The simultaneous occurrence of conditional independences among subvectors of a regular Gaussian vector is examined. All configurations of the conditional independences within four jointly regular Gaussian variables are found and completely characterized in terms of implications involving conditional independence statements. The statements induced by the separation in any simple graph are shown to correspond to such a configuration within a regular Gaussian vector.
We discuss how the choice of the functional setting and the definition of the weak solution affect the existence and uniqueness of the solution to the equation −∆pu = f in Ω, where Ω is a very general domain in RN , including the case Ω = RN .
Some generalizations of the Ostrowski inequality, the Milovanović-Pečarić-Fink inequality, the Dragomir-Agarwal inequality and the Hadamard inequality are given.
An S-closed submodule of a module M is a submodule N for which M/N is nonsingular. A module M is called a generalized CS-module (or briefly, GCS-module) if any S-closed submodule N of M is a direct summand of M. Any homomorphic image of a GCS-module is also a GCS-module. Any direct sum of a singular (uniform) module and a semi-simple module is a GCS-module. All nonsingular right R-modules are projective if and only if all right R-modules are GCS-modules., Qingyi Zeng., and Obsahuje seznam literatury
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.