Let $\overline{{{\mathbb{Q}}_{2}}}$ be an algebraic closure of ${{\mathbb{Q}}_{2}}$ and $K$ be an unramified finite extension of ${{\mathbb{Q}}_{2}}$ included in $\overline{{{\mathbb{Q}}_{2}}}$. Let $E$ be an elliptic curve defined over $K$ with additive reduction over $K$, and having an integral modular invariant. Let us denote by ${{K}_{nr}}$ the maximal unramified extension of $K$ contained in $\overline{{{\mathbb{Q}}_{2}}}$. There exists a smallest finite extension $L$ of ${{K}_{nr}}$ over which $E$ has good reduction. We determine in this paper the degree of the extension $L/{{K}_{nr}}$.

We study the range of the gradients of a ${{C}^{1,\alpha }}$ -smooth bump function defined on a Banach space. We find that this set must satisfy two geometrical conditions: It can not be too flat and it satisfies a strong compactness condition with respect to an appropriate distance. These notions are defined precisely below. With these results we illustrate the differences with the case of ${{C}^{1}}$-smooth bump functions. Finally, we give a sufficient condition on a subset of ${{X}^{*}}$ so that it is the set of the gradients of a ${{C}^{1,1}}$-smooth bump function. In particular, if $X$ is an infinite dimensional Banach space with a ${{C}^{1,1}}$-smooth bump function, then any convex open bounded subset of ${{X}^{*}}$ containing 0 is the set of the gradients of a ${{C}^{1,1}}$-smooth bump function.

We study the Hilbert functions of fat points in ${{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$. If $Z\,\subseteq \,{{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$ is an arbitrary fat point scheme, then it can be shown that for every $i$ and $j$ the values of the Hilbert function ${{H}_{Z}}(l,\,j)$ and ${{H}_{Z}}(i,\,l)$ eventually become constant for $l\,\gg \,0$. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in ${{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$. This enables us to compute all but a finite number values of ${{H}_{Z}}$ without using the coordinates of points. We also characterize the $\text{ACM}$ fat point schemes using our description of the eventual behaviour. In fact, in the case that $Z\,\subseteq \,{{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$ is $\text{ACM}$, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.

Let $\mathcal{H}$ be a complex separable Hilbert space and $\mathcal{L}\left( \mathcal{H} \right)$ denote the collection of bounded linear operators on $\mathcal{H}$. An operator $A$ in $\mathcal{L}\left( \mathcal{H} \right)$ is said to be strongly irreducible, if ${{\mathcal{A}}^{\prime }}(T)$, the commutant of $A$, has no non-trivial idempotent. An operator $A$ in $\mathcal{L}\left( \mathcal{H} \right)$ is said to be a Cowen-Douglas operator, if there exists $\Omega $, a connected open subset of $C$, and $n$, a positive integer, such that

1. (a)

   $$\Omega \,\subset \,\sigma (A)\,=\,\left\{ z\,\in \,C|\,A-z\,\text{not}\,\text{invertible} \right\};$$

2. (b)

   $$\text{ran(A}-z\text{)}\,\text{=}\,\mathcal{H}\text{,}\,\text{for}\,z\,\text{in}\,\Omega \text{;}$$

3. (c)

   $${{\vee }_{z\in \Omega }}\,\ker (A-\,z)\,=\,\mathcal{H}\,\text{and}$$

4. (d)

   $$\dim\,\ker (A-z)\,=\,n\,\text{for}\,z\,\text{in}\,\Omega $$

In the paper, we give a similarity classification of strongly irreducible Cowen-Douglas operators by using the ${{K}_{0}}$-group of the commutant algebra as an invariant.

We explicitly describe the best approximation in geodesic submanifolds of positive definite matrices obtained from involutive congruence transformations on the Cartan-Hadamard manifold $\text{Sym(}n\text{,}\,\mathbb{R}{{\text{)}}^{++}}$ of positive definite matrices. An explicit calculation for the minimal distance function from the geodesic submanifold $\text{Sym(}p\text{,}\,\mathbb{R}{{\text{)}}^{++}}\,\times \,\text{Sym(}q\text{,}\,\mathbb{R}{{\text{)}}^{++}}$ block diagonally embedded in $\text{Sym(}n\text{,}\,\mathbb{R}{{\text{)}}^{++}}$ is given in terms of metric and spectral geometric means, Cayley transform, and Schur complements of positive definite matrices when $p\,\le \,2$ or $q\,\le \,2$.

