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Some additions to the theory of star partitions of graphs

100%

Bell F., Cvetković D., Rowlinson P., Simić S.

Discussiones Mathematicae Graph Theory

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1999

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tom 19

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nr 2

119-134

EN

This paper contains a number of results in the theory of star partitions of graphs. We illustrate a variety of situations which can arise when the Reconstruction Theorem for graphs is used, considering in particular galaxy graphs - these are graphs in which every star set is independent. We discuss a recursive ordering of graphs based on the Reconstruction Theorem, and point out the significance of galaxy graphs in this connection.

2

A Symbolic Test-Function for Describing Electromagnetic Interaction

100%

Toma C.

Waves, Wavelets and Fractals

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2016

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tom 2

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nr 1

EN

This paper presents a symbolic test-function defined on a restricted interval for describing the electromagnetic interaction. An exponential function with the exponent represented by the inverse of a differential operator is put in correspondence with a test function defined on a restricted interval. The lateral limits of this function at the edge of this interval are either zero (when the variable is considered in closer proximity inside this interval) or infinity (when the variable is considered in closer proximity outside this interval). By substituting with a differential operator these edges (stable points) can be put in correspondence with the wave function of charged particles. In a similar manner, the maximum of this function (an unstable point) can be put in correspondence with an electromagnetic propagating wave. As a consequence, the electromagnetic interaction can be described by particles with eigenvalues of differential operators confined within a restricted interval (similar to wavelets presenting a non-zero amplitude on limited space-time intervals).

3

Inequality-based approximation of matrix eigenvectors

88%

Kocsor A., Dombi J., Bálint I.

International Journal of Applied Mathematics and Computer Science

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2002

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tom 12

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nr 4

533-538

EN

A novel procedure is given here for constructing non-negative functions with zero-valued global minima coinciding with eigenvectors of a general real matrix A. Some of these functions are distinct because all their local minima are also global, offering a new way of determining eigenpairs by local optimization. Apart from describing the framework of the method, the error bounds given separately for the approximation of eigenvectors and eigenvalues provide a deeper insight into the fundamentally different nature of their approximations.

4

A Riemann-Hilbert problem with a vanishing coefficient and applications to Toeplitz operators

88%

Perälä A., Virtanen J. A., Wolf L.

Concrete Operators

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2013

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tom 1

28-36

EN

We study the homogeneous Riemann-Hilbert problem with a vanishing scalar-valued continuous coefficient. We characterize non-existence of nontrivial solutions in the case where the coefficient has its values along several rays starting from the origin. As a consequence, some results on injectivity and existence of eigenvalues of Toeplitz operators in Hardy spaces are obtained.

5

Reconstruction of potential and boundary conditions for second order difference equations

88%

Currie S., Love A.

Commentationes Mathematicae

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2018

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tom 58

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nr 1-2

EN

Assume the eigenvalues and the weights are given for a difference boundary value problem and that the form of the boundary conditions at the endpoints is known. In particular, it is known whether the endpoints are fixed (i.e. Dirichlet or non-Dirichlet boundary conditions) or whether the endpoints are free to move (i.e. boundary conditions with affine dependence on the eigenparameter). This work illustrates how the potential as well as the exact boundary conditions can be uniquely reconstructed. The procedure is inductive on the number of unit intervals. This paper follows along the lines of S. Currie and A. Love, Inverse problems for difference equations with quadratic eigenparameter dependent boundary conditions, \emph{Quaestiones Mathematicae}, 40 (2017), no. 7, 861−877. Since the inverse problem considered in this paper contains more unknowns than the inverse problem considered in the above reference, an additional spectrum is required more often than was the case in the unique reconstruction of the potential alone.

6

On composition of signed graphs

88%

Shahul Hameed K., Germina K.

Discussiones Mathematicae Graph Theory

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2012

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tom 32

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nr 3

507-516

EN

A graph whose edges are labeled either as positive or negative is called a signed graph. In this article, we extend the notion of composition of (unsigned) graphs (also called lexicographic product) to signed graphs. We employ Kronecker product of matrices to express the adjacency matrix of this product of two signed graphs and hence find its eigenvalues when the second graph under composition is net-regular. A signed graph is said to be net-regular if every vertex has constant net-degree, namely, the difference of the number of positive and negative edges incident with a vertex. We also characterize balance in signed graph composition and have some results on the Laplacian matrices of this product.

7

Latent Semantic Indexing using eigenvalue analysis for efficient information retrieval

75%

Kumar C. A., Srinivas S.

International Journal of Applied Mathematics and Computer Science

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2006

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tom 16

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nr 4

551-558

EN

Text retrieval using Latent Semantic Indexing (LSI) with truncated Singular Value Decomposition (SVD) has been intensively studied in recent years. However, the expensive complexity involved in computing truncated SVD constitutes a major drawback of the LSI method. In this paper, we demonstrate how matrix rank approximation can influence the effectiveness of information retrieval systems. Besides, we present an implementation of the LSI method based on an eigenvalue analysis for rank approximation without computing truncated SVD, along with its computational details. Significant improvements in computational time while maintaining retrieval accuracy are observed over the tested document collections.

8

Existence and multiplicity results for nonlinear eigenvalue problems with discontinuities

75%

Papageorgiou N. S., Papalini F.

Annales Polonici Mathematici

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2000

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tom 75

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