Critical points for spread-out self-avoiding walk,percolation and the contact process above the upper critical dimensions

We consider self-avoiding walk and percolation in ^d, oriented percolation in ^d×₊, and the contact process in ^d, with p D(·) being the coupling function whose range is proportional to L. For percolation, for example, each bond is independently occupied with probability p D(y–x). The above models are known to exhibit a phase transition when the parameter p varies around a model-dependent critical point p_c. We investigate the value of p_c when d>6 for percolation and d>4 for the other models, and L1. We prove in a unified way that p_c=1+C(D)+O(L^–2d), where the universal term 1 is the mean-field critical value, and the model-dependent term C(D)=O(L^–d) is written explicitly in terms of the random walk transition probability D. We also use this result to prove that p_c=1+cL^–d+O(L^–d–1), where c is a model-dependent constant. Our proof is based on the lace expansion for each of these models.     

The minimum area of a surface with prescribed boundary contour in a pseudoeuclidean space

LetE ^{n, k} be a pseudoeuclidean space with linear elementdx ₁ ² ++dx _n–k ² –dx _n–k ^+1/2 ––dx _n ² . The area of a smooth two-dimensional surface inE ^{n, k} is defined by , whereE, F, andG are the coefficients of the first fundamental form of the surface andD is the region of variation of the parametersu andv. The following theorem is proved: LetL be a piecewise smooth closed curve inE ^{n, k} (1kn–1). Then there exists a two-dimensional piecewise smooth surface of arbitrarily small area bounded by the curveL. 3 figures.Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 18–22.     

On (p a , p b , p a , p a-b )-Relative Difference Sets

This paper provides new exponent and rank conditions for the existence of abelian relative (p ^a,p ^b,p ^a,p ^a–b)-difference sets. It is also shown that no splitting relative (2^2c,2^d,2^2c,2^2c–d)-difference set exists if d > c and the forbidden subgroup is abelian. Furthermore, abelian relative (16, 4, 16, 4)-difference sets are studied in detail; in particular, it is shown that a relative (16, 4, 16, 4)-difference set in an abelian group G Z₈ × Z₄ × Z₂ exists if and only if exp(G) 4 or G = Z₈ × (Z₂)³ with N Z₂ × Z₂.     

Contact Classification of Linear Ordinary Differential Equations: I

It is known that a linear ordinary differential equation of order n3 can be transformed to the Laguerre–Forsyth form y ⁽ⁿ⁾= _i=3 ⁿ a _n–i (x)y ^(n–i) by a point transformation of variables. The classification of equations of this form in a neighborhood of a regular point up to a contact transformation is given.     

Kinetics of formation of the actual contact area in rubbery materials

The kinetics of formation of the actual contact area in rubbery materials is investigated. A procedure for determining the relaxation spectrum of the segmental motion of the macromolecules in the surface layer of a polymer in contact with a solid surface is proposed. The effect of temperature and the applied specific load on H() is determined. The activation energy of segmental motion of the macromolecules in a thin surface layer of the specimen is calculated.Second Moscow Pirogov State Medical Institute. Translated from Mekhanika Polimerov, No. 6, pp. 1047–1052, November–December, 1971.     

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.     

An approximation for the complex normal probability integral

An analytic approximation to the complex normal probability integral, (x+iy)=(2)^–1/2 _–t ^x exp[–(t–iy)²/2]dt, is given together with a study of the error in the approximation.     

A Lower Bound for the Height of a Rational Function at S-unit Points

Let a,b be given, multiplicatively independent positive integers and let >0. In a recent paper jointly with Y. Bugeaud we proved the upper bound exp(n) for g.c.d.(aⁿ–1, bⁿ–1); shortly afterwards we generalized this to the estimate g.c.d.(u–1,v–1)<>u,v) for multiplicatively independent S-units u,vZ. In a subsequent analysis of those results it turned out that a perhaps better formulation of them may be obtained in terms of the language of heights of algebraic numbers. In fact, the purposes of the present paper are: to generalize the upper bound for the g.c.d. to pairs of rational functions other than {u–1,v–1} and to extend the results to the realm of algebraic numbers, giving at the same time a new formulation of the bounds in terms of height functions and algebraic subgroups of G_m².     

