Ultraparabolic equations and unsteady heat transfer

The present paper is concerned with the boundary value problem for the equation _t + u· = k_yy + f. Existence and uniqueness of the generalized solution are proved.     

Asymptotic Formulae and Divisor Problems

We investigate the asymptotic behaviour of the summatory functions of _z(n, ), _k(n, ) z ⁽ⁿ⁾ and _k(n, ) z ⁽ⁿ⁾.     

On the factor-thickness of regular graphs

For eachr-regular graphG, define a binary sequence(G) = ( ₁, ₂,..., _r-1) by _k = 0 ifG has ak-factor, and _k = 1 otherwise. A binary sequence = ( _i |i = 1, 2,...,r – 1) is said to be realizable if there exists anr-regular graphG such that(G) = . In this paper we characterize all binary sequences which are realizable.     

Probabilistic characterizations of the Generalized Domain of Attraction of the multivariate normal law

For independent identically distributed random vectors,X _i , we give necessary and sufficient probabilistic conditions for their common distribution to belong to the Generalized Domain of Attraction of the multivariate normal law. The first condition says that after projecting onto any direction, , the sum of squares, _i ¹⁼¹ X _i , ², properly normalized, converges to one in probability uniformly over the unit sphere. The second condition says that max X _i , ²/ _n ⁱ⁼¹ X _i , ² converges to zero in probability uniformly over the unit sphere.     

Asymptotically minimax tests of composite hypotheses

Summary X ₁,,X> _n are independent, identically distributed random variables with common density function f(x¦ ₁ ,, _k , _k+1 ), assumed to satisfy certain standard regularity conditions. The k+1 parameters are unknown, and the problem is to test the hypothesis that _k+1 =b against the alternative that _k+1 =b+cn ^–1/2 . ₁ ,, _k are nuisance parameters. For this problem, the following artificial problem is temporarily substituted. It is known that ¦ ₁ -a _i ¦n ^–1/2 M(n) for i=1,,k, where a ₁ , ,a _k are known, and M(n) approaches infinity as n increases but n ^–1/2 M(n) approaches zero as n increases. A Bayes decision rule is constructed for this artificial problem, relative to the a priori distribution which assigns weight A to _k+1 =b, and weight 1-A to _k+1 =b+cn ^–1/2 , in each case the weight being spread uniformly over the possible values of ₁ ,, _k in the artificial problem. An analysis of the structure of the Bayes rule shows that if estimates of ₁ ,..., _k are substituted for a ₁ ..., a _k respectively, the resulting rule is a solution to the original problem, and this rule has the same asymptotic properties as a solution to the artificial problem as the Bayes rule for the artificial problem, no matter what the values a ₁ ..., a _k are.Research supported by the U.S. Air Force under Grant AF-AFOSR-68-1472.     

Bayesian estimation in the symmetric location problem

Summary Let X _i =+ _i for i=1, ..., n, where the _i's are i.i.d. F and F is symmetric about 0. F is assumed unknown or only partially known, and the problem is to estimate . Priors are put on the pair (F,). The priors on F are obtained from Doksum's neutral to the right priors, and include symmetrized Dirichlet priors. The marginal posterior distribution of given X ₁, ..., X _nis computed and its general properties studied. It is found that for certain classes of distributions of the _i's, the posterior distribution of is for all large n a point mass at the true value of . If the distribution of the _i's is not exactly symmetric, the Bayes estimates can behave very poorly.     

Nonlinear instability of superposed magnetic fluids in the presence of an oblique magnetic field

The nonlinear evolution of interfacial waves separating two magnetic fluids subjected to an oblique magnetic field is studied in two dimensions, with the use of the method of multiple scales. It is shown that the evolution of the envelope is governed by two partial differential equations. These equations can be combined to yield two alternate Schrödinger equations with cubic nonlinearity; one of them leads to the determination of the cutoff wave number separating stable from unstable deformations while the other Schrödinger equation is used to analyze the stability of the system. The stability of the system is discussed both theoretically and computationally, and the stability diagrams are obtained. It is found in the linear theory that the oblique magnetic field has a stabilizing influence if 0 ₁ + ₂ < /2, or 3/2 < ₁ + ₂ 2 and a destabilizing influence if /2 < ₁ + ₂ < 3/2, where 0 _j , (j=1, 2) and , is the angle between the field and the horizontal axis.In the nonlinear theory, the stability analysis reveals that there exist different regions of stability and instability. It is reported that the oblique magnetic field plays a dual role in the stability criterion and the angles ₁ and ₂ play a distinctive role in this analysis besides the effect of the variation of the magnetic permeabilities.     

