Institute of Mathematics AS CR, v. v. i.
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The Fourier transform in Lebesgue spaces
No full textsummary:For each () it is shown that the Fourier transform is the distributional derivative of a Hölder continuous function. For each , a norm is defined so that the space of Fourier transforms is isometrically isomorphic to . There is an exchange theorem and inversion in norm
Characterization of shadowing for linear autonomous delay differential equations
No full textsummary:A well-known shadowing theorem for ordinary differential equations is generalized to delay differential equations. It is shown that a linear autonomous delay differential equation is shadowable if and only if its characteristic equation has no root on the imaginary axis. The proof is based on the decomposition theory of linear delay differential equations
On products of prime element orders in finite groups
No full textsummary:Let be a finite group. The functions and denote the sum of the element orders and the sum of the prime element orders of , respectively. Significant results related to the study of these functions have been published recently. Further, the function was introduced to denote the product of the element orders of . We introduce , which denotes the product of the prime element orders of a finite group . We find a lower bound for on the set of groups of the same order and deduce a result on nilpotent groups using
New results on additive generator pairs of overlap and grouping functions
No full textsummary:In this article, we deeply reveal the relationship between functions and in an overlap function additively generated by an additive generator pair (,), which is used to characterize the conditions for an overlap function additively generated by the pair being a triangular norm by terms of functions and . We also establish the conditions that an overlap function additively generated by the additive generator pair can be obtained by a distortion of a triangular norm and a (pseudo) automorphism. Finally, we dually give the related results concerned grouping functions
A note to the division by zero
No full textsummary:Pochopení významu čísla nula a používání nuly při operacích s čísly přirozenými i racionálními je pro žáky prvního i druhého stupně základní školy důležité. Je proto nutné hledat přístupy, jak žákům učivo srozumitelně zdůvodňovat. V příspěvku je uveden postup, jak zdůvodnit dělení v případě, že dělitel je číslo nula.summary:Understanding the meaning of the number zero and the use of zero in operations with natural and rational numbers is important for first and second grade elementary school pupils. It is therefore necessary to look for approaches to explain the curriculum to pupils in a comprehensible way. The article describes a procedure for justifying the division if the divisor is the number zero
Discontinuous Galerkin method with Godunov-like numerical fluxes for traffic flows on networks. Part I: stability
No full textsummary:We study the stability of a discontinuous Galerkin (DG) method applied to the numerical solution of traffic flow problems on networks. We discretize the Lighthill-Whitham-Richards equations on each road by DG. At traffic junctions, we consider two types of numerical fluxes that are based on Godunov's numerical flux derived in a previous work of ours. These fluxes are easily constructible for any number of incoming and outgoing roads, respecting the drivers' preferences. The analysis is split into two parts: in Part I, contained in this paper, we analyze the stability of the resulting numerical scheme in the -norm. The resulting estimates allow for a linear-in-time growth of the square of the -norm of the DG solution. This is observed in numerical experiments in certain situations with traffic congestions. Next, we prove that under certain assumptions on the junction parameters (number of incoming and outgoing roads and drivers' preferences) the DG solution satisfies an entropy inequality where the square entropy is nonincreasing in time. Numerical experiments are presented. The work is complemented by the followup paper, Part II, where a maximum principle is proved for the DG scheme with limiters
Infinitely many solutions for Kirchhoff-type equations involving critical growth in Orlicz-Sobolev with negative energies
No full textsummary:We investigate a class of Kirchhoff-type equations characterized by critical growth within Orlicz-Sobolev spaces. The main result establishes the existence of infinitely many solutions with negative energy. Using an adapted concentration-compactness principle and advanced variational methods, we overcome key challenges such as non-compactness and non-differentiability to the associated functionals. This work extends existing results to more general functional spaces, offering new insights into nonlocal nonlinear equations
Hausdorff dimension of some exceptional sets in Lüroth expansions
No full textsummary:We study the metrical theory of the growth rate of digits in Lüroth expansions. More precisely, for , let denote the Lüroth expansion of . We completely determine the Hausdorff dimension of the sets and where is an arbitrary function satisfying as
Local well-posedness of solutions to 2D magnetic Prandtl model in the Prandtl-Hartmann regime
No full textsummary:We consider the 2D magnetic Prandtl equation in the Prandtl-Hartmann regime in a periodic domain and prove the local existence and uniqueness of solutions by energy methods in a polynomial weighted Sobolev space. On the one hand, we have noted that the -derivative of the pressure plays a key role in all known results on the existence and uniqueness of solutions to the Prandtl-Hartmann regime equations, in which the case of favorable or the case of (led by constant outer flow ) was only considered. While in this paper, we have no restriction on the sign of , which has generalized all previous results and definitely gives rise to a difficulty in mathematical treatments. To overcome this difficulty, we shall use the skill of cancellation mechanism which is valid under the monotonicity assumption. One the other hand, we consider the general outer flow , leading to the boundary data at being much more complicated. To deal with these boundary data, some more delicate estimates and mathematical induction method will be used. Therefore, our result also provides an extension of earlier studies by addressing the challenges arising from general outer flow