A network of congruences on an ample semigroup

Semigroup Forum - For an admissible congruence $$\gamma $$ on an ample semigroup S, the minimum admissible congruence $$\gamma _t$$ whose trace is $$\mathrm {tr}\gamma $$ has been determined by the...     

Spectral Shorted Operators

If $$\mathcal{H}$$ is a Hilbert space, $$\mathcal{S}$$ is a closed subspace of $$\mathcal{H},$$ and A is a positive bounded linear operator on $$\mathcal{H},$$ the spectral shorted operator $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ is defined as the infimum of the sequence $$\sum (\mathcal{S},A^n )^{1/n} ,$$ where denotes $$\sum \left( {\mathcal{S},B} \right)$$ the shorted operator of B to $$\mathcal{S}.$$ We characterize the left spectral resolution of $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ and show several properties of this operator, particularly in the case that dim $${\mathcal{S} = 1.}$$ We use these results to generalize the concept of Kolmogorov complexity for the infinite dimensional case and for non invertible operators.     

On congruences on \mathcal{T} -regular semigroups

A nontrivial regular semigroup S with zero in which every interval of idempotents is a finite chain is said to be -regular. The structure of these semigroups is described in terms of trees of completely 0-simple semigroups. For S in this form, we study congruences which we express in terms of congruence aggregates. We determine the inclusion relation, meet and join of congruences, their kernel and trace, and the ends of the intervals which form their classes. We characterize those S for which the kernel relation on the congruence lattice is a congruence, and those for which the operators and are homomorphisms.     

Kernel and Trace Operators for Extensions of Brandt Semigroups

Let S be an (ideal) extension of a Brandt semigroup S₀ by a Brandt semigroup S₁ and let denote the congruence lattice of S. For denote by and the least and the greatest congruences on S with the same kernel as respectively, and let and have the analogous meaning relative to trace. We establish necessary and sufficient conditions on S in order that one or more of the operators

Weak finitely regular relations and applications

In this note, we introduce the concepts of weak finitely regular relations and weak hypercontinuous posets. It is proved that when a binary relation $$\rho :X/\rightarrow Y$$ satisfies a certain condition, $$\rho $$ is weak finitely regular if and only if $$(\varphi _{\rho }(X,Y),\subseteq )$$ is a weak hypercontinuous poset. For a poset P, we get that the relation $$\not \le $$ on P is weak finitely regular if and only if P is weak hypercontinuous.     

A Note on Bifurcation for Harmonic Maps on Annular Domains

Mediterranean Journal of Mathematics - In this paper we consider harmonic maps u(r, θ) from an annular domain $${\Omega _{\rho} = B_{1}\backslash\bar{B}_{\rho}}$$ to S 2 with the boundary...     

Approximation by the Complex form of a Link Operator Between the Phillips and the Szász–Mirakjan Operators

Results in Mathematics - The link operator $${P_{\alpha}^\rho(f,x)=\sum_{k=1}^\infty s_{\alpha,k}(x)\int_0^\infty \theta_{\alpha,k}^\rho(t)f(t)dt+e^{-\alpha x}f(0)}$$ , $${\alpha, \rho >...     

Constants related to operators of class Cρ

Let T be a Hilbert-space operator of class $$\left\| {S^{ - 1} TS} \right\| \leqslant 1\,\,\,\,and\,\,\left\| {S^{ - 1} } \right\| \cdot \left\| S \right\| \leqslant \max (1,\rho )$$ in the sense of Sz.-Nagy and Foia?. Then there exists an invertible operator S such that ∥S^?1TS∥≤1 and ∥S^?1∥·∥S∥≤max(1,ρ). The following estimate is valid for the bilateral limit: $$\mathop {lim}\limits_{n \to \infty } \{ \left\| {T^n h} \right\|^2 + \left\| {T*^n h} \right\|^2 \} \leqslant max(2,\rho )\left\| h \right\|^2 .$$ Here the constants max(1,ρ) and max(2,ρ) are the best possible in their respective cases.     

Nonparametric estimation of Matusita's measure of affinity between absolutely continuous distributions

LetF andG be two distribution functions defined on the same probability space which are absolutely continuous with respect to the Lebesgue measure with probability densitiesf andg, respectively. Matusita [3] defines a measure of the closeness, affinity, betweenF andG as: . Based on two independent samples fromF andG we propose to estimate ρ by , where and are taken to be the kernel estimates off(x) andg(x), respectively, as given by Parzen [5]. In this note sufficient conditions are given such that (i) asx→∞ and (ii) with probability one, asn→∞. Research supported in part by the National Research Council of Canada and by McMaster University Science and Engineering Research Board. The author is presently with the Department of Mathematical Sciences, Memphis State University, Memphis, Tennessee 38152.     

Trigonometric multiplicative chaos and applications to random distributions

Science China Mathematics - The random trigonometric series $$\sum\nolimits_{n = 1}^\infty {{\rho _n}\cos \left( {nt + {\omega _n}} \right)} $$ on the circle $$\mathbb{T}$$ are studied under the...     

一类带有非牛顿位势的可压缩Navier-Stokes方程整体强解的存在性

主要研究了一类带有非牛顿位势的可压缩Navier-Stokes方程:其中粘性系数μ依赖于密度ρ,Φ是非牛顿位势.证明了上述问题的强解的存在性.在相容性条件下,得到了强解的唯一性.     