Exact Solution of the Ising Model on the Cayley Tree with Competing Ternary and Binary Interactions

The exact solution is found for the problem of phase transitions in the Ising model with competing ternary and binary interactions. For the pair of parameters =(J) and ₁=₁(J ₁) in the plane (₁,), we find two critical curves such that a phase transition occurs for all pairs (₁,) lying between the curves.     

Inner functions and related spaces of pseudocontinuable functions

Let be an inner function, let C, ¦¦=1. Then the harmonic function [(+)]/(–)] is the Poisson integral of a singular measure D. N. Clark's known theorem enables us to identify in a natural manner the space H² H² with the space L² ( ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 7–33, 1989.     

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.     

Asymptotic behavior of general M-estimates for regression and scale with random carriers

Summary Let (xini, y _i be a sequence of independent identically distributed random variables, where x _i R ^p and y _i R, and let R ^p be an unknown vector such that y _i =x _i +u _i (^*), where u _i is independent of x _i and has distribution function F(u/), where >0 is an unknown parameter. This paper deals with a general class of M-estimates of regression and scale, ( ^*,^*), defined as solutions of the system: , where r= (y _i –x _i ^1*/)^*, with R ^p ×RR and RR. This class contains estimators of (, ) proposed by Huber, Mallows and Krasker and Welsch. The consistency and asymptotic normality of the general M-estimators are proved assuming general regularity conditions on and and assuming the joint distribution of (x _i , y _i ) to fulfill the model (^*) only approximately.     

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.     

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {X _k , 1 k n} be n independent and real-valued random variables with common subexponential distribution function, and let {k, 1 k n} be other n random variables independent of {X _k , 1 k n} and satisfying a _k b for some 0 < a b < for all 1 k n. This paper proves that the asymptotic relations P (max_{1 m n k=1} ^m _k X _k > x) P (sum _k=1 ⁿ _k X _k > x) sum _k=1 ⁿ P ( _k X _k > x) hold as x . In doing so, no any assumption is made on the dependence structure of the sequence { _k , 1 k n}. An application to ruin theory is proposed.     

Solution of the nonlinear functional equations representing the roll gap relationships in a cold mill

In the development of a roll force model for cold rolling, techniques were developed for solving the system equations which are of general interest. This paper gives a brief introduction to the physical model but concentrates on the solution of the model equations and the simulation. An unusual feature of the model was that the calculated profiles had to satisfy a number of boundary conditions at different points throughout the roll arc. A new method was developed for calculating these profiles and for determining the gradient functions which satisfied the boundary constraints.Nomenclature p() pressure at roll angle - h() gauge - a() roll radius - y() yield stress - g _i () gradient function on iterationi - e() gauge error - (, ) transition function - H(–) Heaviside unit step function at = - (–) unit impulse function at = - H(, ₁, ₂) defined asH( – ₁) –H( – ₂) - angular position from the roll center line - _T angular limits of roll arc represented - _n angular position of the neutral angle - _i angular position ofith strip elastic-plastic boundary - _pi pressure change at the boundaryi - _i , _i , _i constants defined in Appendix A - k ₁,k ₂ elastic region constants - k total number of strip boundaries (elastic-plastic and entry and exit points) - R undeformed work roll radius - R _s roll separation—distance between roll centers - h ₀₁ unstrained gauge in an elastic region - h _in gauge of the strip at the entry to the roll gap - J gauge error cost function - <x, y> inner product ofx andy - x norm ofx - L ₂[0, _T ] the space of Lebesgue square-integrable functions defined on the interval [0, _T ] - JUVY denotes (Dx)() =dx()/d The author would like to acknowledge the help given by Dr. G. F. Bryant, Director, and Mr. M. A. Fuller, Senior Research Engineer, the Industrial Automation Group, Imperial College of Science and Technology, London. He is also grateful to M. J. G. Henderson of the University of Birmingham for his advice and encouragement during the project. He would like to thank the Directors of GEC Electrical Projects Limited for allowing him to undertake the work and also Mr. J. McTaggart and Mr. C. McKenzie (GEC), Professor H. A. Prime of the University of Birmingham, and Dr. G. F. Bryant for arranging the project.     

The Parameters of Bipartite Q-polynomial Distance-Regular Graphs

Let denote a bipartite distance-regular graph with diameter D 3 and valency k 3. Suppose ₀, ₁, ..., _D is a Q-polynomial ordering of the eigenvalues of . This sequence is known to satisfy the recurrence _{i – 1} – _i + _{i + 1} = 0 (0 > i > D), for some real scalar . Let q denote a complex scalar such that q + q ^–1 = . Bannai and Ito have conjectured that q is real if the diameter D is sufficiently large.We settle this conjecture in the bipartite case by showing that q is real if the diameter D 4. Moreover, if D = 3, then q is not real if and only if ₁ is the second largest eigenvalue and the pair (, k) is one of the following: (1, 3), (1, 4), (1, 5), (1, 6), (2, 4), or (2, 5). We observe that each of these pairs has a unique realization by a known bipartite distance-regular graph of diameter 3.     

