On Minimal-Moment Generating Functions

A formula expressing the inverse cumulative distribution function of a non-negative random variable in terms of contour integrals of its minimal-moment generating function is proved as well as an analog of the classical continuity theorem for characteristic functions.     

Equality of Schur and Skew Schur Functions

We determine the precise conditions under which any skew Schur function is equal to a Schur function over both infinitely and finitely many variables. Received May 29, 2004     

On the Distribution of Generating Functions

Investigations concerning the generating function associatedwith the kth powers, originatewith Hardy and Littlewood in their famous series of papers inthe 1920s, ‘On some problems of "Partitio Numerorum"’(see [7, Chapters 2 and 4]). Classical analyses of this andsimilar functions show that when P is large the function approachesP in size only for in a subset of (0, 1) having small measure.Moreover, although it has never been proven, there is some expectationthat for ‘most’ , the generating function is about in magnitude. The main evidence in favour of this expectation comes from mean value estimatesof the form An asymptoticformula of the shape (1.2), with strong error term, is immediatefrom Parseval's identity when s = 2, and follows easily whens = 4 and k > 2 from the work of Hooley [2, 3, 4], Greaves[1], Skinner and Wooley [5] and Wooley [9]. On the other hand,(1.2) is false when s > 2k (see [7, Exercise 2.4]), and whens = 4 and k = 2. However, it is believed that when t < k,the total number of solutions of the diophantine equation with 1 x_j, y_j P (1 j t), is dominatedby the number of solutions in which the x_i are merely a permutationof the y_j, and the truth of such a belief would imply that (1.2)holds for even integers s with 0 s < 2k. The purpose of this paper is to investigate the extent to whichknowledge of the kind (1.2) for an initial segment of even integerexponents s can be used to establish information concerningthe general distribution of f_P(), and the behaviour of the momentsin (1.2) for general real s. 1991 Mathematics Subject Classification11L15.     

Generating Functions in the Knapsack Problem

The knapsack problem with Boolean variables and a single constraint is considered. Combinatorial formulas for calculating and estimating the cardinality of the set of feasible solutions and the values of the functional in various cases depending on given parameters of the problem are derived. The coefficients of the objective function and of the constraint vector and the knapsack size are used as parameters. The baseline technique is the classical method of generating functions. The results obtained can be used to estimate the complexity of enumeration and decomposition methods for solving the problem and can also be used as auxiliary procedures in developing such methods.     

Bilinear Generating Functions for Orthogonal Polynomials

Using realizations of the positive discrete series representations of the Lie algebra su(1,1) in terms of Meixner—Pollaczek polynomials, the action of su(1,1) on Poisson kernels of these polynomials is considered. In the tensor product of two such representations, two sets of eigenfunctions of a certain operator can be considered and they are shown to be related through continuous Hahn polynomials. As a result, a bilinear generating function for continuous Hahn polynomials is obtained involving the Poisson kernel of Meixner—Pollaczek polynomials; this result is also known as the Burchnall—Chaundy formula. For the positive discrete series representations of the quantized universal enveloping algebra U _q (su(1,1)) a similar analysis is performed and leads to a bilinear generating function for Askey—Wilson polynomials involving the Poisson kernel of Al-Salam and Chihara polynomials. July 6, 1997. Date accepted: September 23, 1998.     

两个回归函数相等的Bootstrap检验

两个回归参数相等性检验一直是统计界感兴趣的问题之一．在这篇文章中，四个检验统计量被用于度量两曲线的差异，在原假设下统计量的分布采用向量数据的重复抽样来逼近，并给出了—些模拟结果．     

Generating Functions for Computing the Myerson Value

The complexity of a computational problem is the order of computational resources which are necessary and sufficient to solve the problem. The algorithm complexity is the cost of a particular algorithm. We say that a problem has polynomial complexity if its computational complexity is a polynomial in the measure of input size. We introduce polynomial time algorithms based in generating functions for computing the Myerson value in weighted voting games restricted by a tree. Moreover, we apply the new generating algorithm for computing the Myerson value in the Council of Ministers of the European Union restricted by a communication structure.     

Norms and Generating Functions in Clifford Algebras

Generating functions are commonly used in combinatorics to recover sequences from power series expansions. Convergence of formal power series in Clifford algebras of arbitrary signature is discussed. Given , powers of u are recovered by expanding (1 − tu)⁻¹ as a polynomial in t with Clifford-algebraic coefficients. It is clear that (1 − tu)(1 + tu + t ² u ² + ...) = 1, provided the sum (1 + tu + t ² u ² + ...) exists, in which case u ^m is the Cliffordalgebraic coefficient of t ^m in the series expansion of (1 − tu)⁻¹. In this paper, conditions on for the existence of (1 − tu)⁻¹ are given, and an explicit formulation of the generating function is obtained. Allowing A to be an m × m matrix with entries in , a “Clifford-Frobenius” norm of A is defined. Norm inequalities are then considered, and conditions for the existence of (I − tA)⁻¹ are determined. As an application, adjacency matrices for graphs are defined with vectors of as entries. For positive odd integer k > 3, k-cycles based at a fixed vertex of a graph are enumerated by considering the appropriate entry of A ^k . Moreover, k-cycles in finite graphs are enumerated and expected numbers of k-cycles in random graphs are obtained from the norm of the degree-2k part of tr(1 − tu)⁻¹. Unlike earlier work using commutative subalgebras of , this approach represents a “true” application of Clifford algebras to graph theory.      

