Jordan-wigner type transformation and Hamiltonian circuits in a rectangular lattice

In the present study a generating function is considered for Hamiltonian circuits in a rectangular lattice of dimensionN×M. A generating function Γ(^e) for closed Euler diagrams (with valence number for the vertices δ=0, 2, 4) with constant step is introduced for this lattice. It is proved that these two generating functions coincide as in the case of the corresponding generating functions relative to a single node (in limit as N, M → ∞). To construct the proof, an auxiliary function that is, in fact, the statistical sum for the two-dimensional Ising model is introduced. The two-dimensional Jordan-Wigner type transformations that were introduced in [7] are also used. State University of Poland. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 52–56, October, 1997.     

A remark on the three approaches to 2D quantum gravity

The one-matrix model is considered. The generating function of the correlation numbers is defined in such a way that this function coincides with the generating function of the Liouville gravity. Using the Kontsevich theorem we explain that this generating function is an analytic continuation of the generating function of the Topological gravity. We check the topological recursion relations for the correlation functions in the p-critical Matrix model.     

Adiabatic anholonomy and canonical transformations

Biswas and Soni [4] have surmised a semiclassical formula for Berry’s phase in terms of a generating function. We derive this formula apart from showing that it is not true in general and investigate its domain of validity. We also derive transformation formulae for Berry’s phase (Hannay’s angle) under general canonical transformations. A simpler proof for total angle invariance than hitherto available, is given.     

Dimensional Reduction Formulas for Branched Polymer Correlation Functions

In [BI01] we have proven that the generating function for self-avoiding branched polymers in D+2 continuum dimensions is proportional to the pressure of the hard-core continuum gas at negative activity in D dimensions. This result explains why the critical behavior of branched polymers should be the same as that of the i ³ (or Yang–Lee edge) field theory in two fewer dimensions (as proposed by Parisi and Sourlas in 1981). In this article we review and generalize the results of [BI01]. We show that the generating functions for several branched polymers are proportional to correlation functions of the hard-core gas. We derive Ward identities for certain branched polymer correlations. We give reduction formulae for multi-species branched polymers and the corresponding repulsive gases. Finally, we derive the massive scaling limit for the 2-point function of the one-dimensional hard-core gas, and thereby obtain the scaling form of the 2-point function for branched polymers in three dimensions.     

组合数学方法推引原子谱项

本文引用组合数学方法推引原子谱项，极其简便迅速。所导出的推引公式十分简洁，易于掌握，应用方便灵活。这种推引原子谱项的方法适用于各种等效和非等效电子组态的原子体系。     

Generating functions in two-dimensional quantum gravity

We solve general 1-matrix models without taking the double scaling limit. A method of computing generating functions is presented. We calculate the generating functions for a simple and a double torus. Our method is also applicable to higher genus. Each generating function can be expressed by a “specific heat” function for the sphere. Universal terms, which survived in the double scaling limit can be easily picked out from our exact solutions. We also find that the regular part of the spherical generating function is at most bilinear in the coupling constants of the source terms.     

Forward-backward multiplicity distribution in a two-component model

We calculate stochastic quantities in a two-component model which is a product of a negative binomial and a Poisson distribution. The generating function and the KNO scaling function are obtained. A formula for the forward-backward (F-B) multiplicity correlation is derived from the generating function. We consider the case in which particles are produced in pairs and compare with experiment.     

Generating functions for characters and weight multiplicities of irreducible (4)-modules

Generating functions for the characters of the irreducible representations of simple Lie algebras are rational functions where both the numerator and denominator can be expressed as polynomials in the characters corresponding to the fundamental weights. They encode much information on the representation theory of the algebra, but their explicit expressions are in general very complicated. In fact, it seems that rank three is the highest rank tractable. In this paper, we use a method based on the quantum Calogero-Sutherland model to compute the full generating function for the characters of irreducible modules over the complex Lie algebra (4), and exploit this result to obtain also generating functions giving the multiplicities of some low order weights in all representations. We have applied the same method to compute the generating function for the characters of the modules the other rank three simple Lie algebras, but in these cases the full expressions are very long and appear only in the arXiv version of the paper (arXiv:1705.03711 [math-ph]). Nevertheless, when the generating functions are limited to some particular subsets of characters, the results are quite simple and we present them here.     

Intensity fluctuations in thermal light with orthogonally polarised multiple-peak spectrum

The moment generating function of the integrated light intensity of thermal radiation having multiple peak spectrum is obtained. Cases of two-peak and three-peak spectra where different peaks are in orthogonal states of polarisation are considered. The moment generating function is shown to be the product of two simpler generating functions.     

Normalized Excited Squeezed Vacuum State and Its Applications

By using the intermediate coordinate-momentum representation in quantum optics and generating function for the normalization of the excited squeezed vacuum state （ESVS）, the normalized ESVS is obtained. We find that its normalization constants obtained via two new methods are uniform and a new form which is different from the result obtained by Zhang and Fan [Phys. Lett. A 165 （1992） 14]. By virtue of the normalization constant of the ESVS and the intermediate coordinate-momentum representation, the tomogram of the normalized ESVS and some useful formulae are derived.     

The method of generating functions for the calculation of matrix elements in the deformed-harmonic oscillator basis

The method of generating functions which was previously only employed for the spherical basis of harmonic-oscillator single-particle wave functions is here generalized to the deformed (=cylindrical = asymptotic) basis. One-center and two-center matrix elements which are important in fission or heavy-ion scattering theories are obtained for various operators and potentials, i.e., spin-orbit, l-squared, Gaussian, or Gaussian multiplied by a polynomial, etc. If they cannot be calculated explicitely, recurrence formulae are evaluated. The method circumvents integrating the matrix elements by the introduction of generating parameters and looking for the coefficient of some power of the generating parameter in the expansion of the generating function. The method is further simplified by transforming the operators acting on the wave functions into operators which act onto the generating function or into functions to be multiplied by the generating integral.     

