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1

Recovering the shape of an equilateral quantum tree with the Dirichlet conditions at the pendant vertices

Dudko Anastasia, Lesechko Oleksandr, Pivovarchik Vyacheslav

Opuscula Mathematica

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2024

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Vol. 44, no. 5

689--705

EN

We consider two spectral problems on an equilateral rooted tree with the standard (continuity and Kirchhoff’s type) conditions at the interior vertices (except of the root if it is interior) and Dirichlet conditions at the pendant vertices (except of the root if it is pendant). For the first (Neumann) problem we impose the standard conditions (if the root is an interior vertex) or Neumann condition (if the root is a pendant vertex) at the root, while for the second (Dirichlet) problem we impose the Dirichlet condition at the root. We show that for caterpillar trees the spectra of the Neumann problem and of the Dirichlet problem uniquely determine the shape of the tree. Also, we present an example of co-spectral snowflake graphs.

2

Bochner formula in generalized (k, μ)-space forms

B Shanmukha

Journal of Applied Analysis

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2023

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Vol. 29, nr 2

323--328

EN

In this article, we studied Green’s theorem and the Bochner formula. Further, we apply the Bochner formula to generalized (k, μ)-space forms and show that the generalized (k, μ) space form is either isometric to a sphere or a certain warped product under some geometric conditions.

3

On Ambarzumian type theorems for tree domains

Pivovarchik Vyacheslav

Opuscula Mathematica

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2022

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Vol. 42, no. 3

427--437

EN

It is known that the spectrum of the spectral Sturm–Liouville problem on an equilateral tree with (generalized) Neumann’s conditions at all vertices uniquely determines the potentials on the edges in the unperturbed case, i.e. case of the zero potentials on the edges (Ambarzumian’s theorem). This case is exceptional, and in general case (when the Dirichlet conditions are imposed at some of the pendant vertices) even two spectra of spectral problems do not determine uniquely the potentials on the edges. We consider the spectral Sturm–Liouville problem on an equilateral tree rooted at its pendant vertex with (generalized) Neumann conditions at all vertices except of the root and the Dirichlet condition at the root. In this case Ambarzumian’s theorem can’t be applied. We show that if the spectrum of this problem is unperturbed, the spectrum of the Neumann-Dirichlet problem on the root edge is also unperturbed and the spectra of the problems on the complimentary subtrees with (generalized) Neumann conditions at all vertices except the subtrees’ roots and the Dirichlet condition at the subtrees’ roots are unperturbed then the potential on each edge of the tree is 0 almost everywhere.

4

The weak eigenfunctions of boundary-value problem with symmetric discontinuities

Olğar Hayati, Mukhtarov Oktay S., Muhtarov Fahreddin S., Aydemir Kadriye

Journal of Applied Analysis

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2022

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Vol. 28, nr 2

275--283

EN

The main goal of this study is the investigation of discontinuous boundary-value problems for second-order differential operators with symmetric transmission conditions.We introduce the new notion of weak functions for such type of discontinuous boundary-value problems and develop an operator-theoretic method for the investigation of the spectrum and completeness property of the weak eigenfunction systems. In particular, we define some self-adjoint compact operators in suitable Sobolev spaces such that the considered problem can be reduced to an operator-pencil equation. The main result of this paper is that the spectrum is discrete and the set of eigenfunctions forms a Riesz basis of the suitable Hilbert space.

5

Optimal power system stabilizer design using craziness particle swarm optimization in Sulselrabar system

Djalal Muhammad Ruswandi, Saini Makmur, Yunus A. M. Shiddiq, Kitta Ikhlas

Przegląd Elektrotechniczny

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2021

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R. 97, nr 10

76--81

EN

Power System Stabilizer (PSS) is a supplementary control that provides additional control actions on the excitation side of the generator. In this study a Craziness Particle Swarm Optimization (CRPSO) based tuning method is proposed to optimize the PSS parameters. CRPSO is a development of the conventional PSO method, where in conventional PSO there is a tendency to achieve premature convergence. This condition causes the solution obtained to be the optimum local. With optimal PSS parameters, the optimal PSS performance is obtained. The combination of PSS and excitation is used to reduce the oscillation that occurs in the system. In this research a case study of load addition and load shedding is used. From the simulation results, it is found that system performance is more optimal using CRPSO than using conventional PSO. System performance is shown by the response of the generator speed and rotor angle which results in a small overshoot and a faster settling time when there is an increase in load and also load shedding. Increased system performance is also viewed from the negative system eigenvalue, negative eigenvalue indicates the system is stable.

