Let $K$ be a finite Galois extension of the field of rational numbers with unit rank greater than 3. We prove that the ring of integers of $K$ is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann hypothesis for Dedekind zeta functions. We now prove this unconditionally.

We study the structure of generalized Verma modules over a semi-simple complex finite-dimensional Lie algebra, which are induced from simple modules over a parabolic subalgebra. We consider the case when the annihilator of the starting simple module is a minimal primitive ideal if we restrict this module to the Levi factor of the parabolic subalgebra. We show that these modules correspond to proper standard modules in some parabolic generalization of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$ and prove that the blocks of this parabolic category are equivalent to certain blocks of the category of Harish-Chandra bimodules. From this we derive, in particular, an irreducibility criterion for generalized Verma modules. We also compute the composition multiplicities of those simple subquotients, which correspond to the induction from simple modules whose annihilators are minimal primitive ideals.

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 prove an estimate for the speed of convergence of the transition probability for a symmetric random walk on a nilpotent covering graph. To obtain this estimate, we give a complete proof of the Gaussian bound for the gradient of the Markov kernel.

In this article we are concerned with how to compute the cohomology ring of a symplectic quotient by a circle action using the information we have about the cohomology of the original manifold and some data at the fixed point set of the action. Our method is based on the Tolman-Weitsman theorem which gives a characterization of the kernel of the Kirwan map. First we compute a generating set for the kernel of the Kirwan map for the case of product of compact connected manifolds such that the cohomology ring of each of them is generated by a degree two class. We assume the fixed point set is isolated; however the circle action only needs to be “formally Hamiltonian”. By identifying the kernel, we obtain the cohomology ring of the symplectic quotient. Next we apply this result to some special cases and in particular to the case of products of two dimensional spheres. We show that the results of Kalkman and Hausmann-Knutson are special cases of our result.

Let ${{({{\mathcal{M}}_{i}})}_{i}}{{_{\in }}_{I}},{{({{\mathcal{N}}_{j}})}_{j\in J}}$ be families of von Neumann algebras and $\mathcal{U},\,{\mathcal{U}}'$ be ultrafilters in $I$, $J$, respectively. Let $1\,\le \,p\,<\,\infty $ and $n\,\in \,\mathbb{N}$. Let ${{x}_{1}},...,{{x}_{n}}\,\text{in}\,\prod {{L}_{p}}({{\mathcal{M}}_{i}})$ and ${{y}_{1}},...,{{y}_{n\,}}\text{in }\Pi \,{{L}_{p}}({{\mathcal{N}}_{j}})$ be bounded families. We show the following equality

$$\underset{i,\mathcal{U}}{\mathop{\lim }}\,\,\underset{j,{\mathcal{U}}'}{\mathop{\lim }}\,\,{{\left\| \sum\limits_{k=1}^{n}{{{x}_{k}}(i)\,\otimes \,{{y}_{k}}(j)} \right\|}_{{{L}_{p}}({{\mathcal{M}}_{i}}\otimes {{\mathcal{N}}_{j}})}}=\,\underset{j,{\mathcal{U}}'}{\mathop{\lim }}\,\,\underset{i,\mathcal{U}}{\mathop{\lim }}\,\,{{\left\| \sum\limits_{k=1}^{n}{{{x}_{k}}(i)\,\,\otimes \,{{y}_{k}}(j)} \right\|}_{{{L}_{p}}({{\mathcal{M}}_{i}}\otimes {{\mathcal{N}}_{j}})}}$$

For $p\,=\,1$ this Fubini type result is related to the local reflexivity of duals of ${{C}^{*}}$ -algebras. This fails for $P=\infty $.

We are concerned with a unital separable nuclear purely infinite simple ${{C}^{*}}$-algebra $A$ satisfying UCT with a Rohlin flow, as a continuation of [12]. Our first result (which is independent of the Rohlin flow) is to characterize when two central projections in $A$ are equivalent by a central partial isometry. Our second result shows that the $\text{K}$-theory of the central sequence algebra ${A}'\cap {{A}^{\omega }}$ (for an $\omega \in \beta \text{N }\!\!\backslash\!\!\text{ N}$) and its fixed point algebra under the flow are the same (incorporating the previous result). We will also complete and supplement the characterization result of the Rohlin property for flows stated in [12].

