A note on the Schur multiplier of a nilpotent Lie algebra

For a nilpotent Lie algebra $L$ of dimension $n$ and dim$(L^2)=m$, we find the upper bound dim$(M(L))\leq {1/2}(n+m-2)(n-m-1)+1$, where $M(L)$ denotes the Schur multiplier of $L$. In case $m=1$ the equality holds if and only if $L\cong H(1)\oplus A$, where $A$ is an abelian Lie algebra of dimension $n-3$ and H(1) is the Heisenberg algebra of dimension 3.Comment: Paper in press in Comm. Algebra with small revision

Construction of Self-Dual Integral Normal Bases in Abelian Extensions of Finite and Local Fields

Let $F/E$ be a finite Galois extension of fields with abelian Galois group $\Gamma$. A self-dual normal basis for $F/E$ is a normal basis with the additional property that $Tr_{F/E}(g(x),h(x))=\delta_{g,h}$ for $g,h\in\Gamma$. Bayer-Fluckiger and Lenstra have shown that when $char(E)\neq 2$, then $F$ admits a self-dual normal basis if and only if $[F:E]$ is odd. If $F/E$ is an extension of finite fields and $char(E)=2$, then $F$ admits a self-dual normal basis if and only if the exponent of $\Gamma$ is not divisible by $4$. In this paper we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist. Now let $K$ be a finite extension of \Q_p, let $L/K$ be a finite abelian Galois extension of odd degree and let \bo_L be the valuation ring of $L$. We define $A_{L/K}$ to be the unique fractional \bo_L-ideal with square equal to the inverse different of $L/K$. It is known that a self-dual integral normal basis exists for $A_{L/K}$ if and only if $L/K$ is weakly ramified. Assuming $p\neq 2$, we construct such bases whenever they exist

Generalized iterated wreath products of symmetric groups and generalized rooted trees correspondence

Consider the generalized iterated wreath product $S_{r_1}\wr \ldots \wr S_{r_k}$ of symmetric groups. We give a complete description of the traversal for the generalized iterated wreath product. We also prove an existence of a bijection between the equivalence classes of ordinary irreducible representations of the generalized iterated wreath product and orbits of labels on certain rooted trees. We find a recursion for the number of these labels and the degrees of irreducible representations of the generalized iterated wreath product. Finally, we give rough upper bound estimates for fast Fourier transforms.Comment: 18 pages, to appear in Advances in the Mathematical Sciences. arXiv admin note: text overlap with arXiv:1409.060

D-branes on Singularities: New Quivers from Old

In this paper we present simplifying techniques which allow one to compute the quiver diagrams for various D-branes at (non-Abelian) orbifold singularities with and without discrete torsion. The main idea behind the construction is to take the orbifold of an orbifold. Many interesting discrete groups fit into an exact sequence $N\to G\to G/N$. As such, the orbifold $M/G$ is easier to compute as $(M/N)/(G/N)$ and we present graphical rules which allow fast computation given the $M/N$ quiver.Comment: 25 pages, 13 figures, LaTe

Aspects of ABJM orbifolds with discrete torsion

We analyze orbifolds with discrete torsion of the ABJM theory by a finite subgroup $\Gamma$ of $SU(2)\times SU(2)$ . Discrete torsion is implemented by twisting the crossed product algebra resulting after orbifolding. It is shown that, in general, the order $m$ of the cocycle we chose to twist the algebra by enters in a non trivial way in the moduli space. To be precise, the M-theory fiber is multiplied by a factor of $m$ in addition to the other effects that were found before in the literature. Therefore we got a $\mathbb{Z}_{\frac{k|\Gamma|}{m}}$ action on the fiber. We present a general analysis on how this quotient arises along with a detailed analysis of the cases where $\Gamma$ is abelian

Notes on Orientifolds of Rational Conformal Field Theories

We review and develop the construction of crosscap states associated with parity symmetries in rational conformal field theories. A general method to construct crosscap states in abelian orbifold models is presented. It is then applied to rational U(1) and parafermion systems, where in addition we study the geometrical interpretation of the corresponding parities.Comment: 67 pages, 1 figure, LaTe

Event Webs for Crisis Management

Crises are frequent and varied, and may have profound effects on individuals and organizations. Effectively managing crises, regardless of their scope, requires the ability to quickly process potentially large amounts of information about a rapidly changing world. This paper presents a software architecture, the event web, that can be used to help effectively manage crises. An event web allows for the specification, deployment, observation and management of large numbers of persistent software objects that generate, process or consume streams of events. The system is designed to be independent of any particular programming language or hardware architecture

Event-Driven Architectures for Distributed Crisis Management

This paper describes an approach for developing distributed applications that help deal with rapidly changing situations such as terrorist attacks, hurricanes and supply chain disruptions. Important characteristics of such applications are that they must handle unexpected events and that they are often modified on-the-fly, by multiple people who may belong to different organizations, to deal with changing situations. Abstract and concrete models for specifying, reasoning about, and implementing such systems are presented. In the abstract model, an application is a set of state transition rules over the global state of a distributed system. In the concrete model, an application is a set of messagedriven processes and each computation is a sequence of atomic operations in which a process receives a message, changes its state, and sends messages. An implementation of the concrete model using XML as the message and state format, with state transitions specified using XSLT, is briefly described. A key feature of this implementation is that messages and process states are represented using a format that allows applications to be easily observed and modified during their execution