scholarly journals PEBBLE GAMES AND LINEAR EQUATIONS

2015 ◽  
Vol 80 (3) ◽  
pp. 797-844 ◽  
Author(s):  
MARTIN GROHE ◽  
MARTIN OTTO
Keyword(s):  
Linear Algebra ◽  
Linear Equations ◽  
Graph Isomorphism ◽  
Detailed Account ◽  
Base Level ◽  
Basic Colour ◽  
Higher Dimensional ◽  
Rational Arithmetic ◽  
Pebble Game ◽  
Fractional Isomorphism

AbstractWe give a new, simplified and detailed account of the correspondence between levels of the Sherali–Adams relaxation of graph isomorphism and levels of pebble-game equivalence with counting (higher-dimensional Weisfeiler–Lehman colour refinement). The correspondence between basic colour refinement and fractional isomorphism, due to Tinhofer [22; 23] and Ramana, Scheinerman and Ullman [17], is re-interpreted as the base level of Sherali–Adams and generalised to higher levels in this sense by Atserias and Maneva [1] and Malkin [14], who prove that the two resulting hierarchies interleave. In carrying this analysis further, we here give (a) a precise characterisation of the level k Sherali–Adams relaxation in terms of a modified counting pebble game; (b) a variant of the Sherali–Adams levels that precisely match the k-pebble counting game; (c) a proof that the interleaving between these two hierarchies is strict. We also investigate the variation based on boolean arithmetic instead of real/rational arithmetic and obtain analogous correspondences and separations for plain k-pebble equivalence (without counting). Our results are driven by considerably simplified accounts of the underlying combinatorics and linear algebra.

Numerical Linear Algebra

1966 ◽  
Vol 9 (05) ◽  
pp. 757-801 ◽  
Author(s):  
W. Kahan
Keyword(s):  
Eigenvalue Problem ◽  
Linear Algebra ◽  
Linear Equations ◽  
Numerical Linear Algebra ◽  
System Of Linear Equations ◽  
Image Position

The primordial problems of linear algebra are the solution of a system of linear equations and the solution of the eigenvalue problem for the eigenvalues λk, and corresponding eigenvectors of a given matrix A.

Incremental Hyperplane Partitioning for Classification

2013 ◽  
Vol 4 (2) ◽  
pp. 67-79 ◽  
Author(s):  
Tao Yang ◽  
Sheng-Uei Guan ◽  
Jinghao Song ◽  
Binge Zheng ◽  
Mengying Cao ◽  
...  
Keyword(s):  
Genetic Algorithm ◽  
Linear Equations ◽  
Curse Of Dimensionality ◽  
Experimental Results ◽  
New Method ◽  
Complex Pattern ◽  
Classification Problems ◽  
Encoding Scheme ◽  
Incremental Approach ◽  
Higher Dimensional

The authors propose an incremental hyperplane partitioning approach to classification. Hyperplanes that are close to the classification boundaries of a given problem are searched using an incremental approach based upon Genetic Algorithm (GA). A new method - Incremental Linear Encoding based Genetic Algorithm (ILEGA) is proposed to tackle the difficulty of classification problems caused by the complex pattern relationship and curse of dimensionality. The authors solve classification problems through a simple and flexible chromosome encoding scheme, where the partitioning rules are encoded by linear equations rather than If-Then rules. Moreover, an incremental approach combined with output portioning and pattern reduction is applied to cope with the curse of dimensionality. The algorithm is tested with six datasets. The experimental results show that ILEGA outperform in both lower- and higher-dimensional problems compared with the original GA.

An attempt to reduce learning difficulties in Linear Algebra courses

2016 ◽  
Vol 8 (2) ◽  
pp. 156
Author(s):  
Marta Graciela Caligaris ◽  
María Elena Schivo ◽  
María Rosa Romiti
Keyword(s):  
Linear Algebra ◽  
Linear Equations ◽  
Learning Difficulties ◽  
Vector Spaces ◽  
Analytic Geometry ◽  
Teaching Activities ◽  
Interactive Tools ◽  
Systems Of Linear Equations

In engineering careers, the study of Linear Algebra begins in the first course. Some topics included in this subject are systems of linear equations and vector spaces. Linear Algebra is very useful but can be very abstract for teaching and learning.In an attempt to reduce learning difficulties, different approaches of teaching activities supported by interactive tools were analyzed. This paper presents these tools, designed with GeoGebra for the Algebra and Analytic Geometry course at the Facultad Regional San Nicolás (FRSN), Universidad Tecnológica Nacional (UTN), Argentina.

