Institute of Mathematics AS CR, v. v. i.
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Distributed optimization via active disturbance rejection control: A nabla fractional design
Get PDFsummary:This paper studies distributed optimization problems of a class of agents with fractional order dynamics and unknown external disturbances. Motivated by the celebrated active disturbance rejection control (ADRC) method, a fractional order extended state observer (Frac-ESO) is first constructed, and an ADRC-based PI-like protocol is then proposed for the target distributed optimization problem. It is rigorously shown that the decision variables of the agents reach a domain of the optimal solution when the external disturbance is bounded. In particular, for constant disturbances, the Frac-ESO is Mittag-Leffler convergent and the optimization problem can be solved exactly. Finally, numerical simulations are presented to validate the effective properties of the proposed algorithm
Characterizing finite groups whose enhanced power graphs have universal vertices
No full textsummary:Let be a finite group and construct a graph by taking as the vertex set of and by drawing an edge between two vertices and if is cyclic. Let be the set consisting of the universal vertices of along the identity element. For a solvable group , we present a necessary and sufficient condition for to be nontrivial. We also develop a connection between and when is divisible by two distinct primes and the diameter of is 2
Hypercomplex Numbers and Matrix Algebras
No full textsummary:V článku jsou ukázány tři procesy, kterými z tělesa reálných čísel vznikají algebry komplexních, dvojných, resp. duálních čísel, což jsou jediné neizomorfní algebry dimenze 2, které mají jednotkový prvek. Stejnými procesy vznikají z tělesa komplexních čísel algebry kvaternionů, antikvaternionů, resp. semikvaternionů, a stejnými procesy vznikají z kvaternionů algebry oktáv, antioktáv, resp. semioktáv. Následně je pozornost věnována reprezentacím komplexních, dvojných a duálních čísel, kvaternionů, antikvaternionů a semikvaternionů v reálných, resp. komplexních maticových algebrách. Článek zakončuje obsáhlá historická poznámka
Complete solutions of a Lebesgue-Ramanujan-Nagell type equation
Get PDFsummary:We consider the Lebesgue-Ramanujan-Nagell type equation , where and are unknown integers with . We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all -integral points on a class of elliptic curves
Semiclassical limit of a simplified quantum energy-transport model for bipolar semiconductors
No full textsummary:We are concerned with a simplified quantum energy-transport model for bipolar semiconductors, which consists of nonlinear parabolic fourth-order equations for the electron and hole density; degenerate elliptic heat equations for the electron and hole temperature; and Poisson equation for the electric potential. For the periodic boundary value problem in the torus , the global existence of weak solutions is proved, based on a time-discretization, an entropy-type estimate, and a fixed-point argument. Furthermore, the semiclassical limit is obtained by using a priori estimates independent of the scaled Planck constant
On mean value properties involving a logarithm-type weight
Get PDFsummary:Two new assertions characterizing analytically disks in the Euclidean plane 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
Turán number of two vertex-disjoint copies of cliques
No full textsummary:The Turán number of a given graph , denoted by , is the maximum number of edges in an -free graph on vertices. Applying a well-known result of Hajnal and Szemerédi, we determine the Turán number ) of a vertex-disjoint union of cliques and for all values of
Inequalities involving norm and numeri\nobreak cal radius of Hilbert space operators
No full textsummary:This paper presents several numerical radii and norm inequalities for Hilbert space operators. These inequalities improve some earlier related inequalities. For an operator , we prove that \begin{align*} \omega^{2}(A)\le & \Big\| \frac{A^{*}A+AA^{*}}{2} -\frac{1}{2R}\big(( 1-t){{A}^{*}}A+tA{{A}^{*}} &-((1-t)(A^{*}A)^{1/2}+( AA^{*})^{1/2} )^{2} \big) \Big\| \end{align*} where and