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.     

On the critical percolation probabilities

Summary We prove that the critical probabilities of site percolation on the square lattice satisfy the relation p _c +p _c ^/* =1. Furthermore we prove the continuity of the function percolation probability.Work supported in part by U.S. National Science Foundation Grant PHY78-25390     

On the Aleksandrov-Fenchel inequality involving zonoids

The equality case in the general quadratic inequality V(K, L, K ₁, ..., K _n–2)² V(K, K, K ₁, ..., K _n–2) V(L, L, K ₁, ..., K _n–2) for mixed volumes is settled under the assumption that K and L are centrally symmetric and K ₁, ..., K _n–2 are zonoids. This result partly confirms a conjecture on the general case made in an earlier paper.     

On construction ofk-wise independent random variables

A 0–1probability space is a probability space (, 2,P), where the sample space -{0, 1} ⁿ for somen. A probability space isk-wise independent if, whenY _i is defined to be theith coordinate or the randomn-vector, then any subset ofk of theY _i 's is (mutually) independent, and it is said to be a probability spacefor p ₁,p ₂, ...,p _n ifP[Y _i =1]=p _i .We study constructions ofk-wise independent 0–1 probability spaces in which thep _i 's are arbitrary. It was known that for anyp ₁,p ₂, ...,p _n , ak-wise independent probability space of size always exists. We prove that for somep ₁,p ₂, ...,p _n [0,1],m(n,k) is a lower bound on the size of anyk-wise independent 0–1 probability space. For each fixedk, we prove that everyk-wise independent 0–1 probability space when eachp _i =k/n has size (n _k ). For a very large degree of independence —k=[n], for >1/2- and allp _i =1/2, we prove a lower bound on the size of . We also give explicit constructions ofk-wise independent 0–1 probability spaces.This author was supported in part by NSF grant CCR 9107349.This research was supported in part by the Israel Science Foundation administered by the lsrael Academy of Science and Humanities and by a grant of the Israeli Ministry of Science and Technology.     

Uniqueness of solutions of boundary-value problems for operator-differential equations on a finite segment and on a semiaxis

For the equationL ₀ x(t)+L ₁x(t)+...+L _n x ⁽ⁿ⁾(t)=O, whereL _k,k=0,1,...,n, are operators acting in a Banach space, we establish criteria for an arbitrary solutionx(t) to be zero provided that the following conditions are satisfied:x ^(1–1) (a)=0, 1=1, ..., p, andx ^(1–1) (b)=0, 1=1,...,q, for - <a< b< (the case of a finite segment) orx ^(1–1) (a)=0, 1=1,...,p, under the assumption that a solutionx(t) is summable on the semiaxista with its firstn derivatives.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 279–292, March, 1994.This research was supported by the Ukrainian State Committee on Science and Technology.     

On Piecewise-Constant Approximation of Continuous Functions of n Variables in Integral Metrics

We consider the approximation by piecewise-constant functions for classes of functions of many variables defined by moduli of continuity of the form (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ), where _i ( _i ) are ordinary moduli of continuity that depend on one variable. In the case where _i ( _i ) are convex upward, we obtain exact error estimates in the following cases: (i) in the integral metric L ₂ for (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ); (ii) in the integral metric L _p (p 1) for (₁, ..., _n ) = c ₁₁ + ... + c _{n n} ; (iii) in the integral metric L _{(2, ..., 2, 2r)} (r = 2, 3, ...) for (₁, ..., _n ) = ₁(₁) + ... + _{n – 1}( _{n – 1}) + c _{n n} .     

The existence of a solution of a linear integral equation

A theorem is proved concerning the existence of a solution of a linear integral equation in generalized Lebesgue space L_p (_n),p=(p₁, ..., p_n), where An is an n-dimensional parallelepiped.Translated from Matematicheskie Zametki, Vol. 8, No. 2, pp. 181–185, August, 1970.     

