模上李代数及其基本性质

本文把域上李代数的概念推广到了交换环上模代数 ,给出了同态基本定理等基本性质 ,特别是决定了秩

Duality theorem for smash coproduct over quantum groupoids

The duality theorem of generalized weak smash coproducts of weak module coalgebras and comodule coalgebras over quantum groupoids is studied.Let H be a weak Hopf algebra,C a left weak H-comodule coalgebra and D a left weak H-module coalgebra.First,a weak generalized smash coproduct C×lH D over quantum groupoids is defined and the module and comodule structures on it are constructed.The weak generalized right smash coproduct C×rL D is similar.Then some isomorph-isms between them are obtained.Secondly,by introducing some concepts of a weak convolution invertible element,a weak co-inner coaction and a strongly relative co-inner coaction,a sufficient condition for C×rH D to be isomorphic to Cv D is obtained,where v∈WC（C,H）and the coaction of H on D is right strongly relative co-inner.Finally,the duality theorem for a generalized smash coproduct over quantum groupoids,（C×lH H）×lH H≌Cv（H×lH H）,is obtained.

Relatively Injective Modules with Respect to Torsion Theory

For a hereditary torsion theory τ, this paper mainly discuss properties of A-τ-injective modules, where A is a fixed left R-module. It is proved that if M is an A-τ-injective, B is a submodule of A, then 1） M is A/B-τ-injective; 2） M is B-τ-injective when B is τ-dense in A. Furthermore, we show that if A1,A2,... An, are relatively injective modules, then A1 A2 ... An is self-τ-injective if and only if A1 is self-τ-injective for each i.

Nonlinear Mathematical Simulation and Analysis of Dha Regulon for Glycerol Metabolism in Klebsiella pneumoniae

Glycerol may be converted to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae under anaerobic conditions and glycerol dismutation involves two parallel pathways controlled by the dha regulon. In this study, a fourteen-dimensional nonlinear dynamic system is presented to describe the continuous culture and multiplicity analysis, in which two regulated negative-feedback mechanisms of repression and enzyme inhibition are investigated. The model describing the expression of gene-mRNA-enzyme-product was established according to the repression of the dha regulon by 3-hydroxypropionaldehy (3-HPA). Comparisons between simulated and experimental results indicate that the model can be used to describe the production of 1,3-PD under continuous fermentation. The new model is translated into the corresponding S-system version. The robustness of this model is discussed by using the S-system model and the sensitivity analysis shows that the model is sufficiently robust. The influences of initial glycerol concentration and dilution rate on the biosynthesis of 1,3-PD and the stability of the dha regulon model are investigated. The intracellular concentrations of glycerol, 1,3-PD, 3-HPA, repressor mRNA, repressor, mRNA and protein levels of glycerol dehydratase (GDHt) and 1,3-PD oxydoreductase (PDOR) can be predicted for continuous cultivation. The results of simulation and analysis indicate that 3-HPA accumulation will repress the expression of the dha regulon at the transcriptional level. This model gives new insights into the regulation of glycerol metabolism in K. pneumoniae and explain some of the experimental observations.

Y（sl（2）） Algebra Application in Extended Hydrogen Atom and Monopole Models

We present the extended hydrogen atom and monopole-hydrogen atom theory through generalizing the usual hydrogen atom model and with a monopole model respectively, in which Y (sl(2) ) algebras are realized. We derive the Hamiltonians of the two models based on the Y(sl(2) ) and the generalized Pauli equation. The energy spectra of the systems are also given in terms of Yangian algebra and quantum mechanics.

Transitional Description of Diatomic Molecules in U（4） Vibron Model

U(3)-O(4) transitional description of diatomic molecules in the U(4) vibron model is studied by usingthe algebraic Bethe ansatz, in which the O(4) limit is a special case of the theory. Vibrational band-heads of sometypical diatomic molecules are fitted by both transitional theory and the O(4) limit within the same framework. Theresults show that there are evident deviations from the O(4) limit in description of vibrational spectra of some diatomicmolecules.

