Lie algebras graded by finite reduced root systems have been classified up to isomorphism. In this paper we describe the derivation algebras of these Lie algebras and determine when they possess invariant bilinear forms. The results which we develop to do this are much more general and apply to Lie algebras that are completely reducible with respect to the adjoint action of a finite-dimensional subalgebra.

An odd unbounded (respectively, $p$-summable) Fredholm module for a unital Banach $*$-algebra, $A$, is a pair $(H,D)$ where $A$ is represented on the Hilbert space, $H$, and $D$ is an unbounded self-adjoint operator on $H$ satisfying:

(1) ${{(1+{{D}^{2}})}^{-1}}$ is compact (respectively, Trace $\left( {{\left( 1+{{D}^{2}} \right)}^{-(p/2)}} \right)<\infty $, and

(2) $\{a\in A|\,\,[D,a]\,\,\text{is}\,\,\text{bounded}\}\text{ }$ is a dense $*$- subalgebra of $A$.

If $u$ is a unitary in the dense $*$-subalgebra mentioned in (2) then

$$uD{{u}^{*}}=D+u[D,{{u}^{*}}]=D+B$$

where $B$ is a bounded self-adjoint operator. The path

$$D_{t}^{u}:=(1-t)D+tuD{{u}^{*}}=D+tB$$

is a “continuous” path of unbounded self-adjoint “Fredholm” operators. More precisely, we show that

$$F_{t}^{u}:=D_{t}^{u}{{\left( 1+{{\left( D_{t}^{u} \right)}^{2}} \right)}^{-\frac{1}{2}}}$$

is a norm-continuous path of (bounded) self-adjoint Fredholm operators. The spectral flow of this path $\left\{ F_{t}^{u} \right\}$ (or $\{D_{t}^{u}\}$) is roughly speaking the net number of eigenvalues that pass through 0 in the positive direction as $t$ runs from 0 to 1. This integer,

$$\text{sf}\left( \left\{ D_{t}^{u} \right\} \right):=\text{sf}\left( \left\{ F_{t}^{u} \right\} \right),$$

recovers the pairing of the $K$-homology class $[D]$ with the $K$-theory class $[u]$.

We use I.M. Singer's idea (as did E. Getzler in the $\theta $-summable case) to consider the operator $B$ as a parameter in the Banach manifold, ${{B}_{\text{sa}}}\left( H \right)$, so that spectral flow can be exhibited as the integral of a closed 1-form on this manifold. Now, for $B$ in our manifold, any $X\,\in \,{{T}_{B}}({{B}_{\text{sa}}}(H))$ is given by an $X$ in ${{B}_{\text{sa}}}(H)$ as the derivative at $B$ along the curve $t\,\mapsto \,B\,+\,tX$ in the manifold. Then we show that for $m$ a sufficiently large half-integer:

$$\alpha (X)=\frac{1}{{{{\tilde{C}}}_{m}}}\text{Tr}\left( X{{\left( 1+{{\left( D+B \right)}^{2}} \right)}^{-m}} \right)$$

is a closed 1-form. For any piecewise smooth path $\left\{ {{D}_{t}}=D+{{B}_{t}} \right\}$ with ${{D}_{0}}$ and ${{D}_{1}}$unitarily equivalent we show that

$$\text{sf}\left( \left\{ {{D}_{t}} \right\} \right)=\frac{1}{{{{\tilde{C}}}_{m}}}\int_{\text{0}}^{\text{1}}{\text{T}}\text{r}\left( \frac{d}{dt}\left( {{D}_{t}} \right){{\left( 1+D_{1}^{2} \right)}^{-m}} \right)dt$$

the integral of the 1-form $\alpha $. If ${{D}_{0}}$ and ${{D}_{1}}$ are not unitarily equivalent, wemust add a pair of correction terms to the right-hand side. We also prove a bounded finitely summable version of the form:

$$\text{sf}\left( \left\{ {{F}_{t}} \right\} \right)=\frac{1}{{{C}_{n}}}\int_{\text{0}}^{\text{1}}{\text{T}}\text{r}\left( \frac{d}{dt}\left( {{F}_{t}} \right){{\left( 1-{{F}_{t}}^{2} \right)}^{n}} \right)dt$$

for $n\ge \frac{p-1}{2}$ an integer. The unbounded case is proved by reducing to the bounded case via the map $D\mapsto F=D{{(1+{{D}^{2}})}^{-\frac{1}{2}}}$ We prove simultaneously a type II version of our results.

