ) is not loxodromic; their union is the singular set S(Π) ⊂ ℂ. The closure of the singular set for the two-parabolic group, which is isomorphic to a free subgroup Π of index 4, has been studied extensively.

Structured quasi‐cyclic low‐density parity‐check codes based on cyclotomic cosets

This study considers the construction of four-cycle free quasi-cyclic low-density parity-check (QC-LDPC) codes. These codes are based on the cyclotomic cosets of q modulo n where q is a prime power, n is a prime and (n, q) = 1. If s is the order of q modulo n, then n − 1 = sl for some integer l. Then there are l distinct cyclotomic cosets Cai , 1 ≤ i ≤ l. Concatenation of the l circulant matrices formed using these sets gives an s × sl matrix M co(n, q). In addition, the Kronecker product of the transpose of ∪ i Cai with itself modulo n gives an sl × sl matrix M kr(n, q). Replacing the entries of these matrices with their associated dispersed binary n × n matrices provides binary QC-LDPC codes with girth at least six. Furthermore, for any prime power q′ such that q′ = n + 1 or q′ ≥ 2n, these two matrices can be used to construct q′-ary QC-LDPC codes with girth at least six. These constructions produce large classes of four-cycle free binary and non-binary QC-LDPC codes. Among the structured LDPC codes, the introduced technique, from code-construction-complexity perspective, which is an engineering factor, is at least among the best ones if not the best. Performance results are presented which show that the codes obtained perform well over additive white Gaussian noise channel with the iterative sum-product decoding algorithm.

NEW ICA ALGORITHMS BASED ON SPECIAL LINEAR GROUP

Unlike the traditional independent component analysis (ICA) algorithms and some recently emerging linear ICA algorithms that search for solutions in the space of general matrices or orthogonal matrices, in this paper we propose two new methods which only search for solutions in the space of the matrices with unitary determinant and without whitening. The new algorithms are based on the special linear group SL(n). In order to achieve our target, we first provide a representation theory for any matrix in SL(n), which only simply uses the product of multiple exponentials of traceless matrices. Based on the matrix representation theory, two novel ICA algorithms are developed along with simple analysis on their equilibrium points. Moreover, we apply our methods to the classical problem of signal separation. The experimental results indicate that the superior convergence of our proposed algorithms, which can be expected as two viable alternatives to the ICA algorithms available in publications.

Pseudoprime factorizations of integer matrices

We give a method of factoring integer matrices in Z n × n into components such that the factorization is not unique unless certain information is known. In Section 2, we introduce this method of factorization and provide theorems which establish its well-definedness. In Section 3, we construct a matrix in SL ( n , Z ) as a product of specific types of matrices and establish an algorithm for factoring the result uniquely given an amount of information.

Déformations de réseaux dans certains groupes résolubles

We aim to study local rigidity and deformations for the following class of groups: the semidirect product Γ= Z n ⋊ A Z where n≥2 is an integer and A is a hyperbolic matrix in SL( n,Z ) , considered first as a lattice in the solvable Lie group G= R n ⋊ A R , then as a subgroup of the semisimple Lie group SL( n+1,R ) . We will notably show that, although Γ is locally rigid neither in G nor in H , it is locally SL( n+1,R ) -rigid in G in the sense that every small enough deformation of Γ in G is conjugated to Γ by an element of SL( n+1,R ) .

Disturbed functional connectivity in brain tumour patients: Evaluation by graph analysis of synchronization matrices

Cerebral functions are based on the functional interactions between multiple distinct specialized regions of the brain. Functional interactions require anatomical connections as well as the synchronization of brain oscillations. The present work aims at evaluating the impact of brain tumours on spatial patterns of functional connectivity of the brain measured at rest by MEG. We analyzed the statistical dependency (by computing the synchronization likelihood (SL, a measure of generalized synchronization)) between MEG signals at rest, in 17 patients with a brain tumour and in 15 healthy controls. Following an approach that derives from graph theory, we also analyzed the architectural properties of the networks by computing two parameters from the SL matrix, the cluster coefficient C and the characteristic path length L. Alterations in synchronization levels were found in the patients and were not focal but involved intra-hemispheric connectivity. Effects were different considering the frequencies sub-bands, predominating in a decrease in high frequencies bands for long-distance connections and an increase in slower bands for local connectivity. In addition, graph analysis reveals changes in the normal "small-world" network architecture in addition to changes in synchronization levels with some differences according to the studied frequency sub-bands. Brain tumours alter the functional connectivity and the "network" architecture of the brain. These alterations are not focal and effects are different considering the frequencies sub-bands. These neurophysiological changes may contribute to the cognitive alterations observed in patients with brain tumours.

