On maximal non-selfadjoint reflexive algebras associated with a double triangle lattice

We show that the reflexive algebra Alg(L) given by a double triangle lattice L in a finite factor M(with L" = M) is maximal non-selfadjoint in the class of all weak operator closed subalgebras with the same diagonal subalgebra Alg(L) ∩ Alg(L)~+.Our method can be used to prove similar results in finite-dimensional matrix algebras.As a consequence,we give a new proof to the main theorem by Hou and Zhang(2012).     

The Lifschitz singularity for the density of states on the Sierpinski gasket

Summary We prove the existence of the density of states for the Laplacian on the infinite Sierpinski gasket. Then the Lifschitz-type singularity of the density of states is established. We also investigate the long-time asymptotics of the Brownian trajectory on the Sierpinski gasket, getting bounds similar to those in the ^d-case.     

一种Sierpinski海绵的Hausdorff测度

利用相似分形的几何性质得到了一种Sierpinski海绵的Hausdorff测度为3.     

The number of bound states of a one-particle Hamiltonian on a three-dimensional lattice

We consider the Hamiltonian , describing the motion of one quantum particle on a three-dimensional lattice in an external field. We investigate the number of eigenvalues and their arrangement depending on the value of the interaction energy for μ ≥ 0 and λ ≥ 0. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 425–443, March, 2009.     

The Hausdorff measure of a Sierpinski carpet

The exact value of the Hausdorff measure of a Sierpinski carpet has been obtained. Project partially supported by the Science Foundation of Guangdong Province.     

Nonequilibrium stationary states of noninteracting electrons in a one-dimensional lattice

Nonequilibrium stationary states of a one-dimensional quantum conductor placed between two reservoirs are investigated. Applying the theory of C^*-algebra, as t→+∞, any state including the degrees of freedom of reservoirs is shown to weakly evolve to a quasifree stationary state with nonvanishing currents. The stationary state exhibits transports which are consistent with nonequilibrium thermodynamics and, in this sense, it has broken time symmetry. Particularly, the electric and energy currents are shown to be expressed by two-probe Landauer-type formulas and they reduce to the results by Sivan–Imry and Bagwell–Orlando in appropriate regimes. As a consequence of the time reversal symmetry, there exists another stationary state with anti-thermodynamical transports, which is the t→−∞ limit of the initial state. The consistency between the dynamical reversibility and the irreversibility of the evolution of states is discussed as well.     

Bound states of a two-boson system on a two-dimensional lattice

We consider a Hamiltonian of a two-boson system on a two-dimensional lattice Z². The Schrödinger operator H(k₁, k₂) of the system for k₁ = k₂ = π, where k = (k₁, k₂) is the total quasimomentum, has an infinite number of eigenvalues. In the case of a special potential, one eigenvalue is simple, another one is double, and the other eigenvalues have multiplicity three. We prove that the double eigenvalue of H(π,π) splits into two nondegenerate eigenvalues of H(π, π ? 2β) for small β > 0 and the eigenvalues of multiplicity three similarly split into three different nondegenerate eigenvalues. We obtain asymptotic formulas with the accuracy of β² and also an explicit form of the eigenfunctions of H(π, π ?2β) for these eigenvalues.     

一个广义Siperpinski地毯的Hausdorff测度

§ 1 .　Introduction　　Infractalgeometry ,thestudyingonthetheoryandcalculationofHausdorffdimensionhasmadegreatprogress.ButasfarasHausdorffmeasureisconcerned ,onlyafewresultsareknown .ZHOUZuo linghasobtainedtheexactvalueofHausdorffmeasureforaSierpinskicar pet(see [3]) .MADong kuihasdevelopedthetechniquesin [3](see [4 ]) .ButineverystepofconstructingtheSierpinskicarpetsin [4 ],theremaininglittlesquaressatisfytheconditionthatfourofthemareatthefourvertexanglesoftheirimmediatepredecessorrespe…     

Cubic lines relative to a triangle

An extension of the traditional geometry of the triangle is derived through the construction of two 30-points particular cubics. Two generation procedures founded on triangular quadratic transformations and dual associate properties of the two cubics are presented.     

