Trajectories of the zeros of hypergeometric polynomials F(−n, b; 2b; z) for b < − 1/2

In a previous paper [2] we studied the zeros of hypergeometric polynomials F(−n, b; 2b; z), where b is a real parameter. Making connections with ultraspherical polynomials, we showed that for b > − 1/2 all zeros of F(−n, b; 2b; z) lie on the circle |z − 1| = 1, while for b < 1 − n all zeros are real and greater than 1. Our purpose now is to describe the trajectories of the zeros as b descends below the critical value − 1/2 to 1 − n. The results have counterparts for ultraspherical polynomials and may be said to “explain” the classical formulas of Hilbert and Klein for the number of zeros of Jacobi polynomials in various intervals of the real axis. These applications and others are discussed in a further paper [3].     

Existence and regularity properties of non-isotropic singular elliptic equations

We establish existence and sharp regularity results for solutions to singular elliptic equations of the order u ^−β , 0 < β < 1, with gradient dependence and involving a forcing term λ f(x, u). Our approach is based on a singularly perturbed technique. We show that if the forcing parameter λ > 0 is large enough, our solution is positive. For λ small solutions vanish on a nontrivial set and therefore they exhibit free boundaries. We also establish regularity results for the free boundary and study the asymptotic behavior of the problem as b\searrow 0{\beta\searrow 0} and b\nearrow 1{\beta\nearrow 1}. In the former, we show that our solutions u _β converge to a C ^1,1 function which is a solution to an obstacle type problem. When b\nearrow 1{\beta\nearrow 1} we recover the Alt-Caffarelli theory.     

Weighted least-squares estimators of parametric functions of the regression coefficients under a general linear model

The weighted least-squares estimator of parametric functions K β under a general linear regression model { y, X b, s²S }{\{ {\bf y},\,{\bf X \beta}, \sigma^2{\bf \Sigma} \}} is defined to be K[^(b)]{{\bf K}{\hat{\bf {\beta}}}}, where [^(b)]{\hat{{\bf \beta}}} is a vector that minimizes (y − X β)′V(y − X β) for a given nonnegative definite weight matrix V. In this paper, we study some algebraic and statistical properties of K[^(b)]{{\bf K}\hat{{\bf \beta}}} and the projection matrix associated with the estimator, such as, their ranks, unbiasedness, uniqueness, as well as equalities satisfied by the projection matrices.     

A Higman inequality for regular near polygons

The inequality of Higman for generalized quadrangles of order (s,t) with s>1 states that t≤s ². We generalize this by proving that the intersection number c _i of a regular near 2d-gon of order (s,t) with s>1 satisfies the tight bound c _i ≤(s ²ⁱ −1)/(s ²−1), and we give properties in case of equality. It is known that hemisystems in generalized quadrangles meeting the Higman bound induce strongly regular subgraphs. We also generalize this by proving that a similar subset in regular near 2d-gons meeting the bounds would induce a distance-regular graph with classical parameters (d,b,α,β)=(d,−q,−(q+1)/2,−((−q) ^d +1)/2) with q an odd prime power.     

Global and Finite Convergence of a Generalized Newton Method for Absolute Value Equations

We investigate an efficient method for solving the absolute value equation Ax−|x|=b when the interval matrix [A−I,A+I] is regular. A generalized Newton method which combines the semismooth and the smoothing Newton steps is proposed. We establish global and finite convergence of the method. Preliminary numerical results indicate that the generalized Newton method is promising.     

A Generalized Radon Transform on the Plane

A new generalized Radon transform R _α, β on the plane for functions even in each variable is defined which has natural connections with the bivariate Hankel transform, the generalized biaxially symmetric potential operator Δ _α, β , and the Jacobi polynomials P_k^(b, a)(t)P_{k}^{(\beta,\,\alpha)}(t). The transform R _α, β and its dual R_a, b^*R_{\alpha,\,\beta}^{\ast} are studied in a systematic way, and in particular, the generalized Fuglede formula and some inversion formulas for R _α, β for functions in L_a, b^p(\mathbbR²₊)L_{\alpha,\,\beta}^{p}(\mathbb{R}^{2}_{+}) are obtained in terms of the bivariate Hankel–Riesz potential. Moreover, the transform R _α, β is used to represent the solutions of the partial differential equations Lu:=?_j=1^ma_jD_a, b^ju=fLu:=\sum_{j=1}^{m}a_{j}\Delta_{\alpha,\,\beta}^{j}u=f with constant coefficients a _j and the Cauchy problem for the generalized wave equation associated with the operator Δ _α, β . Another application is that, by an invariant property of R _α, β , a new product formula for the Jacobi polynomials of the type P_k^(b, a)(s)C_2k^a+b+1(t)=còòP_k^(b, a)P_{k}^{(\beta,\,\alpha)}(s)C_{2k}^{\alpha+\beta+1}(t)=c\int\!\!\int P_{k}^{(\beta,\,\alpha)} is obtained.     

