On the Decomposition of Hopf Module Coalgebra

In this paper, the module coradical is introduced. From this, we show that any Hopf module coalgebra which is local finite can be uniquely decomposed into a direct sum of its indecomposable components. This result is a generalization of the decompostion theorem of coalgebra.

On the Random Conjugate Spaces of a Random Locally Convex Module

Theoretically speaking, there are four kinds of possibilities to define the random conjugate space of a random locally convex module. The purpose of this paper is to prove that among the four kinds there are only two which are universally suitable for the current development of the theory of random conjugate spaces. In this process, we also obtain a somewhat surprising and crucial result： if the base （Ω,F, P） of a random normed module is nonatomic then the random normed module is a totally disconnected topological space when it is endowed with the locally L0-convex topology.

On the Finiteness Dimension of Local Cohomology Modules

Let R be a commutative Noetherian ring, α an ideal of R, and M a non-zero finitely generated R-module. Let t be a non-negative integer. In this paper, it is shown that dim Supp Hi a（M） ≤ 1 for all i 〈 t if and only if there exists an ideal b of R such that dimR/b ≤ 1 and Hia（M） ≌ Hi b（M） for all i 〈 t. Moreover, we prove that dimSuppHia（M） 〈≤dim M - i for all i.

Cofiniteness of Local Cohomology Modules

Let M be a non-zero finitely generated module over a commutative Noetherian local ring （R, m）. In this paper we consider when the local cohomology modules are finitely generated. It is shown that if t≥ 0 is an integer and p C Supp H^t_p （M）, then Hm^t＋dim R/p （M） is not p-cofinite. Then we obtain a partial answer to a question raised by Huneke. Namely, if R is a complete local ring, then H^n_m （M） is finitely generated if and only if 0 ≤ n ￠ W, where W ---- {t ＋ dimR/p丨p ∈ SuppH^t_p（M）／V（m）}. Also, we show that if J C I are 1-dimensional ideals of R, then H^t_I（M） is J-cominimax, and H^t_I（M） is finitely generated （resp., minimax） if and only if H}R, （Mp） is finitely generated for all p C Spec R （resp., p ∈ SpecR／MaxR）. Moreover, the concept of the J-cofiniteness dimension cJ（M） of M relative to I is introduced, and we explore an interrelation between c^I_m（M） and the filter depth of M in I. Finally, we show that if R is complete and dim M/IM ≠ 0, then c^I_m （R） ---- inf{depth Mp ＋ dim R/p 丨 P ∈ Supp M/IM／V（m）}.

On Baer-Kaplansky Classes of Modules

We are interested in studying when the class of local modules is Baer- Kaplansky. We provide an example showing that even over a commutative semisimple ring R, we can find two non-isomorphic simple R-modules S1 and S2 such that the rings EndR（S1） and EndR（S2） are isomorphic. We show that over any ring R, the class of semisimple R-modules is Baer Kaplansky if and only if so is the class of simple R-modules.

When Finitely Generated δ-Supplemented Modules Are Supplemented

We investigate the structure of rings over which every finitely generated g- supplemented module is supplemented. Some characterizations of this type of rings are given. We establishe some properties of ⊙-δ-supplemented modules. It is showed that for a ring R, finitely generated δ-supplemented R-modules are supplemented if and only if finitely generated ⊙-δ-supplemented R-modules are ⊙-supplemented.

Modulation of hot regions in waveguide-based evanescent-field-coupled localized surface plasmons for plasmon-enhanced spectroscopy