We study the semi-classical behavior as $h\,\to \,0$ of the scattering amplitude $f(\theta ,\,\omega ,\,\lambda ,\,h)$ associated to a Schrödinger operator $P(h)\,=\,-\,\frac{1}{2}{{h}^{2}}\Delta \,+\,V\,(x)$ with short-range trapping perturbations. First we realize a spatial localization in the general case and we deduce a bound of the scattering amplitude on the real line. Under an additional assumption on the resonances, we show that if we modify the potential $V(x)$ in a domain lying behind the barrier $\left\{ x\,:\,V(x)\,>\,\lambda \right\}$, the scattering amplitude $f(\theta ,\,\omega ,\,\lambda ,\,h)$ changes by a term of order $\mathcal{O}({{h}^{\infty }})$. Under an escape assumption on the classical trajectories incoming with fixed direction $\omega $, we obtain an asymptotic development of $f(\theta ,\,\omega ,\,\lambda ,\,h)$ similar to the one established in the non-trapping case.

Differentiability properties of optimal value functions associated with perturbed optimization problems require strong assumptions. We consider such a set of assumptions which does not use compactness hypothesis but which involves a kind of coherence property. Moreover, a strict differentiability property is obtained by using techniques of Ekeland and Lebourg and a result of Preiss. Such a strengthening is required in order to obtain genericity results.

We study the type decomposition and the rectangular $\text{AFD}$ property for ${{W}^{*}}-\text{TRO }\!\!'\!\!\text{ s}$. Like von Neumann algebras, every ${{W}^{*}}-\text{TRO}$ can be uniquely decomposed into the direct sum of ${{W}^{*}}-\text{TRO }\!\!'\!\!\text{ s}$ of $\text{type}\,I,\,\text{type}\,II$, and $\text{type}\,III$. We may further consider ${{W}^{*}}-\text{TRO }\!\!'\!\!\text{ s}$ of $\text{type}\,{{I}_{m,n}}$ with cardinal numbers $m$ and $n$, and consider ${{W}^{*}}-\text{TRO }\!\!'\!\!\text{ s}$ of $type\,I{{I}_{\lambda ,\mu }}\,\text{with}\,\lambda ,\,\mu \,=\,1\,\text{or}\,\infty $. It is shown that every separable stable ${{W}^{*}}-\text{TRO}$ (which includes $\text{type}\,{{I}_{\infty ,\infty }}$, $\text{type}\,I{{I}_{\infty ,\infty }}$ and $\text{type}\,III$) is $\text{TRO}$-isomorphic to a von Neumann algebra. We also introduce the rectangular version of the approximately finite dimensional property for ${{W}^{*}}-\text{TRO }\!\!'\!\!\text{ s}$. One of our major results is to show that a separable ${{W}^{*}}-\text{TRO}$ is injective if and only if it is rectangularly approximately finite dimensional. As a consequence of this result, we show that a dual operator space is injective if and only if its operator predual is a rigid rectangular $\mathcal{O}{{\mathcal{L}}_{1,{{1}^{+}}}}$ space (equivalently, a rectangular $\mathcal{O}{{\mathcal{L}}_{1,{{1}^{+}}}}$ space).

A coplactic class in the symmetric group ${{\mathcal{S}}_{n}}$ consists of all permutations in ${{\mathcal{S}}_{n}}$ with a given Schensted $Q$-symbol, and may be described in terms of local relations introduced by Knuth. Any Lie element in the group algebra of ${{\mathcal{S}}_{n}}$ which is constant on coplactic classes is already constant on descent classes. As a consequence, the intersection of the Lie convolution algebra introduced by Patras and Reutenauer and the coplactic algebra introduced by Poirier and Reutenauer is the direct sum of all Solomon descent algebras.

This paper is concerned with the Kirillov map for a class of torsion-free nilpotent groups $G$. $G$ is assumed to be discrete, countable and $\pi $-radicable, with $\pi $ containing the primes less than or equal to the nilpotence class of $G$. In addition, it is assumed that all of the characters of $G$ have idempotent absolute value. Such groups are shown to be plentiful.