A Binomial Approximation Method for the Ising Model

A large portion of the computation required for the partition function of the Ising model can be captured with a simple formula. In this work, we support this claim by defining an approximation to the partition function and other thermodynamic quantities of the Ising model that requires no algorithm at all. This approximation, which uses the high temperature expansion, is solely based on the binomial distribution, and performs very well at low temperatures. At high temperatures, we provide an alternative approximation, which also serves as a lower bound on the partition function and is trivial to compute. We provide theoretical evidence and the results of numerical experiments to support the strength of these approximations.     

Partition function data and impact on retrieval quality for an mm/sub-mm limb sounder

The partition function plays a major role as it mainly governs the dependence of the spectral line strength (or line intensity) on the temperature. A wrong partition function will lead to an incorrect line strength, which will lead to a systematic error in the retrieval. Therefore, a careful investigation of the available partition function data and the sensitivity of the retrieval to this is required. Two partition function data sets, the HITRAN and the JPL, are considered in this paper. In a first step, we investigate the spread of the partition function ratios from the simple approximation values given by theory. We find that the HITRAN partition functions are usually higher than the values quoted by the simple approximation. The JPL partition functions are much closer to the values given by the approximation. Comparing directly the two datasets, with some exceptions, a good agreement is found. Only for 16 molecules (from 66 considered molecules), the deviations in the two data sets are larger than 2%. The retrieval error analysis shows that, for the case of the molecular species with strong signatures, an uncertainty in the partition function is directly translated into a retrieval error of the species in question. However, the uncertainty in the partition function of this species can have a high impact on the quality of weak species retrieval.     

Approximation-error-ADP-based optimal tracking control for chaotic systems with convergence proof

In this paper, an optimal tracking control scheme is proposed for a class of discrete-time chaotic systems using the approximation-error-based adaptive dynamic programming （ADP） algorithm. Via the system transformation, the optimal tracking problem is transformed into an optimal regulation problem, and then the novel optimal tracking control method is proposed. It is shown that for the iterative ADP algorithm with finite approximation error, the iterative performance index functions can converge to a finite neighborhood of the greatest lower bound of all performance index functions under some convergence conditions. Two examples are given to demonstrate the validity of the proposed optimal tracking control scheme for chaotic systems.     

Precise implementation of cluster transfer matrix method in the single electron box

The partition function of the single electron box (SEB), a small metallic island connected by a tunnel junction to the source lead and by a gate capacitor to the gate, can be expressed in path-integral form, which contains the effective action of the collective variable, phase, after integrating out the background electron degrees of freedom. The cluster transfer matrix method (CTM) is applied to the SEB. By using an improved numerical algorithm and more intensive calculations with larger cluster size, we obtained a highly accurate result for the effective charging energy of SEB up to a large barrier conductance. With a clear converging tendency and the fact that we do not use any approximation in calculation of the partition function, our CTM calculation is systematic and exact. The result is in excellent agreement with the real time renormalization group method of König and Schoeller.     

Hadronic density of states from string theory

We present an exact calculation of the finite temperature partition function for the hadronic states corresponding to a Penrose-Güven limit of the Maldacena-Nù?ez embedding of the N=1 super Yang-Mills (SYM) into string theory. It is established that the theory exhibits a Hagedorn density of states. We propose a semiclassical string approximation to the finite temperature partition function for confining gauge theories admitting a supergravity dual, by performing an expansion around classical solutions characterized by temporal windings. This semiclassical approximation reveals a hadronic energy density of states of a Hagedorn type, with the coefficient determined by the gauge theory string tension as expected for confining theories. We argue that our proposal captures primarily information about states of pure N=1 SYM theory, given that this semiclassical approximation does not entail a projection onto states of large U(1) charge.     

Funktionalmittelwerte in der statistischen Theorie von quantenmechanischen und klassischen Systemen vieler Teilchen

The grand-canonical partition function of an interacting many-particle-system is represented as a functional integral with Gaussian random variables. The representation can be regarded as a Gaussian average over the partition function of free particles in an external fluctuating potential. The latter partition function is studied by means of diagrammatical techniques. The set of diagrams of a particularly simple structure is summed up by introducing the full scattering amplitude for the scattering in the external potential. The thermodynamicalGibbs' potential proves to be stationary with respect to the true particle density. It is shown that a variational procedure leads directly to an approximation which may be regarded as the renormalized form of the well-known Random-Phase-Approximation (RPA). The main feature of the approximation is thatGibbs' potential is stationary with respect to the two-particle-density correlation function. The classical limit of the renormalized RPA yields the results of the Debye-Hückel theory. In case of an hard-core potential the approximation applies only to the long-range part of the potential. The results are similar to recent developments in the theory of the Ising model and of real gases.     

