It is proved that, given a convex polytope $P$ in $\mathbb{R}^n$, together with a collection of compact convex subsets in the interior of each facet of $P$, there exists a smooth convex body arbitrarily close to $P$ that coincides with each facet precisely along the prescribed sets, and has positive curvature elsewhere.

In this note a proof is given that in the case of finitely generated amenable groups, the classical zero divisor conjecture implies the analytic zero divisor conjecture of Linnell.

The authors of this paper study positive supersolutions to the elliptic equation $-\Delta U=c|x|^{-s}u^p$ in Cone-like domains of $\mathbb{R}^N$ ($N\ge 2$), where $p,s\in\mathbb{R}$ and $c>0$. They prove that in the sublinear case $p<1$ there exists a critical exponent $p_\ast<1$ such that the equation has a positive supersolution if and only if $-\infty<p<p_\ast$. The value of $p_\ast$ is determined explicitly by $s$ and the geometry of the cone.

For a finite group G and a subset S of G with 1∉S and S = S−1, the Cayley graph Cay(G, S) is the graph with vertex set G such that {x, y} is an edge if and only if yx−1∈S. The group G is called a CI-group if, for all subsets S and T of G[setmn ]{1}, Cay(G, S) ≅ Cay(G, T) if and only if Sσ = T for some σ∈Aut(G). In this paper, for each prime p ≡ 1 (mod 4), a symmetric graph Γ(p) is constructed from PSL(2, p) such that Aut Γ(p) = ℤ2 × PSL(2, p); it is then shown that A5 is not a CI-group, and that all finite CI-groups are soluble.

In this paper we find the norm of powers of the indefinite integral operator V, acting on L2(0, 1). This answers a question raised by Halmos, and supplements some recent results of Manakov in [9]. Using results of Stepanov in [13], we show that the operator norm of Vn is asymptotically equal to the Hilbert–Schmidt norm as n→∞.

For the general one-dimensional Schrödinger operator −d2/dx2+q(x) with real q ∈ L1(ℝ), this paper presents a new series representation of the Jost solution which, in turn, implies a new asymptotic representation of the Weyl m-function for locally summable q. This representation is then applied to smooth potentials q to obtain Weyl m-function power asymptotics. The condition q(N) ∈ L1(x0, x0+δ), for N ∈ ℕ0, allows one to derive the (N+1) term for almost all x ∈ [x0, x0+δ), thereby refining a relevant result by Danielyan, Levitan and Simon.

The paper studies an isoperimetric problem for the Gaussian measure and coordinatewise symmetric sets. The notion of boundary measure corresponding to the uniform enlargement is considered, and it is proved that symmetric strips or their complements have minimal boundary measure.

The authors prove in this paper that, given any knot $k$ of genus $g$ , $k$ fails to be strongly $n$ -trivial for all $n$ with $n\ge 6g-3$ .

This paper considers the cone multiplier operator which is defined by $\[\widehat{S^\mu f}(\xi,\tau)=m_\mu(\xi,\tau)\widehat f(\xi,\tau)$, $\qquad (\xi,\tau)\in \mathbb R^2\times \mathbb R\]$ where $m_\mu(\xi,\tau)=\phi(\tau)(1-|\xi|^2/\tau^2)_+^\mu/\Gamma(\mu+1)$ and $\phi\in C_0^\infty(1,2)$. For $-3/2<\mu<-3/14$, sharp $L^p-L^q$ estimates and endpoint estimates for $S^{\mu}$ are obtained.

In the context of Lebesgue integration the Gauss–Green theorem is proved for bounded vector fields with substantial sets of singularities with respect to continuity and differentiability. The resulting integration by parts is applied to removable sets for the Cauchy–Riemann, Laplace, and minimal surface equations. A simple connection between the Gauss–Green theorem and distributional divergence is established.

the degree defined by p. benevieri and m. furi (see ‘a simple notion of orientability for fredholm maps of index zero between banach manifolds and degree theory’, ann. sci. math. québec 22 (1998) 131–148) is used here to obtain some sharp lower bounds for the number of solutions of the $\lambda$-sections of the compact components of the set of non-trivial solutions of $\mf{f}(\lambda,x)=0$, where $\mf{f}$ is a $\cal{c}^1$ fredholm map of index 1 such that $\mf{f}(\lambda,0)=0$ for all $\lambda\in\mathbb{r}$. these bounds are given in terms of the parity of the linearized fredholm family $d_2\mf{f}(\cdot,0)$. the parity is a local invariant measuring the change of the orientation of $d_2\mf{f}(\lambda,0)$ as $\lambda$ crosses an interval. the set of eigenvalues of $d_2\mf{f}(\cdot,0)$ is not assumed to be discrete. therefore, even in the classical situation when $\mf{f}$ is a compact perturbation of the identity, the results presented here are completely new.

it is proved in this paper that super-exponentially decaying, possibly non-selfadjoint perturbations of the free jacobi operator are uniquely determined by the location of all their eigenvalues and resonances.

