Some elementary results are given on restricted sums $2^{\wedge}{\cal A}$ and $3^{\wedge}{\cal A}$ of a subset ${\cal A}\subset {\bb Z}/n{\bb Z}$ in the case when the cardinality of ${\cal A}$ , is close to $n/2$ . In the last section, some of these results are used to derive some new values of a function related to the Erdős–Ginzburg–Ziv problem.

The paper studies a generalized spherical function, or a generalized Whittaker model for generalized principal series representations of $G={\rm Sp}(2,{\bf R})$ induced from the Jacobi maximal parabolic subgroup $P_{\rm J}$ , which is called the Fourier–Jacobi type. In particular, a multiplicity theorem and an explicit formula via the Meijer $G$ -functions for this function are given.

The paper computes the Hausdorff dimension of sets of real numbers which are close to infinitely many real algebraic integers of bounded degree. It also investigates the distribution of real algebraic integers of bounded degree, which are proved to be evenly spaced.

Let $(R,{\frak m})$ be a local ring with prime ideals ${\frak p}$ and ${\frak q}$ such that $\sqrt{{\frak p}+{\frak q}}={\frak m}$ . If $R$ is regular and contains a field, and $\dim(R/{\frak p})+\dim(R/{\frak q})=\dim(R)$ , then it is proved that ${\frak p}^{(m)}\cap {\frak q}^{(n)}\subseteq {\frak m}^{m+n}$ for all positive integers $m$ and $n$ . This is proved using a generalization of Serre's Intersection Theorem which is applied to a hypersurface $R/fR$ . The generalization gives conditions that guarantee that Serre's bound on the intersection dimension $\dim(R/{\frak p})+\dim(R/{\frak q})\le \dim(R)$ holds when $R$ is nonregular.

It is proved that there exists a scheme that represents the functor of line modules over a graded algebra, and it is called the line scheme of the algebra. Its properties and its relationship to the point scheme are studied. If the line scheme of a quadratic, Auslander-regular algebra of global dimension 4 has dimension 1, then it determines the defining relations of the algebra.

Moreover, the following counter-intuitive result is proved. If the zero locus of the defining relations of a quadratic (not necessarily regular) algebra on four generators with six defining relations is finite, then it determines the defining relations of the algebra. Although this result is non-commutative in nature, its proof uses only commutative theory.

The structure of the line scheme and the point scheme of a 4-dimensional regular algebra is also used to determine basic incidence relations between line modules and point modules.

The following autoduality theorem is proved for an integral projective curve $C$ in any characteristic. Given an invertible sheaf ${\cal L}$ of degree 1, form the corresponding Abel map $A_{\cal L}:C\longrightarrow \bar{J}$ , which maps $C$ into its compactified Jacobian, and form its pullback map $A^{\ast}_{\cal L}:{\rm Pic}^0_{\bar{J}}\longrightarrow J$ , which carries the connected component of $0$ in the Picard scheme back to the Jacobian. If $C$ has, at worst, points of multiplicity $2$ , then $A^{\ast}_{\cal L}$ is an isomorphism, and forming it commutes with specializing $C$ .

Much of the work in the paper is valid, more generally, for a family of curves with, at worst, points of embedding dimension $2$ . In this case, the determinant of cohomology is used to construct a right inverse to $A^{\ast}_{\cal L}$ . Then a scheme-theoretic version of the theorem of the cube is proved, generalizing Mumford's, and it is used to prove that $A^{\ast}_{\cal L}$ is independent of the choice of ${\cal L}$ . Finally, the autoduality theorem is proved. The presentation scheme is used to achieve an induction on the difference between the arithmetic and geometric genera; here, special properties of points of multiplicity $2$ are used.

A one-to-one correspondence is described between the set ${\cal J}(m)$ of numerical semigroups with multiplicity $m$ and the set of non-negative integer solutions of a system of linear Diophantine inequalities. This correspondence infers in ${\cal J}(m)$ a semigroup structure and the resulting semigroup is isomorphic to a subsemigroup of ${\bb N}^{m-1}$ . Finally, this result is particularized to the symmetric case.

