Superposition de déformations cylindriques d'axes poissonniens dans le plan

Summary In view of approximating the fractional Brownian process in the plane B. Mandelbrot examined the superimposition of rectilinear faults, centered on the axis of a Poisson process with arbitrary large rate, the profiles of which being of the type Q · sgn(x)¦x¦, ¦¦Q a real valued random variable. In the following, we derive from general hypotheses a formula which caracterises any random profile leading to a Gaussian process of given type and thus providing explicit examples of profiles, thinkably less contrived than the former; some results on the quality of convergence are given.     

The capacity of a channel with arbitrarily varying channel probability functions and binary output alphabet

Summary Let X={1,..., a} be the input alphabet and Y={1,2} be the output alphabet. Let X ^t =X and Y ^t =Y for t=1,2,..., X _n = X ^t and Y _n = Y ^t . Let S be any set, C=={w(·¦·¦)s)¦sS} be a set of (a×2) stochastic matrices w(··¦s), and S ^t=S, t=1,..., n. For every s _n =(s ¹,...,s ⁿ ) S ^t define P(·¦·¦s _n)= w(y ^t ¦x ^t ¦s ^t ) for every x _n=x ¹, , x ⁿX _n and every y _n=(y ¹, , y ⁿ)Y _n. Consider the channel C _n ={P(·¦·¦)s _n )¦s _n S _n } with matrices (·¦·¦s), varying arbitrarily from letter to letter. The authors determine the capacity of this channel when a) neither sender nor receiver knows s _n, b) the sender knows s _n, but the receiver does not, and c) the receiver knows s _n, but the sender does not.Research of both authors supported by the U.S. Air Force under Grant AF-AFOSR-68-1472 to Cornell University.     

Asymptotics of Spectral Data of a Harmonic Oscillator Perturbed by a Potential

Asymptotics of spectral data of a perturbed harmonic oscillator −y″ + x²y + q(x)y are obtained for potentials q(x) such that q′, xq ∈ L²(ℝ). These results are important in the solution of the corresponding inverse spectral problem. Bibliography: 7 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 223–271.     

Automorphisms and isotone self-maps of ordered sets with top and bottom

For an ordered setP letP ^P denote the set of all isotone self-maps on P, that is, all mapsf fromP toP such thatxy impliesf(x)f(y), and let Aut (P) the set of all automorphisms onP, that is, all bijective isotone self-maps inP ^P . We establish an inequality relating ¦P ^P ¦ and ¦Aut(P)¦ in terms of the irreducibles ofP. As a straightforward corollary, we show that Rival and Rutkowski's automorphism conjecture is true for lattices. It is also true for ordered sets with top and bottom whose covering graphs are planar.Supported in part by NSERC (Grant no. A2507).Supported under an NSERC International Research Fellowship.     

Optimal error estimates of a locally one-dimensional method for the multidimensional heat equation

For the multidimensional heat equation in a parallelepiped, optimal error estimates inL ₂(Q) are derived. The error is of the order of +¦h¦² for any right-hand sidef L ₂(Q) and any initial function ; for appropriate classes of less regularf andu ₀, the error is of the order of ((+¦h¦² ), 1/2<1.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 185–197, August, 1996.     

Finite extinction time for some perturbed Hamilton-Jacobi equations

In this paper we study initial value problems likeu _t–R¦u¦^m+u^q=0 in ⁿ× ₊, u(·,0⁺)=u_o(·) in ^N, whereR > 0, 0 <q < 1,m 1, andu _o is a positive uniformly continuous function verifying –R¦u _o¦^m+u ₀ ^q 0 in ^N . We show the existence of the minimum nonnegative continuous viscosity solutionu, as well as the existence of the function t(·) defined byu(x, t) > 0 if 0<t<t (x) andu(x, t)=0 ift t (x). Regularity, extinction rate, and asymptotic behavior of t(x) are also studied. Moreover, form=1 we obtain the representation formulau(x, t)=max{([(u _o(x – t))^1–q –(1–q)t]₊)^1/(1–q): ¦¦R}, (x, t) ₊ ^N+1 .Partially supported by the DGICYT No. 86/0405 project.     

