scholarly journals Generating Asymptotically Non-terminant Initial Variable Values for Linear Diagonalizable Programs.

10.29007/dq11 ◽  
2018 ◽  
Author(s):  
Rachid Rebiha ◽  
Nadir Matringe ◽  
Arnaldo Vieira Moura
Keyword(s):  
Linear System ◽  
Linear Space ◽  
Computational Methods ◽  
Algebraic Methods ◽  
Matrix Encoding ◽  
The Matrix ◽  
Initial Variable ◽  
Loop Condition

We present the key notion of "asymptotically non-terminant initial variable values" for non-terminant loop programs. We show that those specific values are directly associated to inital variable values for which the loop program does not terminate.Considering linear diagonalizable programs, we describe powerful computational methods that generate automatically and symbolically a semi-linear space represented by a linear system of equalities and inequalities. Each element of this space provides us with asymptotically non-terminant initial variable values. Our approach is based on linear algebraic methods and results. We obtain conditions using a decomposition on a specific basis, involving the loop condition and the matrix encoding the instructions of the loop.

scholarly journals EIGENVECTOR LOCALIZATION AS A TOOL TO STUDY SMALL COMMUNITIES IN ONLINE SOCIAL NETWORKS

2010 ◽  
Vol 13 (06) ◽  
pp. 699-723 ◽  
Author(s):  
FRANTIŠEK SLANINA ◽  
ZDENĚK KONOPÁSEK
Keyword(s):  
Online Social Networks ◽  
Large Network ◽  
Bipartite Networks ◽  
Small Communities ◽  
Complementary Approach ◽  
Matrix Encoding ◽  
The Matrix ◽  
Network Modules ◽  
Principal Tool ◽  
Eigenvector Localization

We present and discuss a mathematical procedure for identification of small "communities" or segments within large bipartite networks. The procedure is based on spectral analysis of the matrix encoding network structure. The principal tool here is localization of eigenvectors of the matrix, by means of which the relevant network segments become visible. We exemplified our approach by analyzing the data related to product reviewing on Amazon.com. We found several segments, a kind of hybrid communities of densely interlinked reviewers and products, which we were able to meaningfully interpret in terms of the type and thematic categorization of reviewed items. The method provides a complementary approach to other ways of community detection, typically aiming at identification of large network modules.

Suffix Tree Data Structures for Matrices

1997 ◽  
Author(s):  
R. Giancarlo ◽  
R. Grossi
Keyword(s):  
Linear Space ◽  
Suffix Tree ◽  
Linear Time ◽  
Suffix Trees ◽  
Construction Time ◽  
Matching Problems ◽  
Tree Construction ◽  
The Matrix ◽  
Visual Databases ◽  
Efficient Construction

We discuss the suffix tree generalization to matrices in this chapter. We extend the suffix tree notion (described in Chapter 3) from text strings to text matrices whose entries are taken from an ordered alphabet with the aim of solving pattern-matching problems. This suffix tree generalization can be efficiently used to implement low-level routines for Computer Vision, Data Compression, Geographic Information Systems and Visual Databases. We examine the submatrices in the form of the text’s contiguous parts that still have a matrix shape. Representing these text submatrices as “suitably formatted” strings stored in a compacted trie is the rationale behind suffix trees for matrices. The choice of the format inevitably influences suffix tree construction time and space complexity. We first deal with square matrices and show that many suffix tree families can be defined for the same input matrix according to the matrix’s string representations. We can store each suffix tree in linear space and give an efficient construction algorithm whose input is both the matrix and the string representation chosen. We then treat rectangular matrices and define their corresponding suffix trees by means of some general rules which we list formally. We show that there is a super-linear lower bound to the space required (in contrast with the linear space required by suffix trees for square matrices). We give a simple example of one of these suffix trees. The last part of the chapter illustrates some technical results regarding suffix trees for square matrices: we show how to achieve an expected linear-time suffix tree construction for a constant-size alphabet under some mild probabilistic assumptions about the input distribution. We begin by defining a wide class of string representations for square matrices. We let Σ denote an ordered alphabet of characters and introduce another alphabet of five special characters, called shapes. A shape is one of the special characters taken from set {IN,SW,NW,SE,NE}. Shape IN encodes the 1x1 matrix generated from the empty matrix by creating a square.

