Institute of Mathematics AS CR, v. v. i.
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Rubik's Cube as you don't know it
No full textsummary:Článek je určený k pohledu na Rubikovu kostku okem matematika a také k přiblížení populárních metod pro její skládání. Při čtení je vhodné mít Rubikovu kostku po ruce nebo využít online simulátor na webu alg.cubing.net. Umět již kostku složit není potřeba
The Wold-type decomposition and the kernel condition for quasi-isometries
No full textsummary:This paper investigates the necessary and sufficient conditions under which a quasi-isometry on a Hilbert space admits a Wold-type decomposition in Shimorin's sense. We establish a close connection between this decomposition and the kernel condition , where is the kernel of the adjoint operator of . Additionally, we discuss conditions related to certain cyclic and wandering subspaces, as well as the role of the Cauchy dual operator of . Furthermore, we examine operators similar to contractions, that admit quasi-isometric liftings satisfying the kernel condition. This analysis leads to the identification of a special class of quasicontractions with such liftings, and on the other hand, to the construction of certain expansive quasi-isometric liftings (
A new numerical method for solving neuro-cognitive models via Chebyshev deep neural network (CDNN)
No full textsummary:One of the fundamental applications of artificial neural networks is solving Partial Differential Equations (PDEs) which has been considered in this paper. We have created an effective method by combining the spectral methods and multi-layer perceptron to solve Generalized Fitzhugh-Nagumo (GFHN) equation. In this method, we have used Chebyshev polynomials as activation functions of the multi-layer perceptron. In order to solve PDEs, independent variables, which are collocation points, have been used as input dataset. Furthermore, the loss function has been constructed from the residual of the equation and its boundary condition. Minimizing the loss function has adjusted the appropriate values for the parameters of the network. Hence, the network has shown an outstanding performance not only on the training dataset but also on the unseen data. Some numerical examples and a comparison between the results of our proposed method and other existing approaches have been provided to show the efficiency and accuracy of the proposed method. For this purpose different cases such as linear, nonlinear and multi dimensional equations are considered
Euler–Mascheroni constant and rounding of multiplicative inverses
No full textsummary:V tomto článku se čtenář dozví hned několik zajímavých věcí. Jako první se seznámí s korektním zavedením Eulerova čísla jakožto limity jisté posloupnosti. Poté ukážeme vztah přirozeného logaritmu a částečných součtů harmonické řady, které jsou propojeny jistou \uv{magickou} matematickou konstantou, jež se objevuje ve spoustě jiných problémů. Následně tuto znalost využijeme k sečtení harmonické řady se střídavými znaménky. To nám nakonec, možná trochu překvapivě, umožní vyřešit problém z oblasti teorie pravděpodobnosti týkající se převrácených hodnot čísel
Uniqueness results for differential polynomials sharing a set
Full text linksummary:We investigate the uniqueness results of meromorphic functions if differential polynomials of the form and share a set counting multiplicities or ignoring multiplicities, where is a polynomial of one variable. We give suitable conditions on the degree of and on the number of zeros and the multiplicities of the zeros of . The results of the paper generalize some results due to T. T. H. An and N. V. Phuong (2017) and that of N. V. Phuong (2021)
Two-step Ulm-Chebyshev-like method for inverse singular value problems with multiple singular values
No full textsummary:We study the convergence of two-step Ulm-Chebyshev-like method for solving the inverse singular value problems. We focus on the case when the given singular values are positive and multiple. This work extends the result of W. Ma (2022). We show that the new method is cubically convergent. Moreover, numerical experiments are given in the last section, which show that the proposed method is practical and efficient
A Diophantine equation involving one Linnik prime
No full textsummary:Let denote the integral part of the real number We prove that for , the Diophantine equation is solvable in prime variables , , , , such that with integers and for sufficiently large integer , and we also establish the corresponding asymptotic formula. This result constitutes a refinement upon that of S. Dimitrov (2023)
Generalized semidirect sums of Lie algebras and their modules
No full textsummary:Generalized semidirect sums of Lie algebras and their modules are introduced, which are not necessarily (non)-Abelian extensions and may be applied to construct Lie algebras from modules. Some properties of generalized semidirect sums are described. In particular, it is shown that finite dimensional non-solvable Lie algebras can be realized as generalized semidirect sums. The complete classification up to isomorphism of all generalized semidirect sums of and its finite-dimensional irreducible modules is given