We survey some recent developments in the theory of b-function, spectrum, and multiplier ideals together with certain interesting relations among them including the case of arbitrary subvarieties. Dedicated to Professor Masaki Kashiwara

I. INCREASING CAPACITY ON SPECTRUM THAT IS FLEXIBLY ALLOCATED .................................................................................. 91 II. MOVING SPECTRUM TO BE MORE FLEXIBLY ALLOCATED ............ 93 A. Government spectrum may be able to be used more efficiently .............................................................................. 94 1. Incentives for efficient use by Gov...

let $r$ be a commutative ring and let $m$ be an $r$-module. in this article, we introduce the concept of the zariski socles of submodules of $m$ and investigate their properties. also we study modules with noetherian second spectrum and obtain some related results.

a multicone graph is defined to be join of a clique and a regular graph. a graph $ g $ is  cospectral with  graph $ h $ if their adjacency matrices have the same eigenvalues. a graph $ g $ is  said to be  determined by its spectrum or ds for short, if for any graph $ h $ with $ spec(g)=spec(h)$, we conclude that $ g $ is isomorphic to $ h $. in this paper, we present new classes of  multicone g...

Mathieu and Ruddy proved that if be a unital spectral isometry from a unital C*-algebra Aonto a unital type I C*-algebra B whose primitive ideal space is Hausdorff and totallydisconnected, then is Jordan isomorphism. The aim of this note is to show that if be asurjective spectrum preserving additive map, then is a Jordan isomorphism without the extraassumption totally disconnected.

We investigate the spectrum of the two-dimensional Schrödinger operator H = @ @x ia1(x; y) 2 @ @y ia2(x; y) 2 + V (x; y), where the magnetic field B(x; y) = @ @x a2 @ @y a1 and the electric potential V are spherically symmetric, i.e., B(x; y) = b(r), r = px2 + y2, and b is p-periodic, similarly for V . By considering two different gauges we get the following results: In case R p 0 b(s) ds = 0 t...