应用遗传算法和序列锥规划进行圆阵的方向图综合

为了解决圆阵方向图旁瓣电平较高的问题,提出了一种圆阵方向图综合方法。该方法将圆阵方向图的峰值旁瓣电平作为目标函数,首先采用改进遗传算法对阵元位置和阵元权值进行联合优化,不仅避免了算法的早熟收敛,而且符合理论意义上的全局最优。其次将遗传算法的优化结果作为初始迭代点,在其附近利用一阶泰勒级数将非凸的圆阵方向图综合问题转化为序列锥规划问题,以便采用凸优化理论进行高效求解。由于该算法增加了寻优操作的后期变异能力,因而有效提高了优化性能。最后仿真表明,阵元数一定的情况下,算法在进一步降低峰值旁瓣电平的同时,可有效减小其动态变化范围,使得圆阵方向图综合性能更优。      

Functional averages in the theory of superconductivity

The grand canonical partition function of a superconductor described byGorkov's model Hamiltonian is represented as a functional integral with Gaussian measure. The integrand can be regarded as the partition function of a free Fermi system which interacts with a fluctuating external source potential. Perturbation-theoretic techniques are applied to the latter partition function. TheGibbs' potential proves to be stationary with respect to the energy gap parameterΔ. From the stationarity condition an equation forΔ is obtained which is a generalization of the usual Bardeen-Cooper-Schrieffer (BCS) equation. For the evaluation of the functional integral a variational procedure is employed. It leads to an expression for theGibbs' potential which shows a further remarkable stationarity property. As its simplest approximation this expression contains a result that was firstly derived by Thouless in the ladder graph approximation.     

一维互作用扭折-声子气体的统计理论

本文系统地研究并改进了位移型相变中一维互作用扭折-声子气体模型的统计理论。通过对扭折和声子的配分函数各自采用适当的路径积分表式并改进声子路径积分的计算方法,得到一个较合理且较普遍的巨配分函数表达式。在基态近似下,它可简化为一个与文献[6]相似的结果;在经典近似下,由它计算出的平均扭折密度与计算机模拟实验结果比文献上的符合得要好。     

An approximation for the boundary optimal control problem of a heat equation defined in a variable domain

In this paper, we consider a numerical approximation for the boundary optimal control problem with the control constraint governed by a heat equation defined in a variable domain. For this variable domain problem, the boundary of the domain is moving and the shape of theboundary is defined by a known time-dependent function. By making use of the Galerkin finite element method, we first project the original optimal control problem into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then, based on the aforementioned semi-discrete problem, we apply the control parameterization method to obtain an optimal parameter selection problem governed by a lumped parameter system, which can be solved as a nonlinear optimization problem by a Sequential Quadratic Programming （SQP） algorithm. The numerical simulation is given to illustrate the effectiveness of our numerical approximation for the variable domain problem with the finite element method and the control parameterization method.     

Automatic detection of the belt-like region in an image with variational PDE model

In this paper, we propose a novel method to automatically detect the belt-like object, such as highway,river, etc., in a given image based on Mumford-Shah function and the evolution of two phase curves. The method can automatically detect two curves that are the boundaries of the belt-like object. In fact, this is a partition problem and we model it as an energy minimization of a Mumford-Shah function based minimal partition problem like active contour model. With Eulerian formulation the partial differential equations (PDEs) of curve evolution are given and the two curves will stop on the desired boundary. The stop term does not depend on the gradient of the image and the initial curves can be anywhere in the image. We also give a numerical algorithm using finite differences and present various experimental results. Compared with other methods, our method can directly detect the boundaries of belt-like object as two continuous curves, even if the image is very noisy.     

An Efficient Computational Algorithm for Implementing the Maximum Entropy Method in Deconvolution Inversion Problems

We discuss a theoretical informational approach to solving ill-posed problems of function recovery based on the use of the maximum entropy principle. On this basis, we propose an efficient computational algorithm for implementing the procedure of solving the function-recovery problem and the method of regularization of the problem of function recovery from the convolution. In many cases, the proposed regularization method ensures high recovery quality. If the recovery accuracy is insufficient, then the Largange multipliers obtained by this method can be the best initial approximation (in the sense of the least squares) for their iterative refinement.     

The partition function of a degenerate functional

The partition function of a degenerate quadratic functional is defined and studied. It is shown that Ray-Singer invariants can be interpreted as partition functions of quadratic functionals. In the case of a degenerate non-quadratic functional the semiclassical approximation to the partition function is considered.     

Low depth quantum circuits for Ising models

A scheme for measuring complex temperature partition functions of Ising models is introduced. Two applications of this scheme are presented. First, through appropriate Wick rotations, those amplitudes can be analytically continued to yield estimates for partition functions of Ising models. Bounds on the estimated error are provided through a central-limit theorem whose validity extends beyond the present context; it holds for example for estimations of the Jones polynomial. The kind of state preparations and measurements involved in this application can be made independent of the system size or the parameters of the system being simulated. Second, the scheme allows to accurately estimate non-trivial invariants of links. Another result concerns the computational power of estimations of partition functions for real temperature classical ferromagnetic Ising models. We provide conditions under which estimating such partition functions allows to reconstruct scattering amplitudes of quantum circuits, making the problem BQP-hard. We also show fidelity overlaps for ground states of quantum Hamiltonians, which serve as a witness to quantum phase transitions, can be estimated from classical Ising model partition functions. Finally, we discuss how accurate corner magnetisation measurements on thermal states of two-dimensional Ising models lead to fully polynomial random approximation schemes (FPRAS) for the partition function.     

