The action of translation operators on wavelet subspaces in higher dimensions is investigated. This action defines an equivalence relation on the set of single wavelets of $L^2(\mathbb R^n)$ associated with an arbitrary dilation matrix. The corresponding equivalence classes are characterized in terms of the support of the Fourier transform of the wavelets. Further, examples of wavelets in each of these classes are constructed. This construction shows the existence of wavelets for which the associated wavelet subspaces are invariant under various groups of translation operators.

Under very general conditions, a proof is given, via Morse theory, of the existence of solutions for asymptotically ‘linear’ $p$-Laplacian equations, where the asymptotic limit may be greater than the second eigenvalue. The existence of nonzero solutions is also considered.

Transformation formulas are obtained for generalized Dedekind eta functions; these are simpler to apply than Schoeneberg's formulas. As an application, a list is given of the generators of all the function fields associated with torsion-free genus zero congruence subgroups of $\hbox{PSL}_2(\mathbb R)$.

In this paper, half-dimensional Einstein–Kähler submanifolds or locally reducible Kähler submanifolds are classified in a quaternion projective space.

Frank Smithies was born in Edinburgh, on 10 March 1912. His father, also Frank Smithies, and his mother, Mary (née Blakemore) had met through their involvement in the socialist movement. They had two children, Frank and his sister Violet, some four years younger.

Let $\idemp$ be the set of idempotents in a finite-dimensional real algebra $A$. Let $p$ and $q$ be idempotents that lie in the same component of $\idemp$. Then, among the continuous paths connecting $p$ and $q$ in $\idemp$, there exist a polynomial path of degree at most $3$ and a polygonal path consisting of at most three segments.

The integration-by-parts methods introduced in this paper improve upon the $L^p$ estimates on transport densities given in the recent paper by L. De Pascale and A. Pratelli (Calc. Var. Partial Differential Equations 14 (2002) 249–274).

Simply connected closed symplectic 4-manifolds with $b_{2}^+ \,{=}\,1$ and $K^2 \,{=}\,0$ are investigated. As a result, it is confirmed that most of homotopy elliptic surfaces $\{E(1)_{K} \,{|}\,K\, \mathrm{is\, a\, fibred\, knot}$ in $S^3\}$ constructed by R. Fintushel and R. Stern in Invent. Math. 134 (1998) 363–400 are simply connected closed minimal symplectic 4-manifolds that do not admit a complex structure.This work was supported by grant No. R14-2002-007-01002-0 from KOSEF.

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.

The infimal Heegaard gradient of a 3-manifold was defined and studied by Marc Lackenby in an approach towards proving the well-known virtually Haken conjecture. As instructive examples, Seifert fibered 3-manifolds are considered in this paper. The author shows that a compact orientable Seifert fibered 3-manifold has zero infimal Heegaard gradient if and only if it virtually fibers over either the circle or a surface other than the 2-sphere or, equivalently, if it has infinite fundamental group.

Cet article présente des minorations universelles de la première valeur propre non nulle du laplacien des formes différentielles sur les domaines euclidiens ou hémisphériques convexes ou les hypersurfaces euclidiennes convexes.

Let $s:\Q\,{\longrightarrow}\,\Q$ be the Dedekind sum, given by $s(h/k)=\sum_{\nu=1}^{k-1}({\nu/k}\,{-}\,{1/2})(\{{h\nu/k}\}\,{-}\,{1/2})$ when $\gcd(h,k)\,{=}\,1$. Then for every rational $\alpha\,{\ne}\,1/12$ there are infinitely many rational $x$ such that $s(x)\,{=}\,\alpha x$. Also, the fixed points of $s$ are dense in the real line.

In his curriculum vitae, John Hawkes lists his research interests as geometric measure theory, probability (Lévy processes), and potential theory (probabilistic). In fact, he made significant contributions to all three areas, and there are strong relationships between them. He used both geometric measure theory and potential theory as tools for his study of the trajectories of particular Lévy processes, but in many cases he needed to develop the tool before it was ready to be used. We will summarise his research later, but first we discuss what is known of his life history.

In terms of the upper bounds of a second-order elliptic operator acting on specific Lyapunov-type functions with compact level sets, sufficient conditions are presented for the corresponding Dirichlet form to satisfy the Poincaré and the super-Poincaré inequalities. Here, the elliptic operator is assumed to be symmetric on $L^2(\mu)$ with some probability measure $\mu$. As applications, proofs are given for a class of (non-symmetric) diffusion operators generating $C_0$-semigroups on $L^1(\mu)$: that their $L^p(\mu)$-essential spectrum is empty for $p>1$. This follows since it is proved that their $C_0$-semigroups are compact.Supported in part by the NNSFC for Distinguished Young Scholars (10025105), the 973-Project and TRAPOYT in China.

The following property of a normalized basis in a Banach space is considered: any normalized block sequence of the basis has a subsequence equivalent to the basis. Under uniformity or other natural assumptions, a basis with this property is equivalent to the unit vector basis of $c_0$ or $\ell_p$. An analogous problem concerning spreading models is also addressed.

Let $J$ be a Jacobi real symmetric matrix on $l_{2}$ with zero diagonal and non-diagonal entries of the form $\{1+p_{n}\}$. If $p_{n-1}\pm p_{n}=O(n^{-\alpha})$ with some $\alpha>2/3$, then the existence of bounded solutions of $Ju=\lambda u$ is proved for almost every $\lambda\in(-2,2)$ with the WKB-type asymptotic behavior.

The methods of $p$-adic analysis are used to prove a congruence for $(pn)!/(p^{n}n!)$ modulo a power of a prime $p$.

For $f$ meromorphic in the complex plane and $\varphi$ meromorphic in the unit disc, sharp upper bounds are obtained for $$m\left(r,\frac{f^{(k)}}{f^{(j)}}\right)=\frac{1}{2\pi}\int_0^{2\pi} \log^{+}\left|\frac{f^{(k)}(re^{i\theta})}{f^{(j)}(re^{i\theta})}\right |\, d\theta,\qquad r<\infty,$$ and $$m\left(r,\frac{\vp^{(k)}}{\vp^{(j)}}\right)=\frac{1}{2\pi}\int_0^{2\pi} \log^{+}\left|\frac{\vp^{(k)}(re^{i\theta})}{\vp^{(j)}(re^{i\theta})}\right |\, d\theta,\qquad r<1,$$ where $k$ and $j$ are integers satisfying $k>j\geq 0$. The results generalize the logarithmic derivative estimate due to Gol'dberg and Grinshtein to derivatives of higher order.

Sharp decay estimates are provided in this paper for spherical averages of a certain multilinear extension operator on $L^{2}(\mathbb{S}^{n-1})\times \ldots\times L^{2}(\mathbb{S}^{n-1})$.