It is shown that the adjoint group ${{R}^{{}^\circ }}$ of an arbitrary radical ring $R$ has a series with abelian factors and that its finite subgroups are nilpotent. Moreover, some criteria for subgroups of ${{R}^{{}^\circ }}$ to be locally nilpotent are given.

We study the function spaces ${{Q}_{p}}(R)$ defined on a Riemann surface $R$, which were earlier introduced in the unit disk of the complex plane. The nesting property ${{Q}_{p}}(R)\,\subseteq \,{{Q}_{_{q}}}(R)$ for $0\,<\,p\,<\,q\,<\,\infty $ is shown in case of arbitrary hyperbolic Riemann surfaces. Further, it is proved that the classical Dirichlet space $\text{AD(}R\text{)}\,\subseteq \,{{Q}_{p}}(R)$ for any $p$, $0\,<\,p\,<\,\infty $, thus sharpening T. Metzger's well-known result $\text{AD(}R\text{)}\,\subseteq \,\text{BMOA}(R)$. Also the first author's result $\text{AD}(R)\,\subseteq \,\text{VMOA}(R)$ for a regular Riemann surface $R$ is sharpened by showing that, in fact, $\text{AD(}R\text{)}\,\subseteq \,{{Q}_{p,0}}(R)$ for all $p$, $0\,<\,p\,<\,\infty $. The relationships between ${{Q}_{p}}(R)$ and various generalizations of the Bloch space on $R$ are considered. Finally we show that ${{Q}_{p}}(R)$ is a Banach space for $0\,<\,p\,<\,\infty $.

In a previous paper, we gave a correspondence between certain exact solutions to a (2 + 1)-dimensional integrable Chiral Model and holomorphic bundles on a compact surface. In this paper, we use algebraic geometry to derive a closed-form expression for those solutions and show by way of examples how the algebraic data which parametrise the solution space dictates the behaviour of the solutions.

In this paper we consider Fourier multipliers on local Hardy spaces ${{\mathbf{h}}^{\mathbf{p}}}(0<p\le 1)$ for Chébli-Trimèche hypergroups. The molecular characterization is investigated which allows us to prove a version of Hörmander’s multiplier theorem.

The notion of decompositon class in a semisimple Lie algebra is a common generalization of nilpotent orbits and the set of regular semisimple elements.We prove that the closure of a decomposition class has many properties in common with nilpotent varieties, e.g., its normalization has rational singularities.

The famous Grothendieck simultaneous resolution is related to the decomposition class of regular semisimple elements. We study the properties of the analogous commutative diagrams associated to an arbitrary decomposition class.

The problems on isomorphic classification and quasiequivalence of bases are studied for the class of mixed $F-$, $\text{DF}$-power series spaces, i.e. the spaces of the following kind

$$G(\lambda ,a)=\underset{p\to \infty }{\mathop{\lim }}\,\text{proj}\left( \underset{q\to \infty }{\mathop{\lim }}\,\text{ind}\left( {{\ell }_{1}}\left( {{a}_{i}}\left( p,q \right) \right) \right) \right),$$

where ${{a}_{i}}(p,\,q)\,=\,\exp \left( \left( p\,-\,{{\lambda }_{i}}q \right){{a}_{i}} \right),\,p,\,q\,\in \,\mathbb{N},\,\text{and}\,\lambda \,\text{=}\,{{\left( {{\lambda }_{i}} \right)}_{i\in \mathbb{N}}},\,a=\,{{({{a}_{i}})}_{i\in \mathbb{N}}}$ are some sequences of positive numbers. These spaces, up to isomorphisms, are basis subspaces of tensor products of power series spaces of $F-$ and $\text{DF}$-types, respectively. The m-rectangle characteristic $\mu _{m}^{\lambda ,a}\left( \delta ,\,\varepsilon ;\,\tau ,\,t \right),\,m\,\in \,\mathbb{N}$ of the space $G\left( \text{ }\!\!\lambda\!\!\text{ ,}\,a \right)$ is defined as the number of members of the sequence ${{({{\lambda }_{i}},{{a}_{i}})}_{i\in \mathbb{N}}}$ which are contained in the union of m rectangles ${{P}_{k}}\,=\,\left( {{\delta }_{k}},\,{{\varepsilon }_{k}} \right]\,\times \,\left( {{\tau }_{k}},\,{{t}_{k}} \right]$ , $k\,=\,1,2,\ldots ,m$. It is shown that each m-rectangle characteristic is an invariant on the considered class under some proper definition of an equivalency relation. The main tool are new compound invariants, which combine some version of the classical approximative dimensions (Kolmogorov, Pełczynski) with appropriate geometrical and interpolational operations under neighborhoods of the origin (taken from a given basis).