PL

Stabilizator systemu zasilania (PSS) jest dodatkowym sterowaniem, które zapewnia dodatkowe działania sterujące po stronie wzbudzenia generatora. W tym badaniu zaproponowano metodę strojenia opartą na Craziness Particle Swarm Optimization (CRPSO) w celu optymalizacji parametrów PSS. CRPSO jest rozwinięciem tradycyjnej metody PSO, gdzie w konwencjonalnym PSO istnieje tendencja do osiągnięcia przedwczesnej konwergencji. Stan ten powoduje, że otrzymane rozwiązanie jest optymalne miejscowo. Przy optymalnych parametrach PSS uzyskuje się optymalną wydajność PSS. Połączenie PSS i wzbudzenia służy do zmniejszenia oscylacji występujących w systemie. W tym badaniu wykorzystano studium przypadku dodawania i odciążania. Z symulacji wynika, że wydajność systemu jest bardziej optymalna przy użyciu CRPSO niż przy użyciu konwencjonalnego PSO. Wydajność systemu jest pokazana przez reakcję prędkości generatora i kąta wirnika, co skutkuje niewielkim przeregulowaniem i szybszym czasem ustalania, gdy występuje wzrost obciążenia, a także zmniejszenie obciążenia. Zwiększona wydajność systemu jest również postrzegana z ujemnej wartości własnej systemu, ujemna wartość własna wskazuje, że system jest stabilny.

6

Discrete spectra for some complex infinite band matrices

Malejki Maria

Opuscula Mathematica

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2021

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Vol. 41, no. 6

861--879

EN

Under suitable assumptions the eigenvalues for an unbounded discrete operator A in l2 , given by an infinite complex band-type matrix, are approximated by the eigenvalues of its orthogonal truncations. Let [formula] where [formula] is the set of all limit points of the sequence (λn) and An is a finite dimensional orthogonal truncation of A. The aim of this article is to provide the conditions that are sufficient for the relations σ (A) ⊂ Λ (A) or Λ(A) ) ⊂ σ (A) to be satisfied for the band operator A.

7

On the existence of complex Hadamard submatrices of the Fourier matrices

Bond Bailey Madison, Glickfield R. Alexander, Herr John E.

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

1--9

EN

We use a theorem of Lam and Leung to prove that a submatrix of a Fourier matrix cannot be Hadamard for particular cases when the dimension of the submatrix does not divide the dimension of the Fourier matrix. We also make some observations on the trace-spectrum relationship of dephased Hadamard matrices of low dimension.

8

Algorytmy i pomiary nachylenia i odchyłek prostoliniowości od prostej 3D głowicą pomiarową na CMM

Filipowski R., Lechniak Z., Zawora J.

Przegląd Mechaniczny

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2017

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nr 7-8

60--63

PL

Przedstawiono podstawy matematyczne i obliczenia nachylenia prostych, odchyłek prostoliniowości względem prostej 3D i odchyłki prostoliniowości średniej kwadratowej. Parametry prostej wyznaczono za pomocą regresji ortogonalnej, stosując metodę najmniejszych kwadratów. Prosta aproksymacyjna przechodzi przez środek ciężkości zbioru punktów pomiarowych. Z warunku koniecznego na minimum funkcji aproksymacyjnej Lagrange’a z warunkiem ubocznym uzyskano układy równań jednorodnych o trzech niewiadomych dla prostej 3D. Rozwiązanie tych równań pozwala znaleźć wartości własne λ j, i = 1, 2, 3 oraz wektory własne i ν j, i = 1, 2, 3, będące wektorami kierunkowymi trzech prostych wzajemnie prostopadłych przechodzących przez środek ciężkości. Wektor kierunkowy szukanej prostej odpowiada najmniejszej wartości własnej λ j, i = 1, 2, 3.

EN

A mathematical method and the computer algorithm have been developed for determining the slope, straightness deviation and root-mean square roundness deviation for a line in space. The parameters describing the line location have been determined using the orthogonal regression analysis and the least squares method. The lines approximating true line location have been assumed to include the gravity center for the measured points set. Satisfying the condition for the minimum value of a Lagrange function with side condition, one can derive a system of homogeneous equations with two unknowns for three unknowns for the spatial line case. By solving these equations, the eigenvalue, λ j, i = 1, 2, 3 and the eigenvector i ν j, i = 1, 2, 3, could have been determined; i ν j, i = 1, 2, 3 being the directional vectors of the approximating lines including the gravity center. The directional vector of the line to be determined corresponds to the minimum eigenvalue λ j, i = 1, 2, 3.

9

Higher order Nevanlinna functions and the inverse three spectra problem

Boyko O., Martinyuk O., Pivovarchik V.

