Some variational formulas for Hausdorff dimension,topological entropy,and SRB entropy for hyperbolic dynamical systems

We derive several variational formulas for the topological entropy and SRB entropy of Axiom A flows on compact manifolds and for the Hausdorff dimension of basic sets for Axiom A diffeomorphisms on compact surfaces.     

Two Integrable Couplings of Modified Tu Hierarchy

Two different integrable couplings of the modified Tu hierarchy are obtained under the zero curvature equation by using two higher dimension Lie algebras. Furthermore, a complex Hamiltonian structures of the second integrable couplings is presented by taking use of the variational identity.     

Exciton affinity of quantum dots

An exciton may be attached to a quantum dot if the conduction-band and valence-band offsets, for a given dot size, satisfy certain inequalities. Alternatively, for a set of finite offsets, the dot dimension must exceed a minimal size for an exciton to be bound to the dot. These critical offset or dot size values are determined for a spherical quantum dot by a variational calculation. Both type-I and type-II situations are considered.     

Local Geometry and Dynamical Behavior on Folded Basic Sets

We study new phenomena associated with the dynamics of higher dimensional non-invertible, hyperbolic maps f on basic sets of saddle type; the dynamics in this case presents important differences from the case of diffeomorphisms or expanding maps. We show that the stable dimension (i.e. the Hausdorff dimension of the intersection between local stable manifolds and the basic set) and the unstable dimension (similar definition) give a lot of information about the dynamical/ergodic properties of endomorphisms on folded basic sets. We prove a geometric flattening phenomenon associated to the stable dimension, i.e. we show that if the stable dimension is zero at a point, then the fractal Λ must be contained in a submanifold and f is expanding on Λ. We characterize folded attractors and folded repellers, as those basic sets with full unstable dimension, respectively with full stable dimension. We classify possible dynamical behaviors, and establish when is the system (Λ,f,μ) 1-sided or 2-sided Bernoulli for certain equilibrium measures μ on folded basic sets, for a class of perturbation maps.     

Multifractal Analysis of Asymptotically Additive Sequences

For saturated maps, we effect a complete multifractal analysis of the dimension spectra obtained from asymptotically additive sequences of continuous functions. This includes, for example, the class of maps with the specification property. We consider also the more general cases of ratios of sequences and of multidimensional spectra in which a single sequence is replaced by a vector of sequences. In addition, we establish a conditional variational principle for the topological pressure of a continuous function on the level sets of an asymptotically additive sequence (again in the former general setting). Finally, we apply our results to the dimension spectra of an average conformal repeller. In particular, we obtain almost automatically a conditional variational principle for the Hausdorff dimension of the level sets obtained from an asymptotically additive sequence.     

Variational method in hamiltonian lattice gauge theories

A general expression for the expectation value of the hamiltonian of a d + 1 dimensional lattice gauge theory as a function of the norm of the variational state (that itself has the form of a partition function of a d-dimensional lattice gauge theory) is given. Applications include U(1), SU(2), U(2) and U(N) gauge theories for large N in d = 2 + 1 dimensions. It is also demonstrated that the deconfining phase transition is of first order in every dimension above the critical one, provided it is of first or second order at the critical dimension.     

耗散量子隧道系统中的形变效应与热库维数的关系

本文利用微扰及变分法研究了耗散量子隧道系统中热库声子波函数的形变效应,发现形变与热库的维数有关。随着维数降低,一方面形变效应增大,另一方面形变效应对量子隧道系统的影响变得更为显著。 关键词：     

New renormalization group results for scaling of self-avoiding tethered membranes

The scaling properties of self-avoiding polymerized two-dimensional membranes are studied via renormalization group methods based on a multilocal operator product expansion. The renormalization group functions are calculated to second order. This yields the scaling exponent ν to order ε² Our extrapolations for ν agree with the Gaussian variational estimate for large space dimension d and are close to the Flory estimate for d = 3. The interplay between self-avoidance and rigidity at small d is briefly discussed.     

