An Inequality Involving the Local Eigenvalues of a Distance-Regular Graph

Let denote a distance-regular graph with diameter D 3, valency k, and intersection numbers a _i, b _i, c _i. Let X denote the vertex set of and fix x X. Let denote the vertex-subgraph of induced on the set of vertices in X adjacent X. Observe has k vertices and is regular with valency a ₁. Let _{1 2} ··· _k denote the eigenvalues of and observe ₁ = a ₁. Let denote the set of distinct scalars among ₂, ₃, ..., _k . For let mult denote the number of times appears among ₂, ₃,..., _k . Let denote an indeterminate, and let p ₀, p₁, ...,p _D denote the polynomials in [] satisfying p ₀ = 1 andp _i = c _i+1 p _i+1 + (a _i – c _i+1 + c _i)p _i + b _i p _i–1 (0 i D – 1),where p _–1 = 0. We show where we abbreviate = –1 – b ₁(1+)^–1. Concerning the case of equality we obtain the following result. Let T = T(x) denote the subalgebra of Mat _X ( ) generated by A, E*₀, E*₁, ..., E* _D , where A denotes the adjacency matrix of and E* _i denotes the projection onto the ith subconstituent of with respect to X. T is called the subconstituent algebra or the Terwilliger algebra. An irreducible T-module W is said to be thin whenever dimE* _i W 1 for 0 i D. By the endpoint of W we mean min{i|E* _i W 0}. We show the following are equivalent: (i) Equality holds in the above inequality for 1 i D – 1; (ii) Equality holds in the above inequality for i = D – 1; (iii) Every irreducible T-module with endpoint 1 is thin.     

Maximum of the product of the conformal radii of four nonoverlapping domains

Let a_k, k=1,...,4, be given distinct points of . Let D_k, k=1,...,4, be a system of simply connected domains in the closed plane such that a_kD_k, k=1,...,4, D_kD=for k,=1, ...,4, k. Let R(D_k,a_k) be the conformal radius of the domain D_k relative to the point a_k. In this paper we obtain an explicit expression for the maximum of the product in the family of all indicated system of domains in terms of elliptic functions. In the proof one makes directly use of the property of the extremal family of domains of the problem under consideration in terms of the associated quadratic differential.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 100, pp. 146–165, 1980.     

Computing the Hilbert transform of a Jacobi weight function

We discuss the evaluation of the Hilbert transformf _–1 ¹ (t-)^–1 w(, )(t)dt,–1<<1, of the Jacobi weight functionw(, )(t)=(1–t))(1+t) by analytic and numerical means and also comment on the recursive computation of the quantitiesf _–1 ¹ )(t–)^–1 _n (t;w ^{(, ))} w ^{(, )}(t)dt,n=0, 1, 2, ..., where _n (·;w ^{(, )}) is the Jacobi polynomial of degreen.The work of the first author was supported in part by the National Science Foundation under grant DCR-8320561. The work of the second author was supported by the National Science Foundation under grant DMS-8419086.     

Circles and spheres in pseudo-Riemannian geometry

Summary A totally umbilical pseudo-Riemannian submanifold with the parallel mean curvature vector field is said to be an extrinsic sphere. A regular curve in a pseudo-Riemannian manifold is called a circle if it is an extrinsic sphere. LetM be ann-dimensional pseudo-Riemannian submanifold of index (0n) in a pseudo-Riemannian manifold with the metricg and the second fundamental formB. The following theorems are proved. For ₀ = +1 or –1, ₁ = +1, –1 or 0 (2–2 ₀+ ₁2n–2–2) and a positive constantk, every circlec inM withg(c, c) = ₀ andg( _c c, _c c) = ₁ k ² is a circle in iffM is an extrinsic sphere. For ₀ = +1 or –1 (–₀n–), every geodesicc inM withg(c, c) = ₀ is a circle in iffM is constant isotropic and B(x,x,x) = 0 for anyx T(M). In this theorem, assume, moreover, that 1n–1 and the first normal space is definite or zero at every point. Then we can prove thatM is an extrinsic sphere. When = 0 orn, this fact does not hold in general.     

