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1

An inequality for imaginary parts of eigenvalues of non-compact operators with Hilbert-Schmidt Hermitian components

Gil’ Michael

Opuscula Mathematica

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2024

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Vol. 44, no. 2

241--248

EN

Let A be a bounded linear operator in a complex separable Hilbert space, A∗ be its adjoint one and AI := (A − A∗)/(2i). Assuming that AI is a Hilbert-Schmidt operator, we investigate perturbations of the imaginary parts of the eigenvalues of A. Our results are formulated in terms of the “extended” eigenvalue sets in the sense introduced by T. Kato. Besides, we refine the classical Weyl inequality [formula], where λk(A) (k = 1, 2, . . .) are the eigenvalues of A and N2(·) is the Hilbert-Schmidt norm. In addition, we discuss applications of our results to the Jacobi operators.

2

On spectrum of Metzler matrices

Domka Michał, Mitkowski Wojciech

Przegląd Elektrotechniczny

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2022

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R. 98, nr 12

121--123

EN

In the paper it was proven that spectrum of Metzler matrix must belong to a certain cone in the complex plane. The result is derived from the analytical characterization of spectra of positive matrices obtained by Karpelevich. Furthermore, it was shown that in case of 3x3 matrices this property yields also a sufficient condition that a set of numbers must satisfy in order to be spectrum of some Metzler matrix.

PL

W pracy wykazano, że widmo macierzy każdej Metzlera należy do pewnego stożka na płaszczyźnie zespolonej. Wykorzystano w tym celu analityczną charakteryzację widma macierzy dodatniej wyznaczoną przez Karpielewicza. Ponadto wykazano, że w przypadku macierzy 3x3 własność ta pozwala wyznaczyć również warunek wystarczający, który spełniać musi zbiór liczb zespolonych, aby był widmem pewnej macierzy Metzlera.

3

Energy of cartesian product graph networks

Vivik. J Veninstine, Xavier P., Joshi Afzala R.

Przegląd Elektrotechniczny

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2022

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R. 98, nr 8

28--33

EN

The energy of a graph G is defined as the sum of the absolute values of the eigenvalues of the adjacency matrix of G. The Cartesian product of two graphs namely Path Pm and Double Wheel graph DWn is constructed and its energy values on the formation of adjacency matrix, Laplacian matrix and maximum degree matrix is obtained. The upper bounds for the energy variations of different energies like graph energy, Laplacian energy and maximum degree energy of the initiated product graphs are identified and compared.

PL

Energia grafu G jest zdefiniowana jako suma wartości bezwzględnych wartości własnych macierzy sąsiedztwa G. Konstruowany jest iloczyn kartezjański dwóch grafów, a mianowicie Path Pm i Double Wheel graph DWn oraz jego wartości energii podczas tworzenia sąsiedztwa otrzymuje się macierz, macierz Laplace’a i macierz maksymalnego stopnia. Górne granice dla zmian energii różnych energii, takich jak energia wykresu, energia Laplace’a i maksymalny stopień energii zainicjowanych wykresów produktów, są identyfikowane i porównywane.

4

Eigenvalues assignment in uncontrollable linear systems

Kaczorek Tadeusz

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2022

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Vol. 70, nr 6

art. no. e141987

EN

It is shown that in uncontrollable linear system x = Ax + Bu it is possible to assign arbitrarily the eigenvalues of the closed-loop system with state feedbacks u = Kx, K ∈ ℜn⨉m if rank [A B] = n. The design procedure consists in two steps. In the step 1 a nonsingular matrix M ∈ ℜn⨉m is chosen so that the pair (MA,MB) is controllable. In step 2 the feedback matrix K is chosen so that the closed-loop matrix Ac = A − BK has the desired eigenvalues. The procedure is illustrated by simple example.

5

Axisymmetric free vibration of layered cylindrical shell filled with fluid

Izyan M. D. Nurul, Sabri Nur Ain Ayunni, Hafizah A. K. Nor, Sankar D. S., Viswanathan K. K.

International Journal of Applied Mechanics and Engineering

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2021

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Vol. 26, no. 4

63--76

EN

The aim of the study is to analyse the axisymmetric free vibration of layered cylindrical shells filled with a quiescent fluid. The fluid is assumed to be incompressible and inviscid. The equations of axisymmetric vibrations of layered cylindrical shell filled with fluid, on the longitudinal and transverse displacement components are obtained using Love’s first approximation theory. The solutions of displacement functions are assumed in a separable form to obtain a system of coupled differential equations in terms of displacement functions. The displacement functions are approximated by Bickley-type splines. A generalized eigenvalue problem is obtained and solved numerically for a frequency parameter and an associated eigenvector of spline coefficients. Two layered shells with three different types of materials under clamped-clamped boundary conditions are considered. Parametric studies are made on the variation of the frequency parameter with respect to length-to-radius ratio and length-to-thickness ratio.