Boundary Asymptotics for Orthogonal Rational Functions on the Unit Circle

Let w() be a positive weight function on the unit circle of the complex plane. For a sequence of points { _k } _{k = 1} included in a compact subset of the unit disk, we consider the orthogonal rational functions _n that are obtained by orthogonalization of the sequence { 1, z / ₁, z ² / ₂, ... } where , with respect to the inner product In this paper we discuss the behaviour of _n (t) for t = 1 and n under certain conditions. The main condition on the weight is that it satisfies a Lipschitz–Dini condition and that it is bounded away from zero. This generalizes a theorem given by Szeg in the polynomial case, that is when all _k = 0.     

On the speed of convergence in the central limit theorem of log-likelihood ratio processes

Let , the parameter space, be an open subset ofR ^k ,k1. For each , let the r.v.'sX _n ,n=1, 2,... be defined on the probability space (X, P ) and take values in (S,S,L) whereS is a Borel subset of a Euclidean space andL is the -field of Borel subsets ofS. ForhR _k and a sequence of p.d. normalizing matrices _n = _n ^{k × k} (₀ set _n ^* = ^* = ₀ + _n h, where ₀ is the true value of , such that ^*, . Let _n (^*, _*)( be the log-likelihood ratio of the probability measure with respect to the probability measure , whereP _n is the restriction ofP over _n = (X ₁,X ₂,...,X _n . In this paper we, under a very general dependence setup obtain a rate of convergence of the normalized log-likelihood ratio statistic to Standard Normal Variable. Two examples are taken into account.     

The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms

Given a vector of real numbers=(₁,... _d ) ^d , the Jacobi-Perron algorithm and related algorithms, such as Brun's algorithm and Selmer's algorithm, produce a sequence of (d+1)×(d+1) convergent matrices {C⁽ⁿ⁾():n1} whose rows provide Diophantine approximations to . Such algorithms are specified by two mapsT:[0, 1] ^d [0, 1] ^d and A:[0,1] ^d GL(d+1,), which compute convergent matrices C⁽ⁿ⁾())...A(T())A(). The quality of the Diophantine approximations these algorithms find can be measured in two ways. The best approximation exponent is the upper bound of those values of for which there is some row of the convergent matrices such that for infinitely many values ofn that row of C⁽ⁿ⁾() has . The uniform approximation exponent is the upper bound of those values of such that for all sufficiently large values ofn and all rows of C⁽ⁿ⁾⁽⁾ one has . The paper applies Oseledec's multiplicative ergodic theorem to show that for a large class of such algorithms and take constant values and on a set of Lebesgue measure one. It establishes the formula where are the two largest Lyapunov exponents attached by Oseledec's multiplicative ergodic theorem to the skew-product (T, A,d), whered is aT-invariant measure, absolutely continuous with respect to Lebesgue measure. We conjecture that holds for a large class of such algorithms. These results apply to thed-dimensional Jacobi-Perron algorithm and Selmer's algorithm. We show that; experimental evidence of Baldwin (1992) indicates (nonrigorously) that. We conjecture that holds for alld2.     

On the Minimum Size of Some Minihypers and Related Linear Codes

We denote by m_r,q(s) the minimum value of f for which an {f, _r-2+s ; r,q }-minihyper exists for r 3, 1 s q–1, where _j=(q^j+1–1)/(q–1). It is proved that m_3,q(s)=₁(₁+s) for many cases (e.g., for all q 4 when ) and that m_r,q(s) _r-1+s₁+q for 1 s q – 1,~q 3,~r 4. The nonexistence of some [n,k,n+s–q^k-2]_q codes attaining the Griesmer bound is given as an application.AMS classification: 94B27, 94B05, 51E22, 51E21     

Admissibility of unbiased tests for a composite hypothesis with a restricted alternative

This paper discusses -admissiblility and d-admissiblity which are important concepts in studying the performance of statistical tests for composite hypotheses. A sufficient condition for -admissibility is presented. When =1/m, the Nomakuchi-Sakata test, which is uniformly more powerful than the likelihood ratio test for hypotheses min (₁, ₁) = 0 versus min (₁, ₁) > 0, is generalized for a class of distributions in an exponential family, and its unbiasedness and -admissibility are shown. Finally, the case of 1/m is discussed in brief.