On the convergence of projected gradient processes to singular critical points

The projected gradient methods treated here generate iterates by the rulex _k+1=P (x _k –s _k F(x _k )),x ₁ , where is a closed convex set in a real Hilbert spaceX,s _k is a positive real number determined by a Goldstein-Bertsekas condition,P projectsX into ,F is a differentiable function whose minimum is sought in , and F is locally Lipschitz continuous. Asymptotic stability and convergence rate theorems are proved for singular local minimizers in the interior of , or more generally, in some open facet in . The stability theorem requires that: (i) is a proper local minimizer andF grows uniformly in near ; (ii) –F() lies in the relative interior of the coneK of outer normals to at ; and (iii) is an isolated critical point and the defect P (x – F(x)) –x grows uniformly within the facet containing . The convergence rate theorem imposes (i) and (ii), and also requires that: (iv)F isC ⁴ near and grows no slower than x–⁴ within the facet; and (v) the projected Hessian operatorP _{F 2} F()F is positive definite on its range in the subspaceF orthogonal toK . Under these conditions, {x _k } converges to from nearby starting pointsx ₁, withF(x _k ) –F() =O(k ^–2) and x _k – =O(k ^–1/2). No explicit or implied local pseudoconvexity or level set compactness demands are imposed onF in this analysis. Furthermore, condition (v) and the uniform growth stipulations in (i) and (iii) are redundant in ⁿ .     

Behavior on the real line of entire functions represented by Dirichlet series

Conditions are found under which for an entire function f represented by a Dirichlet series with finite Ritt order on some sequence (x_k), 0 < x_k , as k one has ¦f(x_k)¦=M_t((1 + 0(1) x_k), M_f(x)=sup {¦ f (z) ¦:Re z x}.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 265–269, February, 1991.     

Nonvanishing of central Hecke L-values and rank of certain elliptic curves

Let D 7 mod 8 be a positive squarefree integer, and let h_D be the ideal class number of E_D= . Let d1 mod 4 be a squarefree integer relatively prime to D. Then for any integer k0 there is a constant M=M(k), independent of the pair (D,D), such that if (–1)^k=sign (d), (2k+1,h_D)=1, and >(12/)d² (logd+M(k)), then the central L-value L(k+1, _{D, d} ^2k+1 >0. Furthermore, for k1, we can take M(k)=0. Finally, if D=p is a prime, and d>0, then the associated elliptic curve A(p)^d has Mordell–Weil rank 0 (over its definition field) when >(12/)d² log d.     

On a new class of multiplicative pseudo-random number generators

In the computing literature, there are few detailed analytical studies of the global statistical characteristics of a class of multiplicative pseudo-random number generators.We comment briefly on normal numbers and study analytically the approximately uniform discrete distribution or (j,)-normality in the sense of Besicovitch for complete periods of fractional parts {x _{0 1} ⁱ /p} on [0, 1] fori=0, 1,..., (p–1)p^–1–1, i.e. in current terminology, generators given byx _{n+1 1} x _n mod p wheren=0, 1,..., (p–1)p ^–1–1,p is any odd prime, (x ₀,p)=1, ₁ is a primitive root modp ², and 1 is any positive integer.We derive the expectationsE(X, ),E(X ², ),E(X _nX_n+k); the varianceV(X, ), and the serial correlation coefficient k. By means of Dedekind sums and some results of H. Rademacher, we investigate the asymptotic properties of k for various lagsk and integers 1 and give numerical illustrations. For the frequently used case =1, we find comparable results to estimates of Coveyou and Jansson as well as a mathematical demonstration of a so-called rule of thumb related to the choice of ₁ for small k.Due to the number of parameters in this class of generators, it may be possible to obtain increased control over the statistical behavior of these pseudo-random sequences both analytically as well as computationally.     

Percentage Points of the Largest Among Student's T Random Variable

Let us consider k( 2) independent random variables U₁, . . . ,U_k where U_i is distributed as the Student's t random variable with a degree of freedom m_i, i=1, . . . ,k. Here, m₁, . . . ,m_k are arbitrary positive integers. We denote m=(m₁, . . . ,m_k) and U_k:k=max {U₁, . . . ,U_k}, the largest Student's t random variable. Having fixed 0< <1, let a a(k,) and h_m h_m (k,) be two positive numbers for which we can claim that (i) ^k(a)–^k(–a)=1–, and (ii) P{–h_m U_k:k h_m}=1–. Then, we proceed to derive a Cornish–Fisher expansion (Theorem 3.1) of the percentage point h_m. This expansion involves a as well as expressions such as _i=1 ^k m_i ^–1, _i=1 ^km_i ^–2, and _i=1 ^k m_i ^–3. The corresponding approximation of h_m is shown to be remarkably accurate even when k or m₁, . . . ,m_k are not very large.     