Factoring by Cyclic and Simulated Subsets

Let G be a finite abelian group whose p-component is an elementary p-group for each prime divisor p of ¦G¦. Assume that B, A₁,..., A_n are subsets of G such that each ¦A_i¦ is a prime, ¦B¦ is a product of two primes, and each A_i is either cyclic or simulated. If the product BA₁ ... A_n is direct and is equal to G, then at least one of the factors B, A₁, ..., A_n is periodic.     

Existence and non-existence of periodic solutions of generalized Lineard equations with damping of no lower bound

This paper establishes criteria for the existence and non-existence of nonzero periodic solutions of the generalized Liénard equationx +f(x,x)x +g(x)=0. The main goal is to study to what extent the dampingf can be small so as to guarantee the existence of nonzero periodic solutions of such a system. With some standard additional assumptions we prove that if for a small ¦x¦, ^± ¦f(x,y)¦^–1 dy=±, then the system has at least one nonzero periodic solution, otherwise, the system has no nonzero periodic solution. Many classical and well-known results can be proved as corollaries to ours.Supported by the National Natural Science Foundation of China.     

Superstability of the generalized orthogonality equation on restricted domains

Chmielinski has proved in the paper [4] the superstability of the generalized orthogonality equation |〈f(x), f(y)〉| = |〈x,y〉|. In this paper, we will extend the result of Chmielinski by proving a theorem: LetD _n be a suitable subset of ℝⁿ. If a function f:D _n → ℝⁿ satisfies the inequality ∥〈f(x), f(y)〉| |〈x,y〉∥ ≤ φ(x,y) for an appropriate control function φ(x, y) and for allx, y ∈ D _n, thenf satisfies the generalized orthogonality equation for anyx, y ∈ D _n.     

Roots of the equationf (z)=αf (α) for the class of typically-real functions

Let T_r be the class of functionsf (z)=z+c₂z²+..., regular in the disk ¦z¦ <1, real on the diameter-1f (z) · Im z>0 in the remainder of the disk ¦z¦ <1. Let z _f be the solution off (z)= f (a) on Tr, where is any fixed complex number 0, 1, is any fixed real number, ¦¦< 1. We determine the region of values of the functional z_f on the class T_r. Variation formulas for Stieltjes integrals due to G.M. Goluzin are used.Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 41–52, July, 1971.     

Linear programming by minimizing distances

To solve the linear program (LP): minimizec ^T l subject toA l+b0, for ann×d-matrixA, ann-vectorb and ad-vectorc, the positive orthantS and the planeE(t) are defined by S={(x₁,x)ⁿ⁺¹ ¦(x₁,x)0}, E(t)={(x₁,x)ⁿ⁺¹¦x₁=–c ^c l+t, x=Al+b}. First a geometric algorithm is given to determine d(E(t),S) for fixedt, where d(·,·) denotes euclidean distance. This algorithm is used to construct a second algorithm to find the minimalt with E(t) S , and thus solve LP. It is shown that all algorithms are finite.     

Error-free computation with rational numbers

A method is described for doing error-free computation when the operands are rational numbers. A rational operanda/b is mapped onto the integer ¦a·b ^–1¦ _p and the arithmetic is performed inGF(p). A method is given for taking an integer result and finding its rational equivalent (the one which corresponds to the correct rational result).     

Some relations among group-theoretic invariants of finite p-groups

In this paper there is given a formula for the number of solutions of the equation in an arbitrary finite p-group G (of exponent p ^l , 1 n 1) and a formula for the number of cyclic subgroups of G of any order. A connection is established among ¦G¦, p ^l , and the ranks of those subgroups of G of order greater than p^l; if G is regular, there are analogous relations among the orders of the characteristic subgroups , and the ranks of the subgroups of G of order greater than pⁿ. These results are precise; some of them strengthen the well-known classical theorems of Frobenius and Miller for p-groups.Translated from Matematicheskie Zametki, Vol. 17, No. 4, pp. 571–578, April, 1975.     

