A generalization of Schmüdgen’s Positivstellensatz is given which holds for any basic closed semialgebraic set in ${{\mathbb{R}}^{n}}$ (compact or not). The proof is an extension of Wörmann’s proof.

In this paper the author prove that standard modules of classical groups whose Langlands quotients are generic are irreducible. This establishes a conjecture of Casselman and Shahidi for this important class of groups.

In this paper we find a formula for the Alexander polynomial ${{\Delta }_{p1,...,{{p}_{k}}\left( x \right)}}$ of pretzel knots and links with $\left( {{p}_{1}},...,{{p}_{k}},-1 \right)$ twists, where $k$ is odd and ${{p}_{1}},...,{{p}_{k}}$ are positive integers. The polynomial ${{\Delta }_{2,3,7}}\left( x \right)$ is the well-known Lehmer polynomial, which is conjectured to have the smallest Mahler measure among all monic integer polynomials. We confirm that ${{\Delta }_{2,3,7}}\left( x \right)$ has the smallest Mahler measure among the polynomials arising as ${{\Delta }_{p1,...,{{p}_{k}}\left( x \right)}}$.

The Hamiltonian potentials of the bending deformations of $n$-gons in ${{\mathbb{E}}^{3}}$ studied in $\left[ \text{KM} \right]$ and [Kly] give rise to a Hamiltonian action of the Malcev Lie algebra ${{P}_{n}}$ of the pure braid group ${{P}_{n}}$ on the moduli space ${{M}_{r}}$ of $n$-gon linkages with the side-lengths $r\,=\,\left( {{r}_{1}},\ldots ,{{r}_{n}} \right)$ in ${{\mathbb{E}}^{3}}$. If $e\,\in \,{{M}_{r}}$ is a singular point we may linearize the vector fields in ${{P}_{n}}$ at $e$. This linearization yields a flat connection $\nabla$ on the space $\mathbb{C}_{*}^{n}$ of $n$ distinct points on $\mathbb{C}$. We show that the monodromy of $\nabla$ is the dual of a quotient of a specialized reduced Gassner representation.

Let $\mathcal{E}$ be an ample vector bundle of rank $r$ on a projective variety $X$ with only log-terminal singularities. We consider the nefness of adjoint divisors ${{K}_{X}}\,+\,\left( t-r \right)\,\det \,\mathcal{E}$ when $t\,\ge \,\dim\,X$ and $t\,>\,r$. As an application, we classify pairs $\left( X,\,\mathcal{E} \right)$ with ${{c}_{r}}$-sectional genus zero.

The paper is dealing with determination of the integral ${{\int }_{\gamma }}\,f$ along the fractal arc $\gamma $ on the complex plane by terms of polynomial approximations of the function $f$. We obtain inequalities for polynomials and conditions of integrability for functions from the Hölder, Besov and Slobodetskii spaces.

We give an explicit recipe for the determination of the images associated to the Galois action on $p$-torsion points of elliptic curves. We present a table listing the image for all the elliptic curves defined over $\mathbb{Q}$ without complex multiplication with conductor less than 200 and for each prime number $p$.

Let $G$ be an infinite discrete amenable group or a non-discrete amenable group. It is shown how to construct a net $\left( {{f}_{\alpha }} \right)$ of positive, normalized functions in ${{L}_{1}}\left( G \right)$ such that the net converges weak* to invariance but does not converge strongly to invariance. The solution of certain linear equations determined by colorings of the Cayley graphs of the group are central to this construction.

Nous montrons que la $\text{A}$-catégorie d’un espace simplement connexe de type fini est inférieure ou égale à $n$ si et seulement si son modèle d’Adams-Hilton est un rétracte homotopique d’une algèbre différentielle à $n$ étages. Nous en déduisons que l’invariant Acat augmente au plus de 1 lors de l’attachement d’une cellule à un espace.

Soit ${{H}_{0}}\,=\,\frac{{{x}^{2}}+{{y}^{2}}}{2}$ un hamiltonien isochrone du plan ${{\mathbb{R}}^{2}}$. On met en évidence une classe d’hamiltoniens isochrones qui sont des perturbations polynomiales de ${{H}_{0}}$. On obtient alors une condition nécessaire d’isochronisme, et un critère de choix pour les hamiltoniens isochrones. On voit ce résultat comme étant une généralisation du caractère isochrone des perturbations hamiltoniennes homogènes considérées dans $\left[ \text{L} \right],\,\left[ \text{P} \right],\,\left[ \text{S} \right]$.

The zeta function of a nonsingular pair of quadratic forms defined over a finite field, $k$, of arbitrary characteristic is calculated. A. Weil made this computation when char $k\,\ne \,2$. When the pair has even order, a relationship between the number of zeros of the pair and the number of places of degree one in an appropriate hyperelliptic function field is established.

Motivated by deformation theory of holomorphic maps between almost complex manifolds we endow, in a natural way, the tangent bundle of an almost complexmanifold with an almost complex structure. We describe various properties of this structure.

In this article it is shown that every bounded linear operator on a complex, infinite dimensional, separable Hilbert space is a sum of at most eighteen unilateral (alternatively, bilateral) weighted shifts. As well, we classify products of weighted shifts, as well as sums and limits of the resulting operators.

Irrational rotation ${{C}^{*}}$-algebras have an inductive limit decomposition in terms of matrix algebras over the space of continuous functions on the circle and this decomposition can be chosen to be invariant under the flip automorphism. It is shown that the flip is essentially the only toral automorphism with this property.

Applying the Sullivan conjecture we construct compacta of certain cohomological and extensional dimensions.

We prove an identity between weighted orbital integrals of the unit elements in the Hecke algebras of $\text{GL}\left( r \right)$ and its $n$-fold metaplectic covering, under the assumption that $n$ is relatively prime to any proper divisor of every $1\,\le \,j\,\le \,r$.

Let $p$ be prime and let $\vartheta \,\in \,\mathbb{Z}_{p}^{*}$ be of multiplicative order $t$ modulo $p$. We consider exponential sums of the form

$$S\left( a \right)\,=\,\sum\limits_{x=1}^{t}{\exp \left( 2\pi ia{{\vartheta }^{{{x}^{2}}}}\,/\,p \right)}$$

and prove that for any $\varepsilon \,>\,0$

$$\underset{\gcd (a,\,p)\,=\,1}{\mathop{\max }}\,\,\left| S\left( a \right) \right|\,=\,O\left( {{t}^{5/6+\varepsilon }}\,{{p}^{1/8}} \right)$$

We continue to study the simplest closure conditions for chains of spectral transformations of the Laurent biorthogonal polynomials $\left( \text{LBP} \right)$. It is shown that the 1-1-periodic $q$-closure condition leads to the $\text{LBP}$ introduced by Pastro. We introduce classes of semi-classical and Laguerre-Hahn $\text{LBP}$ associated to generic closure conditions of the chain of spectral transformations.

A challenge by R. Padmanabhan to prove by group theory the commutativity of cancellative semigroups satisfying a particular law has led to the proof of more general semigroup laws being equivalent to quite simple ones.

We give canonical resolutions of singularities of several cone varieties arising from invariant theory. We establish a connection between our resolutions and resolutions of singularities of closure of conjugacy classes in classical Lie algebras.