Opuscula Mathematica

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2016

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Vol. 36, no. 3

301--314

EN

he three spectra problem of recovering the Sturm-Liouville equation by the spectrum of the Dirichlet-Dirichlet boundary value problem on [0, α], the Dirichlet-Dirichlet problem on [0, α/2] and the Neumann-Dirichlet problem on [α/2, α] is considered. Sufficient conditions of solvability and of uniqueness of the solution to such a problem are found.

10

Eigenvalue problems for anisotropic equations involving a potential on Orlicz-Sobolev type spaces

Stancut I. L., Stircu I. D.

Opuscula Mathematica

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2016

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Vol. 36, no. 1

81--101

EN

In this paper we consider an eigenvalue problem that involves a nonhomogeneous elliptic operator, variable growth conditions and a potential Von a bounded domain in Rn (N ≥ 3) with a smooth boundary. We establish three main results with various assumptions. The first one asserts that any λ > 0 is an eigenvalue of our problem. The second theorem states the existence of a constant [formula] such that any [formula] is an eigenvalue, while the third theorem claims the existence of a constant λ* > 0 such that every λ ∈ [λ*∞) is an eigenvalue of the problem.

11

Eigen value approach to two dimensional problem in generalized magneto micropolar thermoelastic medium with rotation effect

Singh R., Kumar V.

International Journal of Applied Mechanics and Engineering

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2016

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Vol. 21, no. 1

205--219

EN

In this study an eigen value approach has been employed to examine the mechanical force applied along with a transverse magnetic field in a two dimensional generalized magneto micropolar thermoelastic infinite space. Results have been obtained by treating rotational velocity to be invariant. Integral transforms have been applied to solve the system of partial differential equations. Components of displacement, normal stress, tangential couple stress, temperature distribution, electric field and magnetic field have been obtained in the transformed domain. Finally numerical inversion technique has been used to invert the result in the physical domain. Graphical analysis has been done to described the study.

12

Analysis of micropolar porous thermoelastic circular plate by eigenvalue approach

Kumar R., Miglani A., Rani R.

Archives of Mechanics

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2016

|

Vol. 68, nr 6

423--439

EN

The present paper examined a two-dimensional axi-symmetric problem of thick circular plate in a micropolar porous thermoelastic medium due to thermomechanical sources. An eigenvalue approach has been employed after applying the Laplace and Hankel transforms to investigate the problem. The expressions of displacements, stresses, microrotation, volume fraction field and temperature distribution are obtained in the transformed domain. A numerical inversion technique has been used to obtain the resulting quantities in the physical domain. The numerical simulated resulting quantities are shown graphically to depict the effects of thermal forces and porosity. Particular cases of interest are also studied and presented.

13

Problematyka stabilności kątowej morskich systemów elektroenergetycznych

Robak S., Piekarz M.

Przegląd Elektrotechniczny

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2015

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R. 91, nr 10

278-283

PL

W artykule zostały omówione zagadnienia dotyczące morskich systemów elektroenergetycznych ze szczególnym uwzględnieniem kwestii stabilności kątowej. Omówiono zasadnicze elementy wchodzące w skład typowych morskich systemów elektroenergetycznych: źródła wytwórcze, odbiory mocy, sieci elektroenergetyczne. Przedstawiono główne czynniki mające wpływ na stabilność morskich systemów elektroenergetycznych. Rozważanie teoretyczne uzupełniono o przykładowe wyniki badań stabilności kątowej lokalnej, bazujące na analizie wartości własnych.

EN

The paper discusses the issues related to the offshore power systems, particularly regarding to rotor angle stability. Main offshore power systems components: generating units, power load, transmission network, have been described and characterized. The paper shows the important factors that affecting stability of the offshore power systems. Theoretical considerations were supplemented with results of rotor angle stability study based on eigenvalues analysis.

14

Determination of Polish Power System model state matrix eigenvalues based on angular speed waveforms

Pruski P., Paszek S.

Przegląd Elektrotechniczny

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2015

|

R. 91, nr 7

21-23

EN

In the paper there are presented the calculation results of the Polish Power System (PPS) state matrix eigenvalues associated with electromechanical phenomena (i.e. electromechanical eigenvalues). These eigenvalues were calculated on the basis of the analysis of generating units angular speed waveforms simulated with the use of a 57-machine PPS model. The method for calculations of electromechanical eigenvalues consists in approximation of these waveforms by the waveforms recovered from searched eigenvalues and their participation factors.

