Smallest graphs with given automorphism group

Journal of Algebraic Combinatorics - For a finite group G, denote by $$\alpha (G)$$ the minimum number of vertices of any graph $$\Gamma $$ having $$\mathrm {Aut}(\Gamma )\cong G$$ . In this paper,...     

Semigroups of partial transformations with kernel and image restricted by an equivalence

Semigroup Forum - For an arbitrary set X and an equivalence relation $$\mu$$ on X, denote by $$P_\mu (X)$$ the semigroup of partial transformations $$\alpha$$ on X such that $$x\mu \subseteq x(\ker...     

THE GLOBAL SMOOTH SOLUTIONS OF SECOND ORDER QUASILINEAR HYPERBOLIC EQUATIONS WITH DISSIPATIVE BOUNDARY CONDITIONS

The paper deals with the following boundary problem of the second order quasilinear hyperbolic equation with a dissipative boundary condition on a part of the boundary:u_(tt)-sum from i,j=1 to n a_(ij)(Du)u_(x_ix_j)=0, in (0, ∞)×Ω,u|Γ_0=0,sum from i,j=1 to n, a_(ij)(Du)n_ju_x_i+b(Du)u_t|Γ_1=0,u|t=0=φ(x), u_t|t=0=ψ(x), in Ω, where Ω=Γ_0∪Γ_1, b(Du)≥b_0>0. Under some assumptions on the equation and domain, the author proves that there exists a global smooth solution for above problem with small data.     

On Ivády’s bounds for the gamma function and related results

Periodica Mathematica Hungarica - We prove that the inequality $$\begin{aligned} \Gamma (x+1)\le \frac{x^2+\beta }{x+\beta } \end{aligned}$$ holds for all $$x\in [0,1]$$ , $$\beta \ge {\beta...     

Minimal time function and viscosity solutions

Two theorems in Ref. 1 are generalized. It is proved that, ifV(A,Γ) is the set of points that can be steered to the origin along a solution of the control systemx′=Ax?c, ifc(t)∈Γ, Γ is a compact subset ofR ⁿ , 0∈ intrelco Γ, and if a rank condition holds, then the minimal time functionT(·) is a viscosity solution of the Bellman equation $$\max \{ \left\langle {DT(x),\gamma - Ax} \right\rangle :\gamma \varepsilon co\Gamma \} - 1 = 0,x\varepsilon V(A,\Gamma )\backslash \{ 0\} ,$$ and of the Hàjek equation $$1 - \max \{ \left\langle {DT(x),\exp [ - AT(x)]} \right\rangle :\gamma \varepsilon co\Gamma \} = 0,x\varepsilon V(A,\Gamma ).$$      

Characterization of numerical radius parallelism in $$C^*$$C∗-algebras

Positivity - Let v(x) denote the numerical radius of an element x in a $$C^*$$-algebra $$\mathfrak {A}$$. First, we prove several numerical radius inequalities in $$\mathfrak {A}$$. Particularly,...     

Limiting Behavior of Infinite Products Scaled by Pisot Numbers

Journal of Fourier Analysis and Applications - For $$\theta >1$$ , the infinite product $$\Gamma _{\theta }(x)=\prod _{n=0}^{\infty }\cos (\pi \theta ^{-j}x)$$ is the Fourier transform of...     

Eisenstein series twisted with non-expanding cusp monodromies

The Ramanujan Journal - Let $$\Gamma $$ be a geometrically finite Fuchsian group and suppose that $$\chi :\Gamma \rightarrow {{\,\mathrm{GL}\,}}(V)$$ is a finite-dimensional representation with...     

On hybrid moments of $$\Delta _2(x)$$ Δ 2 ( x ) and $$\Delta _3(x)$$ Δ 3 ( x )

The Ramanujan Journal - Let $$k\geqslant 2$$ be a fixed natural number and $$d_k(n)$$ denote the number of ways n can be written as a product of k positive integers. We use $$\Delta _k(x)$$ to...     

Sign changes in the prime number theorem

The Ramanujan Journal - Let V(T) denote the number of sign changes in $$\psi (x) - x$$ for $$x\in [1, T]$$ . We show that $$\liminf _{T\rightarrow \infty } V(T)/\log T\ge \gamma _{1}/\pi +...     

The Terwilliger algebra of the halved folded 2n-cube from the viewpoint of its automorphism group action

Journal of Algebraic Combinatorics - Let $$\frac{1}{2}{\overline{H}}(2n,2)$$ denote the halved folded 2n-cube with vertex set X and let $$T{:}{=}T(x)$$ denote the Terwilliger algebra of...     

ON THE ERROR FUNCTION OF THE SQUARE-FULL INTEGERS

Let L(x) denote the number of square-full integers not exceeding x. It is proved in [1] thatL(x)～(ζ(3/2)/ζ(3))x~(1/2) (ζ(2/3)/ζ(2))x~(1/3) as x→∞,where ζ(s) denotes the Riemann zeta function. Let △(x) denote the error function in the asymptotic formula for L(x). It was shown by D. Suryanaryana~([2]) on the Riemann hypothesis (RH) that1/x integral from n=1 to x |△(t)|dt=O(x~(1/10 s))for every ε>0. In this paper the author proves the following asymptotic formula for the mean-value of △(x) under the assumption of R. H.integral from n=1 to T (△~2(t/t~(6/5))) dt～c log T,where c>0 is a constant.     

