Properties of the zeros of the sum of three polynomials

A system of algebraic equations satisfied by the zeros of the sum of three polynomials are reported.     

Correlations between zeros of a random polynomial

We obtain exact analytical expressions for correlations between real zeros of the Kac random polynomial. We show that the zeros in the interval (−1, 1) are asymptotically independent of the zeros outside of this interval, and that the straightened zeros have the same limit-translation-invariant correlations. Then we calculate the correlations between the straightened zeros of theO(1) random polynomial.     

A convenient expression of the time-derivative,of arbitrary order k,of the zero zn(t) of a time-dependent polynomial pN(z;t) of arbitrary degree N in z,and solvable dynamical systems

Let p_N (z; t) be a (monic) time-dependent polynomial of arbitrary degree N in z, and let z_n ≡ z_n (t) be its N zeros: . In this paper we report a convenient expression of the k-th time-derivative of the zero z_n (t). This formula plays a key role in the identification of classes of solvable dynamical systems describing the motion of point-particles moving in the complex z-plane while nonlinearly interacting among themselves; one such example, featuring many arbitrary parameters, is reported, including its variation describing the motion of many particles moving in the real Cartesian xy-plane and interacting among themselves via rotation-invariant Newtonian equations of motion (”accelerations equal forces”).     

Yang-Lee zeros of the Potts model

The Yang-Lee zeros of the three-component ferromagnetic Potts model in one dimension in the complex plane of an applied field are determined. The phase diagram consists of a triple point where three phases coexist. Emerging from the triple point are three lines on which two phases coexist and which terminate at critical points (Yang-Lee edge singularity). The zeros do not all lie on the imaginary axis but along the three two-phase lines. The model can be generalized to give rise to a tricritical point which is a new type of Yang-Lee edge singularity. Gibbs phase rule is generalized to apply to coexisting phases in the complex plane.Supported in part by the National Science Foundation under Grant No. DMR-81-06151.     

The partition function zeros of the anisotropic Ising model with multisite interactions on a zigzag ladder

It is shown that the spin- anisotropic Ising model with multisite interactions on a zigzag ladder may be mapped into the one dimensional spin- Axial-Next-Nearest-Neighbor Ising (ANNNI) model with multisite interactions. The partition function zeros of the ANNNI model with multisite interactions are investigated. A comprehensive analysis of the partition function zeros of the ANNNI model with and without three-site interactions on a zigzag ladder is done using the transfer matrix method. Analytical equations for the distribution of the partition function zeros in the complex magnetic field (Yang-Lee zeros) and temperature (Fisher zeros) planes are derived. The Yang-Lee and Fisher zeros distributions are studied numerically for a variety of values of the model parameters. The densities of the Yang-Lee and Fisher zeros are studied and the corresponding edge singularity exponents are calculated. It is shown that the introduction of three-site interaction terms in the ANNNI model leads to a simpler distribution of the partition function zeros. For example, the Yang-Lee zeros tend to a circular distribution when increasing by modulus the three-site interactions term coefficient. It is found that the Yang-Lee and Fisher edge singularity exponents are universal and equal to each other, .     

Properties of the Zeros of the Sum of two Polynomials

Some properties—including relations having a Diophantine character—of the zeros of the sum of two polynomials are reported.     

Automorphisms of the q-deformed algebra suq(1, 1) and d-Orthogonal polynomials of q-Meixner type

Starting from an operator given as a product of q-exponential functions in irreducible representations of the positive discrete series of the q-deformed algebra su_q(1, 1), we express the associated matrix elements in terms of d-orthogonal polynomials. An algebraic setting allows to establish some properties : recurrence relation, generating function, lowering operator, explicit expression and d-orthogonality relations of the involved polynomials which are reduced to the orthogonal q-Meixner polynomials when d=1. If q ↑ 1, these polynomials tend to some d-orthogonal polynomials of Meixner type.     

Solutions of convex Bethe Ansatz equations and the zeros of (basic) hypergeometric orthogonal polynomials

Via the solutions of systems of algebraic equations of Bethe Ansatz type, we arrive at bounds for the zeros of orthogonal (basic) hypergeometric polynomials belonging to the Askey–Wilson, Wilson and continuous Hahn families.

     

Some new generating function formulae of the two-variable Hermite polynomials and their application in quantum optics

We explore the theoretical possibility of extending the usual squeezed state to those produced by nonlinear singlemode squeezing operators. We derive the wave functions of exp[-（ig/2）（（1-X2）1/2P ＋ P（1-X2）1/2）]｜0 in the coordinate representation. A new operator＇s disentangling formula is derived as a by-product.     

Distribution of roots of random real generalized polynomials

The average density of zeros for monic generalized polynomials, , with real holomorphic ,f _k and real Gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative. We obtain compact expressions for both the regular component (generated by the complex roots) and the singular one (real roots) of the average density of roots. The density of the regular component goes to zero in the vicinity of the real axis like |lmz|. We present the low- and high-disorder asymptotic behaviors. Then we particularize to the large-n limit of the average density of complex roots of monic algebraic polynomials of the form with real independent, identically distributed Gaussian coefficients having zero mean and dispersion . The average density tends to a simple,universal function of =2nlog|z| and in the domain coth(/2)n|sin arg(z)|, where nearly all the roots are located for largen.     

