Let $X$ be a locally finite, connected graph without vertices of degree 1. Non-backtracking random walk moves at each step with equal probability to one of the “forward” neighbours of the actual state, i.e., it does not go back along the preceding edge to the preceding state. This is not a Markov chain, but can be turned into a Markov chain whose state space is the set of oriented edges of $X$. Thus we obtain for infinite $X$ that the $n$-step non-backtracking transition probabilities tend to zero, and we can also compute their limit when $X$ is finite. This provides a short proof of an old result concerning cogrowth of groups, and makes the extension of that result to arbitrary regular graphs rigorous. Even when $X$ is non-regular, but small cycles are dense in$X$, we show that the graph $X$ is non-amenable if and only if the non-backtracking $n$-step transition probabilities decay exponentially fast. This is a partial generalization of the cogrowth criterion for regular graphs which comprises the original cogrowth criterion for finitely generated groups of Grigorchuk and Cohen.

Suppose that we have the unit Euclidean ball in ${{\mathbb{R}}^{n}}$ and construct new bodies using three operations — linear transformations, closure in the radial metric, and multiplicative summation defined by ${{\left\| x \right\|}_{K+0L}}\,=\,\sqrt{{{\left\| x \right\|}_{K}}{{\left\| x \right\|}_{L}}}$. We prove that in dimension 3 this procedure gives all origin-symmetric convex bodies, while this is no longer true in dimensions 4 and higher. We introduce the concept of embedding of a normed space in ${{L}_{0}}$ that naturally extends the corresponding properties of ${{L}_{P}}$-spaces with $p\,\ne \,0$, and show that the procedure described above gives exactly the unit balls of subspaces of ${{L}_{0}}$ in every dimension. We provide Fourier analytic and geometric characterizations of spaces embedding in ${{L}_{0}}$, and prove several facts confirming the place of ${{L}_{0}}$ in the scale of ${{L}_{P}}$-spaces.

Purely simple Kronecker modules $M$, built from an algebraically closed field $K$, arise from a triplet $\left( m,\,h,\,\alpha \right)$ where $m$ is a positive integer, $h:\,K\,\cup \,\{\infty \}\,\to \,\{\infty ,\,0,\,1,\,2,\,3,\,.\,.\,.\,\}$ is a height function, and $\alpha$ is a $K$-linear functional on the space $K\left( X \right)$ of rational functions in one variable $X$. Every pair $\left( h,\,\alpha \right)$ comes with a polynomial $f$ in $K\left( X \right)\left[ Y\, \right]$ called the regulator. When the module $M$ admits non-trivial endomorphisms, $f$ must be linear or quadratic in $Y$. In that case $M$ is purely simple if and only if $f$ is an irreducible quadratic. Then the $K$-algebra End $M$ embeds in the quadratic function field ${K\left( X \right)\left[ Y\, \right]}/{(f\,)}\;$. For some height functions $h$ of infinite support $I$, the search for a functional $\alpha$ for which $\left( h,\,\alpha \right)$ has regulator $0$ comes down to having functions $\eta \,:\,I\,\to \,K$ such that no planar curve intersects the graph of $\eta$ on a cofinite subset. If $K$ has characterictic not $2$, and the triplet $\left( m,\,h,\,\alpha \right)$ gives a purely-simple Kronecker module $M$ having non-trivial endomorphisms, then $h$ attains the value $\infty$ at least once on $\text{K}\,\cup \,\text{ }\!\!\{\!\!\text{ }\infty \text{ }\!\!\}\!\!\text{ }$ and $h$ is finite-valued at least twice on $\text{K}\,\cup \,\text{ }\!\!\{\!\!\text{ }\infty \text{ }\!\!\}\!\!\text{ }$. Conversely all these $h$ form part of such triplets. The proof of this result hinges on the fact that a rational function $r$ is a perfect square in $K\left( X \right)$ if and only if $r$ is a perfect square in the completions of $K\left( X \right)$ with respect to all of its valuations.

Let $K$ be a number field, and let $F$ be a symmetric bilinear form in $2N$ variables over $K$. Let $Z$ be a subspace of ${{K}^{N}}$. A classical theorem of Witt states that the bilinear space $\left( Z,\,F \right)$ can be decomposed into an orthogonal sum of hyperbolic planes and singular and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights of $F$ and $Z$. We also prove a special version of Siegel's lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of the Cartan-Dieudonné theorem. Namely, we show that every isometry $~\sigma$ of a regular bilinear space $\left( Z,\,F \right)$ can be represented as a product of reflections of bounded heights with an explicit bound on heights in terms of heights of $F$, $Z$, and $~\sigma$.