On Sobolev infinitesimal rigidity of linear hyperbolic actions on the 2-torus*

Let A be a symmetric hyperbolic matrix in SL(2, ℤ) and Γ the subgroup of SL(2, ℤ) generated by A. We aim to study the infinitesimal rigidity of the standard action of Γ on the torus $${\Bbb T^2}$$ . More precisely, we will consider the Sobolev W s –infinitesimal rigidity of this action (that is to determine if the cohomology space H1(Γ,W s (T M)) is trivial or not), and show that it is W s –infinitesimally rigid only if 0 ≤ s < 1. A consequence will be that this action is not C∞–infinitesimally rigid.

CRITERION FOR SL(2,Z)-MATRIX TO BE CONJUGATE TO ITS INVERSE

This paper gives a necessary and sufficient condition for a matrix in SL(2, Z) to be conjugate to its inverse. This condition reduces the determination of the conjugation to solving some indeterminate equation of second degree. It yields an algorithm to determine this conjugation in finite steps based on the elementary number theory.

Monodromy of a two degrees of freedom Liouville integrable system with many focus focus singular points

This letter deals with the global monodromy of singular Lagrangian toral fibrations defined by two degrees of freedom Liouville integrable systems with only focus–focus singular points. We show that any global monodromy matrix in Sl(2, ) is realizable by such a system.

An Algorithmic Proof of Suslin′s Stability Theorem for Polynomial Rings

Let k be a field. Then Gaussian elimination over k and the Euclidean division algorithm for the univariate polynomial ring k[ x] allow us to write any matrix in SL n ( k) or SL n ( k[ x]), n ≥ 2, as a product of elementary matrices. Suslin′s stability theorem states that the same is true for SL n ( k[ x l,..., x m ]) with n ≥ 3 and m ≥ 1. In this paper, we present an algorithmic proof of Suslin′s stability theorem, thus providing a method for finding an explicit factorization of a given polynomial matrix into elementary matrices. Gröbner basis techniques may be used in the implementation of the algorithm.

Inductive limit automorphisms of the irrational rotation algebra

It is shown that the flip automorphismU↦U*,V↦V* of the irrational rotation algebra Aθ is an inductive limit automorphism. Here, the algebra Aθ is generated by unitariesU, V satisfyingVU=e2πiθUV, where θ is an irrational number. Recently, Elliott and Evans proved that Aθ can be approximated by unital subalgebras isomorphic to a direct sum of two matrix algebras over\(C(\mathbb{T})\), the algebra of continuous functions on the unit circle. This is the central result which they used to obtain their structure theorem on Aθ; namely, that Aθ is the inductive limit of an increasing sequence of subalgebras each isomorphic to a direct sum of two matrix algebras over\(C(\mathbb{T})\). In their proof, they devised a subtle construction of two complementary towers of projections. In the present paper it is shown that the two towers can be chosen so that each summand of their approximating basic building blocks is invariant under the flip automorphism and, in particular, that the unit projection of the first summand is unitarily equivalent to the complement of the unit of the second by a unitary which is fixed under the flip. Also, an explicit computation of the flip on the approximating basic building blocks of Aθ is given. Further, combining this result along with others, including a theorem of Su and a spectral argument of Bratteli, Evans, and Kishimoto, a two-tower proof is obtained of the fact established by Bratteli and Kishimoto that the fixed point subalgebra Bθ (under the flip) is approximately finite dimensional. Also used here is the fact that Bθ has the cancellation property and is gifted with four basic unbounded trace functionals. The question is raised whether other finite order automorphisms of Aθ (arising from a matrix in SL(2,ℤ)) are inductive limit automorphisms - or evenalmost inductive limit automorphisms in the sense of Voiculescu.

Commutators and companion matrices over rings of stable rank 1

We consider the group GL n A of all invertible n by n matrices over a ring A satisfying the first Bass stable range condition. We prove that every matrix is similar to the product of a lower and upper triangular matrix, and that it is also the product of two matrices each similar to a companion matrix. We use this to show that, when n ⩾3 and A is commutative, every matrix in SL n A is the product of two commutators.

On a question of M. Newman on the number of commutators

M. Newman asked whether there is an absolute constant c such that every matrix in SL n R is the product of at most c commutators, where R ranges over euclidean commutative rings and n ⩾ 3. We give here a negative answer. However, if for the ring R every matrix in SL m R is the product of a bounded number of commutators for some fixed m ⩾ 3, then for all sufficiently large n, every matrix in SL n R is the product of six commutators.