On a Normal Form of the Orthogonal Transformation III

1958 ◽  
Vol 1 (3) ◽  
pp. 183-191 ◽  
Author(s):  
Hans Zassenhaus
Keyword(s):  
Normal Form ◽  
Linear Space ◽  
Bilinear Form ◽  
Matrix Equation ◽  
Orthogonal Transformation ◽  
The Matrix ◽  
Image Position

Under the assumptions of case of theorem 1 we derive from (3.32) the matrix equationso that there corresponds the matrix B to the bilinear form4.1on the linear space4.2and fP,μ, is symmetric if ɛ = (-1)μ+1, anti-symmetric if ɛ = (-1)μ.The last statement remains true in the case a) if P is symmetric irreducible because in that case fP,μ is 0.

scholarly journals Stability and stabilization of linear positive systems on time scales

Positivity ◽  
2020 ◽  
Vol 24 (5) ◽  
pp. 1361-1372
Author(s):  
Zbigniew Bartosiewicz
Keyword(s):  
Control System ◽  
Linear System ◽  
Time Scale ◽  
Sufficient Conditions ◽  
Positive Control ◽  
Positive Systems ◽  
The Matrix ◽  
Exponentially Stable ◽  
Stability And Stabilization ◽  
Necessary And Sufficient

Abstract It is shown that a positive linear system on a time scale with a bounded graininess is uniformly exponentially stable if and only if the characteristic polynomial of the matrix defining the system has all its coefficients positive. Then this fact is used to find necessary and sufficient conditions of positive stabilizability of a positive control system on a time scale.

scholarly journals A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation

2008 ◽  
Vol 2008 ◽  
pp. 1-26 ◽  
Author(s):  
M. Ilić ◽  
I. W. Turner ◽  
V. Anh
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Poisson Equation ◽  
Matrix Function ◽  
Fractional Power ◽  
Analytic Solutions ◽  
Linear System Of Equations ◽  
Exact Analytic Solutions ◽  
The Matrix ◽  
Symmetric Positive Definite Matrices

This study considers the solution of a class of linear systems related with the fractional Poisson equation (FPE) (−∇2)α/2φ=g(x,y) with nonhomogeneous boundary conditions on a bounded domain. A numerical approximation to FPE is derived using a matrix representation of the Laplacian to generate a linear system of equations with its matrix A raised to the fractional power α/2. The solution of the linear system then requires the action of the matrix function f(A)=A−α/2 on a vector b. For large, sparse, and symmetric positive definite matrices, the Lanczos approximation generates f(A)b≈β0Vmf(Tm)e1. This method works well when both the analytic grade of A with respect to b and the residual for the linear system are sufficiently small. Memory constraints often require restarting the Lanczos decomposition; however this is not straightforward in the context of matrix function approximation. In this paper, we use the idea of thick-restart and adaptive preconditioning for solving linear systems to improve convergence of the Lanczos approximation. We give an error bound for the new method and illustrate its role in solving FPE. Numerical results are provided to gauge the performance of the proposed method relative to exact analytic solutions.

Modal Characteristics of Planar Multi-Story Frame Structures

2016 ◽  
Vol 32 (5) ◽  
pp. 501-514 ◽  
Author(s):  
H.-P. Lin ◽  
S.-C. Chang ◽  
C. Chu
Keyword(s):  
Finite Element ◽  
Linear System ◽  
Finite Element Methods ◽  
Analytical Methods ◽  
Theoretical Method ◽  
Frame Structure ◽  
Frame Structures ◽  
Modal Characteristics ◽  
Out Of Plane ◽  
The Matrix

AbstractIn linear system, in-plane motions are decoupled from out-of-plane motions for planar frame structures. A theoretical method is proposed that permits the efficient calculations of modal characteristics of planar multi-story frame structures. There are 3 × m beam components for a planar m-story frame structure. By analyzing the transverse and longitudinal motions of each component simultaneously and considering the compatibility requirements across each frame joint, the undetermined variables of the entire m-story frame structure system can be reduced to six, regardless of the number of stories, and that can be determined by the application of the boundary conditions. The main feature of this method is to decrease the dimensions of the matrix involved in the finite element methods and certain other analytical methods.

Decrypting the matrix: how interspecies interactions influence matrix in multispecies biofilm?

