On démontre un théorème de Voronoï (caractérisation des maxima locaux de l’invariant d’Hermite) pour les familles de réseaux paramétrées par les espaces symétriques irréductibles non exceptionnels de type non compact.

An explicit formula is derived for the logarithmic Mahler measure $m(P)$ of $P(x,\,y)\,=\,P(x)y-q(x)$, where $p(x)$ and $q(x)$ are cyclotomic. This is used to find many examples of such polynomials for which $m(P)$ is rationally related to the Dedekind zeta value ${{\text{ }\!\!\zeta\!\!\text{ }}_{F}}(2)$ for certain quadratic and quartic fields.

We compute the category of perverse sheaves on Hermitian symmetric spaces in types $\text{A}$ and $\text{D}$, constructible with respect to the Schubert stratification. The calculation is microlocal, and uses the action of the Borel group to study the geometry of the conormal variety $\Lambda$.

Nous établissons un résultat d’approximation forte pour des processus bivariés ayant une partie gaussienne et une partie empirique. Ce résultat apporte un nouveau point de vue sur deux théorèmes hongrois bidimensionnels établis précédemment, concernant l’approximation par un processus de Kiefer d’un processus empirique uniforme unidimensionnel et l’approximation par un pont brownien bidimensionnel d’un processus empirique uniforme bidimensionnel. Nous les enrichissons un peu et montrons que sous leur nouvelle forme ils ne sont que deux énoncés d’un même résultat.

We characterize embeddability of algebraic varieties into smooth toric varieties and prevarieties. Our embedding results hold also in an equivariant context and thus generalize a well-known embedding theorem of Sumihiro on quasiprojective $G$-varieties. The main idea is to reduce the embedding problem to the affine case. This is done by constructing equivariant affine conoids, a tool which extends the concept of an equivariant affine cone over a projective $G$-variety to a more general framework.

Given a matrix $A$, let $\mathcal{O}\left( A \right)$ denote the orbit of $A$ under a certain group action such as

1. (1) $U\left( m \right)\,\otimes U\left( n \right)$ acting on $m\,\times \,n$ complex matrices $A$ by $(U,\,V)\,*\,A\,=\,UA{{V}^{t}}$,

2. (2) $O\left( m \right)\otimes O\left( n \right)$ or $\text{SO(}m\text{)}\,\otimes \,\text{SO(}n\text{)}$ acting on $m\,\times \,n$ real matrices $A$ by $(U,\,V)\,*\,A\,=\,UA{{V}^{t}}$,

3. (3) $U(n)$ acting on $n\,\times \,n$ complex symmetric or skew-symmetric matrices $A$ by $U\,*\,A\,=\,UA{{U}^{t}}$,

4. (4) $O(n)$ or $\text{SO(n)}$ acting on $n\,\times \,n$ real symmetric or skew-symmetric matrices $A$ by $U\,*\,A\,=\,UA{{U}^{t}}$.

Denote by

1

$$\mathcal{O}({{A}_{1}},\ldots ,{{A}_{k}})\,=\,\{{{X}_{1\,}}+\cdot \cdot \cdot +\,{{X}_{k}}\,:\,{{X}_{i}}\,\in \,\mathcal{O}({{A}_{i}}),i\,=\,1,\ldots ,k\}$$

the joint orbit of the matrices ${{A}_{1}},\ldots ,{{A}_{k}}$. We study the set of diagonals or partial diagonals of matrices in $\mathcal{O}({{A}_{1}},\ldots ,{{A}_{k}})$, i.e., the set of vectors $({{d}_{1}},\ldots {{d}_{r}})$ whose entries lie in the $(1,\,{{j}_{1}}),\ldots ,(r,\,{{j}_{r}})$ positions of a matrix in $\mathcal{O}({{A}_{1}},\ldots ,{{A}_{k}})$ for some distinct column indices ${{j}_{1}},\ldots ,{{j}_{r}}$. In many cases, complete description of these sets is given in terms of the inequalities involving the singular values of ${{A}_{1}},\ldots ,{{A}_{k}}$. We also characterize those extreme matrices for which the equality cases hold. Furthermore, some convexity properties of the joint orbits are considered. These extend many classical results on matrix inequalities, and answer some questions by Miranda. Related results on the joint orbit $\mathcal{O}({{A}_{1}},\ldots ,{{A}_{k}})$ of complex Hermitian matrices under the action of unitary similarities are also discussed.

