Institute of Mathematics AS CR, v. v. i.
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Quasi-Projection for a class of uninorms (2-uninorms)
Full text linksummary:In 2021, Jayaram et al. demonstrated that a property called Quasi-Projectivity is a necessary condition for Clifford's relation to produce a partial order. Furthermore, their research revealed that although all triangular norms and triangular conorms satisfy and thus can generate posets, their generalized operator, uninorms, does not always possess this property, resulting in not all uninorms being able to generate a poset. In this work, we first investigate the satisfaction of for uninorms with continuous underlying operators, concluding that such uninorms are capable of yielding partial orders if and only if they are locally internal in , and the resulting partially ordered set is a chain. Based on this, we further explore the performance of inducing partial orders within the framework of 2-uninorms, and the results show that it is entirely determined by the underlying uninorms
A note on Kurzweil-Henstock's anticipating non-stochastic integral
Full text linksummary:Motivated by the study of anticipating stochastic integrals using Kurzweil-Henstock approach, we use anticipating interval-point pairs (with the tag as the right-end point of the interval) in studying non-stochastic integral, which we call the Kurzweil-Henstock anticipating non-stochastic integral. We prove the integration-by-parts and integration-by-substitution results, the convergence theorems using our new setting. Using the convergence theorems, we show that the Kurzweil-Henstock's anticipating non-stochastic integral is equivalent to the Lebesgue integral
Nörlund means of the sequence of the iterates of a bounded linear operator, and spectral properties
Full text linksummary:We are concerned here with relating the spectral properties of a bounded linear operator on a Banach space to the behaviour of the means , where is a nondecreasing sequence of positive real numbers, and denotes the inverse of the automorphism on the vector space of scalar sequences which maps each sequence into the sequence of its partial sums. In a previous paper, we obtained a uniform ergodic theorem for the means above, under the hypotheses , , and for a positive integer : indeed, we proved that if converges to zero in the uniform operator topology for such a sequence , then the averages above converge in the same topology if and only if 1 is either in the resolvent set of , or a simple pole of the resolvent function of . In this paper, we prove that if , and the averages above converge in the uniform operator topology, then 1 is either in the resolvent set of , or a simple pole of the resolvent function of . The converse is not true, even if the sequence satisfies all the hypotheses of the theorem recalled above, except membership of in for a positive integer . We also prove that if \lim _{n\rightarrow \infty }\root n\of {s(n)}=1, and the function has no zeros in the open unit disk, then operator norm boundedness of the averages of the sequence induced by implies that the spectral radius of is less than or equal to . This result fails if the assumption about is dropped. Indeed, it may happen that the averages converge in the uniform operator topology for a sequence satisfying , , and for a positive integer , and nevertheless the spectral radius of is strictly larger than 1
Coloring of graph of ring with respect to idempotents
Full text linksummary:Let be a ring with nonzero identity. A graph of with respect to idempotents of has elements of as vertices and distinct vertices , are adjacent if and only if is an idempotent of . In this paper, we prove that is weakly perfect and provide a condition for the perfectness of the same. Further, we characterize finite abelian rings for which the complement of is connected
A projection-free dynamics for nonsmooth composite optimization
No full textsummary:This paper proposes a projection-free primal-dual dynamics for the nonsmooth composite optimization problems with equality and inequality constraints. To deal with optimization constraints, this paper departs from the use of gradient projection method, but resorts to the idea of mirror descent to design a continuous-time smooth optimization dynamics which advantageously leads to easier convergence analysis and more efficient numerical simulation. Also, the strategy of proximal augmented Lagrangian (PAL) is extended to incorporate general convex equality-inequality constraints and the strong convexity-concavity of the primal-dual variables is achieved, ensuring exponential convergence of the resulting algorithm. Furthermore, the convergence result in this paper extends existing exponential convergence which either takes no account of constraints or considers only affine linear constraints, and it also enhances existing asymptotic convergence under convex constraints which unfortunately depends on the complex gradient projection scheme
Interview with Prof. Pavel Lukáč
No full textsummary:V červnu 2025 se dožívá 90 let prof. Pavel Lukáč, významná osobnost české i světové fyziky materiálů. Prof. Lukáč spojil celý svůj profesní život s Matematicko-fyzikální fakultou UK, v letech 1985-1990 byl i jejím děkanem. Následující článek přináší kromě krátkého medailonku rozhovor s prof. Pavlem Lukáčem u příležitosti jeho kulatých narozenin
Memories of Professor Radim Blaheta
No full textsummary:This special issue is a nice opportunity to honor Professor Radim Blaheta, a well-known Czech numerical mathematician. It was supported by his former collaborators, colleagues, friends, and students. Some of them have also contributed to this issue
Algebraic multilevel preconditioning in spectral fractional diffusion
No full textsummary:The numerical solution of linear systems obtained as a result of discretization of a spectral fractional diffusion problem is studied. The finite element method is applied to the considered boundary value problem. The system matrix is a fractional power of the product of the inverse of the mass matrix and the stiffness matrix. The matrix thus defined is symmetric and positive definite (SPD) with respect to the inner product associated with the mass matrix, but is dense, which is consistent with the nonlocal nature of fractional diffusion. The presented results are in the spirit of the BURA (Best Uniform Rational Approximation) method. BURA reduces numerical solution of the dense linear system to the solution of systems with sparse SPD diffusion-reaction matrices, where is the degree of rational approximation. We prove the existence of algebraic multilevel iteration (AMLI) methods for preconditioning such type of emergent matrices that satisfy the conditions for optimal computational complexity. Both multiplicative and additive AMLI preconditioners have been developed, determining the minimum possible degree of the hierarchical -refinement of the mesh
A study of -primary decompositions
No full textsummary:Let be a commutative ring with identity, and let be a multiplicative set. An ideal of (disjoint from ) is said to be -primary if there exists an such that for all with , we have or . Also, we say that an ideal of is -primary decomposable or has an -primary decomposition if it can be written as a finite intersection of -primary ideals. First we provide an example of an -Noetherian ring in which an ideal does not have a primary decomposition. Then our main aim is to establish the existence and uniqueness of -primary decomposition in -Noetherian rings as an extension of a historical theorem of Lasker-Noether