PL

W artykule przedstawiono wyniki obliczeń wartości własnych macierzy stanu Krajowego Systemu Elektroenergetycznego (KSE) związanych ze zjawiskami elektromechanicznymi (tzn. elektromechanicznych wartości własnych) na podstawie analizy przebiegów zakłóceniowych prędkości kątowej zespołów wytwórczych uzyskanych przy użyciu 57- maszynowego modelu KSE. Wykorzystana metoda obliczeń polega na aproksymacji przebiegów zakłóceniowych za pomocą przebiegów wyznaczonych na podstawie z poszukiwanych wartości własnych.

15

Evolution and monotonicity of the first eigenvalue of p-Laplacian under the Ricci-harmonic flow

Abolarinwa A.

Journal of Applied Analysis

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2015

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Vol. 21, nr 2

147--160

EN

We study the evolution and monotonicity of the eigenvalues of p-Laplace operator on an m-dimen-sional compact Riemannian manifold M whose metric g(t) evolves by the Ricci-harmonic flow. The first nonzero eigenvalue is proved to be monotonically nondecreasing along the flow and differentiable almost everywhere. As a corollary, we recover the corresponding results for the usual Laplace-Beltrami operator when p = 2. We also examine the evolution and monotonicity under volume preserving flow and it turns out that the first eigenvalue is not monotone in general.

16

Response due to impulsive force in generalized thermomicrostretch elastic solid

Kumar V., Singh R.

International Journal of Applied Mechanics and Engineering

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2015

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Vol. 20, no. 3

487--502

EN

A two dimensional Cartesian model of a generalized thermo-microstretch elastic solid subjected to impulsive force has been studied. The eigen value approach is employed after applying the Laplace and Fourier transforms on the field equations for L-S and G-L model of the plain strain problem. The integral transforms have been inverted into physical domain numerically and components of normal displacement, normal force stress, couple stress and microstress have been illustrated graphically.

17

Optimization of 3D local orientation map calculation in the Matlab framework

Al Darwich R., Babout L.

Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska

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2015

|

nr 2

22--24

EN

This paper presents the development and evaluation of a new approach toward the optimization of 3D local orientation map calculation in the Matlab framework. This new approach can be detailed as: optimize eigenvector calculation for PCA analysis of X-ray micro tomography images of lamellar Titanium alloys image. We use two different methods to find the eigenvector of the largest eigenvalue and compare them with the Matlab built-in function (eigs). The results show a steep decrease of the calculation time using the authors' method compared to the Matlab built-in function.

PL

W artykule przedstawiono rozwój i ocenę nowego podejścia dotyczącego optymalizacji obliczeń 3D lokalnych orientacji map w środowiska Matlab. Zastosowano dwie różne metody wyznaczania wektora własnego największej wartości własnej. Wyniki są porównywane z wynikami otrzymanymi przy pomocy wbudowanych w pakiecie Matlab funkcji wyznaczające wektory i wartości własne. Wyniki porównania pokazują redukcję czasu obliczeń przy użyciu autorskiej metody w stosunku do funkcji wbudowanej w Matlab.

18

The influence of internal and constructional supports damping on the gamma-type frame vibrations

Sochacki W., Rosikoń P., Topczewska S.

Vibrations in Physical Systems

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2014

|

Vol. 26

257--264

EN

The paper presents the formulation and solution of Γ-type frame damping vibration. The physical system model takes into account the energy dissipation of the vibrating frame due to the internal vibration damping of the viscoelastic frame material and the constructional damping in the place of frame bolt support. As the results of the problem solution, the damping and system geometry effects on the first frame eigenvalue (damped frequencies and coefficients of amplitude decay factor) were presented.

19

Asymptotics of the discrete spectrum for complex Jacobi matrices

Malejki M.

Opuscula Mathematica

|

2014

|

Vol. 34, no. 1

139--160

EN

The spectral properties and the asymptotic behaviour of the discrete spectrum for a special class of infinite tridiagonal matrices are given. We derive the asymptotic formulae for eigenvalues of unbounded complex Jacobi matrices acting in l2(N).

20

Response due to concentrated force in micropolar elastic solid with voids

Singh R., Singh K.

International Journal of Applied Mechanics and Engineering

|

2014

|

Vol. 19, no. 4

755--769

EN

The eigen value approach, following Laplace and Fourier transforms has been employed to find the general solution of the field equation in a micropolar elastic solid with voids for the plane strain problem. An application of an infinite space with impulsive force has been taken to illustrate the utility of the approach. The integral transformations have been inverted by using a numerical inversion technique to get result in physical domain. The result in the form of normal displacement, volume fraction, normal force stress, tangential force stress and tangential couple stress components has been obtained numerically and illustrated graphically to depict the effect of micropolarity and voids.