An accelerated gradient descent method based on a non-monotone backtracking line search scheme for unconstrained optimization and image restoration problems

summary:This study introduces an accelerated gradient descent method based on a non-monotone backtracking line search scheme. A simple adaptive quadratic model is enhanced by utilizing a real, positive definite scalar matrix derived from the Taylor expansion of the objective function, rather than relying on the exact Hessian. The global and superlinear convergence of the defined model is established under appropriate conditions. Numerical experiments on a set of standard unconstrained optimization problems and image restoration problems show that the new algorithm outperforms other comparable methods in terms of efficiency and robustness

Integration and decompositions of weak∗^*-integrable multifunctions

summary:Conditions guaranteeing Pettis integrability of a Gelfand integrable multifunction and a decomposition theorem for the Henstock-Kurzweil-Gelfand integrable multifunctions are presented

On unbounded solutions for differential equations with mean curvature operator

summary:We present necessary and sufficient conditions for the existence of unbounded increasing solutions to ordinary differential equations with mean curvature operator. The results illustrate the asymptotic proximity of such solutions with those of an auxiliary linear equation on the threshold of oscillation. A new oscillation criterion for equations with mean curvature operator, extending Leighton criterion for linear Sturm-Liouville equation, is also derived

On an Old Textbook. Teaching Mathematics “For Life” Two Hundred Years Ago

summary:V článku nejprve stručně načrtneme historii české triviální školy v Dýšině, zejména v 19. století. Dále přiblížíme zajímavé osudy rozvětveného učitelského rodu Rádlů a naznačíme obsah a význam nevelké učebnice počítání zpaměti Jana Rádla, která vyšla takřka před dvěma sty lety. Význam znalostí matematiky demonstruje zásadně na úlohách, v nichž se počítá s penězi, což bylo a stále je pro většinu populace vynikající motivací k učení

The rings whose torsionfree modules have injective dimension at most one

summary:The domains with torsion-free modules of injective dimension at most one have been examined by B. Olberding. A TF-projective module is one that is projective relative to all short exact sequences beginning with torsion-free modules. The rings, where each right ideal of $S$ is TF-projective, are precisely those rings whose torsionfree right modules have injective dimension at most one. The goal of this study is to comprehend the structure of the rings that B. Olberding recently studied. Along the way, we prove for a domain $S$, that if each ideal of $S$ is TF-projective, then $S$ is a Noetherian ring with $\dim (S)\leq 1$. Specifically, we prove that for a commutative domain $S$, each ideal of $S$ is TF-projective if and only if $S$ is a Gorenstein Dedekind domain. A left $P$-coherent ring all of its TF-projective left $S$-modules are projective is precisely left PP ring. Furthermore, we demonstrate that any (cyclic) right $S$-module of $S$ is TF-projective if and only if $S$ is a QF-ring

On the behaviour of the solutions of a kk-order cyclic-type system of max difference equations

summary:We investigate the behaviour of the solutions of a $k$-dimensional cyclic system of difference equations with maximum. More precisely, we study the existence and the number of the equilibria in the case when $k$ is an odd or an even positive integer, but also for the various values of the exponents of the terms of the difference equations of this system. In addition, we find invariant intervals for our system and we invistegate the convergence of the solutions to the unique positive equilibrium. Finally, we study the asymptotic behavior of the positive solutions of the system in the case, where $k=2$ and $k=4$

Adaptive fractional distributed optimization algorithm with directed spanning trees

summary:Distributed optimization has garnered significant attention in past decade, yet existing algorithms mainly rely on Laplacian matrix information for parameter settings, limiting their adaptability and applicability. To design the fully distributed algorithm, this paper uses an adaptive weight framework based on directed spanning trees (DST), which not only solves the consensus optimization problem but also can be extended to solve the resource allocation problem. The innovative integration of Nabla fractional calculus further improves performance, enabling efficient discrete-time distributed optimization. Moreover, The proposed algorithms optimality and convergence properties have been rigorously analyzed, which demonstrates that they can converge to the optimal solution of the problem under consideration. Finally, numerical simulations are conducted to validate the algorithm's feasibility and superiority

Global well-posedness and energy decay for a one dimensional porous-elastic system subject to a neutral delay

summary:We consider a one-dimensional porous-elastic system with porous-viscosity and a distributed delay of neutral type. First, we prove the global existence and uniqueness of the solution by using the Faedo-Galerkin approximations along with some energy estimates. Then, based on the energy method with some appropriate assumptions on the kernel of neutral delay term, we construct a suitable Lyapunov functional and we prove that, despite of the destructive nature of delays in general, the damping mechanism considered provokes an exponential decay of the solution for the case of equal speed of wave propagation. In the case of lack of exponential stability, we show that the solution decays polynomially