Magnetic symmetry and onsager relations

An alternative way of obtaining the generalized Onsager relations (Kleiner, 1966) which are valid for magnetic as well as non-magnetic crystals is given.     

Real hypersurfaces in complex space forms concerned with the local symmetry

This paper consists of two parts. In the first, we find some geometric conditions derived from the local symmetry of the inverse image by the Hopf fibration of a real hypersurface M in complex space form M _m(4ε). In the second, we give a complete classification of real hypersurfaces in M _m(4ε) which satisfy the above geometric facts. The second author was supported by DGICYT research project BFM 2001-2871-C04-01 and the first and the third authors were supported by grant Proj. No. R14-2002-003-01001-0 from Korea Research Foundation, Korea 2006.     

Generalized Serre relations for Lie algebras associated with positive unit forms

Every semisimple Lie algebra defines a root system on the dual space of a Cartan subalgebra and a Cartan matrix, which expresses the dual of the Killing form on a root base. Serre’s Theorem [J.-P. Serre, Complex Semisimple Lie Algebras (G.A. Jones, Trans.), Springer-Verlag, New York, 1987] gives then a representation of the given Lie algebra in generators and relations in terms of the Cartan matrix.In this work, we generalize Serre’s Theorem to give an explicit representation in generators and relations for any simply laced semisimple Lie algebra in terms of a positive quasi-Cartan matrix. Such a quasi-Cartan matrix expresses the dual of the Killing form for a Z-base of roots. Here, by a Z-base of roots, we mean a set of linearly independent roots which generate all roots as linear combinations with integral coefficients.     

Linear relations and congruences for the coefficients of Drinfeld modular forms

We find congruences for the t-expansion coefficients of Drinfeld modular forms for . We give generalized analogies of Siegel’s classical observation on SL ₂(ℤ) by determining all the linear relations among the initial t-expansion coefficients of Drinfeld modular forms for . As a consequence spaces M _k ⁰ are identified, in which there are congruences for the s-expansion coefficients. This work was supported by KOSEF R01-2006-000-10320-0 and by the Korea Research Foundation Grant (KRF-2005-214-M01-2005-000-10100-0)     

Quadratic forms on symmetry classes of tensors

Let V:₁,…,V_m be inner product spaces, and let L be a linear transformation on V₁ ?…?V_m which satisfies (Lz,z)=0 for every decomposable tensor z. It is known that if the field is the complex numbers, then (Lz,z)=0 for every z. This paper contains a short proof of this result, an extension of it to arbitrary symmetry classes of tensors, and an analysis of its failure when the field is the real numbers.     

Multiple commutation relations in the models with gl(2|1) symmetry

We consider quantum integrable models with the gl(2|1) symmetry and derive a set of multiple commutation relations between the monodromy matrix elements. These multiple commutation relations allow obtaining different representations for Bethe vectors.     

Schur orthogonality relations and invariant sesquilinear forms

Important connections between the representation theory of a compact group and are summarized by the Schur orthogonality relations. The first part of this work is to generalize these relations to all finite-dimensional representations of a connected semisimple Lie group The second part establishes a general framework in the case of unitary representations of a separable locally compact group. The key step is to identify the matrix coefficient space with a dense subset of the Hilbert-Schmidt endomorphisms on .

     

Bifurcation of equilibrium forms of a gas column rotating with constant speed around its axis of symmetry

We will be concerned with the problem of deformation of the lateral surface of a column that rotates with constant speed around its axis of symmetry. The column is filled by a gas and our goal is to investigate the deformation of the lateral surface depending on the pressure of the gas.     

