In this paper we are concerned with the following conjecture.

Conjecture. For any positive integers n and k satisfying k < n, and any sequence a1, a2, … ak of not necessarily distinct elements of Zn, there exists a permutation π ∈ Sk such that the elements aπ(i)+i are all distinct modulo n.

We prove this conjecture when 2k [les ] n+1. We then apply this result to tree embeddings. Specifically, we show that, if T is a tree with n edges and radius r, then T decomposes Kt for some t [les ] 32(2r+4)n2+1.

Häggkvist and Scott asked whether one can find a quadratic function q(k) such that, if G is a graph of minimum degree at least q(k), then G contains vertex-disjoint cycles of k consecutive even lengths. In this paper, it is shown that if G is a graph of average degree at least k2+19k+10 with sufficiently many vertices, then G contains vertex-disjoint cycles of k consecutive even lengths, answering the above question in the affirmative. The coefficient of k2 cannot be decreased and, in this sense, this result is best possible.

We consider the problem of sampling ‘unlabelled structures’, i.e., sampling combinatorial structures modulo a group of symmetries. The main tool which has been used for this sampling problem is Burnside’s lemma. In situations where a significant proportion of the structures have no nontrivial symmetries, it is already fairly well understood how to apply this tool. More generally, it is possible to obtain nearly uniform samples by simulating a Markov chain that we call the Burnside process: this is a random walk on a bipartite graph which essentially implements Burnside’s lemma. For this approach to be feasible, the Markov chain ought to be ‘rapidly mixing’, i.e., converge rapidly to equilibrium. The Burnside process was known to be rapidly mixing for some special groups, and it has even been implemented in some computational group theory algorithms. In this paper, we show that the Burnside process is not rapidly mixing in general. In particular, we construct an infinite family of permutation groups for which we show that the mixing time is exponential in the degree of the group.

For a stochastic approximation-type recursion with finitely many possible limit points, we find a lower bound on the probability of converging to a prescribed point in its ‘domain of attraction’. This has implications for the lock-in phenomena in the stochastic models of increasing return economics and the sample complexity of stochastic approximation algorithms in engineering.

Suppose that G is a graph with maximum degree d(G) such that, for every vertex v in G, the neighbourhood of v contains at most d(G)2/f (f > 1) edges. We show that the list chromatic number of G is at most Kd(G)/log f, for some positive constant K. This result is sharp up to the multiplicative constant K and strengthens previous results by Kim [9], Johansson [7], Alon, Krivelevich and Sudakov [3], and the present author [18]. This also motivates several interesting questions.

As an application, we derive several upper bounds for the strong (list) chromatic index of a graph, under various assumptions. These bounds extend earlier results by Faudree, Gyárfás, Schelp and Tuza [6] and Mahdian [13] and determine, up to a constant factor, the strong (list) chromatic index of a random graph. Another application is an extension of a result of Kostochka and Steibitz [10] concerning the structure of list critical graphs.

Let $E$ be a UMD Banach space, and $L$ a positive self-adjoint operator in $\mathrm{L}^2$ of Laplace type, for which the imaginary powers $L^{-\ri t}$ form a $C_0$-group of exponential growth $0\leq\alpha\lt \pi$ on $\mathrm{L}^p(E)$, where $1\lt p\lt\lt \infty$. Suppose $G(z)$ is holomorphic inside and on the boundary of the sector $\{z:z\neq0,\ |\arg z|\leq\phi\}$, and $z^\kappa G(z)\rightarrow0$ uniformly as $z\rightarrow\infty$ for some $\kappa\gt0$ and $\phi\gt\alpha$. Then $G(tL)$ $(t \gt0)$ defines a bounded family of linear operators on $\mathrm{L}^p(E)$; and the maximal operator $f\mapsto\sup_{t \gt 0}\|G(tL)f\|_E$ is bounded on the domain of $\log L$. The proof uses transference methods. These hypotheses hold for the maximal solution operators for the heat, wave and Schrödinger equations, and for Cesàro sums.

