A Remark on the Sobolev Regularity of Classical Solutions of Strongly Elliptic Equations

We prove that C^2,α(Ω ) solutions of problem (1.2) below are in H^m+2(Ω) for all m ∈ ?, if f and the coefficients are in H^m (Ω) n C^0,α (Ω ) Previously, this result was explicitly known only if m> n/2 (or if m = 0). A similar result holds for the quasi-linear equation (1.11) below.     

On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary

We consider nonnegative solutions of initial-boundary value problems for parabolic equationsu _t=u_xx, u_t=(u^m)_xxand (m>1) forx>0,t>0 with nonlinear boundary conditions−u _x=u^p,−(u ^m)_x=u^pand forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove that for each problem there exist positive critical valuesp ₀,p_c(withp ₀<p_c)such that forp∃(0,p ₀],all solutions are global while forp∃(p₀,p_c] any solutionu≢0 blows up in a finite time and forp>p _csmall data solutions exist globally in time while large data solutions are nonglobal. We havep _c=2,p _c=m+1 andp _c=2m for each problem, whilep ₀=1,p ₀=1/2(m+1) andp ₀=2m/(m+1) respectively. This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210.     

The scaling window of the 2‐SAT transition

We consider the random 2‐satisfiability (2‐SAT) problem, in which each instance is a formula that is the conjunction of m clauses of the form x∨y, chosen uniformly at random from among all 2‐clauses on n Boolean variables and their negations. As m and n tend to infinity in the ratio m/n→α, the problem is known to have a phase transition at α_c=1, below which the probability that the formula is satisfiable tends to one and above which it tends to zero. We determine the finite‐size scaling about this transition, namely the scaling of the maximal window W(n, δ)=(α_?(n,δ), α₊(n,δ)) such that the probability of satisfiability is greater than 1?δ for α<α_? and is less than δ for α>α₊. We show that W(n,δ)=(1?Θ(n^?1/3), 1+Θ(n^?1/3)), where the constants implicit in Θ depend on δ. We also determine the rates at which the probability of satisfiability approaches one and zero at the boundaries of the window. Namely, for m=(1+ε)n, where ε may depend on n as long as |ε| is sufficiently small and |ε|n^1/3 is sufficiently large, we show that the probability of satisfiability decays like exp(?Θ(nε³)) above the window, and goes to one like 1?Θ(n^?1|ε|^?3 below the window. We prove these results by defining an order parameter for the transition and establishing its scaling behavior in n both inside and outside the window. Using this order parameter, we prove that the 2‐SAT phase transition is continuous with an order parameter critical exponent of 1. We also determine the values of two other critical exponents, showing that the exponents of 2‐SAT are identical to those of the random graph. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 201–256 2001     

Asymptotic stability of solitons of the gKdV equations with general nonlinearity

We consider the generalized Korteweg-de Vries equation (gKdV)

Optimal lower bounds for cubature error on the sphere

We show that the worst-case cubature error E(Q_m;H^s) of an m-point cubature rule Q_m for functions in the unit ball of the Sobolev space H^s=H^s(S²),s>1, has the lower bound , where the constant c_s is independent of Q_m and m. This lower bound result is optimal, since we have established in previous work that there exist sequences of cubature rules for which with a constant independent of n. The method of proof is constructive: given the cubature rule Q_m, we construct explicitly a ‘bad’ function f_mH^s, which is a function for which Q_mf_m=0 and . The construction uses results about packings of spherical caps on the sphere.     

Monotonie bei den Quadraturverfahren von Gauß und Newton-Cotes

Summary LetQ _n be the quadrature rule of Gauss or Newton-Cotes withn abscissas. It is proven here, thatf ⁽²ⁿ⁾0 impliesQ _n ^G [f]Q _m ^G [f] (for allm>n) andQ _2n–1 ^NC [f]Q _2n ^NC [f]Q _2n+1 ^NC [f]. It follows that the sequenceQ _n[f] (n=1, 2, ...) is monotone, if all derivatives off are positive.

     

C *-algebras generated by orthogonal projections and their applications

We study the following problem: Given a Hilbert spaceH and a set of orthogonal projectionsP, Q ₁, ..., Q_n on it, with the conditionsQ _j ·Q _k = _j,k Q _k , , describe theC ^*-algebraC ^*(P, Q ₁, ..., Q_n) generated by these projections.Applications to Naimark dilation theorems and to Toeplitz operators associated with the Heisenberg group are given.Dedicated to the memory of M. G. Krein.This work was partially supported by CONACYT Project 3114P-E9608, México.     