6

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.

7

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].

8

On spectra of quadratic operator pencils with rank one gyroscopic linear part

Boyko O., Martynyuk O., Pivovarchik V.

Opuscula Mathematica

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2018

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Vol. 38, no. 4

483--500

EN

The spectrum of a selfadjoint quadratic operator pencil of the form [formula] is investigated where M ≥ 0, G ≥ 0 are bounded operators and A is selfadjoint bounded below is investigated. It is shown that in the case of rank one operator G the eigenvalues of such a pencil are of two types. The eigenvalues of one of these types are independent of the operator G. Location of the eigenvalues of both types is described. Examples for the case of the Sturm-Liouville operators A are given. Keywords: q

9

Linear Sturm-Liouville problems with Riemann- Stieltjes integral boundary conditions

Kong Q., George T. E. S.

Opuscula Mathematica

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2018

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Vol. 38, no. 4

557--571

EN

We study second-order linear Sturm-Liouville problems involving general homogeneous linear Riemann-Stieltjes integral boundary conditions. Conditions are obtained for the existence of a sequence of positive eigenvalues with consecutive zero counts of the eigenfunctions. Additionally, we find interlacing relationships between the eigenvalues of such Sturm-Liouville problems and those of Sturm-Liouville problems with certain two-point separated boundary conditions.

10

Reconstruction of potential and boundary conditions for second order difference equations

Currie S., Love A.

Commentationes Mathematicae

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2018

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Vol. 58, No. 1/2

1--9

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, 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.

11

The eigenvalue problem in the dynamics of the steel industrial hall with internal handling system

Pawlak U., Szczecina M.

Structure and Environment

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2016

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Vol. 8, no. 4

229--236

EN

The paper presents the eigenvalue problem in the dynamics of the steel industrial hall with an internal handling system. The aim of the calculations was to determine the eigenfrequencies and eigenvectors for the structure of the hall adapted to a gantry operation. The analysis was performed by using FEM software, namely Robot. Among over a dozen calculated eigenvectors, four ones were chosen as the most representative for a lateral structural arrangement of the hall and its roof.

12

Strong stationary duality for Möbius monotone Markov chains : examples

Lorek P., Szekli R.

Probability and Mathematical Statistics

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2016

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Vol. 36, Fasc. 1

75--97

EN

We construct strong stationary dual chains for nonsymmetric random walks on square lattice, for random walks on hypercube and for some Ising models on the circle. The strong stationary dual chains are all sharp and have the same state space as original chains.We use Möbius monotonicity of these chains with respect to some natural orderings of the corresponding state spaces. This method provides an alternative way to study mixing times for studied models.

13

The approximate location of imperfections in an ellipse using the spectral theory

Brzęk M., Zagórowska M., Mitkowski W.

Measurement Automation Monitoring

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2016

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Vol. 62, No. 4

136--138

EN

In the paper, the authors describe the analysis and results of research on the approximate location of a defect with the help of the spectral theory for the area which is a geometrical ellipse. A computer simulation was conducted in Matlab. At the end of the paper, the authors give an example to illustrate the method of determining the areas in which the ellipse may be damaged

14

On the evolution of academic staff structure in a university setting

Ekhosuehi V. U.

Journal of Mathematics and Applications

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2016

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Vol. 39

47--58

EN

This paper models the academic staff structure in a university as a system of stocks and flows in a three-dimensional space, R3. The stocks are the number of academic staff in a particular state at a given time and the flows are the staff moving between any two states over an interval of time. The paper places emphasis on the grade-specific completion rates of Graduate Assistants, who choose to study in the university in which they are employed for higher degrees. The study describes the evolution of structures in the university as a linear recurrence system. Some aspects of linear algebra are employed as a theoretical underpinning to gain insights into the transformation matrix of the recurrence system. A number of resulting propositions are presented along with their proofs. We provide two theorems to serve as a means of predicting a university manpower structure. Following that a numerical illustration of the theorems and propositions is provided with data which are representative of the kind of data in a Nigerian university system.

15

The Sturm-Liouville eigenvalue problem - a numerical solution using the control volume method

Siedlecki J., Ciesielski M, Błaszczyk T.

