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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.
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].
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
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.
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.
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.
7
Content available remote Strong stationary duality for Möbius monotone Markov chains : examples
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.
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
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.
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.
11
Content available remote Series representation of compact linear operators in Banach spaces
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.
12
Content available On small vibrations of a damped Stieltjes string
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.
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.
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.
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.
EN
Hydrological models are very useful for predictions in many ungauged basins across the world. There are many hydrological models available for discharge data gen-eration with different complexities and varied input parameter requirements. Studies have shown that models with many input parameters do not necessarily perform better than those with few input parameters. Basin morphometric parameters play significant roles in the conversion of rainfall to runoff and obtaining good estimates of these parameters for use in runoff models is sometime challenging as Inaccurate input into models can propagate errors and make the models to perform poorly. This study employs the method of principal component analysis to reduce the number of morphometric parameters required to run a runoff model without losing any major information. Parameters for five selected study basins in central Nigeria were measured and analysed. The result shows that three morphometric parameters (Fitness Ratio, Ruggedness Number and Watershed Eccentricity) can adequately represent other parameters as an input into a runoff model for the basins. This reduces significantly the time and effort needed to compute all the parameters which in actual fact may not improve the quality or efficiency of the runoff model.
17
Content available Eigenvalues of 2-tridiagonal Toeplitz matrix
EN
In this article an explicit formula for eigenvalues of a 2-tridiagonal Toeplitz matrix can be derived on the basis of a certain relation between the determinant of this matrix and the determinant of a pertinent tridiagonal matrix. It can be pointed out that the problem is investigated without imposing any conditions on the elements of matrix.
EN
This study formulates and solves the problem of transverse damped vibration in the system of changing the boom radius in a truck crane with advanced cylinder design for controlling the boom radius. The dissipation of vibration energy in the model adopted in the study occurs as a result of internal damping of the viscoelastic material (rheological Kelvin-Voigt model) of the beams that model the system and movement resistance in the supports of the cylinder and crane boom to the bodywork frame of the crane. Damped frequencies of vibrations and degree of vibration amplitude decay were calculated. The study also presents eigenvalues of system vibration with respect to changes in damping coefficients and system geometry for a selected load.
19
Content available Dynamic balance research of protected systems
EN
The dynamic models of the complex ergatic objects' behavior, presented in the form of differentia equations and their systems were studied. The stability and other properties are researched. The methods of analysis and reduce of harmful factors and their impact on people were theoretically proved. The methods of analysis and critical points removal in dynamic models of hazards distribution are offered. The object of study is the system of the harmful external factors protection. Subject of research is the system of two nonlinear differential equations as a model of technical systems with protection. The object of protection is described by logistic equation. and defense system - by non-linear differential equation with a security functions of rather general form. This paper describes critical modes analysis and stationary states’ stability of protected systems with harmful influences. Numerical solution of general problem and also the analytical solution for the case of fixed expected harmful effects have been obtained. Various types of general models for "Man-machine-environment" systems were studied. Each of describes some kind of the practically important quality of object in an appropriate way. And All together they describe the object in terms of it’s safe operation. Their further detailing process results to either well-known, or some new subsystems’ models. Systems with "fast" protection at a relatively slow dynamics of the object were studied. This leads to the models with small parameter and asymptotic solutions of differentia equations. Some estimates for protection cost in different price-functional and for different functions in the right part of equation, which describes the dynamics of defense were obtained. For calculations, analysis and graphical representations some of mathematical packages was applied.
EN
In the following article we will try to find the dependence between the location of imperfections in a closed domain and the spectrum of the Laplace operator for this region. In the theoretical part we will define the spectral problem which is solved by eigenvalues. These eigenvalues are dependent on location and size of the imperfection. However, we are interested in the inverse task which consists in localizing the imperfection of the domain on a basis of the spectrum of the operator.
PL
W artykule przeanalizowano zastosowanie widma operatora Laplace'a jako narzędzia do przybliżonej lokalizacji uszkodzeń w kole jednostkowym. W części teoretycznej zdefiniowano zagadnienie spektralne rozwiązywane za pomocą wartości własnych. Znalezione wartości zależą od położenia i rozmiaru uszkodzeń. W artykule został zdefiniowany problem odwrotny, który polega na znalezieniu miejsca uszkodzenia na podstawie znanego widma.
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