The paper summarizes and extends the theory of generalized ϕ-entropies Hϕ(X) of random variables X obtained as ϕ-informations Iϕ(X;Y) about X maximized over random variables Y. Among the new results is the proof of the fact that these entropies need not be concave functions of distributions pX. An extended class of power entropies Hα(X) is introduced, parametrized by α∈R, where Hα(X) are concave in pX for α≥0 and convex for α<0. It is proved that all power entropies with α≤2 are maximal ϕ-informations Iϕ(X;X) for appropriate ϕ depending on α. Prominent members of this subclass of power entropies are the Shannon entropy H1(X) and the quadratic entropy H2(X). The paper investigates also the tightness of practically important previously established relations between these two entropies and errors e(X) of Bayesian decisions about possible realizations of X. The quadratic entropy is shown to provide estimates which are in average more than 100 \than those based on the Shannon entropy, and this tightness is shown to increase even further when α increases beyond α=2. Finally, the paper studies various measures of statistical diversity and introduces a general measure of anisotony between them. This measure is numerically evaluated for the entropic measures of diversity H1(X) and H2(X).
Under some conditions we prove that every generalized Jordan triple derivation on a Lie triple system is a generalized derivation. Specially, we conclude that every Jordan triple $\theta $-derivation on a Lie triple system is a $\theta $-derivation.
We give a version of the Moser-Trudinger inequality without boundary condition for Orlicz-Sobolev spaces embedded into exponential and multiple exponential spaces. We also derive the Concentration-Compactness Alternative for this inequality. As an application of our Concentration-Compactness Alternative we prove that a functional with the sub-critical growth attains its maximum.
We first introduce the notion of a right generalized partial smash product and explore some properties of such partial smash product, and consider some examples. Furthermore, we introduce the notion of a generalized partial twisted smash product and discuss a necessary condition under which such partial smash product forms a Hopf algebra. Based on these notions and properties, we construct a Morita context for partial coactions of a co-Frobenius Hopf algebra.
One of the basic questions in the Kleinian group theory is to understand both algebraic and geometric limiting behavior of sequences of discrete subgroups. In this paper we consider the geometric convergence in the setting of the isometric group of the real or complex hyperbolic space. It is known that if $\Gamma $ is a non-elementary finitely generated group and $\rho _{i}\colon \Gamma \rightarrow {\rm SO}(n,1)$ a sequence of discrete and faithful representations, then the geometric limit of $\rho _{i}(\Gamma )$ is a discrete subgroup of ${\rm SO}(n,1)$. We generalize this result by showing that for a sequence of discrete and non-elementary subgroups $\{G_{j}\}$ of ${\rm SO}(n,1)$ or ${\rm PU}(n,1)$, if $\{G_{j}\}$ has uniformly bounded torsion, then its geometric limit is either elementary, or discrete and non-elementary.
The paper describes the general form of functional-differential equations of the first order with $m (m\ge 1)$ delays which allows nontrivial global transformations consisting of a change of the independent variable and of a nonvanishing factor. A functional equation \[ f(t, uv, u_{1}v_{1}, \ldots , u_{m}v_{m}) = f(x, v, v_{1}, \ldots , v_{m})g(t, x, u, u_{1}, \ldots , u_{m})u + h(t, x, u, u_{1}, \ldots , u_{m})v \] for $u\ne 0$ is solved on $\mathbb R$ and a method of proof by J. Aczél is applied.
The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form
\[ f(t,vy,wy+uvz)=f(x,y,z)u^{2}v+g(t,x,u,v,w)vz+h(t,x,u,v,w)y+2uwz \] is solved on $\mathbb R$ for $y\ne 0$, $v\ne 0.$.
Determination of heights with help of GPS in local geodetic networks is still more actual respecting the fact that the GPS technology becames more and more effective with hardware progress, with improvements in measuring and evaluating procedures, and with better modelling of the disturbing influences. In comparison with GPS the employment of classical terrestrial measuring technologies is often more difficult namely in broken mountain environment. In period 1998-2005 authors carried out repeated measurements of GPS baselines of various length and various height differences in local geodynamical network Sněžník and in other experimental areas. On ground of analyses of large GPS data sets the modified procedure for GPS observation was designed. The procedure is based on repetition of shorter static sessions separated by time intervals of optimal length. This technology represents an alternative to the ususal long static sessions, and is offering better effectivity of vertical GPS measurements with minimal loss of accuracy. The paper presents detailed description of the modified procedure together with some statistical analyses of results. The possibilities of elimination or mitigation of some disturbing influences are discussed. Two testing vertical profiles were marked in Sněžník network- longitudinal profile in N-S direction, and transversal profile in E-W direction - which were measured in course of several years by classical method of very precise levelling, and also by modified GPS heighting procedure in repeated sessions. Results obtained contributed to the local quasigeoid model creation., Otakar Švábenský, Josef Weigel and Radovan Machotka., and Obsahuje bibliografii
A proper coloring c : V (G) → {1, 2, . . . , k}, k ≥ 2 of a graph G is called a graceful k-coloring if the induced edge coloring c ′ : E(G) → {1, 2, . . . , k − 1} defined by c ′ (uv) = |c(u) − c(v)| for each edge uv of G is also proper. The minimum integer k for which G has a graceful k-coloring is the graceful chromatic number χg(G). It is known that if T is a tree with maximum degree ∆, then χg(T ) ≤ ⌈ 5⁄3∆⌉ and this bound is best possible. It is shown for each integer ∆ ≥ 2 that there is an infinite class of trees T with maximum degree ∆ such that χg(T ) = ⌈ 5⁄3 ∆⌉. In particular, we investigate for each integer ∆ ≥ 2 a class of rooted trees T∆,h with maximum degree ∆ and height h. The graceful chromatic number of T∆,h is determined for each integer ∆ ≥ 2 when 1 ≤ h ≤ 4. Furthermore, it is shown for each ∆ ≥ 2 that lim h→∞ χg(T∆,h) = ⌈ 5⁄3∆⌉.
Let $G$ be a connected simple graph on $n$ vertices. The Laplacian index of $G$, namely, the greatest Laplacian eigenvalue of $G$, is well known to be bounded above by $n$. In this paper, we give structural characterizations for graphs $G$ with the largest Laplacian index $n$. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on $n$ and $k$ for the existence of a $k$-regular graph $G$ of order $n$ with the largest Laplacian index $n$. We prove that for a graph $G$ of order $n \geq 3$ with the largest Laplacian index $n$, $G$ is Hamiltonian if $G$ is regular or its maximum vertex degree is $\triangle (G)=n/2$. Moreover, we obtain some useful inequalities concerning the Laplacian index and the algebraic connectivity which produce miscellaneous related results.