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).

     

Special point sets inPG(n,q) and the structure of sets with the maximal number of nuclei

The structure of _{n– 1}-sets inPG(n, q) with more thanq – 1 nuclei is investigated. It is shown that classification of these sets with the maximal numberq ^{n– 1}-q ^{n– 2} of nuclei is equivalent to the classification of (q + l)-sets inPG(2,q) havingq –1 nuclei.Dedicated to Professer Walter Benz for his 60^th birthday     

Growth of a mixed derivative of a function ofC^{(l_1 , l_2 )}

For a function f(x, y) with the continuous derivatives we estimate the growth of the mixed derivative D f(₁/l ₁+₂/l ₂=1). We consider generalizations and related problems.Translated from Matematicheskie Zametki, Vol. 15, No. 3, pp. 355–362, March, 1974.In conclusion, the author wishes to thank K. I. Oskolkov for pointing out to him concepts for the discussion of the statement of the problem.     

On the cost of approximating all roots of a complex polynomial

This work examines the computational complexity of a homotopy algorithm in approximating all roots of a complex polynomialf. It is shown that, probabilistically, monotonic convergence to each of the roots occurs after a determined number of steps. Moreover, in all subsequent steps, each rootz is approximated by a complex numberx, where ifx ₀ =x, x _j =x _j–1 –f(x _j–1)/f(x _j–1),j = 1, 2,, then |x _j –z| < (1/|x ₀ –z|)|x _j–1–z|².     

On a class of partially ordered sets and their linear invariants

Let (, <) be a finite partially ordered set with rank function. Then is the disjoint union of the classes _k of elements of rank k and the order relation between elements in _k and _k+1 can be represented by a matrix S _k. We study partially ordered sets which satisfy linear recurrence relations of the type S _k (S _k ^T ) – c _k (S _{k – 1})^T S _{k – 1} = d _k ⁺ – c _k d _k ^– ) Id for all k and certain coefficients d _k ⁺, d _k ^- and c _k.     

Quenching,nonquenching, and beyond quenching for solution of some parabolic equations

Summary In this paper we examine the first initial boundary value problem for u_t=u_xx + (1 – u)^–, > 0, > 0,on (0, 1) × (0, ) from the point of view of dynamical systems. We construct the set of stationary solutions, determine those which are stable, those which are not and show that there are solutions with initial data arbitrarily close to unstable stationary solutions which quench (reach one in finite time). We also examine the related problem u_t=u_xx, 0 <x < 1,t > 0;u(0,t)=0, (1 – u(1, t))^–. The set of stationary solutions for this problem, and the dynamical behavior of solutions of the time dependent problem are somewhat different.This research was sponsored by the U.S. Air Force Office of Scientific Research, Air Forse Systems Command Grants 84-0252 and 88-0031. The United States Government is authorized to reproduce and distribute reprints for Governmental purposes not withstanding any copyright notation therein.     

On the number of solutions to the linear comple-mentarity problem

Given ann × n matrixA and ann-dimensional vectorq letN(A, q) be the cardinality of the set of solutions to the linear complementarity problem defined byA andq. It is shown that ifA is nondegenerate thenN(A, q) + N(A, –q) 2 ⁿ , which in turn impliesN(A, q) 2 ⁿ – 1 ifA is also aQ-matrix.It is then demonstrated that min _q0 N(A, q) 2 ^n–1 – 1, which concludes that the complementary cones cannot spanR ⁿ more than 2 ^n–1 – 1 times around. For anyn, an example of ann × n nondegenerateQ-matrix spanning allR ⁿ , but a subset of empty interior, 2^[n/3] times around is given.     

