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1

Riesz basis of exponential family for a hyperbolic system

Intissar Abdelkader, Jeribi Aref, Walha Ines

Journal of Applied Analysis

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2019

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Vol. 25, nr 1

13--23

EN

This paper studies a linear hyperbolic system with boundary conditions thatwas first studied under someweaker conditions in [8, 11]. Problems on the expansion of a semigroup and a criterion for being a Riesz basis are discussed in the present paper. It is shown that the associated linear system is the infinitesimal generator of a C0-semigroup; its spectrum consists of zeros of a sine-type function, and its exponential system {eλnt}n≥1 constitutes a Riesz basis in L2[0, T]. Furthermore, by the spectral analysis method, it is also shown that the linear system has a sequence of eigenvectors, which form a Riesz basis in Hilbert space, and hence the spectrum-determined growth condition is deduced.

2

Stochastic differential equations for random matrices processes in the nonlinear framework

Stihi S., Boutabia H., Meradji S

Opuscula Mathematica

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2018

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Vol. 38, no. 2

261--283

EN

In this paper, we investigate the processes of eigenvalues and eigenvectors of a symmetric matrix valued process Xt, where Xt is the solution of a general SDE driven by a G-Brownian motion matrix. Stochastic differential equations of these processes are given. This extends results obtained by P. Graczyk and J. Malecki in [Multidimensional Yamada-Watanabe theorem and its applications to particle systems, J. Math. Phys. 54 (2013), 021503].

3

Comprehensive study of free vibration of rectangular mindlin’s plates with rotationally constrained edges using dynamic timoshenko trial functions

Verma Y.

Engineering Transactions

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2018

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Vol. 66, iss. 2

129--160

EN

A comprehensive theoretical study of the free vibration of rotationally restrained rectangular uniform isotropic Mindlin’s plate is presented. The plate mode shape is assumed to be a weighted combination of the product of the Timoshenko beam functions in the either direction, which are previously generated for rotationally constrained boundary conditions. The effect of the uniformly distributed rotational spring constant (modelling the edge) participates in the potential energy of the plate. The Rayleigh-Ritz method has been used to generate the natural frequencies and plate mode shapes for various intermediate boundary conditions, asymptoting to those of the plates with all possible (six) classical boundary conditions. Plates with various thickness ratios have been studied to converge the results to the corresponding Kirchhoff’s frequencies. The eigenvectors from the eigenvalue problem have been scrutinized to establish the beam-wise modal participation from either direction into the final plate mode shape. The square Mindlin’s plate mode shapes have been generated to establish the various types of frequencies; which have been innovatively named and categorized as the (i) single frequencies, (ii) repeated frequencies (identical twins) and (iii) non-repeated frequencies(fraternal twins). Plates with different rectangular aspect ratios have been also analysed to show the deviation in the frequencies and mode shapes from the square plate. Also, their asymptotic behaviour to the corresponding Timoshenko beam at extreme aspect ratios has been established.

4

Continuous Pairwise Comparisons

Saaty T. L.

Fundamenta Informaticae

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2016

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Vol. 144, nr 3/4

213--221

EN

One often assumes that comparisons are discrete and carried out in a matrix of numbers. However, our eyes and other senses perform comparisons in a continuous way by making many simultaneously. Here the mathematics of pairwise comparisons is generalized to the continuous case. It is more likely that in evolution, little creatures had and now still have a "feeling" about the environment in which they find themselves by sensing many things at once. Through evolution, sensing has been transformed to thinking and to discrete comparisons. This paper presents some material from other works written about continuous comparisons.

5

Simple eigenvectors of unbounded operators of the type "normal plus compact"

Gil M.

Opuscula Mathematica

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2015

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Vol. 35, no. 2

161--169

EN

The paper deals with operators of the form A = S + B, where B is a compact operator in a Hilbert space H and S is an unbounded normal one in H, having a compact resolvent. We consider approximations of the eigenvectors of A, corresponding to simple eigenvalues by the eigenvectors of the operators An = S + Bn (n = 1, 2,...), where Bn is an n-dimensional operator. In addition, we obtain the error estimate of the approximation.

6

On the Properties of the Priority Deriving Procedure in the Pairwise Comparisons Method

Kułakowski K.

Fundamenta Informaticae

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2015

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Vol. 139, nr 4

403--419

EN

The pairwise comparisons method can be used when the relative order of preferences among different concepts (alternatives) needs to be determined. There are several popular implementations of this method, including the Eigenvector Method, the Least Squares Method, the Chi Squares Method and others. Each of the above methods comes with one or more inconsistency indices that help to decide whether the consistency of input guarantees obtaining a reliable output, thus taking the optimal decision. This article explores the relationship between inconsistency of input and error of output. An error describes to what extent the obtained results correspond to the single expert’s assessments. On the basis of the inconsistency and the error, two properties of the weight deriving procedure are formulated. These properties are proven for eigenvector method and Koczkodaj’s inconsistency index. Several estimates using Koczkodaj’s inconsistency index for a principal eigenvalue, Saaty’s inconsistency index and the Condition of Order Preservation are also provided.

7

Eigenvalue problem for multi-degree-of-freedom systems with repeated frequencies

Palej R.