Coupling efficiency between the localized surface plasmons(LSPs) of metal nanoparticles(NPs) and incident light dominates the sensitivities of plasmon-based sensing spectroscopies and imaging techniques, e.g., surfaceenhanced Raman scattering(SERS) spectroscopy. Many endogenous features of metal NPs(e.g., size, shape,aggregation form, etc.) that have strong impacts on their LSPs have been discussed in detail in previous studies.Here, the polarization-tuned electromagnetic(EM) field that facilitates the LSP coupling is fully discussed.Numerical analyses on waveguide-based evanescent fields(WEFs) coupled with the LSPs of dispersed silver nanospheres and silver nano-hemispheres are presented and the applicability of the WEF-LSPs to plasmon-enhanced spectroscopy is discussed. Compared with LSPs under direct light excitation that only provide 3–4 times enhancement of the incidence field, the WEF-LSPs can amplify the electric field intensity about 30–90 times(equaling the enhancement factor of 10~6–10~8 in SERS intensity), which is comparable to the EM amplification of the SERS"hot spot" effect. Importantly, the strongest region of EM enhancement around silver nanospheres can be modulated from the gap region to the side surface simply by switching the incident polarization from TM to TE, which widely extends its sensing applications in surface analysis of monolayer of molecule and macromolecule detections. This technique provides us a unique way to achieve remarkable signal gains in many plasmon-enhanced spectroscopic systems in which LSPs are involved.

Some Criteria for the Cohen-Macaulay Property and Local Cohomology

Let a be an ideal of a commutative Noetherian ring R and M be a finitely generated R-module of dimension d. We characterize Cohen-Macaulay rings in term of a special homological dimension. Lastly, we prove that if R is a complete local ring, then the Matlis dual of top local cohomology module Ha^d（M） is a Cohen-Macaulay R-module provided that the R-module M satisfies some conditions.

On Modules Satisfying the Descending Chain Condition on Cyclic Submodules

A module satisfying the descending chain condition on cyclic submodules is called coperfect.The class of coperfect modules lies properly bet ween the class of locally artinian modules and the class of semiartinian modules.Let R be a commutative ring with identity.We show that every semiartinian Ti-module is coperfect if and only if R is a T-ring.It is also shown that the class of coperfect R-modules coincides with the class of locally artinian R-modules if and only if m/m^(2)is a finitely genera ted R-module for every maximal ideal m of R.

Simple Singular Whittaker Modules Over the Schrödinger Algebra

There are no simple singular Whittaker modules over most of important algebras,such as simple complex finite-dimensional Lie algebras,affine Kac-Moody Lie algebras,the Virasoro algebra,the Heisenberg-Virasoro algebra and the Schrödinger-Witt algebra.In this paper,however,we construct simple singular Whittaker modules over the Schrödinger algebra.Moreover,simple singular Whittaker modules over the Schrödinger algebra are classified.As a result,simple modules for the Schrödinger algebrawhich are locally finite over the positive part are completely classified.We also give characterizations of simple highestweight modules and simple singular Whittaker modules.

Recent progress in random metric theory and its applications to conditional risk measures

The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures.This paper includes eight sections.Section 1 is a longer introduction,which gives a brief introduction to random metric theory,risk measures and conditional risk measures.Section 2 gives the central framework in random metric theory,topological structures,important examples,the notions of a random conjugate space and the Hahn-Banach theorems for random linear functionals.Section 3 gives several important representation theorems for random conjugate spaces.Section 4 gives characterizations for a complete random normed module to be random reflexive.Section 5 gives hyperplane separation theorems currently available in random locally convex modules.Section 6 gives the theory of random duality with respect to the locally L0-convex topology and in particular a characterization for a locally L0-convex module to be L0-pre-barreled.Section 7 gives some basic results on L0-convex analysis together with some applications to conditional risk measures.Finally,Section 8 is devoted to extensions of conditional convex risk measures,which shows that every representable L∞-type of conditional convex risk measure and every continuous Lp-type of convex conditional risk measure(1 ≤ p < +∞) can be extended to an L∞F(E)-type of σ,λ(L∞F(E),L1F(E))-lower semicontinuous conditional convex risk measure and an LpF(E)-type of T,λ-continuous conditional convex risk measure(1 ≤ p < +∞),respectively.

Representations of Frobenius-type Triangular Matrix Algebras

The aim of this paper is mainly to build a new representation-theoretic realization of finite root systems through the so-called Frobenius-type triangular matrix algebras by the method of reflection functors over any field. Finally, we give an analog of APR-tilting module for this class of algebras. The major conclusions contains the known results as special cases, e.g., that for path algebras over an algebraically closed field and for path algebras with relations from symmetrizable cartan matrices. Meanwhile, it means the corresponding results for some other important classes of algebras, that is, the path algebras of quivers over Frobenius algebras and the generalized path algebras endowed by Frobenius algebras at vertices.