Diagonal crossed product of multiplier Hopf algebras

Let A and B be two regular multiplier Hopf algebras.First,the notion of diagonal crossed product B#A of multiplier Hopf algebras is constructed using the bimodule algebra,which is a generalization of the diagonal crossed product in the sense of Hopf algebras.The result that the product in B#A is non-degenerate is given.Next,the definition of the comultiplicationΔ#on B#A is introduced,which is composed of the multiplierΔB(b)on B⊗B and the multiplierΔA(a)on A⊗A,and the elementΔ#(b⊗a)is a two-side multiplier of B#A⊗B#A,for any b∈B and a∈A.Then,a sufficient condition for B#A to be a regular multiplier Hopf algebra is described.In particular,Delvaux's main theorem in the case of smash products is generalized.Finally,these integrals on a diagonal crossed product of multiplier Hopf algebras are considered.

On a Result of Pointed Representations of finite Dimensional Simple Lie Algebras

There have been a great many of studies on the pointed representations of fi- nite-dimensional sanple Lie algebras.cf.[1][2]etc.In this paper we give a new proof of an impottant Lemma,and from this we derive our main result:Irreducible pointed modules of finite -dimesional simple Lie algebras are all Harish-Chandra modules.

Extreme points of norm closed nest algebra modules

Suppose that U is a norm closed nest algebra module. Using the characterization of rank one operators in U⊥, a complete description of the extreme points of the unit ball U1 is given.

Some Properties of Fuzzy Filter and Its Category

This paper is devoted to the study of some properties of fuzzy filters in lattice implication algebras. The structure theorem of fuzzy filters and the category of the sets of fuzzy filters were established with some basic properties of it were discussed.

A Comprehensive Study of Algebraic Fault Analysis on PRINCE

PRINCE is a 64-bit lightweight block cipher with a 128-bit key published at ASIACRYPT 2012. Assuming one nibble fault is injected, previous different fault analysis(DFA) on PRINCE adopted the technique from DFA on AES and current results are different. This paper aims to make a comprehensive study of algebraic fault analysis(AFA) on PRINCE. How to build the equations for PRINCE and faults are explained. Extensive experiments are conducted. Under nibble-based fault model, AFA with three or four fault injections can succeed within 300 seconds with a very high probability. Under other fault models such as byte-based, half word-based, word-based fault models, the faults become overlapped in the last round and previous DFAs are difficult to work. Our results show that AFA can still succeed to recover the full master key. To evaluate security of PRINCE against fault attacks, we utilize AFA to calculate the reduced entropy of the secret key for given amount of fault injections. The results can interpret and compare the efficiency of previous work. Under nibble-based fault model, the master key of PRINCE can be reduced to 29.69 and 236.10 with 3 and 2 fault injections on average, respectively.

Matrix Algebras in Non-Hermitian Quantum Mechanics

In principle, non-Hermitian quantum equations of motion can be formulated using as a starting point either the Heisenberg＇s or the Schroedinger＇s picture of quantum dynamics. Here it is shown in both cases how to map the algebra of commutators, defining the time evolution in terms of a non-Hermitian Hamiltonian, onto a non-Hamiltonian algebra with a Hermitian Hamiltonian. The logic behind such a derivation is reversible, so that any Hermitian Hamiltonian can be used in the formulation of non-Hermitian dynamics through a suitable algebra of generalized （non-Hamiltonian） commutators. These results provide a general structure （a template） for non-Hermitian equations of motion to be used in the computer simulation of open quantum systems dynamics.

Detailed Analysis of Quantum Phase Transitions Within the u(2) Algebra

We analyze in detail the quantum phase transitions that arise in models based on the u（2） algebraic description for bosonic systems with two types of scalar bosons. First we discuss the quantum phase transition that occurs in hamiltonians that admix the two dynamical symmetry chains u（2） u（1） and u（2） so（2） by diagonalizing the problem exactly in the u（1） basis. Then we apply the coherent state formalism to determine the energy functioned. Finally we show that a quantum phase transition of a different nature, but displaying similar characteristics, may arise also within a single chain just by including higher order terms in the hamiltonian.