Self-Converse Directed BIBDs with Block Size Four

A directed balanced incomplete block design (or D B(k,;v)) (X,) is called self-converse if there is an isomorphic mapping f from (X,) to (X,^–1), where ^–1={B ^–1:B} and B ^–1=(x _k ,x _{k –1},,x ₂,x ₁) for B=(x ₁,x ₂,,x _{k –1},x _k ). In this paper, we give the existence spectrum for self-converse D B(4,1;v). AMS Classification:05BResearch supported in part by NSFC Grant 10071002 and SRFDP under No. 20010004001     

K-interpolation in problems of uniform approximation of functions

Given an arbitrary metric compactum Q, we compute the K-functional of a pair (C(Q), C(Q)) and derive two-sided bounds for (C_[–,], C² _[–,]). We prove interpolation theorems and study applications to problems of the theory of approximation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 523–533, April, 1992.     

Two-sided polynomial approximation of functions

Conditions are established when the collocation polynomials P_m(x) and P_M(x), m M, constructed respectively using the system of nodes x_j of multiplicities a_j 1, j = O,, n, and the system of nodes x_-r,,x_o,,x_n,,x_n+r1, r O, r₁ O, of multiplicities a_-r,,(a_o + y_o),,(a_n + y_n),,a_n+r1, a_j + y_j 1, are two sided-approximations of the function f on the intervals , x_j[, j = O,...,n + 1, and on unions of any number of these intervals. In this case, the polynomials P_m (x), P_M ^(l) (x) with l a_j are two-sided approximations of the function f⁽¹⁾ in the neighborhood of the node x_j and the integrals of the polynomials P_m(x), P_M(x) over D_j are two-sided approximations of the integral of the function f (over D_j). If the multiplicities a_j a_j + y_j of the nodes x_j are even, then this is also true for integrals over the set _j= _µ ^k D_j µ 1, k n. It is shown that noncollocation polynomials (Fourier polynomials, etc.) do not have these properties.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 31–37, 1989.     

On a variational problem with lack of compactness: the topological effect of the critical points at infinity

We study the subcritical problemsP :–u=u ^p–,u>0 on;u=0 on , being a smooth and bounded domain in ^N,N–3,p+1=2N/N–2 the critical Sobolev exponent and >0 going to zero — in order to compute the difference of topology that the critical points at infinity induce between the level sets of the functional corresponding to the limit case (P₀).

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Résumé Nous étudions les problèmes sous-critiquesP :–u=u ^p–,u > 0 sur;u=0 sur –où est un domaine borné et régulier de ^N,N–3,p + 1=2N/N –2 est l'exposant critique de Sobolev, et >0 tend vers zéro, afin de calculer la différence de toplogie induite par les points critiques à l'infini entre les ensembles de niveau de la fonctionnelle correspondant au cas limite (P₀).

     

Milnor numbers and the topology of polynomial hypersurfaces

Summary LetF: n + 1 be a polynomial. The problem of determining the homology groupsH _q (F ^–1 (c)), c , in terms of the critical points ofF is considered. In the best case it is shown, for a certain generic class of polynomials (tame polynomials), that for allc,F ^–1 (c) has the homotopy type of a bouquet of - ^c n-spheres. Here is the sum of all the Milnor numbers ofF at critical points ofF and ^c is the corresponding sum for critical points lying onF ^–1 (c). A second best case is also discussed and the homology groupsH _q (F ^–1 (c)) are calculated for genericc. This case gives an example in which the critical points at infinity ofF must be considered in order to determine the homology groupsH _q (F ^–1 (c)).     