On the gonality of nodal curves

Here we prove that for every n33 and every t(n ²+3n)/6, the normalization Y of a general plane curve C of degree n and with t nodes has no g _b ¹ with b<n–2 and only g _n–2 ¹ and g _n–1 ¹ induced by a pencil of lines through a point of C. Recently, Coppens and Kato proved stronger results.     

Rational approximations for the cosine function; P-acceptability and order

Properties of order stars corresponding to rational approximations for cos z are derived and are used to prove that the order of accuracy of a P-acceptable approximantR _nm(z ²), with numerator of degreen and denominator of degreem, cannot exceed 2m. It is shown that if the poles ofR _nm(z ²) are restricted to pure-imaginary values ofz the maximum attainable order is 2n+2, whatever the value ofm1. A study of rational approximations for the cosine function produced by symmetric one-step collocation methods, applied to the differential equationy ⁿ =–² y, provides the answer to a question posed by Kramarz [BIT 20 (1980) 215–222]; there are no P-stable methods of that type.     

The Moduli of Flat U(p, 1) Structures on Riemann Surfaces

For X a smooth projective curve over of genus g>1, Hom⁺(₁(X), U(p, 1))/U(p, 1) is the moduli space of flat semi-simple U(p, 1)-connections on X. There is an integer invariant, , the Toledo invariant associated with each element in Hom⁺(₁(X), U(p, 1))/U(p, 1). This paper shows that Hom⁺(₁(X), U(p, 1))/U(p, 1) has one connected component corresponding to each & in 2 with –2(g–1) 2(g–1). Therefore the total number of connected components is 2(g–1)+1.     

On Spin Models, Modular Invariance, and Duality

A spin model is a triple (X, W ⁺, W ^–), where W ⁺ and W ^– are complex matrices with rows and columns indexed by X which satisfy certain equations (these equations allow the construction of a link invariant from(X, W ⁺, W ^–) ). We show that these equations imply the existence of a certain isomorphism between two algebras and associated with (X, W ⁺, W ^–) . When is the Bose-Mesner algebra of some association scheme, and is a duality of . These results had already been obtained in [15] when W ⁺, W ^– are symmetric, and in [5] in the general case, but the present proof is simpler and directly leads to a clear reformulation of the modular invariance property for self-dual association schemes. This reformulation establishes a correspondence between the modular invariance property and the existence of spin models at the algebraic level. Moreover, for Abelian group schemes, spin models at the algebraic level and actual spin models coincide. We solve explicitly the modular invariance equations in this case, obtaining generalizations of the spin models of Bannai and Bannai [3]. We show that these spin models can be identified with those constructed by Kac and Wakimoto [20] using even rational lattices. Finally we give some examples of spin models at the algebraic level which are not actual spin models.     

Isotropic random walks in a tree

Summary Let T be an infinite homogeneous tree of order a+1. We study Markov chains {X _n} in T whose transition functions p(x, y)=A[d(x,y)] depend only on the shortest distance between x and y in the graph. The graph T can be represented as a symmetric space of a p-adic matrix group; we prove a series of results using essentially the spherical functions of this symmetric space. Theorem 1. d(X _n,x) n a.s., where >0 if A(0) 1, X ₀=x. Assuming {X _n} is strongly aperiodic, Theorem 2. p ₂(x, y)CRⁿ/n^3/2 for fixed x, y where R=(d) A(d)<1, and if E[d(X₁, X₀)²]<, Theorem 3. R(1–u, x, y) = (1–u)ⁿp_n(x, y)=Ca^–d[exp(–du/)+o_d(1)] as d=d(x,y) uniformly for 0u2. Using Theorem 3, we calculate the Martin boundary Dirichlet kernel of p(x, y) on T, which turns out to be independent of {itA(d)}. We also consider a stepping-stone model of a randomly-mating-and-migrating population on the nodes of T. If initially all individuals are distinct, then in generation n approximately half of the individuals of a given type are within n of a typical one and essentially all are within 2n.This work was partially supported by the National Science Foundation under grant number MCS 75-08098-A01For the academic year 1977–78: Department of Mathematics GN-50, University of Washington, Seattle, Washington 98195 USA