Institute of Mathematics AS CR, v. v. i.
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A framework to combine vector-valued metrics into a scalar-metric: Application to data comparison
No full textsummary:Distance metrics are at the core of many processing and machine learning algorithms. In many contexts, it is useful to compute the distance between data using multiple criteria. This naturally leads to consider vector-valued metrics, in which the distance is no longer a real positive number but a vector. In this paper, we propose a principled way to combine several metrics into either a scalar-valued or vector-valued metric. We illustrate our framework by reformulating the popular structural similarity (SSIM) index and a simple case of the Wasserstein distance used for optimal transport
Multiscale homogenization of nonlinear hyperbolic-parabolic equations
No full textsummary:The main purpose of the present paper is to study the asymptotic behavior (when ) of the solution related to a nonlinear hyperbolic-parabolic problem given in a periodically heterogeneous domain with multiple spatial scales and one temporal scale. Under certain assumptions on the problem's coefficients and based on a priori estimates and compactness results, we establish homogenization results by using the multiscale convergence method
TPM: Transition probability matrix - Graph structural feature based embedding
Get PDFsummary:In this work, Transition Probability Matrix (TPM) is proposed as a new method for extracting the features of nodes in the graph. The proposed method uses random walks to capture the connectivity structure of a node's close neighborhood. The information obtained from random walks is converted to anonymous walks to extract the topological features of nodes. In the embedding process of nodes, anonymous walks are used since they capture the topological similarities of connectivities better than random walks. Therefore the obtained embedding vectors have richer information about the underlying connectivity structure. The method is applied to node classification and link prediction tasks. The performance of the proposed algorithm is superior to the state-of-the-art algorithms in the recent literature. Moreover, the extracted information about the connectivity structure of similar networks is used to link prediction and node classification tasks for a completely new graph
Consensus of multi-agent systems and stabilization of large-scale systems with time delays and nonlinearities - a comparison of both problems
Get PDFsummary:The problem of stabilization of large-scale systems and the consensus problem of multi-agent systems are related, similar tools for their solution are used. Therefore, they are occasionally confused. Although both problems show similar features, one can also observe important differences. A comparison of both problems is presented in this paper. In both cases, attention is paid to the explanation of the effects of the time delays. The most important fact is that, if the time delays are heterogeneous, full synchronization of the multi-agent systems cannot be achieved; however, stabilization of the large-scale network is reachable. In the case of nonlinear systems, we show that the stabilization of a large-scale nonlinear system is possible under more restrictive assumptions compared to the synchronization of a nonlinear multi-agent system
Teaching material “Varied word problems” as a tool for developing pupils‘ ability to solve word problems
No full textsummary:The paper presents the materials called Varied word problems, whose aim is to develop problem-solving through the student's deeper understanding of the internal structure of word problems. Varied word problems consist of one basic task and two/three variations of the same context and different mathematical structures
Stratified Random Sampling and Prediction of Election Results
No full textsummary:Koncept stratifikovaného náhodného výběru představuje užitečnou alternativu prostého náhodného výběru v situaci, kdy je zkoumaná populace složena z několika částí s různými vlastnostmi. Je tak možné například při stejném rozsahu výběru získat odhad průměru s menším rozptylem. V tomto příspěvku si představíme základy teorie stratifikovaných náhodných výběrů a na příkladu druhého kola prezidentských voleb 2023 si ukážeme jejich praktické použití
Stability result for a thermoelastic Bresse system with delay term in the internal feedback
Get PDFsummary:The studies considered here are concerend with a linear thermoelastic Bresse system with delay term in the feedback. The heat conduction is also given by Cattaneo's law. Under an appropriate assumption between the weight of the delay and the weight of the damping, we prove the well-posedness of the problem using the semigroup method. Furthermore, based on the energy method, we establish an exponential stability result depending of a condition on the constants of the system that was first considered by A. Keddi, T. Apalara, S. A. Messaoudi in 2018
On the average behavior of the Fourier coefficients of th symmetric power -function over certain sequences of positive integers
No full textsummary:We investigate the average behavior of the th normalized Fourier coefficients of the th ( be any fixed integer) symmetric power -function (i.e., ), attached to a primitive holomorphic cusp form of weight for the full modular group over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum where is sufficiently large, and When , the error term which we obtain improves the earlier known result
Eventually positive elements in ordered Banach algebras
No full textsummary:In ordered Banach algebras, we introduce eventually and asymptotically positive elements. We give conditions for the following spectral properties: the spectral radius belongs to the spectrum (Perron--Frobenius property); the spectral radius is the only element in the peripheral spectrum; there are positive (approximate) eigenvectors for the spectral radius. Recently such types of results have been shown for operators on Banach lattices. Our results can be viewed as a complement, since our structural assumptions on the ordered Banach algebra are much weaker