We consider the Sturm–Liouville equation \renewcommand{\theequation}{1.\arabic{equation}} \begin{equation} y^{\prime\prime}(x)+\{\lambda - q(x)\}y(x) = 0\quad (0 \le x < \infty) \end{equation} with a boundary condition at $x = 0$ which can be either the Dirichlet condition \begin{equation} y(0) = 0 \end{equation} or the Neumann condition \begin{equation} y^\prime(0) = 0. \end{equation} As usual, $\lambda$ is the complex spectral parameter with $0 \le \arg \lambda < 2\pi$ , and the potential $q$ is real-valued and locally integrable in $[0, \infty)$ .

Second-order elliptic operators realized under Dirichlet boundary conditions in bounded domains are shown to generate strongly continuous semigroups in the topology of uniform convergence. The stochastic representation and integral representation with kernel for the strongly continuous semigroups are given. Furthermore, several useful formulae for the principal eigenvalue of second-order elliptic operators are obtained.

Fuglede's conjecture states that a set $\Omega\subset{\mathbf R}^n$ tiles ${\mathbf R}$ by translations if and only if $L^2(\Omega)$ has an orthogonal basis of exponentials. New partial results are obtained supporting the conjecture in dimension 1.

Multilinear commutators with vector symbol $\vec{b}=(b_1,\ldots,b_m)$ defined by \[ T_{\vec{b}}(f)(x)=\int_{{\bb R}^n}\Bigg[\prod\limits^m_{j=1}(b_j(x)-b_j(y))\Bigg]K(x,y)f(y)dy \] are considered, where $K$ is a Calderón–Zygmund kernel. The following a priori estimates are proved for $w\in A_\infty$ . For $0 < p < \infty$ , there exists a constant $C$ such that \[ \|\dot{T}_{{\vec{b}}}(f)\|_{L^P(w)}\le C\|\vec{b}\|\|M_{L(\log\,L)^{1/r}}(f)\|_{L^P(w)} \] and \[ \sup_{t>0}\frac{1}{\Phi(\frac{1}{t})}w(\{y\in{\bb R}^n:|T_{\vec{b}}f(y)|>t\})\le C\sup_{t>0}\frac{1}{\Phi(\frac{1}{t})}w(\{y\in{\bb R}^n:M_{L(\log\,L)^{1/r}}(\|\vec{b}\|f)(y)>t\}), \] where \begin{eqnarray*} &\|\vec{b}\|=\prod\limits^m_{j=1}\|b_j\|_{osc_{\exp L}^r j},\\ &\Phi(t)=t\log^{1/r}(e+t),\quad \frac{1}{r}=\frac{1}{r_1}+\cdots+\frac{1}{r_m}, \end{eqnarray*} and $M_{L(\log L)^{\alpha}}$ is an Orlicz type maximal operator. This extends, with a different approach, classical results by Coifman.

As a corollary, it is deduced that the operators $T_{\vec{b}}$ are bounded on $L^p(w)$ when $w\in A_p$ , and that they satisfy corresponding weighted $L(\log\,L)^{1/r}$ -type estimates with $w\in A_1$ .

Let $G$ be a separable locally compact group and let $\hat{G}$ be its dual space with Fell's topology. It is well known that the set $P(G)$ of continuous positive-definite functions on $G$ can be identified with the set of positive linear functionals on the group $C^*$ -algebra $C^*(G)$ . We show that if $\pi$ is discrete in $\hat{G}$ , then there exists a nonzero positive-definite function $\phi_{\pi}$ associated with $\pi$ such that $\phi_{\pi}$ is a $w^*$ -strongly exposed point of $P(G)_0$ , where $P(G)_0=\{f\in p(G):f(e)\le 1\}$ . Conversely, if some nonzero positive-definite function $\phi_\pi$ associated with $\pi$ is a $w^*$ -strongly exposed point of $P(G)_0$ , then $\pi$ is isolated in $\hat{G}$ . Consequently, $G$ is compact if and only if, for every $\pi\in\hat{G}$ , there exists a nonzero positive-definite function associated with $\pi$ that is a $w^*$ -strongly exposed point of $P(G)_0$ . If, in addition, $G$ is unimodular and $\pi\in\hat{G}_\rho$ , then $\pi$ is isolated in $\hat{G}_{\rho}$ if and only if some nonzero positive-definite function associated with $\pi$ is a $w^*$ -strongly exposed point of $P_{\rho}(G)_0$ , where $\rho$ is the left regular representation of $G$ and $\hat{G}_{\rho}$ is the reduced dual space of $G$ . We prove that if $B_{\rho}(G)$ has the Radon–Nikodym property, then the set of isolated points of $\hat{G}_{\rho}$ (so square-integrable if $G$ is unimodular) is dense in $\hat{G}_{\rho}$ . It is also proved that if $G$ is a separable SIN-group, then $G$ is amenable if and only if there exists a closed point in $\hat{G}_{\rho}$ . In particular, for a countable discrete non-amenable group $G$ (for example the free group $F_2$ on two generators), there is no closed point in its reduced dual space $\hat{G}_{\rho}$ .