AMS 2000 Mathematics subject classification: Primary 47D03; 42B25; 47D09

We characterize the Bloch space and the Besov spaces of harmonic functions on the open unit disc $D$ by using the following oscillation:

$$ \sup_\{\beta(z,w)\ltr\}(1-|z|^2)^{\alpha}(1-|w|^2)^{\beta}\biggl|\frac{\hat{D}^{(n-1)}h(z)-\hat{D}^{(n-1)}h(w)}{z-w}\biggr|, $$

where $\alpha+\beta=n$, $\alpha,\beta\in\mathbb{R}$ and $\displaystyle{\hat{D}^{(n)}=(\partial^{n}/\partial^{n}z+\partial^{n}/\partial^{n}\bar{z})}$.

AMS 2000 Mathematics subject classification: Primary 46E15

In 1950 Erdös proved that if $x\equiv2\,036\,812\ (\mo5\,592\,405)$ and $x\equiv3\ (\mo62)$, then $x$ is not of the form $2^n+p$, where $n$ is a non-negative integer and $p$ is a prime. In this note we present a theorem on integers of the form $2^n+cp$, in particular we completely determine all those integers $c$ relatively prime to $5\,592\,405$ such that the residue class $2\,036\,812(\mo5\,592\,405)$ contains integers of the form $2^n+cp$.

AMS 2000 Mathematics subject classification: Primary 11P32. Secondary 11A07; 11B25; 11B75

Let $G$ be a countable discrete group and let $M$ be a proper free $C^r$ $G$-manifold and $N$ a $C^r$ $G$-manifold, where $1\leq r\leq\omega$. We prove that if $G$ acts properly and freely also on $N$ and if $\dim(N)\geq2\dim(M)$, then equivariant immersions form an open dense subset in the space $C^r_G(M,N)$ of all equivariant $C^r$ maps from $M$ to $N$. The space $C^r_G(M,N)$ is equipped with a topology, which coincides with the Whitney $C^r$ topology if $G$ is finite and is suited to studying equivariant maps. We also prove an equivariant version of Thom’s transversality theorem and show that $C^\omega_G(M,N)$ is dense in $C^r_G(M,N)$, for $1\leq r\leq\infty$.

AMS 2000 Mathematics subject classification: Primary 57S20

Let $v$ be a henselian valuation of a field $K$ with value group $G$, let $\bar{v}$ be the (unique) extension of $v$ to a fixed algebraic closure $\bar{K}$ of $K$ and let $(\tilde{K},\tilde{v})$ be a completion of $(K,v)$. For $\alpha\in\bar{K}\setminus K$, let $M(\alpha,K)$ denote the set $\{\bar{v}(\alpha-\beta):\beta\in\bar{K},\ [K(\beta):K] \lt [K(\alpha):K]\}$. It is known that $M(\alpha,K)$ has an upper bound in $\bar{G}$ if and only if $[K(\alpha):K]=[\tilde{K}(\alpha):\tilde{K}]$, and that the supremum of $M(\alpha,K)$, which is denoted by $\delta_{K}(\alpha)$ (usually referred to as the main invariant of $\alpha$), satisfies a principle similar to the Krasner principle. Moreover, each complete discrete rank 1 valued field $(K,v)$ has the property that $\delta_{K}(\alpha)\in M(\alpha,K)$ for every $\alpha\in\bar{K}\setminus K$. In this paper the authors give a characterization of all those henselian valued fields $(K,v)$ which have the property mentioned above.

AMS 2000 Mathematics subject classification: Primary 12J10; 12J25; 13A18

We solve a joint similarity problem for pairs of operators of Foias–Williams/Peller type on weighted Bergman spaces. We show that for the single operator, the Hardy space theory established by Bourgain and Aleksandrov–Peller carries over to weighted Bergman spaces, by establishing the relevant weak factorizations. We then use this fact, together with a recent dilation result due to the first author and Rochberg, to show that a commuting pair of such operators is jointly polynomially bounded if and only if it is jointly completely polynomially bounded. In this case, the pair is jointly similar to a pair of contractions by Paulsen’s similarity theorem.

