On the Embedding of an Affine Space into a Projective Space

Let k, K be fields, and assume that |k| 4 and n, m 2, or |k| = 3 and n 3, m 2. Then, for any embedding of AG(n, k) into PG(m, K), there exists an isomorphism from k into K and an (n+1) × (m+1) matrix B with entries in K such that can be expressed as (x₁,x₂,...,x_n) = [(1,x₁ ,x₂ ,...,x_n )B], where the right-hand side is the equivalence class of (1,x₁ ,x₂ ,...,x_n )B. Moreover, in this expression, is uniquely determined, and B is uniquely determined up to a multiplication of element of K*. Let l 1, and suppose that there exists an embedding of AG(m+l, k) into PG(m, K) which has the above expression. If we put r = dim _k K, then we have r 3 and m > 2 l-1)/(r-2). Conversely, there exists an embedding of AG(l+m, k) into PG(m, K) with the above expression if K is a cyclic extension of k with dim _k K=r 3, and if m 2l/(r-2) with m even or if m 2l/(r-2) +1 with m odd.     

A graded scale of parametric families of distributions,and parameter estimates based on the sample mean

Summary Denote by _k a class of familiesP={P} of distributions on the line R¹ depending on a general scalar parameter , being an interval of R¹, and such that the moments µ₁()=xdP ,...,µ_2k ()=x ^2k dP are finite, ₁ (), ..., _k (), _k+1 () ..., _k () exist and are continuous, with ₁ () 0, and _j +1 ()= ₁ () _j () +[₂() -₁()²] _j ()/ ₁ (), J=2, ..., k. Let ₁=¯x=x ₁ + ... +x _n/n, ₂=x ₁ ² + ... +x _n ²/n, ..., _k =(x ₁ ^k + ... +x _n ^k/n denote the sample moments constructed for a sample x₁, ..., x_n from a population with distribution Pg. We prove that the estimator of the parameter by the method of moments determined from the equation ₁= ₁() and depending on the observations x₁, ..., x_n only via the sample mean ¯x is asymptotically admissible (and optimal) in the class _k of the estimators determined by the estimator equations of the form ₀ () + ₁ () ₁ + ... + _k () _k =0 if and only ifP _k .The asymptotic admissibility (respectively, optimality) means that the variance of the limit, as n (normal) distribution of an estimator normalized in a standard way is less than the same characteristic for any estimator in the class under consideration for at least one 9 (respectively, for every ).The scales arise of classes _{1 2}... of parametric families and of classes _{1 2} ... of estimators related so that the asymptotic admissibility of an estimator by the method of moments in the class _k is equivalent to the membership of the familyP in the class _k .The intersection consists only of the families of distributions with densities of the form h(x) exp {C₀() + C₁() x } when for the latter the problem of moments is definite, that is, there is no other family with the same moments ₁ (), ₂ (), ...Such scales in the problem of estimating the location parameter were predicted by Linnik about 20 years ago and were constructed by the author in [1] (see also [2, 3]) in exact, not asymptotic, formulation.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, pp. 41–47, 1981.     

A Note on a Boundary Value Problem

In this note, we prove that, for Robins boundary value problem, a unique solution exists if f_x(t, x, x), f_x(t, x, x), (t), and (t) are continuous, and f_x -(t), f_x -(t), 4(t) ² + ²(t) ++ 2(t), and 4(t) ² + ²(t) + 2(t).AMS Subject Classification (2000) 34B15     

Canonical Semigroups of States and Cocycles for the Group of Automorphisms of a Homogeneous Tree

Let G=A ut(T) be the group of automorphisms of a homogeneous tree and let d(v,gv) denote the natural tree distance. Fix a base vertex e in T. The function (g)=exp(–d(e,ge)), being positive definte on G, gives rise to a semigroup of states on G whose infinitesimal generator d/d|₌₀=log() is conditionally positive definite but not positive definite. Hence, log() corresponds to a nontrivial cocycle (g): GH in some representation space H . In contrast with the case of PGL(2,), the representation is not irreducible.Let _o (g) be the derivative of the spherical function corresponding to the complementary series of A ut(T). We show that –d(e,ge) and _o (g) come from cohomologous cocycles. Moreover, _o is associated to one of the two (irreducible) special representations of A ut(T).     