scholarly journals USING GPU NVIDIA FOR LINEAR ALGEBRA PROLEMS

2019 ◽  
pp. 144-157
Author(s):  
A. Myasishchev ◽  
S. Lienkov ◽  
V. Dzhulii ◽  
I. Muliar
Keyword(s):  
Linear Algebra ◽  
Linear Equations ◽  
Matrix Multiplication ◽  
Performance Comparison ◽  
Double Precision ◽  
Graphics Processors ◽  
Performance Study ◽  
Software Module ◽  
Maximum Acceleration ◽  
Computational Procedures

Research goals and objectives: the purpose of the article is to study the feasibility of graphics processors using in solving linear equations systems and calculating matrix multiplication as compared with conventional multi-core processors. The peculiarities of the MAGMA and CUBLAS libraries use for various graphics processors are considered. A performance comparison is made between the Tesla C2075 and GeForce GTX 480 GPUs and a six-core AMD processor. Subject of research: the software is developed basing on the MAGMA and CUBLAS libraries for the purpose of the NVIDIA Tesla C2075 and GeForce GTX 480 GPUs performance study for linear equation systems solving and matrix multiplication calculating. Research methods used: libraries were used to parallelize the linear algebra problems solution. For GPUs, these are MAGMA and CUBLAS, for multi-core processors, the ScaLAPACK and ATLAS libraries. To study the operational speed there are used methods and algorithms of computational procedures parallelization similar to these libraries. A software module has been developed for linear equations systems solving and matrix multiplication calculating by parallel systems. Results of the research: it has been determined that for double-precision numbers the GPU GeForce GTX 480 and the GPU Tesla C2075 performance is approximately 3.5 and 6.3 times higher than that of the AMD CPU. And the GPU GeForce GTX 480 performance is 1.3 times higher than the GPU Tesla C2075 performance for single precision numbers. To achieve maximum performance of the NVIDIA CUDA GPU, you need to use the MAGMA or CUBLAS libraries, which accelerate the calculations by about 6.4 times as compared to the traditional programming method. It has been determined that in equations systems solving on a 6-core CPU, it is possible to achieve a maximum acceleration of 3.24 times as compared to calculations on the 1st core using the ScaLAPACK and ATLAS libraries instead of 6-fold theoretical acceleration. Therefore, it is impossible to efficiently use processors with a large number of cores with considered libraries. It is demonstrated that the advantage of the GPU over the CPU increases with the number of equations.

Dapsone is not a Pharmacodynamic Lead Compound for its Aryl Derivatives

2020 ◽  
Vol 16 (3) ◽  
pp. 327-339
Author(s):  
Thomas Scior ◽  
Hassan H. Abdallah ◽  
Kenia Salvador-Atonal ◽  
Stefan Laufer
Keyword(s):  
Linear Equations ◽  
3D Qsar ◽  
Ligand Docking ◽  
Activity Data ◽  
Short Side ◽  
Dihydropteroate Synthase ◽  
Higher Dimensional ◽  
Binding Enthalpy ◽  
The Moment ◽  
2D And 3D

Background: The relatedness between the linear equations of thermodynamics and QSAR was studied thanks to the recently elucidated crystal structure complexes between sulfonamide pterin conjugates and dihydropteroate synthase (DHPS) together with a published set of thirty- six synthetic dapsone derivatives with their reported entropy-driven activity data. Only a few congeners were slightly better than dapsone. Objective : Our study aimed at demonstrating the applicability of thermodynamic QSAR and to shed light on the mechanistic aspects of sulfone binding to DHPS. Methods: To this end ligand docking to DHPS, quantum mechanical properties, 2D- and 3D-QSAR as well as Principle Component Analysis (PCA) were carried out. Results: The short aryl substituents of the docked pterin-sulfa conjugates were outward oriented into the solvent space without interacting with target residues which explains why binding enthalpy (ΔH) did not correlate with potency. PCA revealed how chemically informative descriptors are evenly loaded on the first three PCs (interpreted as ΔG, ΔH and ΔS), while chemically cryptic ones reflected higher dimensional (complex) loadings. Conclusions: It is safe to utter that synthesis efforts to introduce short side chains for aryl derivatization of the dapsone scaffold have failed in the past. On theoretical grounds we provide computed evidence why dapsone is not a pharmacodynamic lead for drug profiling because enthalpic terms do not change significantly at the moment of ligand binding to target.