The Lê varieties,I

Summary For a complex polynomial,f:( ⁿ⁺¹ ,0) (, 0), with a singular set of complex, dimensions at the origin, we define a sequence of varieties—the Lê varieties, _f ^(k) , off at 0. The multiplicities of these varieties, _f ^(k) , generalize the Milnor number for an isolated singularity. In particular, we show that ifsn-2, the Milnor, fibre off is obtained fromB ²ⁿ by successively attaching _f ^{(n – k)} k-handles, wheren-skn Ifs=n-1, the Milnor fibre off is obtained from a2n-manifold with the homotopy type of a bouquet of _f ^{(n – 1)} circles by successively attaching _f ^{(n – k)} k-handles, where 2kn.The author is a National Science Foundation, Postdoctoral Research Fellow supported by grant # DMS-8807216     

Stability and uniqueness of the solution of the inverse kinematic problem of seismology in higher dimensions

The following inverse kinematic problem of seismology is considered. In the compact domain M of dimension ,2 with the metric, we consider the problem of constructing a new metricdu=nds according to the known formula where ,M and K_, is the geodesic in the metric du, connecting the points , . One proves uniqueness and one obtains a stability estimate, where the refraction indices n₁, n₂ are the solutions of the inverse kinematic problem, constructed relative to the functions ₁, ₂, respectively, is the differential form on M×Mwhere =₂–₁,.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 84, pp. 3–6, 1979.     

Sums of linear operators of parabolic type: a priori estimates and strong solutions

Summary We study the equation (A – ) x + (B–)x=y, with unknown x, in a Banach space X. y Xis the datum, > 0, A and B are linear closed unbounded operators in X with domains D_A, D_B. In the non commutative case, under assumptions already considered in the literature (see [7]), we show that for large values of any solution x D_A D_B satisfies an a priori estimate ¦|x¦|c^–1¦|y||and we prove that for any y X there exists a unique strong solution x, i.e. there exist x_nD_A D_B such that x_n x, (A–) x_n+(B–) x_ny in X. We also study regularity properties of strong solutions and we show that they belong to suitable interpolation spaces between D_A (or D_B) and X.     

On improved regularity of weak solutions of some degenerate,Anisotropic elliptic systems

Summary We consider a (possibly) vector-valued function u: R^N, Rⁿ, minimizing the integral , 2-2/(n*1)<p<2, whereD _i u=u/x _i or some more general functional retaining the same behaviour, we prove higher integrability for Du: D₁ u,..., D_n–1 u L^p/(p-1) and D_nu L²; this result allows us to get existence of second weak derivatives: D(D₁ u),...,D(D_n–1u)L² and D(D_n u) L _p.This work has been supported by MURST and GNAFA-CNR.     

G8 estimator of solutions of systems of linear algebraic equations

For linear forms of regularized solutions (x, c)=Re c' · Re[I + i)+A'A_n ^–1]^–1 A'_nb of systems of equations Ax=b, where A is an n×m matrix, x, c, b are vectors, and _n is a sequence of constants, we propose the estimator , where is any measurable solution of the equation ()Re[1+₁a(())]²+ (₁–₂)(1+₁(gq()))=, a(y)=n^–1 Sp[Iy+^–1Z_s'Z^s+ iI]^–1, , _i=n_n ²_n ^–1s_n ^–1, ⁿ=mI_n ²_n ^–1s_n ^–1, X_i are independent observations on the matrix A. Under certain conditions, it is proved that G₈ is a consistent estimator for n and 0.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 66, pp. 111–119, 1988.     

A decomposition for the exponential dispersion model generated by the invariant measure on the hyperboloid

For = ₀, ₁, ₂) andx=(x₀, x₁, x₂) in R³, define [,x] = ₀ x ₀ – ₁ x ₁ – ₂ x ₂,C = {x³:x ₀ > 0 and [x, x]>0},R(x)=([x, x]) ^1/2 forx inC andH ₁={xC: x₀>0,R(x)=1}. Define the measure onH ₁ such that if is inC and =R(), then exp (–[,x])(dx = ( exp )^–1. Therefore, is invariant under the action ofSO (1, 2), the connected component ofO(1, 2) containing the identity. We first prove that there exists a positive measure in ³ such that its Laplace transform is ( exp )^– if and only if >1. Finally, for 1 and inC, denotingP(,)(dx) = ( exp )^– exp (–[,x])(dx, we show that ifY ₀,...,Y _n aren+1 independent variables with densityP(,),j=0,...,n and ifS _k =X ₀ + ... +X _k andQ _k =R(S _k) –R(S _k–1) –R(Y _k),k=1,...,n, then then+1 statisticsD _n = [/,S _k ] –R _{k – 1} ),Q ₁,...,Q _n are independent random variables with the exponential () or gamma (1,1/) distribution.This research has been partially funded by NSERC Grant A8947.