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 use the lace expansion to analyse networks of mutually-avoiding self-avoiding walks, having the topology of a graph. The networks are defined in terms of spread-out self-avoiding walks that are permitted to take large steps. We study the asymptotic behaviour of networks in the limit of widely separated network branch points, and prove Gaussian behaviour for sufficiently spread-out networks on ${{\mathbb{Z}}^{d}}$ in dimensions $d\,>\,4$.

In this article we determine the global geometry of the planar quadratic differential systems with a weak focus of third order. This class plays a significant role in the context of Hilbert's 16-th problem. Indeed, all examples of quadratic differential systems with at least four limit cycles, were obtained by perturbing a system in this family. We use the algebro-geometric concepts of divisor and zero-cycle to encode global properties of the systems and to give structure to this class. We give a theorem of topological classification of such systems in terms of integer-valued affine invariants. According to the possible values taken by them in this family we obtain a total of 18 topologically distinct phase portraits. We show that inside the class of all quadratic systems with the topology of the coefficients, there exists a neighborhood of the family of quadratic systems with a weak focus of third order and which may have graphics but no polycycle in the sense of [15] and no limit cycle, such that any quadratic system in this neighborhood has at most four limit cycles.

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.

For a locally compact group $G$ and $1\,<\,p\,<\,\infty $, let ${{A}_{p}}\left( G \right)$ be the Herz-Figà-Talamanca algebra and let $P{{M}_{p}}\left( G \right)$ be its dual Banach space. For a Banach ${{A}_{p}}\left( G \right)$-module $X$ of $P{{M}_{p}}\left( G \right)$, we prove that the multiplier space $\text{M}\left( {{A}_{p}}\left( G \right),{{X}^{*}} \right)$ is the dual Banach space of ${{Q}_{X}}$, where ${{Q}_{X}}$ is the norm closure of the linear span ${{A}_{p}}\left( G \right)X\,\text{of}\,u\,f\,\text{for}\,u\,\in \,{{A}_{p}}\left( G \right)\,\text{and}\,f\,\in \,X$ in the dual of $\text{M}\left( {{A}_{p}}\left( G \right),{{X}^{*}} \right)$. If $p\,=\,2$ and $P{{F}_{p}}\left( G \right)\subseteq X$, then ${{A}_{p}}\left( G \right)X$ is closed in $X$ if and only if $G$ is amenable. In particular, we prove that the multiplier algebra $M{{A}_{p}}\left( G \right)\,\text{of}\,{{A}_{p}}\left( G \right)$ is the dual of $Q$, where $Q$ is the completion of ${{L}^{1}}\left( G \right)$ in the $||\cdot |{{|}_{M}}$-norm. $Q$ is characterized by the following: $f\,\in \,Q$ if an only if there are ${{u}_{i}}\,\in \,{{A}_{p}}\left( G \right)$ and ${{f}_{i}}\in P{{F}_{p}}\left( G \right)\left( i=1,2,... \right)$ with $\sum\nolimits_{i=1}^{\infty }{||}\,{{u}_{i}}\,|{{|}_{{{A}_{p}}\left( G \right)}}||fi|{{|}_{P{{F}_{p}}\left( G \right)}}\,<\,\infty $ such that $f=\sum{_{i=1}^{\infty }\,{{u}_{i}}{{f}_{i}}}$ on $M{{A}_{p}}\left( G \right)$. It is also proved that if ${{A}_{p}}\left( G \right)$ is dense in $M{{A}_{p}}\left( G \right)$ in the associated ${{w}^{*}}$-topology, then the multiplier norm and $||\cdot |{{|}_{{{A}_{p}}\left( G \right)}}$-norm are equivalent on ${{A}_{p}}\left( G \right))$ if and only if $G$ is amenable.

We introduce and investigate using Hilbert modules the properties of the Fourier algebra$A(G)$ for a locally compact groupoid $G$. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.

The Heisenberg group is considered as the boundary of a manifold. A class of hypersurfaces in this manifold can be regarded as copies of the Heisenberg group. The properties of geodesics in the interior and on the hypersurfaces are worked out in detail. These properties are strongly related to those of the Heisenberg group.

