Let $M$ be a convex body such that the boundary has positive curvature. Then by a well developed theory dating back to Landau and Hlawka for large $\lambda$ the number of lattice points in $\lambda M$ is given by $G\left( \lambda M \right)=V\left( \lambda M \right)+O\left( {{\lambda }^{d-1-\varepsilon \left( d \right)}} \right)$ for some positive $\varepsilon (d)$ . Here we give for general convex bodies the weaker estimate

$$|G\left( \lambda M \right)-V\left( \lambda M \right)|\,\le \,\frac{1}{2}{{S}_{{{Z}^{d}}}}\left( M \right){{\lambda }^{d-1}}+o\left( {{\lambda }^{d-1}} \right)$$

where ${{S}_{{{Z}^{d}}}}\left( M \right)$ denotes the lattice surface area of $M$ . The term ${{S}_{{{Z}^{d}}}}\left( M \right)$ is optimal for all convex bodies and $o\left( {{\lambda }^{d-1}} \right)$ cannot be improved in general. We prove that the same estimate even holds if we allow small deformations of $M$ .

Further we deal with families $\left\{ {{P}_{\lambda }} \right\}$ of convex bodies where the only condition is that the inradius tends to infinity. Here we have

$$|G\left( {{P}_{\lambda }} \right)-V\left( {{P}_{\lambda }} \right)|\,\le \,dV\left( {{P}_{\lambda }},\,K;1 \right)+o\left( S\left( {{P}_{\lambda }} \right) \right)$$

where the convex body $K$ satisfies some simple condition, $V\left( {{P}_{\lambda }},K;1 \right)$ is some mixed volume and $S\left( {{P}_{\lambda }} \right)$ is the surface area of ${{P}_{\lambda }}$ .

In this paper we establish conditions that guarantee, in the setting of a general Banach space, the Painlevé-Kuratowski convergence of the graphs of the subdifferentials of convexly composite functions. We also provide applications to the convergence of multipliers of families of constrained optimization problems and to the generalized second-order derivability of convexly composite functions.

We prove a uniform upper estimate on the number of cuspidal eigenvalues of the $\Gamma$ -automorphic Laplacian below a given bound when $\Gamma$ varies in a family of congruence subgroups of a given reductive linear algebraic group. Each $\Gamma$ in the family is assumed to contain a principal congruence subgroup whose index in $\Gamma$ does not exceed a fixed number. The bound we prove depends linearly on the covolume of $\Gamma$ and is deduced from the analogous result about the cut-off Laplacian. The proof generalizes the heat-kernel method which has been applied by Donnelly in the case of a fixed lattice $\Gamma$ .

It is shown that a morphism of quivers having a certain path lifting property has a decomposition that mimics the decomposition of maps of topological spaces into homotopy equivalences composed with fibrations. Such a decomposition enables one to describe the right adjoint of the restriction of the representation functor along a morphism of quivers having this path lifting property. These right adjoint functors are used to construct injective representations of quivers. As an application, the injective representations of the cyclic quivers are classified when the base ring is left noetherian. In particular, the indecomposable injective representations are described in terms of the injective indecomposable $R$ -modules and the injective indecomposable $R[\text{x},\,{{\text{x}}^{-1}}]$ -modules.

If $\alpha$ is an ordinal, then the space of all ordinals less than or equal to $\alpha$ is a compact Hausdorff space when endowed with the order topology. Let $C(\alpha )$ be the space of all continuous real-valued functions defined on the ordinal interval $[0,\,\alpha ]$ . We characterize the symmetric sequence spaces which embed into $C(\alpha )$ for some countable ordinal $\alpha$ . A hierarchy $\left( {{E}_{\alpha }} \right)$ of symmetric sequence spaces is constructed so that, for each countable ordinal $\alpha$ , ${{E}_{\alpha }}$ embeds into $C\left( {{\omega }^{{{\omega }^{\alpha }}}} \right)$ , but does not embed into $C\left( {{\omega }^{{{\omega }^{\beta }}}} \right)$ for any $\beta \,<\,\alpha$ .

In an earlier paper [10], we studied a generalized Rao bound for ordered orthogonal arrays and $(T,\,M,\,S)$ -nets. In this paper, we extend this to a coding-theoretic approach to ordered orthogonal arrays. Using a certain association scheme, we prove a MacWilliams-type theorem for linear ordered orthogonal arrays and linear ordered codes as well as a linear programming bound for the general case. We include some tables which compare this bound against two previously known bounds for ordered orthogonal arrays. Finally we show that, for even strength, the $\text{LP}$ bound is always at least as strong as the generalized Rao bound.

In this paper, we present a geometric construction of the Moufang quadrangles discovered by Richard Weiss (see Tits & Weiss [18] or Van Maldeghem [19]). The construction uses fixed point free involutions in certain mixed quadrangles, which are then extended to involutions of certain buildings of type ${{F}_{4}}$ . The fixed flags of each such involution constitute a generalized quadrangle. This way, not only the new exceptional quadrangles can be constructed, but also some special type of mixed quadrangles.

We show that a martingale problem associated with a competing species model has a unique solution. The proof of uniqueness of the solution for the martingale problem is based on duality technique. It requires the construction of dual probability measures.