Comparative Analysis for Different Mathematical Models in Electron Optics

The comparative analysis of modem mathematical models for 3D problems in electron optics is presented. The new approach to solve the electron optics problems in three dimensions is presented. It is based on the principal ray method suggested by G. Grinberg in 1948. That perspective approach was not realized before for full three-dimensional electron optic systems, probably because of the complexity of its mathematical apparatus. We describe the analytical technique of the BEM （boundary element method） for the field evaluation, and 3rd order aberration expansion for the trajectory analysis. The first version of such computer code ＂OPTICS-3＂ and some results of numerical simulations with this code were presented.

Derivations of the Schrodinger Algebra on Restricted Simple Modules

Over an algebraically closed field of characteristic p>2,based on the results on the representation theory of special linear Lie algebra sl(2),restricted simple modules L(λ) of the Schrodinger algebra S(1)are determined,and all derivations of S(1)on L(λ)are also obtained.As an application,the first cohomology of S(1)with the coefficient in L(λ)is determined.

Solution to the Yang-Baxter Equation and QuantumYang-Baxter H-comodules

Let H be an arbitrary Hopf algebra over a field k. In this paper, at first wedeal with the relationship between solutions to the Yang-Baxter equation and quantumYang-Baxter H-comodules; then we use the results to give a solution to the Yang-Baxterequation over H.

Regularity of vertex operator superalgebras

We prove that the regularity of a vertex operator superalgebra can be reduced to the semisimplicity of the category of its weak modules.Moreover,the rationality can be replaced by requiring that each 1/2Z+-graded module is a direct sum of irreducible weak modules.

Categorical resolutions of a class of derived categories

We clarify the relation between the subcategory D_(hf)~b(A) of homological finite objects in D^b(A)and the subcategory K^b(P) of perfect complexes in D^b(A), by giving two classes of abelian categories A with enough projective objects such that D_(hf)~b(A) = K^b(P), and finding an example such that D_(hf)~b(A)≠K^b(P). We realize the bounded derived category D^b(A) as a Verdier quotient of the relative derived category D_C^b(A), where C is an arbitrary resolving contravariantly finite subcategory of A. Using this relative derived categories, we get categorical resolutions of a class of bounded derived categories of module categories of infinite global dimension.We prove that if an Artin algebra A of infinite global dimension has a module T with inj.dimT <∞ such that ~⊥T is finite, then D^b(modA) admits a categorical resolution; and that for a CM(Cohen-Macaulay)-finite Gorenstein algebra, such a categorical resolution is weakly crepant.

Projections of weak braided Hopf algebras

In this paper we study the projections of weak braided Hopf algebras using the notion of Yetter-Drinfeld module associated with a weak braided Hopf algebra. As a consequence, we complete the study ofthe structure of weak Hopf algebras with a projection in a braiding setting obtaining a categorical equivalencebetween the category of weak Hopf algebra projections associated with a weak Hopf algebra H living in abraided monoidal category and the category of Hopf algebras in the non-strict braided monoidal category ofleft-left Yetter-Drinfeld modules over H.

On Hopf Galois Extension of Separable Algebras

In this paper, the classical Galois theory to the H＊-Galois case is developed. Let H be a semisimple and cosemisimple Hopf algebra over a field k, A a left H-module algebra, and A/An a right H＊-Galois extension. The authors prove that, if An is a separable kalgebra, then for any right coideal subalgebra B of H, the B-invariants AB = {a ∈ A ｜ b · a = ε（b）a, Ab ε B} is a separable k-algebra. They also establish a Galois connection between right coideal subalgebras of H and separable subalgebras of A containing AH as in the classical case. The results are applied to the case H = （kG）＊ for a finite group G to get a Galois 1-1 correspondence.