On the finiteness and stability of certain sets of associated primes ideals of local cohomology modules

Let $(R,\frak{m})$ be a Noetherian local ring, $I$ an ideal of $R$ and $N$ a finitely generated $R$-module. Let $k{\ge}-1$ be an integer and r=\depth_k(I,N) the length of a maximal $N$-sequence in dimension $>k$ in $I$ defined by M. Brodmann and L. T. Nhan ({Comm. Algebra, 36 (2008), 1527-1536). For a subset S\subseteq \Spec R we set S_{{\ge}k}={\p\in S\mid\dim(R/\p){\ge}k}. We first prove in this paper that \Ass_R(H^j_I(N))_{\ge k} is a finite set for all $j{\le}r$}. Let \fN=\oplus_{n\ge 0}N_n be a finitely generated graded \fR-module, where \fR is a finitely generated standard graded algebra over $R_0=R$. Let $r$ be the eventual value of \depth_k(I,N_n). Then our second result says that for all $l{\le}r$ the sets \bigcup_{j{\le}l}\Ass_R(H^j_I(N_n))_{{\ge}k} are stable for large $n$.Comment: To appear in Communication in Algebr

Localized and complete resonance in plasmonic structures

This paper studies a possible connection between the way the time averaged electromagnetic power dissipated into heat blows up and the anomalous localized resonance in plasmonic structures. We show that there is a setting in which the localized resonance takes place whenever the resonance does and moreover, the power is always bounded and might go to $0$. We also provide another setting in which the resonance is complete and the power goes to infinity whenever resonance occurs; as a consequence of this fact there is no localized resonance. This work is motivated from recent works on cloaking via anomalous localized resonance

Cosmological Constant and Gravitational Repulsion Effect: 1. Homogeneous models with radiation

Within the framework of the minimum quadratic Poincare gauge theory of gravity in the Riemann-Cartan spacetime we study the influence of gravitational vacuum energy density (a cosmological constant) on the dynamics of various gravitating systems. It is shown that the inclusion of the cosmological term can lead to gravitational repulsion. For some simple cases of spatially homogeneous cosmological models with radiation we obtain non-singular solutions in form of elementary functions and elliptic integrals.Comment: REVTEX, 11 pages, Syracuse University preprint SU-GP-93/5-