We extend the basic theory of Lie algebras of affine algebraic groups to the case of pro-affine algebraic groups over an algebraically closed field $K$ of characteristic 0. However, some modifications are needed in some extensions. So we introduce the pro-discrete topology on the Lie algebra $\mathcal{L}(G)$ of the pro-affine algebraic group $G$ over $K$, which is discrete in the finite-dimensional case and linearly compact in general. As an example, if $L$ is any sub Lie algebra of $\mathcal{L}(G)$, we show that the closure of $\left[ L,\,L \right]$ in $\mathcal{L}(G)$ is algebraic in $\mathcal{L}(G)$.

We also discuss the Hopf algebra of representative functions $H(L)$ of a residually finite dimensional Lie algebra $L$. As an example, we show that if $L$ is a sub Lie algebra of $\mathcal{L}(G)$ and $G$ is connected, then the canonical Hopf algebra morphism from $K\left[ G \right]$ into $H(L)$ is injective if and only if $L$ is algebraically dense in $\mathcal{L}(G)$.

We give conditions which determine if cat of a map go up when extending over a cofibre. We apply this to reprove a result of Roitberg giving an example of a $\text{CW}$ complex $Z$ such that $\text{cat}(Z)\,=\,2$ but every skeleton of $Z$ is of category 1. We also find conditions when $\text{cat}(f\,\times \,g)\,<\,\text{cat}(f)\,+\,\text{cat}(g)$. We apply our result to show that under suitable conditions for rational maps $f,\,\text{mcat}(f)\,<\,\text{cat}(f)$ is equivalent to $\text{cat(}f)\,=\,\text{cat(}f\,\times \,\text{i}{{\text{d}}_{{{S}^{n}}}})$. Many examples with $\text{mcat}(f)\,<\,\text{cat}(f)$ satisfying our conditions are constructed. We also answer a question of Iwase by constructing $p$-local spaces $X$ such that $\text{cat(}X\ \times \,{{S}^{1}}\text{)}\,\text{=}\,\text{cat(}X\text{)}\,\text{=2}$. In fact for our spaces and every $Y\,\not{\simeq }\,*,\,\text{cat}(X\,\times \,Y)\,\le \,\text{cat}(Y)\,+\,1\,\text{cat}(Y)\,+\,\text{cat}(X)$. We show that this same $X$ has the property $\text{cat}(X)=\,\text{cat}(X\,\times \,X)\,=\,\text{cl}(X\,\times \,X)\,=\,2$.

We consider a unitary representation of a discrete countable abelian group on a separable Hilbert space which is associated to a cyclic generalized frame multiresolution analysis. We extend Robertson’s theorem to apply to frames generated by the action of the group. Within this setup we use Stone’s theorem and the theory of projection valued measures to analyze wandering frame collections. This yields a functional analytic method of constructing a wavelet from a generalized frame multiresolution analysis in terms of the frame scaling vectors. We then explicitly apply our results to the action of the integers given by translations on ${{L}^{2}}(\mathbb{R})$.

Consider the sixth Painlevé equation $({{\text{P}}_{6}})$ below where $\alpha ,\beta ,\gamma$ and $\delta$ are complex parameters. We prove the necessary and sufficient conditions for the existence of rational solutions of equation $({{\text{P}}_{6}})$ in term of special relations among the parameters. The number of distinct rational solutions in each case is exactly one or two or infinite. And each of them may be generated by means of transformation group found by Okamoto [7] and Bäcklund transformations found by Fokas and Yortsos [4]. A list of rational solutions is included in the appendix. For the sake of completeness, we collected all the corresponding results of other five Painlevé equations $({{\text{P}}_{1}})-({{\text{P}}_{5}})$ below, which have been investigated by many authors [1]–[7].