CR-quadrics with a symmetry property

For non-degenerate CR-quadrics ${Q \subset \mathbb{C}^{n}}$ it is well known that the real Lie algebra ${\mathfrak{g} = \mathfrak{hol}(Q)}$ of all infinitesimal CR-automorphisms has a canonical grading ${\mathfrak{g} = \mathfrak{g}^{-2} \oplus\mathfrak{g}^{-1} \oplus\mathfrak{g}^{0} \oplus\mathfrak{g}^{1} \oplus\mathfrak{g}^{2}}$ . While the first three spaces in this grading, responsible for the affine automorphisms of Q, are always easy to describe this is not the case for the last two. In general, it is even difficult to determine the dimensions of ${\mathfrak{g}^{1}}$ and ${\mathfrak{g}^{2}}$ . Here we consider a class of quadrics with a certain symmetry property for which ${\mathfrak{g}^{1}, \mathfrak{g}^{2}}$ can be determined explicitly. The task then is to verify that there exist enough interesting examples. By generalizing the ?ilov boundaries of irreducible bounded symmetric domains of non-tube type we get a collection of basic examples. Further examples are obtained by ‘tensoring’ any quadric having the symmetry property with an arbitrary commutative (associative) unital *-algebra A (of finite dimension). For certain quadrics this also works if A is not necessarily commutative.     

两相同部件并联部分可修复系统的可靠性分析

用C_0半群理论,研究了一类两相同部件并联部分可修复系统解的存在惟一性及指数稳定性,并从本征向量的角度讨论了此系统的一些主要可靠性指标,给出了瞬态可用度的数值模拟.     

Union and symmetry preserving homomorphisms of semigroups of closed relations

The symbol β_X denotes the semigroup of all binary relations on a nonempty set X under composition which is defined by α_oβ={(x,y)} ∈ X×X: (x,z) ∈ β and (z,y) ∈ α for some z∈X} for all α,β ∈ β_x . In a recent paper [1, Theorem 3, p. 310], A. H. Clifford and D. D. Miller initiated a study of the endomorphisms of β_X when they completely determined those which preserve unions and take symmetric relations into symmetric relations. The purpose here is to place the theorem of Clifford and Miller in a topological setting and to discuss some of the problems which then arise naturally. The full results will appear in [9]. Partial financial support from Australian National University and the research foundation of State University of New York is gratefully acknowledged.     

Eichler integrals, period relations and Jacobi forms

This paper contains three main results: the first one is to derive two “period relations” and the second one is a complete characterization of period functions of Jacobi forms in terms of period relations. These are done by introducing a concept of “Jacobi integrals” on the full Jacobi group. The last one is to show, for the given holomorphic function P(τ, z) having two period relations, there exists a unique Jacobi integral, up to Jacobi forms, with a given function P(τ, z) as its period function. This is done by constructing a generalized Jacobi Poincaré series explicitly. This is to say that every holomorphic function with “period relations” is coming from a Jacobi integral. It is an analogy of Eichler cohomology theory studied in Knopp (Bull Am Math Soc 80:607–632, 1974) for the functions with elliptic and modular variables. It explains the functional equations satisfied by the “Mordell integrals” associated with the Lerch sums (Zwegers in Mock theta functions, PhD thesis, Universiteit Utrecht, 2002) or, more generally, with the higher Appell functions (Semikhatov et?al. in Commun Math Phys 255(2):469–512, 2005). Developing theories of Jacobi integrals with elliptic and modular variables in this paper is a natural extension of the Eichler integral with modular variable. Period functions can be explained in terms of the parabolic cohomology group as well.     

Limit cycles for a perturbation of a quadratic center with symmetry

To estimate the number of limit cycles appearing under a perturbation of a quadratic system that has a center with symmetry, we use the method of generalized Dulac functions. To this end, we reduce the perturbed system to a Liénard system with a small parameter, for which we construct a Dulac function depending on the parameter. This permits one to estimate the number of limit cycles in the perturbed system for all sufficiently small parameter values. We find the Dulac function by solving a linear programming problem. The suggested method is used to analyze four specific perturbed systems that globally have exactly three limit cycles [i.e., the limit cycle distribution 3 or (3, 0)] and two systems that have the limit cycle distribution (3, 1) (i.e., one nest around each of the two foci).     

Defining relations for a viscoelastic medium with microrotation

The results of [1, 2] are extended to the case of a Cosserat medium with a memory (the force stress tensor and the couple stress tensor depend on the history of deformations and rotations of a particle in the medium). In the linear approximation the defining relations have the form of convolutions with some relaxation kernels with respect to time. Restrictions for the kernels are obtained, which follow from the general principles of thermodynamics. The propagation of weak perturbations is studied. The general functional form of the ken nels corresponding to experimental data on the viscoelasticity of rock formations is given.