Lie perfect, Lie central extension and generalization of nilpotency in multiplicative Lie algebras

summary:This paper aims to introduce and explore the concept of Lie perfect multiplicative Lie algebras, with a particular focus on their connections to the central extension theory of multiplicative Lie algebras. The primary objective is to establish and provide proof for a range of results derived from Lie perfect multiplicative Lie algebras. Furthermore, the study extends the notion of Lie nilpotency by introducing and examining the concept of local nilpotency within multiplicative Lie algebras. The paper presents an innovative adaptation of the Hirsch-Plotkin theorem specifically tailored for multiplicative Lie algebras.\looseness -

Polyanalytic Besov spaces and approximation by dilatations

summary:Using partial derivatives $\partial f / \partial z$ and $\partial f / \partial \bar {z}$, we introduce Besov spaces of polyanalytic functions in the open unit disk, as well as in the upper half-plane. We then prove that the dilatations of functions in certain weighted polyanalytic Besov spaces converge to the same functions in norm. When restricted to the open unit disk, we prove that each polyanalytic function of degree $q$ can be approximated in norm by polyanalytic polynomials of degree at most $q$

New equivalent conditions for Hardy-type inequalities

summary:We consider a Hardy-type inequality with Oinarov's kernel in weighted Lebesgue spaces. We give new equivalent conditions for satisfying the inequality, and provide lower and upper estimates for its best constant. The findings are crucial in the study of oscillation and non-oscillation properties of differential equation solutions, as well as spectral properties

On mean value properties involving a logarithm-type weight

summary:Two new assertions characterizing analytically disks in the Euclidean plane $\mathbb {R}^2$ are proved. Weighted mean value property of positive solutions to the Helmholtz and modified Helmholtz equations are used for this purpose; the weight has a logarithmic singularity. The obtained results are compared with those without weight that were found earlier

On sparsity of approximate solutions to max-plus linear systems

summary:When a system of one-sided max-plus linear equations is inconsistent, the approximate solutions within an admissible error bound may be desired instead, particularly with some sparsity property. It is demonstrated in this paper that obtaining the sparsest approximate solution within a given $L_{\infty}$ error bound may be transformed in polynomial time into the set covering problem, which is known to be NP-hard. Besides, the problem of obtaining the sparsest approximate solution within a given $L_1$ error bound may be reformulated as a polynomial-sized mixed integer linear programming problem, which may be regarded as a special scenario of the facility location-allocation problem. By this reformulation approach, this paper reveals some interesting connections between the sparsest approximate solution problems in max-plus algebra and some well known problems in discrete and combinatorial optimization

Packing of non-blocking four-dimensional cubes into the unit cube

summary:Any collection of non-blocking four-dimensional cubes, whose total volume does not exceed 17/81, can be packed into the unit four-dimensional cube. This bound is tight for the parallel packing

The Beginnings of Solar Observations at the Skalnaté Pleso Observatory

summary:Článok sa detailne venuje prvému pozorovaniu Slnka na Observatóriu Skalnaté Pleso, ktoré vykonal 19. septembra 1943 dr. Antonín Bečvář - zakladateľ a prvý riaditeľ observatória. Pretože v príslušnom protokole je ako pozorovacie miesto uvedené Štrbské Pleso a Skalnaté Pleso, zvláštna pozornosť je venovaná lokalizácii a okolnostiam pozorovaní pred a po 19. septembri 1943 s dôrazom na argumenty jednoznačne svedčiace o tom, že pozorovanie v tomto a nasledujúcich dňoch boli získané na Observatóriu Skalnaté Pleso

C∗C^*-basic construction between non-balanced quantum doubles

summary:For finite groups $X$, $G$ and the right $G$-action on $X$ by group automorphisms, the non-balanced quantum double $D(X;G)$ is defined as the crossed product $(\Bbb {C}X^{\rm op})^*\rtimes \Bbb {C}G$. We firstly prove that $D(X;G)$ is a finite-dimensional Hopf $C^*$-algebra. For any subgroup $H$ of $G$, $D(X;H)$ can be defined as a Hopf $C^*$-subalgebra of $D(X;G)$ in the natural way. Then there is a conditonal expectation from $D(X;G)$ onto $D(X;H)$ and the index is $[G;H]$. Moreover, we prove that an associated natural inclusion of non-balanced quantum doubles is the crossed product by the group algebra