A generalized Newton method for absolute value equations

A direct generalized Newton method is proposed for solving the NP-hard absolute value equation (AVE) Ax − |x| = b when the singular values of A exceed 1. A simple MATLAB implementation of the method solved 100 randomly generated 1,000-dimensional AVEs to an accuracy of 10⁻⁶ in less than 10 s each. Similarly, AVEs corresponding to 100 randomly generated linear complementarity problems with 1,000 × 1,000 nonsymmetric positive definite matrices were also solved to the same accuracy in less than 29 s each.     

Triangle-free distance-regular graphs

Let Γ denote a distance-regular graph with diameter d≥3. By a parallelogram of length 3, we mean a 4-tuple xyzw consisting of vertices of Γ such that ∂(x,y)=∂(z,w)=1, ∂(x,z)=3, and ∂(x,w)=∂(y,w)=∂(y,z)=2, where ∂ denotes the path-length distance function. Assume that Γ has intersection numbers a ₁=0 and a ₂≠0. We prove that the following (i) and (ii) are equivalent. (i) Γ is Q-polynomial and contains no parallelograms of length 3; (ii) Γ has classical parameters (d,b,α,β) with b<−1. Furthermore, suppose that (i) and (ii) hold. We show that each of b(b+1)²(b+2)/c ₂, (b−2)(b−1)b(b+1)/(2+2b−c ₂) is an integer and that c ₂≤b(b+1). This upper bound for c ₂ is optimal, since the Hermitian forms graph Her₂(d) is a triangle-free distance-regular graph that satisfies c ₂=b(b+1). Work partially supported by the National Science Council of Taiwan, R.O.C.     

Sur la singularité des produits de Riesz et des mesures spectrales associées à la somme des chiffres

Using a property of generalized characters, we first prove that two Riesz products with constant coefficients and distinct Fourier spectra are mutually singular. IfS _r (n) denotes the sum of digits in ther-adic representation of the integern, the same technique allows us to establish the mutual singularity of the spectral measures of the sequences: α(n)=exp[2iπaS _p (n)],β(n)=exp[2iπbS _q (n)], for every pair of integersp≠q, a, b being real numbers such thata(p−1)∉ {tiZ} andb(q−1)∉Z. This result has been proved by T. Kamae whenp andq are two relatively prime integers.      

Reciprocal polynomials with all zeros on the unit circle

Let f(x)=a _d x ^d +a _d−1 x ^d−1+⋅⋅⋅+a ₀∈ℝ[x] be a reciprocal polynomial of degree d. We prove that if the coefficient vector (a _d ,a _d−1,…,a ₀) or (a _d−1,a _d−2,…,a ₁) is close enough, in the l ¹-distance, to the constant vector (b,b,…,b)∈ℝ ^d+1 or ℝ ^d−1, then all of its zeros have moduli 1.     

On a difference equation arising in a learning-theory model

An analysis is presented of the equationf(x+a)−f(x)=e ^−x {f(x)−f(x−b)}. Herea andb denote arbitrary positive constants, and a solution is sought which satisfies the following conditions:f(−∞)=0,f(+∞)=1, 0≦f(x)≦1. Existence and uniqueness of solution are established, and then an analytical form of the solution is obtained by use of bilateral Laplace transform. Research supported by the National Science Foundation, Grant GP-2558.     

A minimal area problem in conformal mapping II

LetS denote the usual class of functionsf holomorphic and univalent in the unit diskU. For 0<r<1 andr(1+r)⁻²<b<r(1−r)⁻², letS(r, b) be the subclass of functionsf∈S such that |f(r)|=b. In Theorem 1, we solve the problem of minimizing the Dirichlet integral inS(r, b). The first main ingredient of the solution is the establishment of sufficient regularity of the domains onto whichU is mapped by extremal functions, and here techniques of symmetrization and polarization play an essential role. The second main ingredient is the identification of all Jordan domains satisfying a certain kind of functional equation (called “quadrature identities”) which are encountered by applying variational techniques. These turn out to be conformal images ofU by mappings of a special form involving a logarithmic function. In Theorem 2, this aspect of our work is generalized to encompass analogous minimal area problem when a larger number of initial data are prescribed. The third author thanks for its hospitality the Mittag-Leffler Institute of Royal Swedish Academy of Sciences where this work was finalized. This author was supported in part by the Swedish Institute and by the Russian Fund for Fundamental Research, grant no. 97-01-00259.     