Oscillations of Higher Order Neutral Differential Equations with Variable Coefficients

We obtain sufficient conditions for the oscillation of all solutions of the higher order neutral differential equation dⁿ/d^m[y(t) + P(t) y(t - μ)] + Q(t) y(t ?σ) = 0, t ≧ t₀ where n ≧ 1, P ? C[t₀, ∞), R ], Q ? C[t₀, ∞), R ] and τ, μ ? R ⁺. Our results extend and improve several known results in the literature.     

Hamilton decompositions of cartesian products of multicycles

We define the multicycle C^(r)_m as a cycle on m vertices where each edge has multiplicity r. So C^(r)_m can be decomposed into r Hamilton cycles. We provide a complete answer to the following question: for which positive integers m, n, r, s with m, n ≥ 3 can the Cartesian product of two multicycles C^(r)_m x C^(s)_n be decomposed into r + s Hamilton cycles? We find some interesting characterizations of Hamilton cycles of C_m x C_n while answering the above question. © 1997 John Wiley & Sons, Inc.     

Die Singularitätenfunktionen der gespannten Platte und der Kreiszylinderschale

Zusammenfassung Es wird eine Darstellung einer Fundamentallösung des OperatorsP()=Q()²–(c₁)^2m durch Fundamentallösungen der OperatorenQ()±(c₁) ^m angegeben. Als Anwendung berechnen wir die Singularitätenfunktionen der gespannten Platte und der Kreiszylinderschale.

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Summary A method is given, which allows to derive a fundamental solution of the operatorP()=Q()²–(c₁)^2m from some fundamental solutions of the operatorsQ()±(c₁) ^m . As an application we easily obtain the singular solutions of the unidirectionally stretched plate and of the circular cylindrical shell.

     

Order and chaos in Hofstadter's Q(n) sequence

A number of observations are made on Hofstadter's integer sequence defined by Q(n) = Q(n − Q(n − 1)) + Q(n − Q(n − 2)), for n > 2, and Q(1) = Q(2) = 1. On short scales, the sequence looks chaotic. It turns out, however, that the Q(n) can be grouped into a sequence of generations. The k‐th generation has 2^k members that have “parents” mostly in generation k − 1 and a few from generation k − 2. In this sense, the sequence becomes Fibonacci type on a logarithmic scale. The variance of S(n) = Q(n) − n/2, averaged over generations, is ≅2^αk, with exponent α = 0.88(1). The probability distribution p*(x) of x = R(n) = S(n)/n^α, n ≫ 1, is well defined and strongly non‐Gaussian, with tails well described by the error function erfc. The probability distribution of x_m = R(n) − R(n − m) is given by p_m(x_m) = λ_m p*(x_m/λ_m), with λ_m → √2 for large m. © 1999 John Wiley & Sons, Inc.     

Schauder estimates for oblique derivative problems

We prove that the solution of the oblique derivative parabolic problem in a noncylindrical domain Ω^T belongs to the anisotropic Holder space C^{2+α, 1+α/2}(gwT) 0 < α < 1, even if the nonsmooth “lateral boundary” of Ω^T is only of class C^{1+α, (1+α)/2}). As a corollary, we also obtain an a priori estimate in the Hölder space C^2+α(Ω₀) for a solution of the oblique derivative elliptic problem in a domain Ω₀ whose boundary belongs only to the classe C^1+α.     

A lower bound for the worst-case cubature error on spheres of arbitrary dimension

This paper is concerned with numerical integration on the unit sphere S^r of dimension r≥2 in the Euclidean space ℝ^r+1. We consider the worst-case cubature error, denoted by E(Q_m;H^s(S^r)), of an arbitrary m-point cubature rule Q_m for functions in the unit ball of the Sobolev space H^s(S^r), where s>, and show that The positive constant c_s,r in the estimate depends only on the sphere dimension r≥2 and the index s of the Sobolev space H^s(S^r). This result was previously only known for r=2, in which case the estimate is order optimal. The method of proof is constructive: we construct for each Q_m a `bad' function f_m, that is, a function which vanishes in all nodes of the cubature rule and for which Our proof uses a packing of the sphere S^r with spherical caps, as well as an interpolation result between Sobolev spaces of different indices.     