A sharp function estimate is established for multilinear singular integral operators. As applications, some weighted norm inequalities are obtained for such operators.This research was supported by NSF of Henan Province and NNSF (No. 10271015) of China.

A classical theorem of Zimmermann describes the relation between almost split sequences in the category of finitely presented modules and those in the category of all modules over some fixed ring. An analogue of Auslander–Reiten triangles in triangulated categories is proved in this paper. This is used to explain the relation between different existence results for Auslander–Reiten triangles, which are based either on Brown's representability theorem, or on the existence of Serre functors.

In this paper, a number of results are deduced on the arithmetic structure of products of integers in short intervals. By way of an example, work of Saradha and Hanrot, and of Saradha and Shorey, is completed by the provision of an answer to the question of when the product of $k$ out of $k+1$ consecutive positive integers can be an ‘almost’ perfect power. The main new ingredient in these proofs is what might be termed a practical method for resolving high-degree binomial Thue equations of the form $ax^n - by^n = \pm 1$, based upon results from the theory of Galois representations and modular forms.

In the model of the OK Corral formulated by Williams and McIlroy [2]: ‘Two lines of gunmen face each other, there being initially m on one side, n on the other. Each person involved is a hopeless shot, but keeps firing at the enemy until either he himself is killed or there is no one left on the other side.’ They are interested in the number S of survivors when the shooting ceases, and the surprising result, for which they give both numerical and heuristic evidence, is that when m=n,

formula here

as m→∞ (where E denotes expectation).

It is the occurrence of this curious power of m, rather than any application to real gunfights, which makes the Williams–McIlroy process of interest. The purpose of this paper is to give an essentially elementary proof of the fact that if m and n are not too different, then S/m3/4 has a simple asymptotic distribution that leads at once to (1.1).

Let ${\cal L}$ be a uniformly elliptic operator in divergence form on ${\bb R}^d$ , and let $p(t,x,y)$ be the fundamental solution to the heat equation for ${\cal L}$ . A new proof is given of Aronson's upper bound: \[ p(t,x,y)\le c_1t^{-d/2}\exp(-c_2|x-y|^2/t). \]

The relative rank $\mbox{\rm rank}(S:U)$ of a subsemigroup $U$ of a semigroup $S$ is the minimum size of a set $V\subseteq S$ such that $U$ together with $V$ generates the whole of $S$. As a consequence of a result of Sierpiński, it follows that for $U\leq \mathcal{T}_{X}$, the monoid of all self-maps of an infinite set $X$, $\mbox{\rm rank}(\mathcal{T}_{X}:U)$ is either 0, 1 or 2, or uncountable. In this paper, the relative ranks $\mbox{\rm rank}(\mathcal{T}_{X}:\mathcal{O}_{X})$ are considered, where $X$ is a countably infinite partially ordered set and $\mathcal{O}_{X}$ is the endomorphism monoid of $X$. We show that $\mbox{\rm rank}(\mathcal{T}_{X}:\mathcal{O}_{X})\leq 2$ if and only if either: there exists at least one element in $X$ which is greater than, or less than, an infinite number of elements of $X$; or $X$ has $|X|$ connected components. Four examples are given of posets where the minimum number of members of $\mathcal{T}_{X}$ that need to be adjoined to $\mathcal{O}_{X}$ to form a generating set is, respectively, 0, 1, 2 and uncountable.

The values taken by $\Gamma_0(n)$-modular functions at elliptic points of order 2 for the Fricke group $\Gamma_0(n)^\dagger$ that lie outside $\Gamma_0(n)$ are studied. In the case of a principal modulus (‘Hauptmodul’) for $\Gamma_0(n)$ or $\Gamma_0(n)^\dagger$, the class fields generated by these values are determined.

Let X be a compact manifold with a smooth action of a compact connected Lie group G. Let L → X be a complex line bundle. Using the Cartan complex for equivariant cohomology, we give a new proof of a theorem of Hattori and Yoshida which says that the action of G lifts to L if and only if the first Chern class c1(L) of L can be lifted to an integral equivariant cohomology class in H2G(X; ℤ), and that the different lifts of the action are classified by the lifts of c1(L) to H2G(X; ℤ). As a corollary of our method of proof, we prove that, if the action is Hamiltonian and ∇ is a connection on L which is unitary for some metric on L, and which has a G-invariant curvature, then there is a lift of the action to a certain power Ld (where d is independent of L) which leaves fixed the induced metric on Ld and the connection ∇[otimes ]d. This generalises to symplectic geometry a well-known result in geometric invariant theory.