Institute of Mathematics AS CR, v. v. i.
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On the stability analysis of Darboux problem on both bounded and unbounded domains
No full textsummary:In this paper, we first investigate the existence and uniqueness of solution for the Darboux problem with modified argument on both bounded and unbounded domains. Then, we derive different types of the Ulam stability for the proposed problem on these domains. Finally, we present some illustrative examples to support our results
The generalized Toeplitz operators on the Fock space
No full textsummary:Let be a positive Borel measure on the complex plane and let with . We study the generalized Toeplitz operators on the Fock space . We prove that is bounded (or compact) on if and only if is a Fock-Carleson measure (or vanishing Fock-Carleson measure). Furthermore, we give a necessary and sufficient condition for to be in the Schatten -class for
Lie perfect, Lie central extension and generalization of nilpotency in multiplicative Lie algebras
No full textsummary:This paper aims to introduce and explore the concept of Lie perfect multiplicative Lie algebras, with a particular focus on their connections to the central extension theory of multiplicative Lie algebras. The primary objective is to establish and provide proof for a range of results derived from Lie perfect multiplicative Lie algebras. Furthermore, the study extends the notion of Lie nilpotency by introducing and examining the concept of local nilpotency within multiplicative Lie algebras. The paper presents an innovative adaptation of the Hirsch-Plotkin theorem specifically tailored for multiplicative Lie algebras.\looseness -
Complete monotonicity of the remainder in an asymptotic series related to the psi function
No full textsummary:Let \ with , and , and let where We establish the asymptotic expansion \mathcal {D}_{0} ( x;p,q ) \sim -\sum _{n=1}^{\infty } \frac {B_{2n} ( s ) }{2n ( x+\sigma ) ^{2n}} \quad \text {as} \^^Mx\rightarrow \infty , where stands for the Bernoulli polynomials. Further, we prove that the functions and are completely monotonic in on for every if and only if and , respectively. This not only unifies the two known results but also yields some new results
Exploring the impact of post-training rounding in regression models
No full textsummary:Post-training rounding, also known as quantization, of estimated parameters stands as a widely adopted technique for mitigating energy consumption and latency in machine learning models. This theoretical endeavor delves into the examination of the impact of rounding estimated parameters in key regression methods within the realms of statistics and machine learning. The proposed approach allows for the perturbation of parameters through an additive error with values within a specified interval. This method is elucidated through its application to linear regression and is subsequently extended to encompass radial basis function networks, multilayer perceptrons, regularization networks, and logistic regression, maintaining a consistent approach throughout
Extremal inverse eigenvalue problem for matrices described by a connected unicyclic graph
No full textsummary:In this paper, we deal with the construction of symmetric matrix whose corresponding graph is connected and unicyclic using some pre-assigned spectral data. Spectral data for the problem consist of the smallest and the largest eigenvalues of each leading principal submatrices. Inverse eigenvalue problem (IEP) with this set of spectral data is generally known as the extremal IEP. We use a standard scheme of labeling the vertices of the graph, which helps in getting a simple relation between the characteristic polynomials of each leading principal submatrix. Sufficient condition for the existence of the solution is obtained. The proof is constructive, hence provides an algorithmic procedure for finding the required matrix. Furthermore, we provide the condition under which the same problem is solvable when two particular entries of the required matrix satisfy a linear relation
Degenerate elliptic equations with variable exponents and multiple types of terms
No full textsummary:We present a comprehensive analysis of the existence and regularity of distributional solutions for a given class of nonlinear degenerate elliptic equations. The equations under consideration contain singular gradient lower order terms and are related to the data, where the exponent satisfies . To deal with this problem, we use a functional framework that includes Lebesgue--Sobolev spaces with variable exponents. Our findings provide valuable additions to the previous research discussed in Nonlinear degenerate -Laplacian equation with singular gradient and lower order term (2023) by H. Khelifi and M. A. Zouatini
Optimal error estimates for finite elements on meshes containing bands of caps
No full textsummary:In this short note we provide an optimal analysis of finite element convergence on meshes containing a so-called band of caps. These structures consist of a zig-zag arrangement of `degenerating' triangles which violate the maximum angle condition. A necessary condition on the geometry of such a structure for various -convergence rates was previously given by Kučera. Here we prove that the condition is also sufficient, providing an optimal analysis of this special case of meshes. In the special case of optimal -convergence of finite elements, the analysis states that such optimal convergence is possible if and only if the height of the band of caps is at least for some constant . Numerical experiments confirm this result
Comparison of preconditioning and deflation techniques of FETI methods for problem of 2D linear elasticity
No full textsummary:This paper deals with the basic preconditioning and deflation variants of the FETI-1 and TFETI-1 methods, with (T)FETI-1 with deflation being called (T)FETI-2. It also presents the results of numerical experiments performed on a simple benchmark 2D problem of linear elasticity to compare the computational efficiency of FETI-1 and TFETI-1 and each variant of their preconditioning or deflation in terms of number of executed CG iterations
Spherical RBF interpolation employing particular geodesic metrics and trend functions
No full textsummary:The paper is concerned with spherical radial basis function (SRBF) interpolation. We introduce particular SRBF interpolants employing several different geodesic metrics and a single trend function. Interpolation on a sphere is an important tool serving to processing data measured on the Earth's surface by satellites. Nevertheless, our model physical quantity is the magnetic susceptibility of rock measured in different directions. We construct a general SRBF formula and prove conditions sufficient for its existence. Particular formulae with specified geodesic metrics, trend and SRBFs are then constructed and tested on a series of magnetic susceptibility examples. The results show that this interpolation is sufficiently robust in general