2020 ◽  
Author(s):  
Cristina Amador ◽  
Henriette L. Røder ◽  
Ute Kuhlicke ◽  
Thomas Neu ◽  
Mette Burmølle
Keyword(s):  
Single Species ◽  
Matrix Composition ◽  
Emergent Properties ◽  
Interspecies Interactions ◽  
Matrix Encoding ◽  
The Matrix ◽  
Static Conditions ◽  
Encoding Genes ◽  
Multispecies Biofilms ◽  
Biofilm Matrix

<p>The biofilm matrix contributes to the establishment of microbial cells on very diverse surfaces, stabilizing biofilms and providing cells with protection against multiple hostile conditions. Moreover, the biofilm matrix can also retain nutrients, enzymes or quorum sensing molecules, favoring the establishment of social interactions among biofilm cells. Functional bacterial amyloids are part of the biofilm structural components of various species, and they were previously proven to bind QS molecules and strengthen the matrix. Multiple studies have been conducted to characterize matrix determinants and their regulation in single species biofilms, while these remain scarcely understood in multispecies biofilms. We have previously isolated and characterized a soil-derived consortium composed of Xanthomonas retroflexus, Stenotrophomonas rhizophila, Microbacterium oxydans and Paenibacillus amylolyticus showing enhanced biofilm biomass and differential gene/protein expression specific of the four-species biofilm.</p> <p>This study aimed at exploring the effect of interspecies interactions on biofilm matrix production in the four-species biofilm. We hypothesize that interspecies interactions may result in differential expression of matrix-encoding genes responsible for biofilm emergent properties.</p> <p>We searched for matrix determinant homologues in X.retroflexus and combined different techniques for characterizing the matrix identity and expression in mono-, dual- and multispecies biofilms.</p> <p>The fap amyloid operon, described in Pseudomonas as a biofilm-scaffold contributing element, was deleted in X. retroflexus, replaced in the four-species model and compared to the parental community for biofilm structure and adhesion capability. The fap mutant displayed poor substrate colonization in flow cells in both mono- and multispecies biofilms with relative filamentous structure compared to the parental strain/ consortium. However, adhesion did not significantly change under static conditions. To characterize matrix composition, we tested 78 different lectins in multispecies biofilms and identified five that bound to our samples. Interestingly, some matrix glycoconjugates were only produced in the consortium.</p> <p>Our data suggest that loss of matrix components, such as the Fap amyloid, and the presence of other species, influences synergistic biofilm properties in the four-species consortium. Ongoing approaches involving localized expression of matrix-encoding genes and matrix proteomes will aid in identifying the mechanisms underlying emergent properties in the four-species biofilm.</p> <p> </p>

New Computational Methods for Solving Problems of the Astronomical Vessel Position

2005 ◽  
Vol 58 (2) ◽  
pp. 315-335 ◽  
Author(s):  
Tien-Pen Hsu ◽  
Chih-Li Chen ◽  
Jiang-Ren Chang
Keyword(s):  
Coordinate System ◽  
Computational Methods ◽  
Matrix Method ◽  
Iteration Scheme ◽  
Computation Method ◽  
Direct Computation ◽  
Celestial Navigation ◽  
Vector Algebra ◽  
The Matrix ◽  
Triangle Method

In this paper, a simplified and direct computation method formulated by the fixed coordinate system and relative meridian concept in conjunction with vector algebra is developed to deal with the classical problems of celestial navigation. It is found that the proposed approach, the Simultaneous Equal-altitude Equation Method (SEEM), can directly calculate the Astronomical Vessel Position (AVP) without an additional graphical procedure. The SEEM is not only simpler than the matrix method but is also more straightforward than the Spherical Triangle Method (STM). Due to tedious computation procedures existing in the commonly used methods for determining the AVP, a set of optimal computation procedures for the STM is also suggested. In addition, aimed at drawbacks of the intercept method, an improved approach with a new computation procedure is also presented to plot the celestial line of position without the intercept. The improved approach with iteration scheme is used to solve the AVP and validate the SEEM successfully. Methods of solving AVP problems are also discussed in detail. Finally, a benchmark example is included to demonstrate these proposed methods.

scholarly journals Solving rank one revised linear systems by the scaled ABS method

2004 ◽  
Vol 46 (2) ◽  
pp. 225-236
Author(s):  
Keyvan Amini
Keyword(s):  
Mathematical Programming ◽  
Linear System ◽  
General Solution ◽  
Linear Systems ◽  
Matrix Coefficient ◽  
Active Set ◽  
Constraint Matrix ◽  
Active Set Strategies ◽  
The Matrix ◽  
Rank One

AbstractIn mathematical programming, an important tool is the use of active set strategies to update the current solution of a linear system after a rank one change in the constraint matrix. We show how to update the general solution of a linear system obtained by use of the scaled ABS method when the matrix coefficient is subjected to a rank one change.

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