The usual constructions of classifying spaces for monoidal categories produce $\text{CW}$ -complexes with many cells that,moreover, do not have any proper geometric meaning. However, geometric nerves of monoidal categories are very handy simplicial sets whose simplices have a pleasing geometric description: they are diagrams with the shape of the 2-skeleton of oriented standard simplices. The purpose of this paper is to prove that geometric realizations of geometric nerves are classifying spaces for monoidal categories.

Let $M$ be a complete flat Lorentz 3-manifold $M$ with purely hyperbolic holonomy $\Gamma $ . Recurrent geodesic rays are completely classified when $\Gamma $ is cyclic. This implies that for any pair of periodic geodesics ${{\gamma }_{1}}$ , ${{\gamma }_{2}}$ , a unique geodesic forward spirals towards ${{\gamma }_{1}}$ and backward spirals towards ${{\gamma }_{2}}$ .

We construct new examples of non-nil algebras with any number of generators, which are direct sums of two locally nilpotent subalgebras. Like all previously known examples, our examples are contracted semigroup algebras and the underlying semigroups are unions of locally nilpotent subsemigroups. In our constructions we make more transparent than in the past the close relationship between the considered problem and combinatorics of words.

We prove an integral formula on closed oriented manifolds equipped with a codimension two foliation whose leaves are compact.

If $A$ is a set of $M$ positive integers, let $G(A)$ be the maximum of ${{a}_{i}}/\,\gcd ({{a}_{i}},\,\,{{a}_{j}}\,)$ over ${{a}_{i}},\,{{a}_{j}}\,\,\in \,\,A.$ We show that if $G(A)$ is not too much larger than $M$ , then $A$ must have a special structure.

We characterize surjective linear maps on $B\text{(X)}$ that are spectrally bounded and spectrally bounded below.

We show that the product of four or five consecutive positive terms in arithmetic progression can never be a perfect power whenever the initial term is coprime to the common difference of the arithmetic progression. This is a generalization of the results of Euler and Obláth for the case of squares, and an extension of a theorem of Győry on three terms in arithmetic progressions. Several other results concerning the integral solutions of the equation of the title are also obtained. We extend results of Sander on the rational solutions of the equation in $n,\,y$ when $b\,=\,d\,=\,1$ . We show that there are only finitely many solutions in $n,\,d,\,b,\,y$ when $k\,\ge \,3,\,l\,\ge \,2$ are fixed and $k\,+\,l\,>\,6$ .

In this paper we give an inversion formula of the Radon transform on the Heisenberg group by using the wavelets defined in [3]. In addition, we characterize a space such that the inversion formula of the Radon transform holds in the weak sense.

Let $V$ be a $K3$ surface defined over a number field $k$ . The Batyrev-Manin conjecture for $V$ states that for every nonempty open subset $U$ of $V$ , there exists a finite set ${{Z}_{U}}$ of accumulating rational curves such that the density of rational points on $U\,-\,{{Z}_{U}}$ is strictly less than the density of rational points on ${{Z}_{U}}$ . Thus, the set of rational points of $V$ conjecturally admits a stratification corresponding to the sets ${{Z}_{U}}$ for successively smaller sets $U$ .

In this paper, in the case that $V$ is a Kummer surface, we prove that the Batyrev-Manin conjecture for $V$ can be reduced to the Batyrev-Manin conjecture for $V$ modulo the endomorphisms of $V$ induced by multiplication by $m$ on the associated abelian surface $A$ . As an application, we use this to show that given some restrictions on $A$ , the set of rational points of $V$ which lie on rational curves whose preimages have geometric genus 2 admits a stratification of Batyrev-Manin type.

We apply the method of complex scaling to give a natural proof of a formula relating the multiplicity of a resonance to the multiplicity of a pole of the scattering matrix.

In this paper we establish a Hopf type lemma and a $\text{CR}$ type inversion for the generalized Greiner operator. Some nonlinear Liouville type results are given.

For certain real quadratic number fields, we prove density results concerning 4-ranks of tame kernels. We also discuss a relationship between 4-ranks of tame kernels and 4-class ranks of narrow ideal class groups. Additionally, we give a product formula for a local Hilbert symbol.

The stable basin theorem was introduced by Basmajian and Miner as a key step in their necessary condition for the discreteness of a non-elementary group of complex hyperbolic isometries. In this paper we improve several of Basmajian and Miner’s key estimates and so give a substantial improvement on the main inequality in the stable basin theorem.

For a locally compact group $G$ , the convolution product on the space $N({{L}^{p}}\ (G))$ of nuclear operators was defined by Neufang [11]. We study homological properties of the convolution algebra $N({{L}^{p}}\ (G))$ and relate them to some properties of the group $G$ , such as compactness, finiteness, discreteness, and amenability.

In this paper we shall give an affirmative solution to a problem, posed by Berger, Coburn and Lebow, for ${{C}^{*}}$ -algebras on Hardy submodules.

We investigate the problem of determining elements in the Selberg class by means of their Dirichlet series coefficients at primes.

D. J. Grubb [3] has shown that uniqueness holds, under a mild growth condition, for Vilenkin series which converge almost everywhere to zero. We show that, under even less restrictive growth conditions, one can replace the limit function 0 by an arbitrary $f\,\in \,{{L}^{q}}$ , when $q\,>\,1$ .