There are infinitely many triplets of primes $p,\,q,\,r$ such that the arithmetic means of any two of them, $\frac{p+q}{2},\,\frac{p+r}{2},\,\frac{q+r}{2}$ are also primes. We give an asymptotic formula for the number of such triplets up to a limit. The more involved problem of asking that in addition to the above the arithmetic mean of all three of them, $\frac{p+q+r}{3}$ is also prime seems to be out of reach. We show by combining the Hardy-Littlewood method with the sieve method that there are quite a few triplets for which six of the seven entries are primes and the last is almost prime.

Let $R$ be a countable, principal ideal domain which is not a field and $A$ be a countable $R$-algebra which is free as an $R$-module. Then we will construct an ${{\aleph }_{1}}$-free $R$-module $G$ of rank ${{\aleph }_{1}}$ with endomorphism algebra $\text{En}{{\text{d}}_{R}}\,G=A$. Clearly the result does not hold for fields. Recall that an $R$-module is ${{\aleph }_{1}}$-free if all its countable submodules are free, a condition closely related to Pontryagin’s theorem. This result has many consequences, depending on the algebra $A$ in use. For instance, if we choose $A\,=\,R$, then clearly $G$ is an indecomposable ‘almost free’module. The existence of such modules was unknown for rings with only finitelymany primes like $R={{\mathbb{Z}}_{\left( p \right)}}$, the integers localized at some prime $p$. The result complements a classical realization theorem of Corner’s showing that any such algebra is an endomorphism algebra of some torsionfree, reduced $R$-module $G$ of countable rank. Its proof is based on new combinatorialalgebraic techniques related with what we call rigid tree-elements coming from a module generated over a forest of trees.

Etant donnée une partie $D$ dénombrable et dense de $\mathbb{R}$, nous construisons une infinité de fonctions Lipschitziennes définies sur $\mathbb{R}$, s'annulant en zéro, dont le sous-différentiel proximal est égal à ]–1, 1[ en tout point de $D$ et est vide en tout point du complémentaire de $D$. Nous déduisons que deux fonctions dont la différence n'est pas constante peuvent avoir les mêmes sous-différentiels.

Let $K$ be a convex body in ${{\mathbf{E}}^{d}}$ and denote by ${{C}_{n}}$ the set of centroids of $n$ non-overlapping translates of $K$. For $\varrho \,>\,0$, assume that the parallel body conv ${{C}_{n}}\,+\,\varrho K$ of conv ${{C}_{n}}$ has minimal volume. The notion of parametric density (see [21]) provides a bridge between finite and infinite packings (see [4] or [14]). It is known that there exists a maximal ${{\varrho }_{s}}(K)\,\ge \,1/(32{{d}^{2}})$ such that conv ${{C}_{n}}$ is a segment for $\varrho \,<\,{{\varrho }_{s}}$ (see [5]). We prove the existence of a minimal ${{\varrho }_{c}}(K)\,\le \,d\,+\,1$ such that if $\varrho \,>\,{{\varrho }_{c}}$ and $n$ is large then the shape of conv ${{C}_{n}}$ can not be too far from the shape of $K$. For $d\,=\,2$, we verify that ${{\varrho }_{s\,}}\,=\,{{\varrho }_{c}}$. For $d\,\ge \,3$, we present the first example of a convex body with known ${{\varrho }_{s}}$ and ${{\varrho }_{c}}$; namely, we have ${{\varrho }_{s}}\,=\,{{\varrho }_{c}}\,=\,1$ for the parallelotope.

We introduce a distributional Ricci curvature on complete smooth manifolds with Lipschitz continuous metrics. Under an assumption on the volume growth of geodesics balls, we obtain a gradient estimate for weakly harmonic functions if the distributional Ricci curvature is bounded below.