厄米多项式的定义及其他

分析了关于厄米多项式的争论.介绍了厄米多项式的两种定义.     

SU(1, 1) Random Polynomials

We study statistical properties of zeros of random polynomials and random analytic functions associated with the pseudoeuclidean group of symmetries SU(1, 1), by utilizing both analytical and numerical techniques. We first show that zeros of the SU(1, 1) random polynomial of degree N are concentrated in a narrow annulus of the order of N ^–1 around the unit circle on the complex plane, and we find an explicit formula for the scaled density of the zeros distribution along the radius in the limit N. Our results are supported through various numerical simulations. We then extend results of Hannay⁽¹⁾ and Bleher et al. ⁽²⁾ to derive different formulae for correlations between zeros of the SU(1, 1) random analytic functions, by applying the generalized Kac–Rice formula. We express the correlation functions in terms of some Gaussian integrals, which can be evaluated combinatorially as a finite sum over Feynman diagrams or as a supersymmetric integral. Due to the SU(1, 1) symmetry, the correlation functions depend only on the hyperbolic distances between the points on the unit disk, and we obtain an explicit formula for the two point correlation function. It displays quadratic repulsion at small distances and fast decay of correlations at infinity. In an appendix to the paper we evaluate correlations between the outer zeros |z _j |>1 of the SU(1, 1) random polynomial, and we prove that the inner and outer zeros are independent in the limit when the degree of the polynomial goes to infinity.     

On the Zeros of Polynomials Satisfying Certain Linear Second-Order ODEs Featuring Many Free Parameters

Certain techniques to obtain properties of the zeros of polynomials satisfying second-order ODEs are reviewed. The application of these techniques to the classical polynomials yields formulas which were already known; new are instead the formulas for the zeros of the (recently identified, and rather explicitly known) polynomials satisfying a (recently identified) second-order ODE which features many free parameters and only polynomial solutions. Some of these formulas have a Diophantine connotation. Techniques to manufacture infinite sequences of second-order ODEs featuring only polynomial solutions are also reported.     

Integrability of extended (2+1)-dimensional shallow water wave equation with Bell polynomials

We investigate the extended (2+1)-dimensional shallow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Bäcklund transformation, Lax pair, and Darboux covariant Lax pair for this equation. Moreover, the infinite conservation laws of this equation are found by using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas. The N-soliton solutions are also presented by means of the Hirota bilinear method.     

两维位置灵敏探测器的能量和动量响应函数刻度

用弹性散射对多通道(e,2e)谱仪两维位置灵敏探测器的能量和动量响应函数进行了刻度,从位置灵敏探测器输出的两维位置信息中获得了弹性散射的能谱和角度谱,估计了在当前实验条件下的能量分辨和角度分辨.同时,我们用正交多项式的最小二乘法拟合得到了谱仪单路的能量和动量响应函数.     

The Clebsch-Gordan coefficients ofsl(2,R) in the parabolic basis

We realize the Lie algebra sl(2,R) in terms of second-order differential operators defined on a dense common domain of square-integrable functions on a two-chart space, where the self-adjoint extension(s) (families) lead to all (and only) self-adjoint irreducible representations of the algebra, single- as well as multi-valued over the group. This allows for a rather straight-forward evaluation of the Clebsch-Gordan coefficients of sl(2,R) in the parabolic subalgebra basis.Invited talk presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.I would like to thank the hospitality of Prof. J. A. de Azcárraga, at the Instituto de Fí'sica Teórica, Universidad de Valencia, where this review was written.     

The Clebsch-Gordan coefficients of SU(4)

A set of recurrence relations connecting the matrix elements of finite transformation belonging to the same irreducible representation of SU(4) is used to obtain a wide class of matrix elements. An expression for the Clebsch-Gordan coefficient is obtained by integrating the product of three matrix elements belonging to three different irreducible representations of the group. The symmetry properties of the matrix elements and the Clebsch-Gordan coefficients are discussed.The author is grateful to Professors S.Datta Majumdar and G.Bandyopadhyay of the Department of Physics, I.I.T., Kharagpur, for many helpful discussions. This work was supported by the C.S.I.R., Government of India.     

The probability representation of quantum states and integral relations for Hermite and Laguerre polynomials

Some integral relations for orthogonal polynomials are elucidated. We review the generic scheme of the star-product construction and study in detail the star-product scheme based on the tomographic map. The dual star-product operator symbols are also considered and studied. Some integral kernels related to the star-product are calculated and new integral formulas for special functions are derived.     

A Law of Large Numbers for the Zeroes of Heine–Stieltjes Polynomials

We determine the limiting density of the zeroes of Heine–Stieltjes polynomials (or of any set of points satisfying the conclusion of Heine–Stieltjes Theorem) in the thermodynamic limit and use this to prove a strong law of large numbers for the zeroes.     

New generating function formulae of even- and odd-Hermite polynomials obtained and applied in the context of quantum optics

By combining the operator Hermite polynomial method and the technique of integration within an ordered product of operators, for the first time we derive the generating function of even- and odd-Hermite polynomials which will be useful in constructing new optical field states. We then show that the squeezed state and photon-added squeezed state can be expressed by even- and odd-Hermite polynomials.