A symmetry problem in the calculus of variations

Let be a ball in ^N, centered at zero, and letu be a minimizer of the nonconvex functional over one of the classesC _M := {w W _loc ^1, () 0 w(x) M in,w concave} orE _M := {w W _loc ^1,2 () 0 w(x) M in,w 0 inL()}of admissible functions. Thenu is not radial and not unique. Therefore one can further reduce the resistance of Newton's rotational body of minimal resistance through symmetry breaking.     

On cubature formulas on the basis of lattices

Q (.. , L). Q . P(S_r(2)) — 2 (S _r(2) (r — ). , M(P(S _r(m=sup{t(·)t(·)₁:t P(S _r(2)),t 0}. , /4+(1)M(P(S _r(2)))/r ²15/17+(1)(r+). (Q), Q L.     

Asymptotics of the partial sums of a set of integral transforms

In this paper, we explore the asymptotic distribution of the zeros of the partial sums of the family of entire functions of order 1 and type 1, defined by G(,,z)=₀ ¹(t)t ^–1×(1–t)^–1e ^zt dt, where Re,Re>0, is Riemann-integrable on [0,1], continuous at t=0, 1 and satisfies (0)(1)0.     

Large deviations for Markov processes with discontinuous statistics,II: random walks

Summary Let ¹ and ² be Borel probability measures on ^d with finite moment generating functions. The main theorem in this paper proves the large deviation principle for a random walk whose transition mechanism is governed by ¹ when the walk is in the left halfspace ¹ = {x ^d :x ₁0} and whose transition mechanism is governed by ² when the walk is in the right halfspace ² = {x ^d :x ₁>0}. When the measures ¹ and ² are equal, the main theorem reduces to Cramér's Theorem.This research was supported in part by a grant from the National Science Foundation (NSF-DMS-8902333)This research was supported in part by a grant from the National Science Foundation (NSF-DMS-8901138) and in part by a Lady Davis Fellowship while visiting the Faculty of Industrial Engineering and Management at the Technion during the spring semester of 1989     

Lattice points in high-dimensional spheres

LetN(x, n, ) denote the number of integer lattice points inside then-dimensional sphere of radius (an)^1/2 with center at x. This numberN(x,n, ) is studied for fixed,n , andx varying. The average value (asx varies) ofN(x,n, ) is just the volume of the sphere, which is roughly of the form (2 e, ) ^n/2. it is shown that the maximal and minimal values ofN (x,n, ) differ from the everage by factors exponential inn, which is in contrast to the usual lattice point problems in bounded dimensions. This lattice point problem arose separately in universal quantization and in low density subset sum problems.     

Projectivities in Moufang-Klingenberg planes

Let = = (,,) be a Moufang-Klingenberg plane coordinatized by a local alternative ring R. We define the projectivities of a line g in geometrically as products of perspectivities. It is shown that under certain conditions the group of projectivities of g is generated by the algebraically defined permutations xx+t (tR), xcx (cR a unit), xx ^–.     

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.     

Singular Eigenvalue Problems for the Equation y (n) + λp(x)y = 0

The paper studies singular eigenvalue problems for the equation y ⁽ⁿ⁾ +p(x)y=0 with boundary conditions imposed on the derivatives y ⁽ⁱ⁾ at the points x=a and x=. We look for singular problems which are analogous to regular problems on a finite interval. It is characterized when each eigenfunction has a finite number of zeros and when the spectrum is discrete or continuous, respectively.     

On the Existence of <Emphasis Type="Italic">φ</Emphasis>-Moments of the Limit of a Normalized Supercritical Galton–Watson Process

Let (Z _n ) _{n 0} be a supercritical Galton–Watson process with finite re-production mean  and normalized limit W=lim _n ^–n Z _n . Let further : [0,) [0,) be a convex differentiable function with (0)=(0)=0 and such that ( ) is convex with concave derivative for some n 0. By using convex function inequalities due to Topchii and Vatutin, and Burkholder, Davis and Gundy, we prove that 0 < E (W) < if, and only if, , where

A generalization of the results of Pillai

In a recent article Pillai (1990,Ann. Inst. Statist. Math.,42, 157–161) showed that the distribution 1–E (–x ), 0<1; 0x, whereE (x) is the Mittag-Leffler function, is infinitely divisible and geometrically infinitely divisible. He also clarified the relation between this distribution and a stable distribution. In the present paper, we generalize his results by using Bernstein functions. In statistics, this generalization is important, because it gives a new characterization of geometrically infinitely divisible distributions with support in (0, ).     

