In this work, the behaviour of solutions for the Dirichlet problem of the non-local equation

$$ u_t=\varDelta(\kappa(u))+\frac{\lambda f(u)}{(\int_{\varOmega}f(u)\,\mathrm{d}x)^p},\quad \varOmega\subset\mathbb{R}^N,\quad N=1,2, $$

is studied, mainly for the case where $f(s)=\mathrm{e}^{\kappa(s)}$. More precisely, the interplay of exponent $p$ of the non-local term and spatial dimension $N$ is investigated with regard to the existence and non-existence of solutions of the associated steady-state problem as well as the global existence and finite-time blow-up of the time-dependent solutions $u(x,t)$. The asymptotic stability of the steady-state solutions is also studied.

AMS 2000 Mathematics subject classification: Primary 35K60. Secondary 35B40

We prove the following.

(1) The inequalities

$$ \biggl(2-\frac{1}{\varGamma(x)}\biggr)^{\!a}+\biggl(2-\frac{1}{\varGamma(1/x)}\biggr)^{\!a}\leq 2\leq\biggl(2-\frac{1}{\varGamma(x)}\biggr)^{\!b}+\biggl(2-\frac{1}{\varGamma(1/x)}\biggr)^{\!b} $$

hold for all $x>0$ if and only if

$$ -1.204\,64\ldots=2+\frac{1}{\gamma}-\frac{1}{6}\biggl(\frac{\pi}{\gamma}\biggr)^{\!2}\leq a\leq0\leq b. $$

(2) For all real numbers $x\in(0,1]$ we have

$$ x^{\alpha}\leq\frac{1}{2}\biggl(\frac{1}{\varGamma(x)}+\frac{1}{\varGamma(1/x)}\biggr)\leq x^{\beta}, $$

with the best possible constants

$$ \alpha=1.321\,76\dots\link{and}\beta=0. $$

These theorems extend and complement a result of Gautschi (from 1974), who proved that for all $x>0$ the harmonic mean of $\varGamma(x)$ and $\varGamma(1/x)$ is greater than or equal to $1$.

AMS 2000 Mathematics subject classification: Primary 33B15; 26D15

We investigate the first and second moments of shifted convolutions of the generalized divisor function d3(n).

Two n-planes Γ and Δ in real Euclidean r-space Rr are called isoclinic with parameter λ if the angle θ between any x in Γ and its orthogonal projection Px on Δ is unique, with cos2 θ = λ. Let vλ(n, r) denote the maximum number of equi-isoclinic (i.e. pairwise isoclinic) n-planes in Rr with parameter λ.

Strongly prime rings were introduced by Handelman and Lawrence [5] and in [2] Groenewald and Heyman investigated the upper radical determined by the class of all strongly prime rings. In this paper we extend the concept of strongly prime to near-rings. We show that the class M of distributively generated near-rings is a special class in the sense of Kaarli [6]. We also show that if N is any distributively generated near-ring, then UM(N), UM denotes the upper radical determined by the class M, coincides with the intersection of all the strongly prime ideals of N.

We prove the birational rigidity of Fano complete intersections of index 1 with a singular point of high multiplicity, which can be close to the degree of the variety. In particular, the groups of birational and biregular automorphisms of these varieties are equal, and they are non-rational. The proof is based on the techniques of the method of maximal singularities, the generalized 4n2-inequality for complete intersection singularities and the technique of hypertangent divisors.

In the representation theory of finite groups, the minimal idempotents of the group algebra play a central role. In this case the minimal idempotents determine irreducible modules over the group algebra, which in turn are in direct correspondence with the irreducible matrix representations of the group; see Chapter IV of the book of C. Curtis and I. Reiner (2). Many of the same ideas generalise to the situation where the group is compact. In addition, minimal idempotents are involved in some important parts of the theory of Hubert algebras; see M. Rieffel's paper (20).

The system $\dot{X}=TX+Q(X)$ (in $\mathbb{R}^{n}$), where $T$ is linear and $Q$ is quadratic, is considered via commutative algebras. The case of the linearized system having a centre manifold spanned on vectors $E_{1}$, $E_{2}$ (and $TE_{1}=\omega E_{2}$, $TE_{2}=-\omega E_{1}$) is studied. It is shown that for $\Span(E_{1},E_{2})$ being a subalgebra (of the algebra corresponding to the form $Q(X)$), the system is stable. Necessary and sufficient conditions are given for stability of the system in the case where $\mathrm{span}(E_{1},E_{2})$ is not a subalgebra.

AMS 2000 Mathematics subject classification: Primary 34A34; 34C15; 34C27; 34C35. Secondary 13M99

We describe a class of soluble groups with a finite complete rewriting system which includes all the soluble groups known to have such a system. It is an open question, related to deep questions in the theory of groups, whether it includes all soluble groups with such a system.

