We characterize generalized extreme points of compact convex sets. In particular, we show that if the polyconvex hull is convex in R m×n , min(m, n) ≤ 2, then it is constructed from polyconvex extreme points via sequential lamination. Further, we give theorems ensuring equality of the quasiconvex (polyconvex) and the rank-1 convex envelopes of a lower semicontinuous function without explicit convexity assumptions on the quasiconvex (polyconvex) envelope.
In this paper we present a result that relates merging of closed convex sets of discrete probability functions respectively by the squared Euclidean distance and the Kullback-Leibler divergence, using an inspiration from the Rényi entropy. While selecting the probability function with the highest Shannon entropy appears to be a convincingly justified way of representing a closed convex set of probability functions, the discussion on how to represent several closed convex sets of probability functions is still ongoing. The presented result provides a perspective on this discussion. Furthermore, for those who prefer the standard minimisation based on the squared Euclidean distance, it provides a connection to a probabilistic merging operator based on the Kullback-Leibler divergence, which is closely connected to the Shannon entropy.
Let G be a group. If every nontrivial subgroup of G has a proper supplement, then G is called an aS-group. We study some properties of aS-groups. For instance, it is shown that a nilpotent group G is an aS-group if and only if G is a subdirect product of cyclic groups of prime orders. We prove that if G is an aS-group which satisfies the descending chain condition on subgroups, then G is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group is an aS-group. Finally, it is shown that if G is an aS-group and |G| ≠ pq, p, where p and q are primes, then G has a triple factorization., Reza Nikandish, Babak Miraftab., and Obsahuje seznam literatury
Controversial aspects of the conventional and widely used concept of the integral vortex strength are briefly discussed. The strength of a vortex is usually calculated as the circulation along the vortex boundary, or equivalently due to Green’s theorem, as the surface integral of vorticity over the planar vortex cross section. However, the local effect of an arbitrary ''superimposed shear'' is fully absorbed by vorticity what makes the circulation a shear-biased vortex characteristic. The present paper shows that different vortexstrength models can be derived on the basis of different local vortex intensities proposed in the literature. The outcome of these models naturally differs, even for an ideally axisymmetric vortex. Three different vortex-strength models are compared and discussed by examining the unsteady Taylor vortex. and V práci jsou stručně diskutovány sporné stránky konvenčního a široce užívaného pojetí integrální síly víru. Síla víru je obvykle počítána jako cirkulace podél hranice víru nebo ekvivalentně podle Greenovy věty jako plošný integrál vířivosti přes příčný rovinný řez vírem. Lokální efekt libovolného ''superponovaného smyku'' je však plně absorbován vířivostí, což činí z cirkulace smykově zkreslenou vírovou charakteristiku. Tento článek ukazuje, že lze odvodit různé modely síly víru na základě různých lokálních intenzit víru navržených v odborné literatuře. Výsledky těchto modelů se přirozeně liší, dokonce i pro ideálně osově symetrický vír. Na podkladě zkoumání nestacionárního Taylorova víru jsou porovnány a diskutovány tři různé modely síly víru.
Integration by parts results concerning Stieltjes integrals for functions with values in Banach spaces are presented. The background of the theory is the Kurzweil approach to integration based on Riemann type integral sums, which leads to the Perron integral.
Recall that a space X is a c-semistratifiable (CSS) space, if the compact sets of X are Gδ-sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a T2-space X is a k-c-semistratifiable space if and only if X has a g function which satisfies the following conditions: (1) For each x ∈ X, {x} = ∩ {g(x, n): n ∈ ℕ} and g(x, n + 1) ⊆ g(x, n) for each n ∈ N. (2) If a sequence {xn}n∈N of X converges to a point x ∈ X and yn ∈ g(xn, n) for each n ∈ N, then for any convergent subsequence {ynk }k∈N of {yn}n∈N we have that {ynk }k∈N converges to x. By the above characterization, we show that if X is a submesocompact locally k-csemistratifiable space, then X is a k-c-semistratifible space, and the countable product of k-c-semistratifiable spaces is a k-c-semistratifiable space. If X = ∪ {Int(Xn): n ∈ N} and Xn is a closed k-c-semistratifiable space for each n, then X is a k-c-semistratifiable space. In the last part of this note, we show that if X = ∪ {Xn : n ∈ N} and Xn is a closed strong β-space for each n ∈ ℕ, then X is a strong β-space.
In recent papers Henrard and Lemaître have studied what they call "The Second Fundamental Model for Resonance" and higher order generalizations of it. The action integral ("area index") was computed analytically, but the phase space and the action integral as a function of the parameter δ were only plotted on scale by a computer. By using properties of quartic equations, however, the mathematically special values of δ were found. For third order resonances, one of these turned out to correspond to a minimum
in the value of the "area index" A2, but since it is very shallow and very close to the starting point of the function, this feature was invisible in Lemaître's plots, This has some theoretical implications for the process of capture into a third order resonance, although numerically the effect will be small due to the shallowness of the minimum. A similar exercise on first and second order resonances revealed no new features.
Let $H$ be an infinite-dimensional almost separable Hilbert space. We show that every local automorphism of $\mathcal B(H)$, the algebra of all bounded linear operators on a Hilbert space $H$, is an automorphism.
In stochastic partial differential equations it is important to have pathwise regularity properties of stochastic convolutions. In this note we present a new sufficient condition for the pathwise continuity of stochastic convolutions in Banach spaces.