Groupoids,Discrete Mechanics,and Discrete Variation

After introducing some of the basic definitions and results from the theory of groupoid and Lie algebroid, we investigate the discrete Lagrangian mechanics from the viewpoint of groupoid theory and give the connection between groupoids variation and the methods of the first and second discrete variational principles.     

On the equivalence of theKMS condition and the variational principle for quantum lattice systems

For quantum spin systems on a lattice of an arbitrary dimension, theKMS condition and the variational principle are shown to be equivalent at an arbitrary temperature for translationally invariant states.     

Equations of motion,commutation relations and ambiguities in the Lagrangian formalism

The inverse problem of the calculus of variations is analysed in the case of Newtonian mechanics. Its connection with the question: “Does the equation of motion determine commutation relations?” raised some time ago by Wigner, is exhibited. All the Lagrangians that yield variational equations equivalent to a given set of second order differential equations are explicitly determined for some simple classes of forces.     

由Gurtin变分原理求解策动力下一维动力学初值问题的半解析法

基于Gurtin变分原理,以梁为对象研究了在策动力下动力学初值问题的半解析解法。该方法在空间域作有限元离散,在时间域上取级数。     

光子晶体理论研究的新方法——混合变分法

利用混合变分法研究了二维光子晶体的能带结构,得到了通带、禁带和群速度,并详细分析光子晶体中的电磁场分布和能流密度分布.该方法方便实用,理论上能够应用于任意维度任意周期结构的光子晶体的计算.     

Approximation methods for stationary solutions of discrete master equations

Two methods for finding stationary solutions of discrete master equation in general cases without detailed balance are set up and compared for the case of the Laser equation. The first method starts from a reduced master equation obtained for appropriate forms of the Liouvillean, the second from a variational principle for the stationary solution of the full master equation. In the Laser case, both methods lead to forms of the stationary solution essentially equivalent to the result of Dohm.     

A variational treatment of the periodic Anderson model with antiferromagnetic order

We discuss a variational ground state wave function for the symmetric periodic Anderson model with commensurate spin density order. The energy of this ansatz is evaluated in closed form. Our approach generalizes a variational treatment proposed recently by Strack and Vollhardt and results in significantly lower energy. Contrary to the Gutzwiller ansatz the wave function recovers the Schrieffer-Wolff limit for large Coulomb repulsion. We clarify the relation of our approach to unrestricted Hartree-Fock and present a comparison with existing quantum Monte Carlo calculations for one dimension.     

An alternative analytic treatment of the Gutzwiller ansatz

An alternative analytic approach is formulated for treating Gutzwiller's variational ansatz. In contrast to the diagrammatic treatment recently published by Metzner and Vollhardt, our approach is purely algebraic and mainly based on expansions in terms of cluster integrals. The present theory is carried out for arbitrary dimension and band filling, but is applied here only to the Hubbard interaction term. In one dimension we find exact agreement between our results and those obtained by Metzner and Vollhardt.     

Polaron effects on excitons in parabolic quantum wells: fractional-dimension variational approach

Polaron effects on excitons in parabolic quantum wells are studied theoretically by using a variational approach with the so-called fractional dimension model. The numerical results for the exciton binding energies and longitudinal-optical phonon contributions in GaAs/Al_0.3Ga_0.7As parabolic quantum well structures are obtained as functions of the well width. It is shown that the exciton binding energies are obviously reduced by the electron (hole)-phonon interaction and the polaron effects are un-negligible. The results demonstrate that the fractional-dimension variational theory is effectual in the investigations of excitonic polaron problems in parabolic quantum wells.     

Tense Operators on Basic Algebras

The concept of tense operators on a basic algebra is introduced. Since basic algebras can serve as an axiomatization of a many-valued quantum logic (see e.g. Chajda et al. in Algebra Univer. 60(1):63–90, 2009), these tense operators are considered to quantify time dimension, i.e. one expresses the quantification “it is always going to be the case that” and the other expresses “it has always been the case that”. We set up the axiomatization and basic properties of tense operators on basic algebras and involve a certain construction of these operators for left-monotonous basic algebras. Finally, we relate basic algebras with tense operators with another quantum structures which are the so-called dynamic effect algebras.