$D^2(\Omega,\rho)$的再生核与再生核诱导的度量

An important property of the reproducing kernel of D^2（Ω, ρ） is obtained and the reproducing kernels for D^2（Ω, ρ） are calculated when Ω = Bn× Bn and ρ are some special functions. A reproducing kernel is used to construct a semi-positive definite matrix and a distance function defined on Ω×Ω. An inequality is obtained about the distance function and the pseudodistance induced by the matrix.     

Third Derivative of the One-Electron Density at the Nucleus

We study electron densities of eigenfunctions of atomic Schr?dinger operators. We prove the existence of , the third derivative of the spherically averaged atomic density at the nucleus. For eigenfunctions with corresponding eigenvalue below the essential spectrum in any symmetry subspace we obtain the bound , where Z denotes the nuclear charge. This bound is optimal. ? 2008 by the authors. This article may be reproduced in its entirety for non-commercial purposes. Submitted: April 22, 2008. Accepted: July 7, 2008.     

A nonstandard Cauchy problem for the heat equation

Mathematical Notes - General necessary conditions on the real parameters α, β, C, D for the function $${e^{ - \alpha \rho \left( x \right)}}\left( {C\cos \beta \rho \left( x \right) +...     

Capacity solution for a perturbed nonlinear coupled system

We shall give the existence of a capacity solution to a nonlinear elliptic coupled system, whose unknowns are the temperature inside a semiconductor material, u, and the electric potential, $$\varphi $$, the model problem we refer to is $$\begin{aligned} \left\{ \begin{array}{l} \Delta _p u+g(x,u)= \rho (u)|\nabla \varphi |^2 \quad \mathrm{in} \quad \Omega ,\\ {{\,\mathrm{div}\,}}(\rho (u)\nabla \varphi ) =0 \quad \mathrm{in} \quad \Omega ,\\ \varphi =\varphi _0 \quad \text{ on } \quad {\partial \Omega },\\ u=0 \quad \mathrm{on} \quad {\partial \Omega }, \end{array} \right. \end{aligned}$$where $$\Omega \subset \mathbb {R}^N$$, $$N\ge 2$$ and $$\Delta _p u=-{\text {div}}\left( |\nabla u|^{p-2} \nabla u\right) $$ is the so-called p-Laplacian operator, and g a nonlinearity which satisfies the sign condition but without any restriction on its growth. This problem may be regarded as a generalization of the so-called thermistor problem, where we consider the case of the elliptic equation is non-uniformly elliptic.     

THE BEHAVIOR OF SOLUTIONS IN THE VICINITY OF A BOUNDED SOLUTION TO AUTONOMOUS DIFFERENTIAL EQUATIONS

By using the exponential dichotomy,this paper investigates the behavior of solutionsin the vicinity of a bounded solution to the autonomous differential systemdx/dt=f(x).(1)Suppose x=u(t)is a nontrivial bounded solution of system(1).By discussing theequivalent equations of system(1)dθ/dt=1 (p,θ)dp/dt=A(θ)p (p,θ)(2)with respect to the moving orthonormal transformationx=u(θ) s(θ)p,the author proves that if linear system corresponding to(2)admits exponential dichotomy,then the given bounded solution x=u(t)should be periodic.The author also discusses thestadility of the obtained periodic solution.Finally,this paper discusses perturbation of thebounded solution of autonomous system(1).     

Packing chromatic number, $$\mathbf (1, 1, 2, 2) $$ ( 1 , 1 , 2 , 2 ) -colorings,and characterizing the Petersen graph

The packing chromatic number $$\chi _{\rho }(G)$$     

Pro- $$\mathcal {C}$$ C congruence properties for groups of rooted tree automorphisms

We propose a generalisation of the congruence subgroup problem for groups acting on rooted trees. Instead of only comparing the profinite completion to that given by level stabilizers, we also compare pro- $$\mathcal {C}$$ completions of the group, where $$\mathcal {C}$$ is a pseudo-variety of finite groups. A group acting on a rooted, locally finite tree has the $$\mathcal {C}$$ -congruence subgroup property ( $$\mathcal {C}$$ -CSP) if its pro- $$\mathcal {C}$$ completion coincides with the completion with respect to level stabilizers. We give a sufficient condition for a weakly regular branch group to have the $$\mathcal {C}$$ -CSP. In the case where $$\mathcal {C}$$ is also closed under extensions (for instance the class of all finite p-groups for some prime p), our sufficient condition is also necessary. We apply the criterion to show that the Basilica group and the GGS-groups with constant defining vector (odd prime relatives of the Basilica group) have the p-CSP.     

Positivity of the spherically averaged atomic one-electron density

We investigate the positivity of the spherically averaged atomic one-electron density . For a which stems from a physical ground state we prove that for r ≥ 0. This article may be reproduced in its entirety for non-commercial purposes.     

Transformation Operators in the Problems of Controllability for the Degenerate Wave Equation with Variable Coefficients

Ukrainian Mathematical Journal - We study a control system $$ {w}_{tt}=\frac{1}{\rho }{\left(k{w}_x\right)}_x+\gamma w,w\left(0,t\right)=u(t),x\in \left(0,l\right),t\in \left(0,T\right), $$ in...