How to best fold a triangle

We fold a triangle once along a straight line and study how small the area of the folded figure can be. It can always be as small as the fraction \(2-\sqrt{2}\) of the area of the original triangle.This is best possible: For every positive number \(\varepsilon\) there are triangles that cannot be folded better than \(2-\sqrt{2}-\varepsilon\).     

一类Sierpinski垫的Hausdorff测度

设S_λ为压缩比为λ(λ≤1/3)的一类Sierpinski垫,s=-log_λ3为S_λ的Hausdorff维数,N为产生S_λ的所有基本三角形的集合.本文使用网测度方法,获得了S_λ的s-维Hausdorff测度的精确值H~s(S_λ)=1,同时证明了H~s(S_λ)可由S_λ关于网N的s-维Hausdorff测度H_N~s(S_λ)确定,获得了S_λ的非平凡的最佳覆盖.     

The Hausdorff measure of a class of Sierpinski carpets

In this paper we obtain the exact value of the Hausdorff measure of a class of Sierpinski carpets with Hausdorff dimension no more than 1 and show the fact that the Hausdorff measure of such Sierpinski carpets can be determined by coverings which only consist of basic squares.     

The triangle closure is a polyhedron

Recently, cutting planes derived from maximal lattice-free convex sets have been studied intensively by the integer programming community. An important question in this research area has been to decide whether the closures associated with certain families of lattice-free sets are polyhedra. For a long time, the only result known was the celebrated theorem of Cook, Kannan and Schrijver who showed that the split closure is a polyhedron. Although some fairly general results were obtained by Andersen et al. (Math Oper Res 35(1):233–256, 2010) and Averkov (Discret Optimiz 9(4):209–215, 2012), some basic questions have remained unresolved. For example, maximal lattice-free triangles are the natural family to study beyond the family of splits and it has been a standing open problem to decide whether the triangle closure is a polyhedron. In this paper, we show that when the number of integer variables $m=2$ the triangle closure is indeed a polyhedron and its number of facets can be bounded by a polynomial in the size of the input data. The techniques of this proof are also used to give a refinement of necessary conditions for valid inequalities being facet-defining due to Cornuéjols and Margot (Math Program 120:429–456, 2009) and obtain polynomial complexity results about the mixed integer hull.     

Bound states of a system of two fermions on a one-dimensional lattice

We consider the Hamiltonian of a system of two fermions on a one-dimensional integer lattice. We prove that the number of bound states N(k) is a nondecreasing function of the total quasimomentum of the system k ∈ [0, π]. We describe the set of discontinuity points of N(k) and evaluate the jump N(k +0) − N(k) at the discontinuity points. We establish that the bound-state energy z _n (k) increases as the total quasimomentum k ∈ [0, π] increases. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 1, pp. 47–57, April, 2006.     

Spatiotemporal quasiperiodicity transition to chaos in the one-way coupled bistability lattice system

One-way coupled optical bistability lattice (OCBL) system for open flow is investigated. By using numerical simulations spatiotemporal quasiperiodicity is found. A rough phase diagram indicating the main spatiotemporal dynamic behavior in weakly coupled lattice is given. The transitions between spatiotemporal quasiperiodicity and chaos are observed. This result is important for the understanding of turbulence. Project supported by the Natural Science Foundation of Hebei Province.     

一类Sierpinski地毯的Hausdorff测度的估计

设Sr是压缩比为r(0.250≤r≤0.292)的Sierpinski地毯,该文证明了Sr的Hausdorff测度满足公式:21-s/2≤Hs(Sr)≤2s/2,其中s=-logr4.     

Generalized coherent states and nonlinear dynamics of a lattice quasispin model

The nonlinear lattice equation of the ϕ⁶ theory is studied by using the technique of generalized coherent states associated to a SU(2) Lie group. We analyze the discrete nonlinear equation with weak interaction between sites. The existence of saddles and centers is shown. The qualitative parametric domains which contain kinks, bubbles and plane waves were obtained. The specific implications of saddles and centers to the parametric first- and second-order phase transitions are identified and analyzed.