Symmetric Functions and Generating Functions for Descents and Major Indices in Compositions

In [18], Mendes and Remmel showed how Gessel’s generating function for the distributions of the number of descents, the major index, and the number of inversions of permutations in the symmetric group could be derived by applying a ring homomorphism defined on the ring of symmetric functions to a simple symmetric function identity. We show how similar methods may be used to prove analogues of that generating function for compositions.     

检验两个部分线性模型中非参函数相等

We propose the test statistic to check whether the nonparametric func-tions in two partially linear models are equality or not in this paper. We estimate the nonparametric function both in null hypothesis and the alternative by the local linear method, where we ignore the parametric components, and then estimate the parameters by the two stage method. The test statistic is derived, and it is shown to be asymptotically normal under the null hypothesis.     

Some Generating Functions for the Generalized Bessel Polynomials

The authors begin by presenting a systematic (historical) account of some linear generating functions for the generalized Bessel polynomials. It is then shown how these linear generating functions can be applied with a view to obtaining various new families of bilinear, bilateral, or mixed multilateral generating functions for the generalized Bessel polynomials. Some interesting generalizations of these classes of generating functions, involving hypergeometric functions, are also considered.     

Neighborhood Complexes and Generating Functions for Affine Semigroups

Given a₁,a₂,...,a_n ∈ ℤ^d$, we examine the set, G, of all non-negative integer combinations of these a_i. In particular, we examine the generating function f(z) = ∑_{b ∈ G}z_b. We prove that one can write this generating function as a rational function using the neighborhood complex (sometimes called the complex of maximal lattice-free bodies or the Scarf complex) on a particular lattice in ℤ_n. In the generic case, this follows from algebraic results of Bayer and Sturmfels. Here we prove it geometrically in all cases, and we examine a generalization involving the neighborhood complex on an arbitrary lattice.     

Generating Functions for Spherical Harmonics and Spherical Monogenics

In this paper, we study generating functions for the standard orthogonal bases of spherical harmonics and spherical monogenics in \({\mathbb{R}^{m}}\) . Here spherical monogenics are polynomial solutions of the Dirac equation in \({\mathbb{R}^{m}}\) . In particular, we obtain the recurrence formula which expresses the generating function in dimension m in terms of that in dimension m–1. Hence we can find closed formulæ of generating functions in \({\mathbb{R}^{m}}\) by induction on the dimension m.     

Generating Functions for Permutations Avoiding a Consecutive Pattern

Given a permutation τ of length j, we say that a permutation σ has a τ-match starting at position i, if the elements in positions i, i+1, . . . , i+j−1 in σ have the same relative order as the elements of τ. We have been able to take advantage of the results of Mendes and Remmel [1] to obtain a generating function for the number of permutations with no τ-matches for several new classes of permutations. These new classes include a large class of permutations which are shuffles of an increasing sequence 1 2 · · · n with an arbitrary permutation σ of the integers {n + 1, . . . , n + m}. Finally we give a formula for the generating function for the number of permutations which have no 1 3 2 4-matches.     

Characterization of Generating Ideals in Some Rings of Entire Functions

Let E be a ring of entire functions on with the operation of pointwise multiplication, and let f ₁,...,f _m be a set of nonzero elements in E. The ideal E(f ₁,...,f _m ) in E with generators f ₁,...,f _m is said to be generating if E(f ₁,...,f _m ) = E. The generating ideals in rings of entire functions on determined by the growth of their maximum moduli are characterized in terms of the distribution of the zero sets of their generators. Under the additional condition of rapid variation of the weight sequences determining the ring, criteria for generating ideals are established; they are stated in terms of d(z) max _{1 j m} d _j (z), where d _j (z) is the distance from a point to the zero set of f _j for 1 j m. It is shown that, in rings of entire functions of finite or minimal type with respect to a given order, a similar characterization (i.e., in terms of d(z)) cannot be given.     

Toeplitz Preconditioners for Toeplitz Systems with Nonnegative Generating Functions

The preconditioned conjugate gradient method is employed tosolve Toeplitz systems T_nx = b where the generating functionsof the n-by-n Toeplitz matrices T_n are continuous nonnegativeperiodic functions defined in [–,]. The preconditionedC_n are band Toeplitz matrices with band-widths independent ofn. We prove that the spectra of C_n^-1T_n are uniformly boundedby constants independent of n. In particular, we show that thesolutions of T_nx = b can be obtained in O(nlogn) operations.     

Generating Functions for Alternating Descents and Alternating Major Index

In 2008, Chebikin introduced the alternating descent set, AltDes(??), of a permutation ?? =??? ₁ ··· ?? _n in the symmetric group S _n as the set of all i such that either i is odd and ?? _i >??? _i+1 or i is even and ?? _i <??? _i+1. We can then define altdes(??) =?|AltDes(??)| and ${{\rm altmaj}(\sigma) = \sum_{i \in AltDes(\sigma)}i}$ . In this paper, we compute a generating function for the joint distribution of altdes(??) and altmaj(??) over S _n . Our formula is similar to the formula for the joint distribution of des and maj over the symmetric group that was first proved by Gessel. We also compute similar generating functions for the groups B _n and D _n and for r-tuples of permutations in S _n . Finally we prove a general extension of these formulas in cases where we keep track of descents only at positions r, 2r, . . ..     

Free Distributional Data of Arithmetic Functions and Corresponding Generating Functions Determined by Gaps Between Primes

In this paper, we study free probability on the algebra $\mathcal{A }$ consisting of all arithmetic functions, determined by the gaps between primes. As a continuation and as an application of Cho (Classification on Arithmetic Functions and Free-Moment $L$ -Functions. Submitted to B. Korean Math Soc, 2013), we study moment series and R-transforms induced by both arithmetic functions and gaps between primes. Also, we consider R-transform calculus on the arithmetic algebra.