Institute of Mathematics AS CR, v. v. i.
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Explicit control law for a heat equation based on output tracking
No full textsummary:Boundary control to track the output reference of the heat equation is considered. A control input is implemented at one boundary while requiring the other boundary to track the output reference. By introducing the error system and backstepping transformation, the control law is designed. The undetermined coefficient method and two identities are used to obtain the analytical solution of the kernel function by complex mathematical calculation. This establishes an explicit control law and ensures that the error system can effectively achieve the desired closed-loop stability. Simulation results validate the proposed theoretical results
An adaptive mesh refinement scheme for hierarchical hybrid grids
No full textsummary:This work introduces an adaptive mesh refinement technique for hierarchical hybrid grids with the goal to reach scalability and maintain excellent performance on massively parallel computer systems. On the block-structured hierarchical hybrid grids, this is accomplished by using classical, unstructured refinement only on the coarsest level of the hierarchy, while keeping the number of structured refinement levels constant over the whole domain. This leads to a compromise, where the excellent performance characteristics of hierarchical hybrid grids can be maintained at the price that the flexibility of generating locally refined meshes is constrained. Furthermore, the mesh adaptivity often relies on a posteriori error estimators or error indicators, which tend to become computationally expensive. Again, with the goal of preserving scalability and performance, a method is proposed that leverages the grid hierarchy and the full multigrid scheme. Utilizing the sequence of approximations on the nested hierarchy of grids permits the computation of a cheap error estimator that is well-suited for large-scale parallel computing. We present the theoretical foundations for both global and local error estimates, and present a rigorous analysis of their effectivity. The proposed method, including the error estimator and the adaptive coarse grid refinement, is implemented in the finite element framework HyTeG. Extensive numerical experiments are conducted to validate the effectiveness, as well as performance and scalability
On classes of groups characterized by classes of lattices
No full textsummary:Classes of groups are identified and characterized in lattice-theoretic terms, i.e., by common properties of the weak congruence lattices of groups in the class. In particular, the necessary and sufficient condition for a class of groups to be a variety has been given in terms of the lattice of weak congruences of groups
rings
No full textsummary:We intend to unviel a new class of rings, called rings, if the units of a ring equal the sum of 1 and an element from . Recall that is a subset of , not necessarily a subring, which equals for some . Both and rings are rings credited to the fact that nilpotents and are subsets of . The properties exhibited by rings are explored in a thorough manner following which its relations with other rings are observed. For instance, rings are and no matrix ring, when is . We have focused on extensions of rings like , , , Morita context and group rings
Semiclean and quasi-clean elements in rings
No full textsummary:We prove uniquely semiclean rings are reduced and hence Abelian. The semiclean elements of , , and are compared. In particular, we show that the indeterminate is never semiclean. The 2-primality of a ring is characterized via the semiclean elements of . We also consider the quasi-clean elements of a ring and compare them with the semiclean elements of
Some Didactic Approaches to Euler's Number e in Secondary-School Mathematics Teaching
No full textsummary:Článek shrnuje některé didaktické přístupy k výuce Eulerova čísla e na středních školách. Kromě transmisivního a formálně algebraického přístupu k výuce tohoto obtížného abstraktního matematického pojmu ukazujeme také konstruktivisticko-heuristický přístup, který se může opírat o reálnou situaci, vizualizace či experimenty a simulace. Do výuky je možné doporučit takovou didaktickou transformaci učiva o čísle e, která bude odpovídat učebním schopnostem studentů.summary:The article summarizes some didactic approaches to teaching Euler's number e in secondary schools. In addition to the transmissive and formally algebraic approach to teaching this difficult abstract mathematical concept, we also show a constructivist-heuristic approach, which can be based on a real situation, visualizations or experiments and simulations. It is possible to recommend such a didactic transformation of the subject matter about the number e that will correspond to the learning abilities of students
Exponential and Logarithmic Functions
No full textsummary:Článek zkoumá exponenciální a logaritmické funkce, jejich matematický význam a různé aplikace. Diskutuje jejich historický vývoj, význam v matematice a přírodních vědách, řešení rovnic a grafické znázornění. Mezi praktické aplikace patří růst populace, radioaktivní rozpad, stupnice pH a Richterova stupnice. Článek také propojuje tyto funkce s moderními technologiemi, jako jsou algoritmy, šifrování a vizualizace dat. V neposlední řadě vyzdvihuje jejich přínos k vědeckému pochopení přírodních jevů.summary:The article explores exponential and logarithmic functions, their mathematical significance, and various applications. It discusses their historical development, importance in mathematics and natural sciences, equation solving, and graphical representation. Practical applications include population growth, radioactive decay, the pH scale, and the Richter scale. The article also connects these functions to modern technologies, such as algorithms, encryption, and data visualization. Finally, it highlights their contribution to scientific understanding of natural phenomena
ET-DMGing: Event-triggered distributed momentum-gradient tracking optimization algorithm for multi-agent systems
No full textsummary:This paper proposes an event-triggered distributed momentum-gradient tracking optimization algorithm (ET-DMGing) for the collaborative optimization problem of minimizing the sum of all agents' local objective functions in multi-agent systems. Firstly, gradient tracking is employed to precisely track the average momentum gradient for updating agent states, which effectively reduces their dwell time in flat and oscillatory regions. The proposed ET-DMGing exhibits enhanced directional consistency and dynamic stability during optimization by leveraging momentum accumulation effects, achieving a linear convergence rate. Secondly, a new event-triggered condition is proposed, which considers the dual metrics of state error and momentum gradient error. This allows for a more comprehensive assessment of the agents' triggering needs, avoiding instability caused by single-dimensional triggering, and improving the triggering threshold. This event-triggered condition reduces the communication frequency among agents. Thirdly, we rigorously proved that the proposed ET-DMGing converges to the global optimum at a linear convergence rate by employing the small-gain theorem. Furthermore, explicit convergence conditions have been derived for parameter selection, including step size parameters and event-triggered weighting coefficients. Finally, numerical simulations are performed to verify the effectiveness and accuracy of the theoretical results
New methods to construct uninorms by extending uninorms with closure operators and t-superconorms
No full textsummary:In this paper, we provide new methods to construct uninorms by extending given uninorms on a subinterval of a bounded lattice with closure operators (resp. interior operators) and t-superconorms (resp. t-subnorms). Meanwhile, these methods for uninorms generalize some known methods for uninorms in the literature. An example is also provided to show our method