Let $\mathcal{D}$ be a division algebra over a nonarchimedean local field. Given an irreducible representation $\pi $ of $G{{L}_{2}}(\mathcal{D})$, we describe its restriction to the diagonal subgroup ${{\mathcal{D}}^{*}}\,\times \,{{\mathcal{D}}^{*}}$. The description is in terms of the structure of the twisted Jacquet module of the representation $\pi $. The proof involves Kirillov theory that we have developed earlier in joint work with Dipendra Prasad. The main result on restriction also shows that $\pi $ is ${{\mathcal{D}}^{*}}\,\times \,{{\mathcal{D}}^{*}}$-distinguished if and only if $\pi $ admits a Shalika model. We further prove that if $\mathcal{D}$ is a quaternion division algebra then the twisted Jacquet module is multiplicity-free by proving an appropriate theorem on invariant distributions; this then proves a multiplicity-one theorem on the restriction to ${{\mathcal{D}}^{*}}\,\times \,{{\mathcal{D}}^{*}}$ in the quaternionic case.

In this paper, we characterize unitary representations of $\pi \text{ }:=\text{ }\pi {{\text{ }}_{1}}({{S}^{2}}\backslash \{{{s}_{1}},\ldots ,{{s}_{l}}\})$ whose generators ${{u}_{1}},\,\ldots ,\,{{u}_{l}}$ (lying in conjugacy classes fixed initially) can be decomposed as products of two Lagrangian involutions ${{u}_{j}}\,=\,{{\sigma }_{j}}{{\sigma }_{j+1}}$ with ${{\sigma }_{l+1}}\,=\,{{\sigma }_{1}}$. Our main result is that such representations are exactly the elements of the fixed-point set of an anti-symplectic involution defined on the moduli space ${{\mathcal{M}}_{C}}\,:\,=\,\text{Ho}{{\text{m}}_{C}}(\pi ,\,U(n))\,/\,U(n)$. Consequently, as this fixed-point set is non-empty, it is a Lagrangian submanifold of ${{\mathcal{M}}_{C}}$. To prove this, we use the quasi-Hamiltonian description of the symplectic structure of ${{\mathcal{M}}_{C}}$ and give conditions on an involution defined on a quasi-Hamiltonian $U$-space $(M,\,\omega ,\,\mu :\,M\to \,U)$ for it to induce an anti-symplectic involution on the reduced space $M//U:=\,{{\mu }^{-1}}\,(\{1\})/U$.

We use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of $p$-convex, $p$-concave and positive $p$-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators.

Dans cet article on étudie la différence entre les deux premières valeurs propres, le splitting, d’un opérateur de Klein–Gordon semi-classique unidimensionnel, dans le cas d’un potentiel symétrique présentant un double puits. Dans le cas d’une petite barrière de potentiel, B. Helffer et B. Parisse ont obtenu des résultats analogues à ceux existant pour l’opérateur de Schrödinger. Dans le cas d’une grande barrière de potentiel, on obtient ici des estimations des tranformées de Fourier des fonctions propres qui conduisent à une conjecture du splitting. Des calculs numériques viennent appuyer cette conjecture.

It is known that the Brandt–Lickorish–Millett–Ho polynomial $Q$ contains Casson's knot invariant. Whether there are (essentially) other Vassiliev knot invariants obtainable from $Q$ is an open problem. We show that this is not so up to degree 9. We also give the (apparently) first examples of knots not distinguished by 2-cable HOMFLY polynomials which are not mutants. Our calculations provide evidence of a negative answer to the question whether Vassiliev knot invariants of degree $d\,\le \,10$ are determined by the HOMFLY and Kauffman polynomials and their 2-cables, and for the existence of algebras of such Vassiliev invariants not isomorphic to the algebras of their weight systems.

We prove dispersive and Strichartz inequalities for the solution of the wave equation related to the full Laplacian on the Heisenberg group, by means of Besov spaces defined by a Littlewood–Paley decomposition related to the spectral resolution of the full Laplacian. This requires a careful analysis due also to the non-homogeneous nature of the full Laplacian. This result has to be compared to a previous one by Bahouri, Gérard and Xu concerning the solution of the wave equation related to the Kohn Laplacian.