DLR measures for one-dimensional harmonic systems

Summary A one-dimensional chain of nearest neighbor linearly interacting oscillators {q _x } _x is studied. The set of all its extremal DLR measures is characterized in terms of a parameter ². For each there is a Gaussian DLR measure with support on the set of configurations determined by the rate of growth of¦q _x¦. It is then finally proved that there is only one translationally invariant DLR measure. This proves the following conjecture: invariant DLR measures give uniformly finite first moment to ¦q _x¦.     

On the asymptotic and invariantσ-algebras of random walks on locally compact groups

Summary Consider a random walk of law on a locally compact second countable groupG. Let the starting measure be equivalent to the Haar measure and denote byQ the corresponding Markov measure on the space of pathsG . We study the relation between the spacesL (G , ^a ,Q) andL (G , ⁱ ,Q) where ^a and ⁱ stand for the asymptotic and invariant -algebras, respectively. We obtain a factorizationL (G , ^a ,Q) L (G , ⁱ ,Q)L (C) whereC is a cyclic group whose order (finite or infinite) coincides with the period of the Markov shift and is determined by the asymptotic behaviour of the convolution powers ⁿ.     

Interpolation of generalized analytical functions of complex variable

A general approach is proposed to the interpolation of x -analytical functions of a complex variable with an arbitrary –,+[Basis x -analytical functions whose imaginary pan is a polynomial in x, and y are obtained in explicit form.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 3–9, 1986.     

Analytic Approximations for a Singularly Perturbed Convection—Diffusion Problem with Discontinuous Data in a Half-Infinite Strip

We consider a singularly perturbed convection—diffusion equation, –u+v u=0, defined on a half-infinite strip, (x,y)(0,)×(0,1) with a discontinuous Dirichlet boundary condition: u(x,0)=1, u(x,1)=u(0,y)=0. Asymptotic expansions of the solution are obtained from an integral representation in two limits: (a) as the singular parameter 0⁺ (with fixed distance r to the discontinuity point of the boundary condition) and (b) as that distance r0⁺ (with fixed ). It is shown that the first term of the expansion at =0 contains an error function or a combination of error functions. This term characterizes the effect of discontinuities on the -behavior of the solution and its derivatives in the boundary or internal layers. On the other hand, near the point of discontinuity of the boundary condition, the solution u(x,y) is approximated by a linear function of the polar angle at the point of discontinuity (0,0).     

Superconvergence for galerkin methods for the two point boundary problem via local projections

The two point boundary problemy'-a(x)y–b(x)y=-f(x), o<x<1,y(0)=y(1)=0, is first solved approximately by the standard Galerkin method, (Y, ) + (aY+bY, )=(f, ), ₁ ⁰ (r, ), for a function Y ₁ ⁰ (r, ), the space ofC ¹-piecewise--degree-polynomials vanishing atx=0 andx=1 and having knots at {x ₀ ,x ₁ , ...,x _M }=. ThenY is projected locally into a polynomial of higher degree by means of one of several projections. It is then shown that higher-order convergence results locally, provided thaty is locally smooth and is quasi-uniform.This research was supported in part by the National Science Foundation.     

Random Logistic Maps II. The Critical Case

Let (X _n ) ₀ be a Markov chain with state space S=[0,1] generated by the iteration of i.i.d. random logistic maps, i.e., X _n+1=C _n+1 X _n (1–X _n ),n0, where (C _n ) ₁ are i.i.d. random variables with values in [0, 4] and independent of X ₀. In the critical case, i.e., when E(log C ₁)=0, Athreya and Dai⁽²⁾ have shown that X _n ^P 0. In this paper it is shown that if P(C ₁=1)<1 and E(log C ₁)=0 then(i) X _n does not go to zero with probability one (w.p.1) and in fact, there exists a 0<<1 and a countable set (0,1) such that for all xA(0,1), P _x (X _n for infinitely many n1)=1, where P _x stands for the probability distribution of (X _n ) ₀ with X ₀=x w.p.1. A is a closed set for (X _n ) ₀.(ii) If is the supremum of the support of the distribution of C ₁, then for all xA (a)