The exact solution of the mutator model with directed mutations,linear fitness function,and finite genome length

In this study, we considered the model by Beckman and Loeb [Proc. Natl. Acad. Sci. U.S.A. 103 (2006) 14140] for the mutator phenomena. We construct an infinite population Crow-Kimura model with a mutator gene, directed mutations, a linear fitness function, and a finite genome length. We solved analytically the dynamics of the model using the generating function method. Such models provide realistic predictions for finite population sizes and have been widely discussed recently. The analytical formulas provided can be used to calculate the advantage of the mutator mechanism for the accumulation of mutations in cancer biology.     

Bounds on average multiplicity

Bound on average multiplicity is derived using polynomial boundedness of multiparticle generating function. Consequences of distribution of zeros of the generating function and its connection with improvement of the bound is studied.     

Localization and Gluing of Topological Amplitudes

We develop a gluing algorithm for Gromov-Witten invariants of toric Calabi-Yau threefolds based on localization and gluing graphs. The main building block of this algorithm is a generating function of cubic Hodge integrals of special form. We conjecture a precise relation between this generating function and the topological vertex at fractional framing.     

A stochastic theory for clustering of quenched-in vacancies—2. A solvable model

The model introduced for clustering of quenched-in vacancies in the first part of this series of papers is considered. Using a generating function, the rate equations are converted into a first order partial differential equation for the generating function coupled to a differential equation for the rate of change of the concentration of single vacancy units. A decoupling scheme is effected which gives an exponentially decaying solution with a very short time constant for the concentration of single vacancy units. The differential equation for the generating function is solved for times larger than the time required for the concentration of single vacancy units to reach its asymptotic value. The distribution for the size of the clusters is obtained by inverting the solution thus obtained. Several results that follow are shown to be in reasonably good agreement with the experimental results.     

Autonomous generation of all Wigner functions and marginal probability densities of Landau levels

Generation of Wigner functions of Landau levels and determination of their symmetries and generic properties are achieved in the autonomous framework of deformation quantization. Transformation properties of diagonal Wigner functions under space inversion, time reversal and parity transformations are specified and their invariance under a four-parameter subgroup of symplectic transformations are established. A generating function for all Wigner functions is developed and this has been identified as the phase-space coherent state for Landau levels. Integrated forms of generating function are used in generating explicit expressions of marginal probability densities on all possible two dimensional phase-space coordinate planes. Phase-space realization of unitary similarity and gauge transformations as well as some general implications for the Wigner function theory are presented.     

Analytic continuation of the multiple Daehee q-l-functions associated with Daehee numbers

In this paper, we present a new generating function which is related to Daehee numbers. By using the Mellin transformation formula and the Cauchy theorem for this generating function, we define multiple Daehee q-l-functions and q-zeta functions We also give the values of this function at negative integers.     

Fluctuation symmetry in a two-state Markov model

We show that the scaled cumulant generating and large deviation function, associated to a two-state Markov process involving two processes, obey a symmetry relation reminiscent of the fluctuation theorem, independent from any conditions on the transition rates. The Legendre transform leading from the scaled cumulant generating function to the large deviation function is performed in an ingenious way, avoiding the sign problem associated to taking a square root. Applications to the theory of random walks and to the stochastic thermodynamics for a quantum dot are presented.     

A Method for q-Calculus

Abstract

We present a notation for q-calculus, which leads to a new method for computations and classifications of q-special functions. With this notation many formulas of q-calculus become very natural, and the q-analogues of many orthogonal polynomials and functions assume a very pleasant form reminding directly of their classical counterparts.

The first main topic of the method is the tilde operator, which is an involution operating on the parameters in a q-hypergeometric series. The second topic is the q-addition, which consists of the Ward–AlSalam q-addition invented by Ward 1936 [102, p. 256] and Al-Salam 1959 [5, p. 240], and the Hahn q-addition.

In contrast to the the Ward–AlSalam q-addition, the Hahn q-addition, compare [57, p. 362] is neither commutative nor associative, but on the other hand, it can be written as a finite product.

We will use the generating function technique by Rainville [76] to prove recurrences for q-Laguerre polynomials, which are q-analogues of results in [76]. We will also find q-analogues of Carlitz’ [26] operator expression for Laguerre polynomials. The notation for Cigler’s [37] operational calculus will be used when needed. As an application, q-analogues of bilinear generating formulas for Laguerre polynomials of Chatterjea [33, p. 57], [32, p. 88] will be found.     

交替寻优生成元素幅值结合混沌随机相位构造循环测量矩阵

循环矩阵由于其对应离散卷积且具有快速算法被广泛应用于压缩测量矩阵. 本文从循环测量矩阵生成元素的幅值和相位两个方面探索循环测量矩阵的优化构造, 提出交替寻优生成元素的幅值并结合混沌随机相位实现循环测量矩阵的最优构造. 由一维和二维信号循环测量矩阵的不同表示形式出发, 将等价字典列向量之间互相干系数的Welch界作为逼近目标, 推导出了一维和二维信号循环测量矩阵生成元素幅值优化的统一数学模型, 提出采用交替寻优方法求解生成元素幅值的最优解. 利用混沌序列构造循环测量矩阵生成元素的随机相位. 与已有的典型循环测量矩阵相比, 本文优化构造的循环测量矩阵所对应的等价字典列向量之间具有更低的互相干性, 这正是所构造的循环测量矩阵优越性的本质所在.