Institute of Mathematics AS CR, v. v. i.
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Bertrand partner trajectories related to PAFORS
Full text linksummary:In this study, we consider the concept of Bertrand partner trajectories related to Positional Adapted Frame on Regular Surfaces (shortly PAFORS) for the particles moving on the different regular surfaces in Euclidean 3-space. The relations between the PAFORS elements of the aforesaid trajectories are given. Also, the relations between Darboux basis vectors of them are found. Furthermore, some characterizations are given for some special cases of these trajectories with the aid of their PAFORS elements
A roller coaster approach to integration and Peano's existence theorem
No full textsummary:This is a didactic proposal on how to introduce the Newton integral in just three or four sessions in elementary courses. Our motivation for this paper were Talvila's work on the continuous primitive integral and Koliha's general approach to the Newton integral. We introduce it independently of any other integration theory, so some basic results require somewhat nonstandard proofs. As an instance, showing that continuous functions on compact intervals are Newton integrable (or, equivalently, that they have primitives) cannot lean on indefinite Riemann integrals. Remarkably, there is a very old proof (without integrals) of a more general result, and it is precisely that of Peano's existence theorem for continuous nonlinear ODEs, published in 1886. Some elements in Peano's original proof lack rigor, and that is why his proof has been criticized and revised several times. However, modern proofs are based on integration and do not use Peano's original ideas. In this note we provide an updated correct version of Peano's original proof, which obviously contains the proof that continuous functions have primitives, and it is also worthy of remark because it does not use the Ascoli-Arzelà theorem, uniform continuity, or any integration theory
Stieltjes differential problems with general boundary value conditions. Existence and bounds of solutions
No full textsummary:We are concerned with first order set-valued problems with very general boundary value conditions \begin{cases} u'_g(t)\in F(t,u(t)),\quad \mu _g \text {-a.e.} \t\in [0,T] , \\ L(u(0),\u(T))=0 \end{cases} involving the Stieltjes derivative with respect to a left-continuous nondecreasing function , a Carathéodory multifunction and a continuous . Using appropriate notions of lower and upper solutions, we prove the existence of solutions via a fixed point result for condensing mappings. In the periodic single-valued case, combining an existence theory for the linear case with a recent result involving lower and upper solutions (which can be seen as a consequence of our existence theorem mentioned before), we get not only the existence of solutions, but also lower and upper bounds, respectively, by imposing an estimation for the right-hand side
Elementary construction of Hölder functions such that the Kurzweil-Stieltjes integral does not exist
No full textsummary:For any with we provide a simple construction of an -Hölde function and a -Hölder function such that the integral fails to exist even in the Kurzweil-Stieltjes sense
On -expansive tuples of commuting operators on a Banach space
No full textsummary:We present -expansive tuples of commuting operators in a complex Banach space, expanding upon the concept of -isometric tuples. We provide a characterization of the joint approximate point spectrum of these tuples. Furthermore, we investigate a multivariable extension of these single-variable -expansive operators discussed in M. Chō, I. Hur, J. E. Lee (2024) and delve into several fundamental properties associated with them
On the existence of nontrivial solutions for modified fractional Schrödinger-Poisson systems via perturbation method
No full textsummary:The existence of nontrivial solutions is considered for the fractional Schrödinger-Poisson system with double quasi-linear terms: where is the fractional Laplacian for , with . Under assumptions on and , we prove the existence of positive solutions and negative solutions for the above system by using perturbation method and the mountain pass theorem
Fuzzy clustering of fuzzy data considering the shape of the membership functions using a novel representation learning technique
No full textsummary:Most existing distance measures for fuzzy data do not capture differences in the shapes of the left and right tails of membership functions. As a result, they may calculate a distance of zero between fuzzy data even when these differences exist. Additionally, some distance measures cannot compute distances between fuzzy data when their membership functions differ in type. In this paper, inspired by human visual perception, we propose a fuzzy clustering method for fuzzy data using a novel representation technique that is capable of detecting small differences in the shapes of the left and right tails of membership functions. Moreover, it effectively clusters fuzzy data even when their membership functions differ in type. By utilizing the pre-trained ResNet50 network as a feature extractor and applying the FCM clustering method to the output from the last convolutional layer, our approach achieves high accuracy in clustering both synthetic and real data sets. Experimental results demonstrate that our method achieved a Rand Index of 0.9965, outperforming state-of-the-art methods, making it particularly suitable for applications that require high clustering accuracy
Possibilities of developing pupils' logical thinking in mathematics lessons – Part 2
No full textsummary:V článku se věnujeme druhé ze čtyř kategorií logických úloh – úlohám založeným na uplatnění vztahů. Tuto kategorii reprezentujeme pomocí sedmnácti ukázkových úloh, které prošli pilotním testováním v hodinách matematiky. Důležité postřehy či náměty z tohoto testování zahrnujeme do krátkých komentářů k jednotlivým úlohám.summary:In this article, we focus on the second of four categories of logic tasks – tasks based on the application of relationships. This category is represented by seventeen sample tasks, which underwent pilot testing in mathematics lessons. Key insights and suggestions from this testing are included in brief comments accompanying the individual tasks
How about a fractal? – Part 2
No full textsummary:The article begins by briefly answering the question of what a fractal is and provides reasons why fractals are worth exploring in education. Various methods of constructing fractals suitable for classroom use are then presented. Fractals can be easily integrated into the curriculum as early as upper primary school, either as part of a comprehensive course or as individual thematic units. Constructions are primarily created using GeoGebra, but also through traditional methods (pencil, paper) or other digital tools (PowerPoint, web applications). The conclusion demonstrates that the concept of fractal dimension can be accessible to high school students