Institute of Mathematics AS CR, v. v. i.
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First- and second-order adjoint methods for stochastic identification problems
No full textsummary:We present a unified framework for estimating stochastic parameters in general variational problems. This nonlinear inverse problem is formulated as a stochastic optimization problem using the output least-squares (OLS) objective, which minimizes the discrepancy between observed data and the computed solution. A key challenge in OLS-based formulations is the efficient computation of first- and second-order derivatives of the OLS functional, which depend on the corresponding derivatives of the parameter-to-solution map often costly and difficult to evaluate, especially in stochastic settings. To address this, we develop a rigorous computational approach based on first- and second-order adjoint methods for inverse problems governed by stochastic variational problems. Specifically, we propose a new first-order adjoint method for computing the gradient of the OLS objective and introduce two novel second-order adjoint methods for Hessian evaluation. A stochastic Galerkin discretization framework is employed, enabling efficient implementation of the adjoint-based derivative computations. Numerical experiments demonstrate the accuracy and efficiency of the proposed computational framework
Lie -pseudoalgebras with Rota-Baxter -operators
No full textsummary:A Lie conformal algebra is defined as a -module ( is an indeterminate), endowed with a -linear map , satisfying axioms similar to those of Lie algebra. Then Bakalov, D'Andrea and Kac introduced the notion of Lie -pseudoalgebras by replacing the above polynomial algebra with any cocommutative Hopf algebra . We first classify solvable Lie -pseudoalgebras of rank two. Then we consider the Rota-Baxter -operators on such Lie -pseudoalgebras. Finally, we study the relationship between Rota-Baxter -operators on Lie -pseudoalgebra and Rota-Baxter operators on its annihilation algebra
Logarithm in Mathematics Teaching
No full textsummary:Článek stručně charakterizuje genetický přístup ve výuce a ukazuje, jak by bylo možné tento přístup využít v případě zavedení pojmu logaritmus ve výuce matematiky na střední škole.summary:The introduction of the article briefly characterizes the genetic principle in teaching and recalls the memorandum of American mathematics teachers from 1962. The focus of the article is an attempt to explain the concept of logarithm using the genetic principle. The explanation is divided into the following steps: Problem-Idea-Logarithm-History-Logarithmic function. These steps are briefly described
New constructions of nullnorms on bounded trellises
No full textsummary:In this paper, we focus on the construction of nullnorms on bounded trellises. The features of the element that acts as the annihilator of a nullnorm are discussed and the relevant results show that the element acting as the annihilator must not be included in any cycle. Drawing upon this revelation, we propose some new methods for constructing nullnorms on bounded trellises, which are different from those given by Xiu et al. Additionally, some illustrative examples are provided to facilitate a more comprehensive understanding
Schofield sequences for weighted projective lines of type
No full textsummary:We focus on the Schofield sequences over weighted projective lines in the sense of Geigle and Lenzing. We give a classification of all the Schofield sequences whose middle terms are line bundles or extension bundles for any weight type . In particular, for the latter case such Schofield sequences are related to the distinguished exact structure on the subcategory of vector bundles. More precisely, the projective covers and injective hulls of extension bundles will provide Schofield sequences, and all the Schofield sequences in the subcategory of vector bundles can be obtained in this way
Commuting Toeplitz operators with harmonic symbols on the Fock-Sobolev space
No full textsummary:We make a progress towards describing the commuting Toeplitz operators with harmonic symbols on the Fock-Sobolev space. For the certain symbol space, we obtain two Toeplitz operators with harmonic symbols commuting only in the obvious cases, which is different from the known result in the Fock space
Finite groups with a small number of cyclic subgroups
No full textsummary:A finite group is called an -cyclic group if it has exactly cyclic subgroups (including the identity subgroup). For , the -cyclic groups have been classified in a series of papers. We push the above research work further to classify the finite \hbox {13-cyclic} groups, which could be considered as a step to answer the open problem posed by M. Tărnăuceanu (2015). The detailed structure of many groups of ``small'' orders is also analyzed. The following main theorem is proved: Let be a finite 13-cyclic group. Then , and one of the following holds: \begin {itemize} \item [(1)] , or with a prime. \item [(2)] , , , , , , , or . \end {itemize
Constructing mixed uninorms on bounded lattices
Full text linksummary:In this paper, we present the definition of mixed uninorms and propose several methods for constructing two special classes of mixed uninorms on bounded lattices through t-subnorms and t-superconorms. These methods generalize and on bounded lattices that have been previously discussed in the literature. Some examples are given to construct mixed uninorms on bounded lattices