The SVD-Fundamental Theorem of Linear Algebra

2006 ◽  
Vol 11 (2) ◽  
pp. 123-136 ◽  
Author(s):  
A. G. Akritas ◽  
G. I. Malaschonok ◽  
P. S. Vigklas
Keyword(s):  
Linear System ◽  
Linear Algebra ◽  
Fundamental Theorem ◽  
Gaussian Elimination ◽  
Linear Equations ◽  
Orthogonal Complement ◽  
Schmidt Method ◽  
Orthonormal Bases ◽  
Svd Analysis ◽  
Systems Of Linear Equations

Given an m × n matrix A, with m ≥ n, the four subspaces associated with it are shown in Fig. 1 (see [1]). Fig. 1. The row spaces and the nullspaces of A and AT; a1 through an and h1 through hm are abbreviations of the alignerframe and hangerframe vectors respectively (see [2]). The Fundamental Theorem of Linear Algebra tells us that N(A) is the orthogonal complement of R(AT). These four subspaces tell the whole story of the Linear System Ax = y.  So, for example, the absence of N(AT) indicates that a solution always exists, whereas the absence of N(A) indicates that this solution is unique. Given the importance of these subspaces, computing bases for them is the gist of Linear Algebra. In “Classical” Linear Algebra, bases for these subspaces are computed using Gaussian Elimination; they are orthonormalized with the help of the Gram-Schmidt method. Continuing our previous work [3] and following Uhl’s excellent approach [2] we use SVD analysis to compute orthonormal bases for the four subspaces associated with A, and give a 3D explanation. We then state and prove what we call the “SVD-Fundamental Theorem” of Linear Algebra, and apply it in solving systems of linear equations.

scholarly journals Linear Equations in Singular Moduli

2018 ◽  
Vol 2020 (21) ◽  
pp. 7617-7643
Author(s):  
Yuri Bilu ◽  
Lars Kühne
Keyword(s):  
Linear Equations ◽  
Modular Curves ◽  
Linear Subspaces ◽  
Effective Version ◽  
Singular Moduli ◽  
Higher Dimensional

Abstract We establish an effective version of the André–Oort conjecture for linear subspaces of $Y(1)^n_{\mathbb{C}} \approx \mathbb{A}_{\mathbb{C}}^n$. This gives the first effective nontrivial results of André–Oort type for higher-dimensional varieties in products of modular curves.

Nonlinear Dynamics of Flexible L-Shaped Beam Based on Exact Modes Truncation

2017 ◽  
Vol 27 (03) ◽  
pp. 1750035 ◽  
Author(s):  
Tian-Jun Yu ◽  
Wei Zhang ◽  
Xiao-Dong Yang
Keyword(s):  
Nonlinear Dynamics ◽  
Linear Equations ◽  
Elliptic Functions ◽  
Governing Equations ◽  
Modal Interactions ◽  
Partial Differential ◽  
Shaped Beam ◽  
Higher Dimensional ◽  
Theoretical Predictions ◽  
Energy Exchanges

Nonlinear dynamics of flexible multibeam structures modeled as an L-shaped beam are investigated systematically considering the modal interactions. Taking into account nonlinear coupling and nonlinear inertia, Hamilton’s principle is employed to derive the partial differential governing equations of the structure. Exact mode functions are obtained by the coupled linear equations governing the horizontal and vertical beams and the results are verified by the finite element method. Then the exact modes are adopted to truncate the partial differential governing equations into two coupled nonlinear ordinary differential equations by using Galerkin method. The undamped free oscillations are studied in terms of Jacobi elliptic functions and results indicate that the energy exchanges are continual between the two modes. The saturation and jumping phenomena are then observed for the forced damped multibeam structure. Further, a higher-dimensional, Melnikov-type perturbation method is used to explore the physical mechanism leading to chaotic behaviors for such an autoparametric system. Numerical simulations are performed to validate the theoretical predictions.

Solution of Linear Equations With Coefficients and Right-Hand Members in An Arithmetic Sequence

1977 ◽  
Vol 70 (2) ◽  
pp. 170-172
Author(s):  
Murli M. Gupta
Keyword(s):  
General Solution ◽  
Linear Algebra ◽  
Linear Equations ◽  
Arithmetic Sequence ◽  
Right Hand

A general solution of a problem in linear algebra.

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