On the Elementary Retarded,Ultrahyperbolic Solution of the Klein-Gordon Operator,Iterated k Times

Let t = (t₁,…,t_n) be a point of ?ⁿ. We shall write . We put, by the definition, W_α(u, m) = (m^?2u)^{(α ? n)/4}[π^{(n ? 2)/2}2^{(α + n ? 2)/2}Г(α/2)]J_{(α ? n)/2}(m²u)^1/2; here α is a complex parameter, m a real nonnegative number, and n the dimension of the space. W_α(u, m), which is an ordinary function if Re α ≥ n, is an entire distributional function of α. First we evaluate {□ + m²}W_{α + 2}(u, m) = W_α(u, m), where {□ + m²} is the ultrahyperbolic operator. Then we express W_α(u, m) as a linear combination of R_α(u) of differntial orders; R_α(u) is Marcel Riesz's ultrahyperbolic kernel. We also obtain the following results: W_?2k(u, m) = {□ + m²}^kδ, k = 0, 1,…; W₀(u, m) = δ; and {□ + m²}^kW_2k(u, m) = δ. Finally we prove that W_α(u, m = 0) = R_α(u). Several of these results, in the particular case µ = 1, were proved earlier by a completely different method.     

On the Kronecker pairing for mixed cusp forms

LetG⊃PSL(2,R) be a Fuchsian group of the first kind with no elements of finite order, and letS ^2m V be the 2m-fold symmetric power of the standard representationV ofSL(2,R) on C². We determine the value of the Kronecker pairing between the canonical image of a mixed cusp formf of type (2,2m) inH ¹(G, S ^2m V) and a cycleg ⊗Q _g ^m inH ₁ (G, (S ^2m V)^*) for eachg inG, whereQ _g ^m is an element of (S ^2m V)^* associated tog, m and a monodromy representation ofG.     

Lower bound of overall exponential sums for polynomialsϕ(x d )

LetS ⁽¹⁾ (n, Q) denote the maximum module of exponential sums for polynomials of degree n over the Galois fieldF _Q . In a previous paper the transition to the multiple exponential sums allowed us to obtain a good lower bound of the valueS ⁽¹⁾ (n, Q), which coincides with Weil's bound whenn = q ^(m-1)/2 + 1, whereq, m are odd andm 3. Here the same approach is used for the estimation of the valueS ^(d) (n, Q), which corresponds to polynomials(x ^d ) overF _Q , whered is any divisor ofq – 1.     

The Virtual Poincaré Polynomials of Homogeneous Spaces

We factor the virtual Poincaré polynomial of every homogeneous space G/H, where G is a complex connected linear algebraic group and H is an algebraic subgroup, as t^2u (t²–1)^r Q_G/H(t²) for a polynomial Q_G/H with nonnegative integer coefficients. Moreover, we show that Q_G/H(t²) divides the virtual Poincaré polynomial of every regular embedding of G/H, if H is connected.     

COMPARATIVE PROBABILITY ON THE C* ALGEBRA OF COMPACT OPERATORS

Abstract

Let A be a von Neumann algebra on a Hilbert space H and let P(A) denote the projections of A. A comparative probability (CP) on A (or more correctly on P(A)) is a preorder ? on P(A) satisfying:

0 ? P ? P ε P(A) with Q ≠ 0 for some Q ε P(A).

If P, Q ε P(A) then either P ? Q or Q ? P.

If P, Q and R are all in P(A) and P⊥R, Q⊥R, then P ? Q ? P + R ? Q + R.

Let τ be any of the usual locally convex topologies on A. We say ? is τ continuous if the interval topology induced on P(A) by ? is weaker than the τ topology on P(A). If μ an additive (completely additive) measure on P(A) then μ induces a uniformly (weakly) continuous CP ?μ on P(A) given by P ?_μ Q if μ(P) ? μ(Q). We show that if A is the C* algebra C(H) of compact operators on an infinite dimensional Hilbert space H, the converse is true under an extra boundedness condition on the CP which is automatically satisfied whenever the identity is present in A = P(C(H)).     

Convergence of ray sequences of Padé approximants for 2 f 1(a, 1; c; z), (c > a > 0)

The Padé table of ₂ F ₁(a, 1; c; z) is normal for c > a > 0 (cf. [4]). For m ≥ n - 1 and c ? Z^-, the denominator polynomial Q _mn (z) in the [m/n] Padé approximant P _mn (z)/Q _mn (z) for ₂ F ₁(a, 1; c; z) and the remainder term Q _mn (z)₂ F ₁(a, 1; c; z)-P_mn (z) were explicitly evaluated by Padé (cf. [2], [6] or [9]). We show that for c > a > 0 and m ≥ n - 1, the poles of P_mn (z)/Q_mn (z) lie on the cut (1,∞). We deduce that the sequence of approximants P_mn (z)/Q_mn (z) converges to ₂ F ₁(a, 1; c; z) as m → ∞, n/m → ρ with 0 < ρ ≤ 1, uniformly on compact subsets of the unit disc |z| < 1 for c > a > 0.