let $r$ be a commutative ring with identity and $m$ be a finitely generated unital $r$-module. in this paper, first we give necessary and sufficient conditions that a finitely generated module to be a multiplication module. moreover, we investigate some conditions which imply that the module $m$ is the direct sum of some cyclic modules and free modules. then some properties of fitting ideals of...

We study sums of the form ∑ ζ R(ζ), where R is a rational function and the sum is over all nth roots of unity ζ (often with ζ = 1 excluded). We call these generalized Dedekind sums, since the most well-known sums of this form are Dedekind sums. We discuss three methods for evaluating such sums: The method of factorization applies if we have an explicit formula for ∏ ζ(1− xR(ζ)). Multisection ca...

We present an infrared/optical study of the dynamics of the strongly correlated electron system YbIn1−xAgxCu4 as a function of composition and temperature for x ranging from 0 to 1, and T between 20 and 300 K. YbIn1−xAgxCu4 evolves from a mixed-valent system at low x to a heavy-Fermion system at high x, and exhibits an unusual electronic phase transition in the low x region. This paper reveals ...

In the weak coupling limit of QED, it is shown how to express the parton distributions in particular the photon distribution in terms of the nonrelativistic wavefunctions. A sum rule is derived, relating the momentum carried by the photons to the binding energy, thus allowing quantitative predictions about the modification of the photon distribution as molecules are formed from atoms. The physi...

The mass parameter of orbitally excited Λc baryons is calculated by using QCD sum rule in the framework of heavy quark effective theory. Two kinds of interpolating current for the excited heavy baryons are introduced. It is obtained that Λ̄ = 1.08 −0.104 GeV for the non-derivative current and Λ̄ = 1.06 +0.090 −0.107 GeV for the current with derivative. These results are consistent with experiment...