The billiard flow in the plane has a simple geometric definition; the movement along straight lines of points except where elastic reflections are made with the boundary of the billiard domain. We consider a class of open billiards, where the billiard domain is unbounded, and the boundary is that of a finite number of strictly convex obstacles. We estimate the Hausdorff dimension of the nonwandering set ${{M}_{0}}$ of the discrete time billiard ball map, which is known to be a Cantor set and the largest invariant set. Under certain conditions on the obstacles, we use a well-known coding of ${{M}_{0}}$ [Mor91] and estimates using convex fronts related to the derivative of the billiard ball map [Sto03] to estimate the Hausdorff dimension of local unstable sets. Consideration of the local product structure then yields the desired estimates, which provide asymptotic bounds on the Hausdorff dimension's convergence to zero as the obstacles are separated.

Let $K$ be a knot in ${{S}^{3}}$ . This paper is devoted to Dehn surgeries which create 3-manifolds containing a closed non-orientable surface $\hat{S}$ . We look at the slope $p/q$ of the surgery, the Euler characteristic $\mathcal{X}(\hat{S})$ of the surface and the intersection number $s$ between $\hat{S}$ and the core of the Dehn surgery. We prove that if $\mathcal{X}(\hat{S})\,\ge \,15\,-3q$, then $s\,=\,1$. Furthermore, if $s\,=\,1$ then $q\,\le \,4\,-\,3\,\mathcal{X}(\hat{S})$ or $K$ is cabled and $q\,\le \,8\,-5\mathcal{X}(\hat{S})$ . As consequence, if $K$ is hyperbolic and $\mathcal{X}(\hat{S})\,=\,-1$ , then $q\,\le \,7$.

Every norm $\mathcal{V}$ on ${{\text{C}}^{\text{n}}}$ induces two norm numerical ranges on the algebra ${{M}_{n}}\,\text{of}\,\text{all}\,\text{n}\,\times \,\text{n}$ complex matrices, the spatial numerical range

$$W\left( A \right)=\left\{ {{x}^{*}}Ay:x,y\in {{\text{C}}^{n}},{{v}^{D}}\left( x \right)=v\left( y \right)={{x}^{*}}y=1 \right\},$$

where ${{\mathcal{V}}^{D}}$ is the norm dual to $\mathcal{V}$, and the algebra numerical range

$$V\left( A \right)=\left\{ f\left( A \right):f\in S \right\},$$

where $S$ is the set of states on the normed algebra ${{M}_{n}}$ under the operator norm induced by $\mathcal{V}$. For a symmetric norm $\mathcal{V}$, we identify all linear maps on ${{M}_{n}}$ that preserve either one of the two norm numerical ranges or the set of states or vector states. We also identify the numerical radius isometries, i.e., linear maps that preserve the (one) numerical radius induced by either numerical range. In particular, it is shown that if $\mathcal{V}$ is not the ${{\ell }_{1}},{{\ell }_{2}},\,\text{or}\,{{\ell }_{\infty }}$ norms, then the linear maps that preserve either numerical range or either set of states are “inner”, i.e., of the form $A\mapsto {{Q}^{*}}AQ$, where $Q$ is a product of a diagonal unitary matrix and a permutation matrix and the numerical radius isometries are unimodular scalar multiples of such inner maps. For the ${{\ell }_{1}}$ and the ${{\ell }_{\infty }}$ norms, the results are quite different.

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.

This paper contains a comparison of several definitions of equivariant formality for actions of torus groups. We develop and prove some relations between the definitions. Focusing on the case of the circle group, we use ${{S}^{1}}$ -equivariant minimal models to give a number of examples of ${{S}^{1}}$-spaces illustrating the properties of the various definitions.

Let $a$ be a natural number greater than 1. Let ${{f}_{a}}\left( n \right)$ be the order of $a\,\bmod \,n$. Denote by $\omega \left( n \right)$ the number of distinct prime factors of $n$. Assuming a weak form of the generalised Riemann hypothesis, we prove the following conjecture of Erdös and Pomerance:

The number of $n\,\le \,x$ coprime to a satisfying

$$\alpha \le \frac{\omega \left( {{f}_{a}}\left( n \right) \right)-{{\left( \log \,\log \,n \right)}^{2}}/2}{{{\left( \log \,\log \,n \right)}^{3/2}}/\sqrt{3}}\le \beta $$

is asymptotic to $\left( \frac{1}{\sqrt{2\pi }}\int_{\alpha }^{\beta }{{{e}^{-{{t}^{2}}/2}}}dt \right)\frac{x\phi \left( a \right)}{a}$ as $x$ tends to infinity.