A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral ${{\mathbb{Z}}_{2}}$-lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by-product, some fundamental representations of affine Kac–Moody Lie algebra of type $A_{n}^{\left( 2 \right)}$ are recovered by the new method.

In this paper, we give a general explicit form of Cassels’ $p$-adic embedding theorem for number fields. We also give its refined form in the case of cyclotomic fields. As a byproduct, given an irreducible polynomial $f$ over $\mathbb{Z}$, we give a general unconditional upper bound for the smallest prime number $p$ such that $f$ has a simple root modulo $p$.

Let $\theta \,\in \,\left[ 0,\,1 \right]$ be any irrational number. It is shown that the extended rotation algebra ${{B}_{\theta }}$ introduced by the authors in J. Reine Angew. Math. 665(2012), pp. 1–71, is always an $\text{AF}$ algebra.

In this paper, we study the stability in the Lyapunov sense of the equilibrium solutions of discrete or difference Hamiltonian systems in the plane. First, we perform a detailed study of linear Hamiltonian systems as a function of the parameters. In particular we analyze the regular and the degenerate cases. Next, we give a detailed study of the normal form associated with the linear Hamiltonian system. At the same time we obtain the conditions under which we can get stability (in linear approximation) of the equilibrium solution, classifying all the possible phase diagrams as a function of the parameters. After that, we study the stability of the equilibrium solutions of the first order difference system in the plane associated with mechanical Hamiltonian systems and Hamiltonian systems defined by cubic polynomials. Finally, we point out important differences with the continuous case.

In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form $H$ and extend the result to $n$-dimensions. We also obtain an estimate for the gradient of the smaller principal curvature in 2 dimensions.

Associated with two commutative Banach algebras $A$ and $B$ and a character $\theta $ of $B$ is a certain Banach algebra product $A\,{{\times }_{\theta }}\,B$, which is a splitting extension of $B$ by $A$. We investigate two topics for the algebra $A\,{{\times }_{\theta }}\,B$ in relation to the corresponding ones of $A$ and $B$. The first one is the Bochner–Schoenberg–Eberlein property and the algebra of Bochner–Schoenberg–Eberlein functions on the spectrum, whereas the second one concerns the wide range of spectral synthesis problems for $A\,{{\times }_{\theta }}\,B$.

In this article we study exponential trichotomy for infinite dimensional discrete time dynamical systems. The goal of this article is to prove that finite time exponential trichotomy conditions allow us to derive exponential trichotomy for arbitrary times. We present an application to the case of pseudo orbits in some neighborhood of a normally hyperbolic set.

An integer is said to be $y$-friable if its greatest prime factor is less than $y$. In this paper, we obtain estimates for exponential sums over $y$-friable numbers up to $x$ which are non-trivial when $y\,\ge \,\exp \left\{ c \right.\sqrt{\log \,x\,}\log \,\log \,\left. x \right\}$. As a consequence, we obtain an asymptotic formula for the number of $y$-friable solutions to the equation $a\,+\,b\,=\,c$ which is valid unconditionally under the same assumption. We use a contour integration argument based on the saddle point method, as developped in the context of friable numbers by Hildebrand and Tenenbaum, and used by Lagarias, Soundararajan and Harper to study exponential and character sums over friable numbers.

We study a multimarginal optimal transportation problem in one dimension. For a symmetric, repulsive cost function, we show that, given a minimizing transport plan, its symmetrization is induced by a cyclical map, and that the symmetric optimal plan is unique. The class of costs that we consider includes, in particular, the Coulomb cost, whose optimal transport problem is strictly related to the strong interaction limit of Density Functional Theory. In this last setting, our result justifies some qualitative properties of the potentials observed in numerical experiments.

We demonstrate that the notions of bi-free independence and combinatorial-bi-free independence of two-faced families are equivalent using a diagrammatic view of bi-non-crossing partitions. These diagrams produce an operator model on a Fock space suitable for representing any two-faced family of non-commutative random variables. Furthermore, using a Kreweras complement on bi-non-crossing partitions we establish the expected formulas for the multiplicative convolution of a bi-free pair of two-faced families.

Let $\psi$ be a generic Drinfeld module of rank $r\,\ge \,2$ . We study the first elementary divisor ${{d}_{1,\,\wp }}\,\left( \psi\right)$ of the reduction of $\psi$ modulo a prime $\wp $ , as $\wp $ varies. In particular, we prove the existence of the density of the primes $\wp $ for which ${{d}_{1,\,\wp }}\,\left( \psi\right)$ is fixed. For $r\,=\,2$ , we also study the second elementary divisor (the exponent) of the reduction of $\psi$ modulo $\wp $ and prove that, on average, it has a large norm. Our work is motivated by J.-P. Serre's study of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's study developed by the first author and M. R. Murty.

We obtain results on the unitary equivalence of weak contractions of class ${{C}_{0}}$ to their Jordan models under an assumption on their commutants. In particular, our work addresses the case of arbitrary finite multiplicity. The main tool in this paper is the theory of boundary representations due to Arveson. We also generalize and improve previously known results concerning unitary equivalence and similarity to Jordan models when the minimal function is a Blaschke product.

It is proved that if $G\,=\,{{G}_{1}}\,{{*}_{{{G}_{3}}}}\,{{G}_{2}}$ is free product of probability measure preserving $s$–regular ergodic discrete groupoids amalgamated over an amenable subgroupoid ${{G}_{3}}$, then the sofic dimension $s(G)$ satisfies the equality

$$s(G)=\mathfrak{h}(G_{1}^{0})s({{G}_{1}})+\mathfrak{h}(G_{2}^{0})s\left( {{G}_{2}} \right)-\,\mathfrak{h}(G_{3}^{0})s({{G}_{3}}),$$

where $\mathfrak{h}$ is the normalized Haar measure on $G$.