A circle similarity algorithm for an automatic circular decomposition of blood cell images

This paper presents an iterative algorithm for circular decomposition which investigates the separation of overlapped circular particles of a binary image, in order to locate their center coordinates and to estimate their radii. Since this algorithm is based on the measure of a circle similarity of an object in an image to execute a search for concavities, object segmentation and circle recognition, its implementation is simpler than the algorithm based on polygonal approximation. In this work we compare the accuracy and robustness of the proposed circle similarity algorithm with a polygonal approximation based algorithm using synthetic images and real blood cell images. Both the algorithms are able to decompose connected blood cells. However, the input parameters of the polygonal approximation method shows high sensitivity to the shape or agglomeration of an object, while the proposed algorithm demonstrates more stable performance.     

一种改进的开关电流滤波器实现小波变换的方法

提出了一种小波变换的开关电流滤波器实现的新方案,基于网络函数逼近理论,采用一种函数逼近的系统算法将基本小波函数综合为有理分式和,并利用开关电流并联结构实现该基本小波函数滤波器.以高斯一阶导数函数为例,给出了逼近网络的具体构造过程和开关电流滤波器实现结构.该滤波器网络由六个以S²I存储单元为核的双二次开关电流滤波器并联而成.理论分析和仿真结果表明,新方案比原有实现方法在逼近精度、系统稳定性、电路性能方面均有明显改善. 关键词： 小波变换 开关电流 网络函数逼近 有理分式综合     

Modularity-maximizing graph communities via mathematical programming

In many networks, it is of great interest to identify communities, unusually densely knit groups of individuals. Such communities often shed light on the function of the networks or underlying properties of the individuals. Recently, Newman suggested modularity as a natural measure of the quality of a network partitioning into communities. Since then, various algorithms have been proposed for (approximately) maximizing the modularity of the partitioning determined. In this paper, we introduce the technique of rounding mathematical programs to the problem of modularity maximization, presenting two novel algorithms. More specifically, the algorithms round solutions to linear and vector programs. Importantly, the linear programing algorithm comes with an a posteriori approximation guarantee: by comparing the solution quality to the fractional solution of the linear program, a bound on the available “room for improvement” can be obtained. The vector programming algorithm provides a similar bound for the best partition into two communities. We evaluate both algorithms using experiments on several standard test cases for network partitioning algorithms, and find that they perform comparably or better than past algorithms, while being more efficient than exhaustive techniques.     

任意阶标度分形格分抗与非正则格型标度方程

Carlson分形格电路是分抗的理想逼近情形,但仅具有负半阶运算性能,逼近效益随着电路节次数的增加逐渐降低.虽然可嵌套得到-1/2~n阶(n为大于或等于2的整数)分抗逼近电路,但结构复杂,无法实现任意分数阶运算.通过类比拓展Carlson分形格电路,获得具有高逼近效益的任意实数阶微积算子的分抗逼近电路——标度分形格分抗,并用非正则格型标度方程进行数学描述.分别探讨非正则格型标度方程的近似求解和真实解.通过调节电阻递进比α与电容递进比β的取值,可构造出具有任意运算阶的标度分形格分抗逼近电路.标度拓展极大地提高了标度分形格分抗电路的逼近效益.随着标度因子的增加,负半阶标度分形格分抗的逼近效益逐渐增大并明显高于Carlson分形格分抗.设计了基于五节Carlson分形格分抗与负半阶标度分形格分抗的半阶微分运算电路,并对周期三角波和周期方波信号进行半阶微分运算,实验测试结果与理论分析一致.     

一类时变时滞混沌系统的参数辨识方法

时变的未知时滞参数普遍存在于混沌系统中,它使得混沌系统同步控制变得非常困难. 针对时滞混沌系统中参数时变且未知的问题, 提出了一种新颖的辨识方法. 该方法首先将未知时变参数用分段常数函数来近似, 把求解非线性函数的问题转化为参数向量选择问题, 其中分段常数函数的高度向量成为待求解参数向量; 然后推导了目标函数对分段常数高度向量的梯度信息, 结合序列二次规划法求解得最优分段函数; 随着分段数的增加, 最优分段函数将逼近原非线性时变函数. 数值实例结果验证了该方法的有效性.     

Partition function for the Anderson model

A new approach to the Anderson localized magnetic moment model is given, using the functional integral method. The partition function is explicitely given in an approximation suitable for strong coupling.