On the difference between the number of prime divisors at consecutive integers

Suppose thatg(n) is equal to the number of divisors ofn, counting multiplicity, or the number of divisors ofn, a≠0 is an integer, andN(x,b)=|{n∶n≤x, g(n+a)−g(n)=b orb+1}|. In the paper we prove that sup _b N(x,b)≤C(a)x)(log log 10 _x )^−1/2 and that there exists a constantC(a,μ)>0 such that, given an integerb |b|≤μ(log logx)^1/2,x≥x _o, the inequalityN(x,b)≥C(a,μ)x(log logx(^−1/2) is valid. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 579–595, October, 1999.     

The Lambert W-functions and some of their integrals: a case study of high-precision computation

The real-valued Lambert W-functions considered here are w ₀(y) and w _− 1(y), solutions of we ^w  = y, − 1/e < y < 0, with values respectively in ( − 1,0) and ( − ∞ , − 1). A study is made of the numerical evaluation to high precision of these functions and of the integrals ò₁^￥ [-w₀(-xe^-x)]^a x^-bx\int_1^\infty [-w_0(-xe^{-x})]^\alpha x^{-\beta}\d x, α > 0, β ∈ ℝ, and ò₀¹ [-w_-1(-x e^-x)]^a x^-bx\int_0^1 [-w_{-1}(-x e^{-x})]^\alpha x^{-\beta}\d x, α > − 1, β < 1. For the latter we use known integral representations and their evaluation by nonstandard Gaussian quadrature, if α ≠ β, and explicit formulae involving the trigamma function, if α = β.     

Groups of typeF a,b,−c

For integersa, b andc, the groupF ^a,b,−c is defined to be the group 〈R, S : R ²=RS ^aRS^bRS^−c=1〉. In this paper we identify certain subgroups of the group of affine linear transformations of finite fields of orderp ⁿ (for certainp andn) as groups of typeF ^a,b,−c for certain (not unique) choices ofa, b andc.     

On (a, b)-consecutive petersen graphs

The generalized Petersen graphsP(n,k), n≥3, 1≤k<n/2, consist of an outern-cyclex _o x ₁ x ₂...x _n−1 , a set ofn spokesx _i y _i (0≤i≤n−1), andn inner edgesy _i y _i +k with indices taken modulon. This paper deals with (a,b)-consecutive labelings of generalized Petersen graphP(n,k).     

Extraction of smooth solutions of a class of regular equations in a rectangle

This work is devoted to the study of two-dimensional, regular, almost hypoelliptic operators P(D) = P(D ₂, D ₂) with regular Newton polyhedrons. It is proved that all generalized (weak) solutions of the equation P(D)u = f from a several weighted Sobolev space are infinitely differentiable functions in the rectangle {x ∈ E ²: −a < x ₁ < a, −b < x ₂ < b} in the variable x ₂, in which the function f is infinitely differentiable.     

Invariant sets for a class of perturbed differential equations of retarded type

LetX be a Banach space and leta, b, q be real numbers such thata<b,q>0. Denote byD a locally closed subset ofX. A necessary and sufficient condition for the existence of a mild solutionu∈C([a−q, b ₁],X),a<b ₁<b, to the differential equationdu(t)/dt=Au(t)+f(t, u _t), such thatu:[a,b ₁]→D, u _a=ϕ is given. The linear operatorA is the generator of aC ₀ semigroupT(t), t≧0, withT(t) compact fort>0,f: [a, b)×C([−q,0],D _λ)→X is continuous and ϕ∈C([−q,0],D _λ) with ϕ(0)∈D. D _λ is a neighbourhood ofD. Applications to parabolic partial differential equations with retarded argument are given.     

Penultimate limiting forms in extreme value theory

Summary Let {X _n}_n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM _n=max (X ₁,…,X _n), suitably normalized with attraction coefficients {α_n}_n≧1(α_n>0) and {b _n}_n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s which better approximate the d.f. of(M _n−b_n)/a_n thanG(x), for moderaten. We thus consider differences of the formF ⁿ(a_nx+b_n)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф _α(x)=exp (−x^−α), x>0] or a type III [Ψ _α(x)= exp (−(−x)^α), x<0] d.f. of largest values, much closer toF ⁿ(a_nx+b_n) than the ultimate itself.     

A limiting case of a generalized beta integral on two intervals

Akhiezer Polynomials orthogonal on several intervals are used to define a generalization of the beta integral where the integral is over two disjoint intervals of the real line, [−1,−β]∪[β,1]. An explicit evaluation of the integral is given in the limiting case as β→1.