In this paper we construct a bounded strictly positive function $\sigma $ such that the Liouville property fails for the divergence form operator $L\,=\,\nabla ({{\sigma }^{2}}\nabla )$. Since in addition $\Delta \sigma /\sigma $ is bounded, this example also gives a negative answer to a problem of Berestycki, Caffarelli and Nirenberg concerning linear Schrödinger operators.

Let $X$ be a graph with vertex set $V$ and let $A$ be its adjacency matrix. If $E$ is the matrix representing orthogonal projection onto an eigenspace of $A$ with dimension $m$, then $E$ is positive semi-definite. Hence it is the Gram matrix of a set of $\left| V \right|$ vectors in ${{R}^{m}}$. We call the convex hull of a such a set of vectors an eigenpolytope of $X$. The connection between the properties of this polytope and the graph is strongest when $X$ is distance regular and, in this case, it is most natural to consider the eigenpolytope associated to the second largest eigenvalue of $A$. The main result of this paper is the characterisation of those distance regular graphs $X$ for which the 1-skeleton of this eigenpolytope is isomorphic to $X$.

Let $G$ be an affine Kac-Moody group, ${{\pi }_{0}},...,{{\pi }_{r}},{{\pi }_{\delta }}$ its fundamental irreducible representations and ${{\chi }_{0}},...,{{\chi }_{r}},{{\chi }_{\delta }}$ their characters. We determine the set of all group elements $x$ such that all ${{\pi }_{i}}(x)$ act as trace class operators, i.e., such that ${{\chi }_{i}}(x)$ exists, then prove that the ${{\chi }_{i}}$ are class functions. Thus, ($\chi \,:=\,({{\chi }_{0}},...,{{\chi }_{r}},\,{{\chi }_{\delta }})$)factors to an adjoint quotient $\bar{\chi }$ for $G$. In a second part, following Steinberg, we define a cross-section $C$ for the potential regular classes in $G$. We prove that the restriction $\chi \text{ }\!\!|\!\!\text{ }c$ behaves well algebraically. Moreover, we obtain an action of ${{\mathbb{C}}^{\times }}$ on $C$, which leads to a functional identity for $\text{ }\chi |\text{ c}$ which shows that $\text{ }\chi |\text{ c}$ is quasi-homogeneous.

Central to the study of simple infinite dimensional $g\ell (n,\,\mathbb{C})$-modules having finite dimensional weight spaces are the torsion free modules. All degree 1 torsion free modules are known. Torsion free modules of arbitrary degree can be constructed by tensoring torsion free modules of degree 1 with finite dimensional simple modules. In this paper, the central characters of such a tensor product module are shown to be given by a Pieri-like formula, complete reducibility is established when these central characters are distinct and an example is presented illustrating the existence of a nonsimple indecomposable submodule when these characters are not distinct.

Given function $\Omega $ on ${{\mathbb{R}}^{n}}$, we define the fractional maximal operator and the fractional integral operator by

$${{M}_{\Omega ,\,\alpha }}f(x)\,=\,_{r>0}^{\sup }\frac{1}{{{r}^{n-\alpha }}}\,\int{{{_{|y|}}_{<r}}\,}|\Omega (y)|\,|f(x-y)|\,dy$$

$${{T}_{\Omega ,\,\alpha }}f(x)\,=\,\int{_{{{\mathbb{R}}^{n}}}}\,\frac{\Omega (y)}{{{\left| y \right|}^{n-\alpha }}}f(x-y)dy$$

$0\,<\,\alpha \,<\,n$

${{M}_{\Omega ,\,\alpha }}$

${{T}_{\Omega ,\,\alpha }}$

$\alpha ,\,s$

$A(p,\,\,q)$

$\Omega \,\in \,{{L}^{s}}({{S}^{n-1}})(s>1)$

In this paper, we solve the isomorphism problem for metacirculant graphs of order $pq$ that are not circulant. To solve this problem, we first extend Babai’s characterization of the $\text{CI}$-property to non-Cayley vertex-transitive hypergraphs. Additionally, we find a simple characterization of metacirculant Cayley graphs of order $pq$, and exactly determine the full isomorphism classes of circulant graphs of order $pq$.

This paper deals with periodic solutions for the billiard problem in a bounded open set of ${{\mathbb{R}}^{N}}$ which are limits of regular solutions of Lagrangian systems with a potential well. We give a precise link between the Morse index of approximate solutions (regarded as critical points of Lagrangian functionals) and the properties of the bounce trajectory to which they converge.