Exterior differential systems, Lie algebra cohomology, and the rigidity of homogenous varieties

These are expository notes from the 2008 Srni Winter School. They have two purposes: (1) to give a quick introduction to exterior differential systems (EDS), which is a collection of techniques for determining local existence to systems of partial differential equations, and (2) to give an exposition of recent work (joint with C. Robles) on the study of the Fubini-Griffiths-Harris rigidity of rational homogeneous varieties, which also involves an advance in the EDS technology.Comment: To appear in the proceedings of the 2008 Srni Winter School on Geometry and Physic

A Remark on Almost Umbilical Hypersurfaces

In this article, we prove new stability results for almost-Einstein hypersurfaces of the Euclidean space, based on previous eigenvalue pinching results. Then, we deduce some comparable results for almost umbilical hypersurfaces.Comment: 8 page

On connections between hypergraphs and algebras

summary:The aim of the present paper is to translate some algebraic concepts to hypergraphs. Thus we obtain a new language, very useful in the investigation of subalgebra lattices of partial, and also total, algebras. In this paper we solve three such problems on subalgebra lattices, other will be solved in [[Pio4]]. First, we show that for two arbitrary partial algebras, if their directed hypergraphs are isomorphic, then their weak, relative and strong subalgebra lattices are isomorphic. Secondly, we prove that two partial algebras have isomorphic weak subalgebra lattices iff their hypergraphs are isomorphic. Thirdly, for an arbitrary lattice $\mathbf {L}$ and a partial algebra $\mathbf {A}$ we describe (necessary and sufficient conditions) when the weak subalgebra lattice of $\mathbf {A}$ is isomorphic to $\mathbf {L}$

Relation of the spectra of symplectic Rarita-Schwinger and Dirac operators on flat symplectic manifolds

summary:Consider a flat symplectic manifold $(M^{2l},\omega )$, $l\ge 2$, admitting a metaplectic structure. We prove that the symplectic twistor operator maps the eigenvectors of the symplectic Dirac operator, that are not symplectic Killing spinors, to the eigenvectors of the symplectic Rarita-Schwinger operator. If $\lambda$ is an eigenvalue of the symplectic Dirac operator such that $-\imath l \lambda$ is not a symplectic Killing number, then $\frac{l-1}{l}\lambda$ is an eigenvalue of the symplectic Rarita-Schwinger operator

τ\tau -supplemented modules and τ\tau -weakly supplemented modules

summary:Given a hereditary torsion theory $\tau = (\mathbb {T},\mathbb {F})$ in Mod-$R$, a module $M$ is called $\tau$-supplemented if every submodule $A$ of $M$ contains a direct summand $C$ of $M$ with $A/C$ $\tau -$torsion. A submodule $V$ of $M$ is called $\tau$-supplement of $U$ in $M$ if $U+V=M$ and $U\cap V\le \tau (V)$ and $M$ is $\tau$-weakly supplemented if every submodule of $M$ has a $\tau$-supplement in $M$. Let $M$ be a $\tau$-weakly supplemented module. Then $M$ has a decomposition $M=M_1\oplus M_2$ where $M_1$ is a semisimple module and $M_2$ is a module with $\tau (M_2)\le _e M_2$. Also, it is shown that; any finite sum of $\tau$-weakly supplemented modules is a $\tau$-weakly supplemented module

Approximation of solutions of the forced duffing equation with nonlocal discontinuous type integral boundary conditions

summary:A generalized quasilinearization technique is applied to obtain a sequence of approximate solutions converging monotonically and quadratically to the unique solution of the forced Duffing equation with nonlocal discontinuous type integral boundary conditions

Pivoting algorithm in class of ABS methods

summary:Summary: The paper deals with a pivoting modification of the algorithm in the class of ABS methods. Numerical experiments compare this pivoting modification with the fundamental version. A hybrid algorithm for the solution of the linear system with the Hankel matrix is introduced

On the HH-property of some Banach sequence spaces

summary:In this paper we define a generalized Cesàro sequence space $\operatorname{ces\,}(p)$ and consider it equipped with the Luxemburg norm under which it is a Banach space, and we show that the space $\operatorname{ces\,}(p)$ posses property (H) and property (G), and it is rotund, where $p = (p_k)$ is a bounded sequence of positive real numbers with $p_k > 1$ for all $k \in N$

Variational method and conjugacy criteria for half-linear differential equations

summary:We establish new conjugacy criteria for half-linear second order differential equations. These criteria are based on the relationship between conjugacy of the investigated equation and nonpositivity of the associated energy functional