It is proved that the Bloch norm of an arbitrary $C^1$-function defined on the unit ball $\mathbb{B}_n\subset\mathbb{R}^n$ is equal to

$$ \sup_{x,y\in\mathbb{B}_n,\,x\neq y}{(1-|x|^2)^{1/2}(1-|y|^2)^{1/2}}\frac{|f(x)-f(y)|}{|x-y|}. $$

Soient K un anneau commutatif à élémént unité, P un K-module unitaire, SK(P) l'algèbre symétrique de P et le sous-K-module de SK(P) des éléments homogénes de degré m≧0. Rappelons que si P est un K-module libre de rang n + 1, l'algèbre SK(P) est isomorphe à l'algèbre des polynômes K[X0,X1,…,Xn] en les indéterminéesX0,X1,…,Xn à coefficients dans K et est le K-module libre dont une base est formée par les monômes tels que ie., Par les moôomes homogènes de degré m.

It is well-known that the number of irreducible characters of a finite group G is equal to the number of conjugate classes of G. The purpose of this article is to give some analogous properties between these basic concepts.

Groups for which the distributively generated near-ring generated by the endomorphisms is in fact a ring are known as E-groups and are discussed in (3). R. Faudree in (1) has given the only published examples of non-abelian E-groups by presenting defining relations for a family of p–groups. However, as shown in (3), Faudree's group does not have the desired property when p = 2.

The class of non-associative algebras over a field of characteristic zero named in the title is studied using a result of Ouattara [9]. As an application, the differential equation for overlapping generations in the time-continuous model is solved.

In many biological diffusion-reaction studies, it was found that one should include the effect of density dependent rates, drift terms and spatially varying growth rates, in order to obtain more accurate results. (See e.g. [7],[10], [8] , [3]). On the other hand, many recent mathematical results on reaction-diffusion systems do not include such general setting. This article investigates the behaviour of competing-species reaction-diffusion model under this more general situation. Efforts are made to obtain results concerning coexistence, survival and extinction, by methods similar to that in [5], [6].

We answer the following questions negatively: Does there exist a simple locally finite barely transitive group (LFBT-group)? More precisely we have: There exists no simple LFBT -group. We also deal with the question, whether there exists a LFBT-group G acting on an infinite set Ω so that G is a group of finitary permutations on Ω. Along this direction we prove: If there exists a finitary LFBT-group G, then G is a minimal non-FC p-group. Moreover we prove that: If a stabilizer of a point in a LFBT-group G is abelian, then G is metabelian. Furthermore G is a p-group for some prime p, G/G′ ≅ Cp∞, and G′ is an abelian group of finite exponent.

Let be a finite dimensional toric variety over an algebraically closed field of characteristic zero, k. Let be the sheaf of differential operators on . We show that the ring of global sections, is a finitely generated Noetherian k-algebra and that its generators can be explicitly found. We prove a similar result for the sheaf of differential operators with coefficients in a line bundle.

Chladni figures are formed when particles scattered across a plate move due to an external harmonic force resonating with one of the natural frequencies of the plate. Chladni figures are precisely the nodal set of the vibrational mode corresponding to the frequency resonating with the external force. We propose a plausible model for the movement of the particles that explains the formation of Chladni figures in terms of the stochastic stability of the equilibrium solutions of stochastic differential equations.

This paper considers Banach algebras with properties 𝔸 or 𝔹, introduced recently by Alaminos et al. The class of Banach algebras satisfying either of these two properties is quite large; in particular, it includes C *-algebras and group algebras on locally compact groups. Our first main result states that a continuous orthogonally additive n-homogeneous polynomial on a commutative Banach algebra with property 𝔸 and having a bounded approximate identity is of a standard form. The other main results describe Banach algebras A with property 𝔹 and having a bounded approximate identity that admit non-zero continuous symmetric orthosymmetric n-linear maps from An into ℂ.

Let G be an arbitrary non-zero additive subgroup of the complex number field ℂ, and let Vir[G] be the corresponding generalized Virasoro algebra over ℂ. In this paper we determine all irreducible weight modules with finite-dimensional weight spaces over Vir[G]. The classification strongly depends on the index group G. If G does not have a direct summand isomorphic to ℤ (the integers), then such irreducible modules over Vir[G] are only modules of intermediate series whose weight spaces are all one dimensional. Otherwise, there is one further class of modules that are constructed by using intermediate series modules over a generalized Virasoro subalgebra Vir[G0] of Vir[G] for a direct summand G0 of G with G = G0 ⊕ ℤb, where b ∈ G \ G0. This class of irreducible weight modules do not have corresponding weight modules for the classical Virasoro algebra.