Institute of Mathematics AS CR, v. v. i.
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Symmetric interior penalty discontinuous Galerkin method for nonlinear fully coupled quasi-static thermo-poroelasticity problems
No full textsummary:We propose a symmetric interior penalty discontinuous Galerkin (DG) method for nonlinear fully coupled quasi-static thermo-poroelasticity problems. Firstly, a fully implicit nonlinear discrete scheme is constructed by adopting the DG method for the spatial approximation and the backward Euler method for the temporal discretization. Subsequently, the existence and uniqueness of the solution of the numerical scheme is proved, and then we derive the a priori error estimate for the three variables, i.e., the displacement, the pressure and the temperature. Lastly, we carry out numerical experiments to confirm the theoretical findings of our suggested approach
Solvability of (max,+) and (min,+)-equation systems
Get PDFsummary:Properties of (max,+)-linear and (min,+)-linear equation systems are used to study solvability of the systems. Solvability conditions of the systems are investigated. Both one-sided and two-sided systems are studied. Solvability of one class of (max,+)-nonlinear problems will be investigated. Small numerical examples illustrate the theoretical results
Efficient Karhunen-Loève expansions via Legendre-Galerkin discretization and tensor structure
No full textsummary:We develop an efficient framework for Karhunen-Loève expansions of isotropic Gaussian random fields on hyper-rectangular domains. The approach approximates the covariance kernel by a positive mixture of squared-exponentials, fitted via Newton optimization with a theoretically informed initialization; we also provide convergence estimates for this Gaussian-mixture approximation. The resulting separable kernel enables a Legendre-Galerkin discretization with a Kronecker product structure across dimensions, together with submatrices exhibiting even/odd parity. For assembly, we employ a Duffy-type transformation followed by Gaussian quadrature. These structural properties substantially reduce memory usage and arithmetic cost compared with naive formulations. All algorithms and numerical experiments are released in an open-source repository that reproduces every figure and table. For completeness, a concise derivation of the three-term recurrence for Legendre polynomials is included in appendix
Mittag-Leffler modules and definable subcategories II
No full textsummary:A countably generated module is Mittag-Leffler if and only if it is pure-projective, i.e., a direct summand of a direct sum of finitely presented modules. Trying to generalize this description to countably generated relative Mittag-Leffler modules, one runs into serious obstacles. The last theorem of Part I of the same title describes them as what was called uniform relative pure epimorphic images of a specific kind of -limits of finitely presented modules. A second part of this theorem attempted to make the -limits in question more concrete. In the formulation the application of those uniform pure epimorphisms was erroneously omitted. Correcting this led to a better result to be presented here. It states that under a mild assumption on the context, every countably generated relative Mittag-Leffler module `in the context' is a direct summand of a certain preenvelope of a union of a relatively pure -chain of finitely presented modules. In conclusion a number of examples are presented that start with and grew out of the study of -purity of monomorphisms in -Mod for , the definable subcategory of divisible Abelian groups. Rings that get particular attention in this are RD-rings
Remarks on retracts of polynomial rings in three variables in any characteristic
No full textsummary:Let be a retract of the polynomial ring in three variables over a field . It is known that if or , then is a polynomial ring. We give some sufficient conditions for to be the polynomial ring in two variables over when and
An Unexpected Encounter with Euler’s Number
No full textsummary:Článek pojednává o případech, ve kterých se vyskytuje Eulerovo číslo. Jedná se o situace, ve kterých hledáme specifické společné body grafů exponenciální a logaritmické funkce. Dále se zabýváme úlohou dělení daného čísla tak, aby součin sčítanců byl maximální možný. Poslední část se věnuje chování tzv. mocninných věží.summary:The article discusses the cases where Euler's constant appears. These are situations in which we are looking for specific common points of the graphs of exponential and logarithmic functions. The next part deals with the problem of how to divide a given number into individual summands so that their product is as large as possible. The last part is devoted to the behavior of the so-called power towers
Anděl na útěku
No full textsummary:V tomto článku se zabýváme hrou dvou hráčů na nekonečné šachovnici. Jeden z hráčů (ďábel) odebírá pole a snaží se soupeře polapit, zatímco druhý hráč (anděl) se pohybuje dle předepsaných pravidel a snaží se do nekonečna unikat. Je známo, že anděl dokáže unikat, je-li jeho předem stanovená rychlost dostatečná, a je naopak chycen, když je jeho rychlost příliš malá
New equivalent conditions for Hardy-type inequalities
Get PDFsummary:We consider a Hardy-type inequality with Oinarov's kernel in weighted Lebesgue spaces. We give new equivalent conditions for satisfying the inequality, and provide lower and upper estimates for its best constant. The findings are crucial in the study of oscillation and non-oscillation properties of differential equation solutions, as well as spectral properties