The Beurling algebras $l^1({\cal D},\omega)\;({\cal D}={\bb N},{\bb Z})$ that are semi-simple, with compact Gelfand transform, are considered. The paper gives a necessary and sufficient condition (on $\omega$ ) such that $l^1({\cal D},\omega)$ possesses a uniform quantitative version of Wiener's theorem in the sense that there exists a function $\phi:]0,+\infty[\longrightarrow ]0,+\infty[$ such that, for every invertible element $x$ in the unit ball of $l^1({\cal D},\omega)$ , we have \[ \|x^{-1}\|\le \phi(r(x^{-1}))\quad r(x^{-1})\hbox{ is the spectral radius of }x^{-1}. \]

Time-variant filters based on Calderón and Gabor reproducing formulas are important tools in time-frequency analysis. The paper studies the behavior of the eigenvalues of these filters. Optimal two-sided estimates of the number of eigenvalues contained in the interval $(\delta_1,\delta_2)$ , where $0<\delta_1<\delta_2<1$ , are obtained. The estimates cover large classes of localization domains and generating functions.

The interaction between harmonic analysis and operator theory has been fruitful. In [1] Arveson related results in the theory of operator algebras to spectral synthesis. He defined synthesis for subspace lattices and proved that certain classes of lattices are synthetic. The main result in this paper is a generalization of a result of Arveson for the case of subspace maps. In order to describe in more detail the content of the present work, we need to introduce some definitions and facts from [2] and [9].

Let ${\cal H}_1$ and ${\cal H}_2$ be separable complex Hilbert spaces, and ${\cal P}_i$ be the lattice of all (orthogonal) projections on ${\cal H}_i, i = 1, 2$ . Following Erdos [1], we let ${\cal M}({\cal P}_1, {\cal P}_2)$ denote the set of all maps $\varphi : {\cal P}_1 \longrightarrow {\cal P}_2$ which are 0-preserving and $\vee$ -continuous (that is, they preserve arbitrary suprema). We will call such maps subspace maps. It was shown in [2] that each $\varphi \in {\cal M}({\cal P}_1, {\cal P}_2)$ uniquely defines semi-lattices ${\cal S}_{1\varphi} \subseteq {\cal P}_1$ and ${\cal S}_{2\varphi} = \varphi({\cal P}_1) \subseteq {\cal P}_{2}$ such that $\varphi$ is a bijection between ${\cal S}_{1\varphi}$ and ${\cal S}_{2\varphi}$ and is uniquely determined by its restriction to ${\cal S}_{1\varphi}$ . Moreover, ${\cal S}_{1\varphi}$ is meet-complete and contains the identity projection while ${\cal S}_{2\varphi}$ is join-complete and contains the zero projection. If ${\cal S}_{1\varphi}$ and ${\cal S}_{2\varphi}$ are commutative, we say that $\varphi$ is a commutative subspace map.

A new invariant of (integrable) homotopy type for foliations is introduced: the tangential category of a foliated manifold.

The classical Lusternik–Schnirelmann theory is generalized to foliations and the relations of the tangential category with other known invariants such as the fibrewise and the equivariant category are studied. Cohomological lower bounds are provided in terms of foliated cohomology.

If the foliation is a product, the tangential category coincides with the ordinary category of the leaves. In general it is just bounded below. Estimates are given of the tangential category for compact-Hausdorff foliations and suspensions. Examples show that the lower and upper bounds are realized.

The following theorem is proved: if $(M^{4n+1},g)$ is a complete Riemannian manifold and $\Sigma\subset M$ is an oriented hypersurface partitioning $M$ and with non-zero signature, then the spectrum of the Hodge–deRham Laplacian is $[0,\infty[$ . This result is obtained by a new Callias-type index. This new formula links half-bound harmonic forms (that is, nearly $L^2$ but not in $L^2$ ) with the signature of $\Sigma$ .