Given a faithful action of a finite group $G$ on an algebraic curve $X$ of genus $gx\,\ge \,2$, we give explicit criteria for the induced action of $G$ on the Riemann–Roch space ${{H}^{0}}\left( X,\,{{\mathcal{O}}_{X}}\left( D \right) \right)$ to be faithful, where $D$ is a $G$-invariant divisor on $X$ of degree at least ${{2}_{gX}}\,-\,2$. This leads to a concise answer to the question of when the action of $G$ on the space ${{H}^{0}}\left( X,\,\Omega _{X}^{\otimes m} \right)$ of global holomorphic polydifferentials of order $m$ is faithful. If $X$ is hyperelliptic, we provide an explicit basis of ${{H}^{0}}\left( X,\,\Omega _{X}^{\otimes m} \right)$. Finally, we give applications in deformation theory and in coding theory and discuss the analogous problem for the action of $G$ on the first homology ${{H}_{1}}\left( X,\,\mathbb{Z}/m\mathbb{Z} \right)$ if $X$ is a Riemann surface.

Let $\mathbf{TB}$ be the category of totally bounded, locally compact metric spaces with the ${{C}_{0}}$ coarse structures. We show that if $X$ and $Y$ are in $\mathbf{TB}$, then $X$ and $Y$ are coarsely equivalent if and only if their Higson coronas are homeomorphic. In fact, the Higson corona functor gives an equivalence of categories $\mathbf{TB}\,\to \,\mathbf{K}$, where $\mathbf{K}$ is the category of compact metrizable spaces. We use this fact to show that the continuously controlled coarse structure on a locally compact space $X$ induced by some metrizable compactification $\widetilde{X}$ is determined only by the topology of the remainder $\widetilde{X}\,\backslash \,X$.

The algebraic cobordism group of a scheme is generated by cycles that are proper morphisms from smooth quasiprojective varieties. We prove that over a field of characteristic zero the quasiprojectivity assumption can be omitted to get the same theory.

Let $p$ be an odd prime. We study the variation of the $p$-rank of the Selmer groups of Abelian varieties with complex multiplication in certain towers of number fields.

We show that if $u$ and $v$ are any two unitaries in a unital ${{C}^{*}}$–algebra such that $\left\| uv\,-\,vu \right\|\,<\,2$ and $uv{{u}^{*}}{{v}^{*}}$ commutes with $u$ and $v$, then the ${{C}^{*}}$–subalgebra ${{A}_{u,v}}$ generated by $u$ and $v$ is isomorphic to a quotient of some rotation algebra ${{A}_{\theta }}$, provided that ${{A}_{u,v}}$ has a unique tracial state. We also show that the Exel trace formula holds in any unital ${{C}^{*}}$–algebra. Let $\theta \,\in \,\left( -1/2,\,1/2 \right)$ be a real number. For any $\in \,>\,0$, we prove that there exists $\delta \,>\,0$ satisfying the following: if $u$ and $v$ are two unitaries in any unital simple ${{C}^{*}}$–algebra $A$ with tracial rank zero such that

$$\left\| uv\,-\,{{e}^{2\pi i\theta }}vu \right\|\,<\,\delta \,\,\,\text{and}\,\,\frac{1}{2\pi i}\tau \left( \log \left( uv{{u}^{*}}{{v}^{*}} \right) \right)\,=\,\theta ,$$

for all tracial states $\tau$ of $A$, then there exists a pair of unitaries $\widetilde{u}$ and $\widetilde{v}$ in $A$ such that

$$\widetilde{u}\widetilde{v}\,=\,{{e}^{2\pi i\theta }}\widetilde{v}\widetilde{u},\,\,\,\,\,\,\,\left\| u\,-\,\widetilde{u} \right\|\,<\,\in \,\,\,\text{and}\,\,\left\| v\,-\,\widetilde{v} \right\|\,<\,\in.$$

We consider random i.i.d. samples of absolutely continuous measures on bounded connected domains. We prove an upper bound on the $\infty $-transportation distance between the measure and the empirical measure of the sample. The bound is optimal in terms of scaling with the number of sample points.

Let $\beta:\,{{S}^{2n+1}}\,\to \,{{S}^{2n+1}}$ be a minimal homeomorphism $\left( n\,\ge \,1 \right)$. We show that the crossed product $C\left( {{S}^{2n+1}} \right)\,{{\rtimes }_{\beta}}\mathbb{Z}$ has rational tracial rank at most one. Let $\Omega $ be a connected, compact, metric space with finite covering dimension and with ${{H}^{1}}\left( \Omega ,\,\mathbb{Z} \right)\,=\,\left\{ 0 \right\}$. Suppose that ${{K}_{i}}\left( C\left( \Omega \right) \right)\,=\,\mathbb{Z}\,\oplus \,{{G}_{i}}$, where ${{G}_{i}}$ is a finite abelian group, $i\,=\,0,\,1$. Let $\beta :\,\Omega \,\to \,\Omega $ be a minimal homeomorphism. We also show that $A\,=\,C\left( \Omega \right)\,{{\rtimes }_{\beta}}\,\mathbb{Z}$ has rational tracial rank at most one and is classifiable. In particular, this applies to the minimal dynamical systems on odd dimensional real projective spaces. This is done by studying minimal homeomorphisms on $X\,\times \,\Omega $, where $X$ is the Cantor set.