Vibrations in Physical Systems

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2010

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Vol. 24

307--312

EN

The paper deals with the eigenvalue problem related with discrete systems, consisting of n identical masses connected with springs in such a way that the stiffness matrix has the form of a multiband symmetric matrix. The eigenvalue problem formulated for such systems is characterized by repeated eigenvalues to which linearly independent eigenvectors correspond. The solution of the eigenvalue problem has been found for an arbitrary, finite number of degrees of freedom for the fully coupled systems and the systems in which masses are connected exclusively with the nearest neighbours.

PL

Praca dotyczy zagadnienia własnego macierzy związanych z układami dyskretnymi, zbudowanymi z n identycznych mas połączonych sprężynami w taki sposób, by macierz sztywności miała budowę wielopasmowej macierzy symetrycznej. Zagadnienie własne tego typu macierzy charakteryzuje się wielokrotnymi wartościami własnymi, którym odpowiadają układy liniowo niezależnych wektorów własnych. W pracy podano analityczne rozwiązanie zagadnienia własnego macierzy dla układu w pełni sprzężonego i układu, w którym każda masa połączona jest wyłącznie z dwiema sąsiednimi.

8

Crack detection in beam like structures. Part I. Identification of crack location

Majkut L.

Diagnostyka

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2008

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nr 2(46)

17-20

EN

This paper deals with detection of a transverse crack location in beam like structure. An input quantity for detection a natural mode of vibrations was chosen. The crack is substituted by rotational spring, which flexibility is calculated by using Castigliano theorem and laws of the fracture mechanics. Based on this model of crack the first natural mode of Bernoulli-Euler beam was determined then analysed. The modal assurance criterion (MAC coefficients) were determined as a function of crack depth and location. These quantities can be used for crack detection in beam like structures. In order to crack location the discrete and continuous wavelet analysis of mode shapes was used.

PL

W pracy opisano pewien sposób detekcji pęknięcia oraz identyfikacji jego lokalizacji. Jako wielkości wejściowe procesu identyfikacji wybrano wektory własne belki z pęknięciem. Jako model pęknięcia przyjęto przegub sprężysty, którego podatność wyznaczono na podstawie praw mechaniki pękania i twierdzenia Castigliano. Wykorzystując taki opis pęknięcia wyznaczono wektory własne pękniętej belki opisanej modelem Bernoulliego-Eulera. Wyznaczono współczynniki MAC (modal assurance criterion) w funkcji głębokości i miejsca pęknięcia. Wielkości te pozwalają jedynie na detekcję pęknięcia. W celu wyznaczenia lokalizacji uszkodzenia wykorzystano dyskretną i ciągłą analizę falkową wektorów własnych.

9

Approximation of a straight line in software for coordinate measuring machine.

Filipowski R., Dąbrowski L., Zawora J., Kossowski R.

Archive of Mechanical Engineering

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2006

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Vol. LIII, nr 1

73-88

EN

The software for coordinate measuring machine (CMM) is used for approximating geometric elements in plane (2d) and in space (3d) by approximation algorithms when number of measuring points obtained is larger than the minimum number of points necessary for element definition. Orthogonal regression methods are discussed in this paper. For 2d and 3d cases and they are compared with methods for defining straight line using linear regression. The advantage of orthogonal regression over linear regression is shown by comparing variance of measuring point displacement from both approximation line types. Algorithms for orthogonal regression make it possible to determine optimum position of straight line for which value of variance of point distance from the line is the lowest possible. The tests have been performed on the Carl Zeiss CMM.

PL

Oprogramowanie współrzędnościowych maszyn pomiarowych (CMM) przybliża twory geometryczne na płaszczyźnie (2D) i w przestrzeni (3D) algorytmami aproksymacyjnymi , gdy liczba punktów pomiarowych jest większa od matematycznie najmniejszej liczby punktów niezbędnych do ich zdefiniowania. W artykule omówiono metody regresji ortogonalnej definiowania prostych 2D i 3D i porównano je z metodami definiowania prostych metodami regresji prostoliniowej. Przewagę prostych regresji ortogonalnej nad prostymi regresji prostoliniowej wykazano poprzez porównanie wariancji odchyleń położenia punktów pomiarowych względem obu typów prostych. Algorytmy regresji ortogonalnej pozwalają ustalić optymalne położenie prostej, przy którym wariancja odległości punktów pomiarowych względem tej prostej przyjmuje wartość najmniejszą. Testowanie przeprowadzono na CMM VISTA firmy Carl Zeiss.

10

Data dimensionality reduction for face recognition

Kukharev G., Forczmański P.

Machine Graphics and Vision

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2004

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Vol. 13, No. 1/2

99-121

EN

In the process of image recognition in most of the applications there is a problem with gathering, processing and storing large amounts of data. A possible solution for reducing this amounts and speeding--up computations is to use some sort of data reduction. Efficient reduction of the stored data without losing any important part of it requires an adaptive method, which works without any supervision. In this article we discuss a few variants of a two--step approach, which involves Karhunen--Loeve Transform (KLT) and Linear Discriminant Analysis (LDA). The KLT gives a good approximation of the input data, however it requires a large number of eigenvalues. The second step reduces data dimensionality futher using LDA. The efficiency of KLT depends on the quality and quantity of the input data. In the case when only one image in a class is given as input, its features are not stable in comparison with other images in other classes. In this article we present a few methods for solving this problem, which improve on the ideas presented in [6, 9].

11

Inequality-Based Approximation of Matrix Eigenvectors

Kocsor A., Dombi J., Balint I.

International Journal of Applied Mathematics and Computer Science

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2002

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Vol. 12, no 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.