AMS 2000 Mathematics subject classification: Primary 47B35; 47B47

We study a three parameter deformation $\mathcal{U}_{abc}$ of $\mathcal{U}(\mathfrak{sl}_2)$ introduced by Le Bruyn in 1995. Working over an arbitrary algebraically closed field of characteristic zero, we determine the centres, the finite-dimensional irreducible representations, and, when the parameter $a$ is not a non-trivial root of unity, the prime ideals of those $\mathcal{U}_{abc}$, with $ac\neq0$, which are conformal as ambiskew polynomial rings.

AMS 2000 Mathematics subject classification: Primary 16W35; 17B37. Secondary 16S36; 16S80

We study the geometry of surfaces in $\mathbb{R}^4$ associated to contact with hyperplanes. We list all possible transitions that occur on the parabolic and so-called $A_3$-set, and analyse the configurations of the asymptotic curves and their bifurcations in generic one-parameter families.

AMS 2000 Mathematics subject classification: Primary 58C27. Secondary 53A05

We establish a generalization of the Widder–Arendt theorem from Laplace transform theory. Given a Banach space $E$, a non-negative Borel measure $\mea$ on the set $\Rplus$ of all non-negative numbers, and an element $\bnd$ of $\R\cup\{-\infty\}$ such that $\natres{-\coefl}$ is $\mea$-integrable for all $\coefl>\bnd$, where $\natres{-\coefl}$ is defined by $\natres{-\coefl}(t)=\exp(-\coefl t)$ for all $t\in\Rplus$, our generalization gives an intrinsic description of functions $\f\colon\Set\to E$ that can be represented as $\f(\coefl)=T(\natres{-\coefl})$ for some bounded linear operator $T\colon\Ma\to E$ and all $\coefl> \bnd$; here $\Ma$ denotes the Lebesgue space based on $\mea$. We use this result to characterize pseudo-resolvents with values in a Banach algebra, satisfying a growth condition of Hille–Yosida type.

AMS 2000 Mathematics subject classification: Primary 44A10; 47A10. Secondary 43A20; 46B22; 46G10; 46J25; 47D06

In this paper we give a relation between the Futaki invariant for a compact complex manifold $M$ and the holonomy of a determinant line bundle over a loop in the base space of any principal $G$-bundle, where $G$ is the identity component of the maximal compact subgroup of the complex Lie group consisting of all biholomorphic automorphisms of $M$. Using the property of the Futaki invariant, we show that the holonomy is an obstruction to the existence of the Einstein–Kähler metrics on $M$. Our main result is Theorem 2.1.

AMS 2000 Mathematics subject classification: Primary 32Q20. Secondary 58J28; 58J52

We consider a parabolic matrix–vector system in which the diffusion matrix may be time dependent. For the time-independent case we construct approximate solutions with guaranteed error bounds using spectral information from certain matrix–vector Sturm–Liouville problems. For the time-dependent case we employ an approximation procedure which reduces the problem, on each time-step, to the time-independent case. We give an algorithm which may be used a priori at each time-step in the time-dependent case to guarantee accuracy to a specified tolerance.

AMS 2000 Mathematics subject classification: Primary 35C10; 35M10. Secondary 15A

It is proved in this paper that for any non-elementary subgroup $G$ of $\mathrm{PSL}(2,\sGa_n)$, which has no elliptic element, to be not strict, there is a minimal generating system of $G$ consisting of loxodromic elements, and that if $G$ is a non-elementary subgroup of $\mathrm{PSL}(2,\sGa_n)$ of which each loxodromic element is hyperbolic, then $G$ is conjugate to a subgroup of $\mathrm{PSL}(2,\mathbb{R})$.

AMS 2000 Mathematics subject classification: Primary 30F40. Secondary 20H10

We present an alternative proof of the existence of density-conserving solutions to the discrete coagulation–fragmentation equations when the coagulation rates grow at most linearly. The proof relies on the study of the propagation of some moments of the solutions to approximating equations and simplifies the previous argument of Ball and Carr which involves rather delicate estimates. The case of multiple fragmentation is also considered, and the question of uniqueness as well.

AMS 2000 Mathematics subject classification: Primary 34A34; 82C22