Ranges of certain system of functionals in classes of functions with a positive real part

We give the algebraic characteristics of the range of the system C_p, C_2p, ..., c_np f() ( fixed, 0<¦¦<1, n1, P=1, 2, ...) on certain subclasses C_m,p, of the class C of functions, regular in the circle ¦z¦<1 and satisfying in it the condition Re f(z)>0. As an application one finds the range of f() on the subclasses C _m,p, ⁽ⁿ⁾ of functions from C_m,p, with prescribed coefficients c_p c_2p, ..., c_np.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 100, pp. 17–25, 1980.     

Integral approximation sequences

Letn linear formsL _i onm variables be given, normalized so that all coefficients have absolute value at most unity. Letw ₁, ...,w _m be real numbers andx ₁, ...,x _m be integers. We sayE _i =L _i (w ₁, ...,w _m )-L _i (x ₁, ...,x _m ) is the error in approximating thew's by thex's with respect to formL _i It is shown that given anyw's there is an integral approximation ofx's so that the errorsE _i are small-roughly that simultaneously for alli.     

Hermite Interpolation and Sobolev Orthogonality

In this paper, we study orthogonal polynomials with respect to the bilinear form (f, g) _S = V(f) A V(g) ^T + <u, f ^(N) g ^(N)V(f) =(f(c ₀), f "(c ₀), ..., f ^{(n – 1)} ₀(c ₀), ..., f(c _p ), f "(c _p ), ..., f ^{(n – 1)} _p(c _p )) u is a regular linear functional on the linear space P of real polynomials, c ₀, c ₁, ..., c _p are distinct real numbers, n ₀, n ₁, ..., n _p are positive integer numbers, N=n ₀+n ₁+...+n _p , and A is a N × N real matrix with all its principal submatrices nonsingular. We establish relations with the theory of interpolation and approximation.     

Approximation by Fourier sums of classes of functions with several bounded derivatives

An ordered estimate is obtained for the approximation by Fourier sums, in the metric ofd=(d ₁ , ...,d _n ), 1<dj<,j=1, ...,_n of classes of periodic functions of several variables with zero means with respect to all their arguments, having m mixed derivatives of order a¹..., a^m., aⁱ rⁿ. which are bounded in the metrics ofp ⁱ =p ₁ ⁱ , ..., p _n ⁱ , i

j ⁱ <,i=i, ...,n, j=1, ...,n by the constants ₁, ., _m, respectively.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 197–212, February, 1978.     

(C,α) Summability of Walsh—Fourier Series

The one-dimensional dyadic martingale Hardy spaces H _p are introduced and it is proved that the maximal operator of the (C,) means of a Walsh—Fourier series is bounded from H _p to L _p (1/( + 1) < p < ) and is of weak type (L ₁,L ₁). As a consequence, we obtain the summability result due to Fine; more exactly, the (C,) means of the Walsh—Fourier series of a function f L ₁ converge a.e. to the function in question. Moreover, we prove that the (C,) means are uniformly bounded on H _p whenever 1/( + 1) < p < . We define the two-dimensional dyadic hybrid Hardy space H ₁ and verify that the maximal operator of the (C,,) means of a two-dimensional function is of weak type H ₁ ,L ₁). Consequence, the Walsh—Fourier series of every function f H ₁ is (C,,) summable to the function f.     

Mean-field critical behaviour for correlation length for percolation in high dimensions

Summary Extending the method of [27], we prove that the corrlation length of independent bond percolation models exhibits mean-field type critical behaviour (i.e. (p(p _c –p)^–1/2 aspp _c ) in two situations: i) for nearest-neighbour independent bond percolation models on ad-dimensional hypercubic lattice ^d , withd sufficiently large, and ii) for a class of spread-out independent bond percolation models, which are believed to belong to the same universality class as the nearest-neighbour model, in more than six dimensions. The proof is based on, and extends, a method developed in [27], where it was used to prove the triangle condition and hence mean-field behaviour of the critical exponents , , , and ₂ for the above two cases.     

The deviation of polygonal functions in the Lp metric

The precise value is given of the upper bound of the deviation in the L_p metric (1 < p < ) of a function f(x) in the class H , given by a convex modulus of continuity(t), from its polygonal approximation at the points x_k=k/n (k=0, 1 ...,n).Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 31–37, 1969.I would like to express my appreciation to A. A. Nudel'man for suggesting the problem considered here and for his help.     

