Institute of Mathematics AS CR, v. v. i.
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On the class of positive disjoint weak -convergent operators
Get PDFsummary:We introduce and study the disjoint weak -convergent operators in Banach lattices, and we give a characterization of it in terms of sequences in the positive cones. As an application, we derive the domination and the duality properties of the class of positive disjoint weak -convergent operators. Next, we examine the relationship between disjoint weak -convergent operators and disjoint -convergent operators. Finally, we characterize order bounded disjoint weak -convergent operators in terms of sequences in Banach lattices
On the lattice of pronormal subgroups of dicyclic, alternating and symmetric groups
Get PDFsummary:In this paper, the structures of collection of pronormal subgroups of dicyclic, symmetric and alternating groups are studied in respect of formation of lattices and sublattices of . It is proved that the collections of all pronormal subgroups of and S do not form sublattices of respective and , whereas the collection of all pronormal subgroups of a dicyclic group is a sublattice of . Furthermore, it is shown that and ) are lower semimodular lattices
A note on linear derivations
No full textsummary:At first we prove some results on a general polynomial derivation using few results of linear derivation. Then we study the ring of constants of a linear derivation for some rings. We know that any linear derivation is a nonsimple derivation. In the last section we find the smallest integer such that the polynomial ring in variables is -differentially simple, all derivations are nonsimple and the derivations set contains a linear derivation
Cotorsion pairs in comma categories
No full textsummary:Let and be abelian categories with enough projective and injective objects, and a left exact additive functor. Then one has a comma category . It is shown that if is -exact, then is a (hereditary) cotorsion pair in and ) is a (hereditary) cotorsion pair in if and only if is a (hereditary) cotorsion pair in and and are closed under extensions. Furthermore, we characterize when special preenveloping classes in abelian categories and can induce special preenveloping classes in
The operation and operation of Cohen-Macaulay bipartite graphs
No full textsummary:Let be a finite simple graph with the vertex set and let be its edge ideal in the polynomial ring . We compute the depth and the Castelnuovo-Mumford regularity of when or is a graph obtained from Cohen-Macaulay bipartite graphs , by the operation or operation, respectively
A remark on a Diophantine equation of S. S. Pillai
No full textsummary:S. S. Pillai proved that for a fixed positive integer , the exponential Diophantine equation , , has only finitely many solutions in integers and . We prove that when is of the form , the above equation has no solution in integers and with
Mathematizing student thinking: connecting problem solving to everyday life and building capable and confident math learners
No full textsummary:Recenzia knihy sa venuje postupom riešenia matematických problémových typov úloh v kontexte skutočného života. Približuje: 1. úlohu učiteľa a žiaka pri riešení matematických úloh; 2. dve kategórie problémových typov úloh; 3. postup pri riešení úloh; 4. modifikovaný postup - matematické modelovanie; 4. hodnotenie vo vyučovaní
A note on the shift theorem for the Laplacian in polygonal domains
No full textsummary:We present a shift theorem for solutions of the Poisson equation in a finite planar cone (and hence also on plane polygons) for Dirichlet, Neumann, and mixed boundary conditions. The range in which the shift theorem holds depends on the angle of the cone. For the right endpoint of the range, the shift theorem is described in terms of Besov spaces rather than Sobolev spaces
Exponential expressivity of neural networks on Gevrey classes with point singularities
No full textsummary:We analyze deep Neural Network emulation rates of smooth functions with point singularities in bounded, polytopal domains , . We prove exponential emulation rates in Sobolev spaces in terms of the number of neurons and in terms of the number of nonzero coefficients for Gevrey-regular solution classes defined in terms of weighted Sobolev scales in , comprising the countably-normed spaces of I. M. Babuška and B. Q. Guo. \endgraf As intermediate result, we prove that continuous, piecewise polynomial high order (``-version'') finite elements with elementwise polynomial degree on arbitrary, regular, simplicial partitions of polyhedral domains , , can be \emph {exactly emulated} by neural networks combining ReLU and ReLU activations. \endgraf On shape-regular, simplicial partitions of polytopal domains , both the number of neurons and the number of nonzero parameters are proportional to the number of degrees of freedom of the finite element space of I. M. Babuška and B. Q. Guo