For each real number $\xi$, let $\widehat{{{\lambda }_{2}}}\left( \xi \right)$ denote the supremum of all real numbers $\text{ }\!\!\lambda\!\!\text{ }$ such that, for each sufficiently large $X$ , the inequalities $\left| {{x}_{0}} \right|\,\le \,X,\,\left| {{x}_{0}}\xi \,-\,{{x}_{1}} \right|\,\le \,{{X}^{-\lambda \text{ }}}$ and $\left| {{x}_{0}}{{\xi }^{2}}\,-\,{{x}_{2}} \right|\,\le \,{{X}^{-\lambda \text{ }}}$ admit a solution in integers ${{x}_{0}},\,{{x}_{1}}$ and ${{x}_{2}}$ not all zero, and let $\widehat{{{\omega }_{2}}}\left( \xi \right)$ denote the supremum of all real numbers $\omega $ such that, for each sufficiently large $X$, the dual inequalities $\left| {{x}_{0}}\,+\,{{x}_{1}}\xi \,+\,{{x}_{2}}{{\xi }^{2}} \right|\,\le \,{{X}^{-\omega }}$, $\left| {{x}_{1}} \right|\,\le \,X$ and $\left| {{x}_{2}} \right|\,\le \,X$ admit a solution in integers ${{x}_{0}},\,{{x}_{1}}$ and ${{x}_{2}}$ not all zero. Answering a question of Y. Bugeaud and M. Laurent, we show that the exponents $\widehat{{{\lambda }_{2}}}\left( \xi \right)$ where $\xi$ ranges through all real numbers with $[\mathbb{Q}(\xi )\,:\mathbb{Q}]\,>\,2$ form a dense subset of the interval $\left[ 1/2,\,\left( \sqrt{5}\,-\,1 \right)/2 \right]$ while, for the same values of $\xi$, the dual exponents $\widehat{{{\omega }_{2}}}\left( \xi \right)$ form a dense subset of $\left[ 2,\,\left( \sqrt{5}\,+\,3 \right)/2 \right]$. Part of the proof rests on a result of V. Jarník showing that $\widehat{{{\lambda }_{2}}}\left( \xi \right)=1-{{\hat{\omega }}_{2}}{{\left( \xi \right)}^{-1}}$ for any real number $\xi$ with $[\mathbb{Q}(\xi )\,:\mathbb{Q}]\,>\,2$.

Le diagramme d’Eisenbud et Neumann d’un germe est un arbre qui représente ce germe et permet d’en calculer les invariants. On donne une démonstration algébrique d’un résultat caractérisant l’ensemble des quotients jacobiens d’un germe d’application $(f,\,g)$ à partir du diagramme d’Eisenbud et Neumann de $fg$.

An ideal $I$ of a ring $R$ is called a radical ideal if $I\,=\,\mathcal{R}(R)$ where $\mathcal{R}$ is a radical in the sense of Kurosh–Amitsur. The main theorem of this paper asserts that if $R$ is a valuation domain, then a proper ideal $I$ of $R$ is a radical ideal if and only if $I$ is a distinguished ideal of $R$ (the latter property means that if $J$ and $K$ are ideals of $R$ such that $J\,\subset \,I\,\subset \,K$ then we cannot have $I/J\,\cong \,K/I$ as rings) and that such an ideal is necessarily prime. Examples are exhibited which show that, unlike prime ideals, distinguished ideals are not characterizable in terms of a property of the underlying value group of the valuation domain.

We give bounds on the distance from a non-zero idempotent to the set of nilpotents in the set of $n\,\times \,n$ matrices in terms of the norm of the idempotent. We construct explicit idempotents and nilpotents which achieve these distances, and determine exact distances in some special cases.

In this paper we study ruled surfaces which appear as exceptional surface in a succession of blowing-ups. In particular we prove that the $e$-invariant of such a ruled exceptional surface $E$ is strictly positive whenever its intersection with the other exceptional surfaces does not contain a fiber (of $E$). This fact immediately enables us to resolve an open problem concerning an intersection configuration on such a ruled exceptional surface consisting of three nonintersecting sections. In the second part of the paper we apply the non-vanishing of $e$ to the study of the poles of the well-known topological, Hodge and motivic zeta functions.

Let $p$ be an odd prime number, and let $F$ be a field of characteristic not $p$ and not containing the group ${{\mu }_{p}}$ of $p$-th roots of unity. We consider cyclic $p$-algebras over $F$ by descent from $L\,=\,F\left( {{\mu }_{p}} \right)$. We generalize a theorem of Albert by showing that if ${{\mu }_{{{p}^{n}}}}\,\subseteq \,L$, then a division algebra $D$ of degree ${{p}^{n}}$ over $F$ is a cyclic algebra if and only if there is $d\,\in \,D$ with ${{d}^{{{P}^{n}}}}\,\in \,F\,-\,{{F}^{P}}$. Let $F(p)$ be the maximal $p$-extension of $F$. We show that $F(p)$ has a noncyclic algebra of degree $p$ if and only if a certain eigencomponent of the $p$-torsion of $\text{Br(F(p)(}{{\mu }_{p}}\text{))}$ is nontrivial. To get a better understanding of $F(p)$, we consider the valuations on $F(p)$ with residue characteristic not $p$, and determine what residue fields and value groups can occur. Our results support the conjecture that the $p$ torsion in $\text{Br}(F(p))$ is always trivial.

We prove two spectral identities. The first one relates the relative trace formula for the spherical variety $\text{GSpin(4,}\,\text{3)/}{{G}_{2}}$ with a weighted trace formula for $G{{L}_{2}}$. The second relates a spherical variety pertaining to ${{F}_{4}}$ to one of $GSp\left( 6 \right)$. These identities are in accordance with a conjecture made by Jacquet, Lai, and Rallis, and are obtained without an appeal to a geometric comparison.