Eventually areally meanp-valent functions

Theorems concerning areally meanp-valent functions are extended to eventually areally meanp-valent functions. In particular, suppose is eventually areally meanp-valent in the unit disc,b, c are positive integers,a≧max {p−1, 0}. If |a _n|≦Cn ^α for alln=bm+c,m=1, 2, …, then |a _n|≦C′n ^α for alln. This is a marked extension of results due to Goluzin and to Hayman.     

Certain Convolution Operators for Meromorphic Functions

Let (p N) be the class of functions analytic in 0 < |z| < 1. A convolution operator L_p(a, c) on p is introduced. This paper gives some sharp inequalities for f(z) satisfying Re{(1 – )z^pL_p(a, c) f(z) + z^pL_p(a + 1, c) f(z)} > , where 0, < 1, a > 0 and c 0, –1, –2,....AMS Subject Classification (1991) 30C45 30A10     

Relative MilnorK-theory

LetR be a commutative, semi-local ring,I ₁, ...,I _s ideals. In this paper, we define therelative Milnor K-groups of (R;I ₁, ...,I _s ),K _p ^M (R;I ₁, ...,I _s ), and show that these groups have many of the properties of the usual MilnorK-groups of a field. In particular, assuming a weak condition on the ideals, we show thatK _p ^M (R;I ₁, ...,I _s ) is isomorphic to the weightp portion of the relative QuillenK-groupK _p (R;I ₁, ...,I _s ), after inverting (p–1)!. We also define the relative group homology of GL _n (R;I ₁, ...,I _s ), and show thatK _p ^M (R;I ₁, ...,I _s ) is isomorphic toH _p (GL_p(R;I ₁, ...,I _s ))/Im(H _p (GL _p–1 (R;I ₁, ...,I _s ))). Finally, we consider a generalization to the relative setting of Kato's conjecture asserting that the Galois symbol gives an isomorphism fromK _p ^M (F)/l ^v to , and show that this relative version of Kato's conjecture implies the Quillen-Lichtenbaum conjectures asserting the Chern class:

On theL 1 exact penalty function with locally lipschitz functions

In this paper, we discuss the following inequality constrained optimization problem (P) minf(x) subject tog(x)0,g(x)=(g ₁(x), ...,g _r (x)) , wheref(x),g _j (x)(j=1, ...,r) are locally Lipschitz functions. TheL ₁ exact penalty function of the problem (P) is (PC) minf(x)+cp(x) subject tox R ⁿ , wherep(x)=max {0,g ₁(x), ...,g _r (x)},c>0. We will discuss the relationships between (P) and (PC). In particular, we will prove that under some (mild) conditions a local minimum of (PC) is also a local minimum of (P).     

On surface area measures of convex bodies

The set L _j of jth-order surface area measures of convex bodies in d-space is well known for j=d–1. A characterization of L _j was obtained by Firey and Berg. The determination of L _j, for j{2, ..., d–2}, is an open problem. Here we show some properties of L _j concerning convexity, closeness, and size. Especially we prove that the difference set L _j–L _j is dense (in the weak topology) in the set of signed Borel measures on the unit sphere which have barycentre 0.     

A generalization of the Bernoulli model occurring in order statistics. I.

Let X=(x₁, ..., x_n) and Y=(y₁, ..., y_m) be independent samples from populations G_x and G_y, x⁽¹⁾ ,... x⁽ⁿ⁾ be ordered statistics constructed from the sample X. A model of trials associated with the occurrence of dependent events A_k={y_k (x⁽ⁱ⁾}, x^(j), i < j, k=1, 2, ..., m, where x⁽ⁱ⁾, x^(j) are order statistics, is considered. This model is a generalization of the Bernoulli model. Distribution of frequencies of occurrences of events A_k and the limit theorems which describe asymptotic properties of these frequencies are investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 518–528, April, 1990.