Journal of Applied Mathematics and Computational Mechanics

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2016

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Vol. 15, nr 2

127--136

EN

The solution of the 1D Sturm-Liouville problem using the Control Volume Method is discussed. The second order linear differential equation with homogeneous boundary conditions is discretized and converted to the system of linear algebraic equations. The matrix associated with this system is tridiagonal and eigenvalues of this system are an approximation of the real eigenvalues of the boundary value problem. The numerical results of the eigenvalues for various cases and the experimental rate of convergence are presented.

16

Series representation of compact linear operators in Banach spaces

Edmunds D., Lang J.

Commentationes Mathematicae

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2016

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Vol 56, No. 1

17--27

EN

Let p∈(1,∞) and I=(0,1); suppose that T:Lp(I)→Lp(I) is a~compact linear map with trivial kernel and range dense in Lp(I). It is shown that if the Gelfand numbers of T decay sufficiently quickly, then the action of T is given by a series with calculable coefficients. The special properties of Lp(I) enable this to be established under weaker conditions on the Gelfand numbers than in earlier work set in the context of more general spaces.

17

On small vibrations of a damped Stieltjes string

Boyko O., Pivovarchik V.

Opuscula Mathematica

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2015

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Vol. 35, no. 2

143--159

EN

Inverse problem of recovering masses, coefficients of damping and lengths of the intervals between the masses using two spectra of boundary value problems and the total length of the Stieltjes string (an elastic thread bearing point masses) is considered. For the case of point-wise damping at the first counting from the right end mass the problem of recovering the masses, the damping coefficient and the lengths of the subintervals by one spectrum and the total length of the string is solved.

18

Reduction of linear electrical circuits with complex eigenvalues to linear electrical circuits with real eigenvalues

Kaczorek T., Borawski K.

Measurement Automation Monitoring

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2015

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Vol. 61, No. 4

115--117

EN

The problem of reduction of linear electrical circuits with complex eigenvalues to linear electrical circuits with real eigenvalues is analyzed. Methods for finding the transformation matrix are presented. Considerations are illustrated by numerical examples.

19

Zagadnienie własne w dynamice stalowej hali przemysłowej z transportem wewnętrznym

Pawlak U., Szczecina M.

Logistyka

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2015

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nr 3

3816--3824, CD 1

PL

W referacie przedstawiono zagadnienie własne w dynamice hali stalowej z transportem wewnętrznym. Celem prowadzonych obliczeń było ustalenie częstości drgań własnych i wektorów własnych dla konstrukcji hali dostosowanej do pracy suwnicy. Analizę wykonano za pomocą MES w oprogramowaniu Robot. Spośród kilkunastu obliczonych postaci drgań, wyodrębniono te najbardziej reprezentatywne dla układu poprzecznego hali oraz jej dachu.

EN

The paper presents the eigenvalue problem of the dynamics of the steel industrial hall with internal transport. The aim of the calculations was to determine the eigenfrequencies and eigenvectors for the structure of the hall adapted to a gantry. The analysis was performed in FEM software, namely Robot. Among over a dozen calculated eigenvectors, four ones were chosen as the most representative for a cross section of the hall and its roof.

20

Dynamiczne zagadnienie własne płyty drogowej o nawierzchni betonowej

Pawlak U., Szczecina M.

Logistyka

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2015

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nr 3

3808--3815, CD 1

PL

W referacie zaprezentowano analizę dynamiczną płyty drogowej o nawierzchni betonowej. Zastosowano metodę elementów skończonych modelując przykładową płytę betonową w programie Autodesk Robot Structural Analysis. Płytę zadano jako spoczywającą na jednoparametrowym podłożu sprężystym , dla którego osobno policzono współczynnik sprężystości. Zaprezentowano wartości częstości drgań własnych oraz postacie drgań jakimi są wektory własne, w postaci maksymalnych pionowych przemieszczeń węzłowych. Na podstawie wyników obliczeń sformułowane zostały podstawowe zalecenia dla projektantów betonowych nawierzchni drogowych.

EN

The paper presents an analysis of the dynamic eigenvalue problem of concrete slab road surface. A sample concrete slab was modelled in Autodesk Robot Structural Analysis software and calculated with finite element method. The slab was set on a one-parameter elastic subsoil, for which the modulus of elasticity was separately calculated. The eigenfrequencies and eigenvectors (as maximal vertical nodal displacements) were presented. Based on the results of calculations, some basic recommendations for designers of concrete road surface were formulated.