Sufficient conditions for local equivalence of mappings

Let $$f,g:({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^m,0)$$ be $$C^{r+1}$$ mappings and let $$Z=\{x\in \mathbf {\mathbb {R}}^n:\nu (df (x))=0\}$$ , $$0\in Z$$ , $$m\le n$$ . We will show that if there exist a neighbourhood U of $$0\in {\mathbb {R}}^n$$ and constants $$C,C'>0$$ and $$k>1$$ such that for $$x\in U$$ $$\begin{aligned}&\nu (df(x))\ge C{\text {dist}}(x,Z)^{k-1}, \\&\left| \partial ^{s} (f_i-g_i)(x) \right| \le C'\nu (df(x))^{r+k-|s|}, \end{aligned}$$ for any $$i\in \{1,\dots , m\}$$ and for any $$s \in \mathbf {\mathbb {N}}^n_0$$ such that $$|s|\le r$$ , then there exists a $$C^r$$ diffeomorphism $$\varphi :({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^n,0)$$ such that $$f=g\circ \varphi $$ in a neighbourhood of $$0\in {\mathbb {R}}^n$$ . By $$\nu (df)$$ we denote the Rabier function.     

Finite-dimensional quasi-variational inequalities associated with discontinuous functions

In this paper, given a nonempty closed convex setX ? ⁿ , a functionf: X→? ⁿ , and a multifunction Γ:X→2^X, we deal with the problem of finding a point \(\hat x\) ∈X such that $$\hat x \in \Gamma (\hat x) and \langle f(\hat x), \hat x - y\rangle \leqslant 0, for all y \in \Gamma (\hat x).$$ For such problem, we establish a result where, in particular, the functionf is not assumed to be continuous. More precisely, we extend to the present setting a finite-dimensional version of a result by Ricceri on variational inequalities (Ref. 1).     

The universal minimal one parameter system with quasi-discrete spectrum

The universal minimal one parameter system will be characterized as the space $$\Gamma ^{\infty }$$, in which $$\Gamma$$ is the Bohr compactification of the additive group $${\mathbb {R}}$$ of real numbers. In this way, we need to show that $$\Gamma ^\infty$$ is isomorphic to the spectrum of $$W({\mathbb {R}})$$, the norm closure of the invariant algebra generated by the maps $$\exp q(t)$$, where q(t) is a real polynomial on $${\mathbb {R}}$$.     

A Delocalization Property for Assembly Maps and an Application in Algebraic K-Theory of Group Rings

Let be a group, F the free -module on the set of finite order elements in , with acting by conjugation, and the ring extension of by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaWaaiWaaeaada% WcaaqaaiaaigdaaeaatCvAUfKttLearyGqLXgBG0evaGqbciab-5ga% UbaaieaacaGFLbGaaGOmaiaabc8acqWFPbqAcaqGVaGae8NBa42aaq% qaaeaacqGHdicjcqaHZoWzcqGHiiIZcqqHtoWrcaqGGaGaae4Baiaa% bAgacaqGGaGaae4BaiaabkhacaqGKbGaaeyzaiaabkhacaqGGaGae8% NBa4gacaGLhWoaaiaawUhacaGL9baaaaa!563E!\[\left\{ {\frac{1}{n}e2{\text{\pi }}i{\text{/}}n\left| {\exists \gamma \in \Gamma {\text{ of order }}n} \right.} \right\}\]. For a ring R with , we build an injective assembly map , detected by the Dennis trace map. This is proved by establishing a delocalization property for the assembly map in Hochschild homology, namely providing a gluing of simpler assembly maps (i.e. localized at the identity of ) to build , and by delocalizing a known assembly map in K-theory to define . We also prove the delocalization property in cyclic homology and in related theories.     

NONLINEAR BOUNDARY PROBLEMS WITH NONLOCAL BOUNDARY CONDITIONS

By means of the supersolution and subsolution method and monotone iteration technique, the following nonlinear elliptic boundary problem with the nonlocal boundary conditions is considerd. The sufficient conditions which ensure at least one solution are given. Furthermore, the estimate of the first nonzero eigenvalue for the following linear eigenproblem is obtained, that is λ_1≥2α/(nd~2).     

On graphs and Lie rings

From a finite oriented graph , finite-dimensional graded nilpotent Lie rings () and () are naturally constructed; these rings are related to subtrees and connected subgraphs of , respectively. Diverse versions of these constructions are also suggested. Moreover, an embedding of Lie rings of the form () in the adjoint Lie rings of finite-dimensional associative rings (also determined by the graph ) is indicated.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 449–459.Original Russian Text Copyright © 2005 by Yu. S. Sernenov.This revised version was published online in April 2005 with a corrected issue number.     

The First-Order Nonholonomic Connections with the Galilean Groups of Local Transformations. III

In the canonical smooth fiber bundles endowed with the metric tensor fields of relevant structure, we consider natural representations of the Galilean groups (1, n) and construct (1, n)-invariant generalized differential-geometric connections. In both regular and special cases of the representations of the considered groups (1, n), we find all affine nonholonomic , and _1,2-connections of the first order (see [1]–[3]) possessing the local Lie groups of transformations (1, n) and also describe the corresponding (1, n)invariant planar connections.     

On the Existence of a Generalized Solution to a Three-Dimensional Elliptic Equation with Radiation Boundary Condition

For a second order elliptic equation with a nonlinear radiation-type boundary condition on the surface of a three-dimensional domain, we prove existence of generalized solutions without explicit conditions (like ) on the trace of solutions. In the boundary condition, we admit polynomial growth of any fixed degree in the unknown solution, and the heat exchange and emissivity coefficients may vary along the radiating surface. Our generalized solution is contained in a Sobolev space with an exponent q which is greater than for the fourth power law.