Intrinsic metrics preserving maps on Abelian lattice-ordered groups

Swamy studied the natural metric ¦x–y¦ on Abelian lattice-ordered groupsG, and he proved that the stable isometries which preserve this metric have to be automorphisms ofG. Holland proved that the only intrinsic metrics on lattice-ordered groups, i.e., invariant and symmetric metrics, are the multiples n¦x –y¦ for some integern. We show that iff is an arbitrary surjection from an Abelian lattice-ordered groupG ₁ onto an Archimedean Abelian lattice-ordered groupG ₂ such that f(0)]0 and, for some non-zero intrinsic metricsD andd, D(f(x),f(y)) depends functionally on d(x,y), thenf is a homomorphism of G₁ onto G₂.Presented by R. S. Pierce.     

Return probabilities for random walk on a half-line

A random walk with reflecting zone on the nonnegative integers is a Markov chain whose transition probabilitiesq(x, y) are those of a random walk (i.e.,q(x, y)=p(y–x)) outside a finite set {0, 1, 2,...,K}, and such that the distributionq(x,·) stochastically dominatesp(·–x) for everyx{0, 1, 2,..., K}. Under mild hypotheses, it is proved that when xp _x>0, the transition probabilities satisfyq ⁿ(x, y)C_xyR^–nn^–3/2 asn, and when xp _x=0,q ⁿ(x, y)C_xyn^–1/2.Supported by National Science Foundation Grant DMS-9307855.     

Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type

LetX be a Banach space,K a nonempty, bounded, closed and convex subset ofX, and supposeT:K→K satisfies: for eachx∈K, lim sup _i→∞{sup _y∈K ‖t ⁱx−Tⁱy∼−‖x−y‖}≦0. IfT ^N is continuous for some positive integerN, and if either (a)X is uniformly convex, or (b)K is compact, thenT has a fixed point inK. The former generalizes a theorem of Goebel and Kirk for asymptotically nonexpansive mappings. These are mappingsT:K→K satisfying, fori sufficiently large, ‖Tⁱx−Tⁱy‖≦k _i‖x−y∼,x,y∈K, wherek _i→1 asi→∞. The precise assumption in (a) is somewhat weaker than uniform convexity, requiring only that Goebel’s characteristic of convexity, ɛ₀ (X), be less than one. Research supported by National Science Foundation Grant GP 18045.     

Parallel Multiplication in GF(2k) using Polynomial Residue Arithmetic

We present a novel method of parallelization of the multiplication operation in GF(2^k) for an arbitrary value of k and arbitrary irreducible polynomial n(x) generating the field. The parallel algorithm is based on polynomial residue arithmetic, and requires that we find L pairwise relatively prime modulim _i(x) such that the degree of the product polynomialM(x)=m ₁(x)m ₂(x)··· m_L(x) is at least 2k. The parallel algorithm receives the residue representations of the input operands (elements of the field) and produces the result in its residue form, however, it is guaranteed that the degree of this polynomial is less than k and it is properly reduced by the generating polynomial n(x), i.e., it is an element of the field. In order to perform the reductions, we also describe a new table lookup based polynomial reduction method.     

Aleksandrov reflection and nonlinear evolution equations,I: The n-sphere and n-ball

We consider the (degenerate) parabolic equationu _t =G(u + ug, t) on then-sphereS ⁿ . This corresponds to the evolution of a hypersurface in Euclidean space by a general function of the principal curvatures, whereu is the support function. Using a version of the Aleksandrov reflection method, we prove the uniform gradient estimate ¦u(·,t)¦ <C, whereC depends on the initial conditionu(·, 0) but not ont, nor on the nonlinear functionG. We also prove analogous results for the equationu _t =G(u +cu, ¦x¦,t) on then-ballB ⁿ , wherec ₂(B ⁿ ).     

类对称函数的Schur凸性和不等式

对x = (x₁, x₂,···, x_n) ∈ (0,1)ⁿ 和 r ∈ {1, 2,···, n} 定义对称函数 F_n(x, r) = F_n(x₁, x₂,···, x_n; r) =∏_{1≤i1∑j=1^r(1+xi3/1- xi3)^1/r, 其中i1, i2, ···, ir 是整数. 该文证明了Fn(x, r) 是(0,1)ⁿ 上的Schur凸、Schur乘性凸和Schur调和凸函数. 作为应用,利用控制理论建立了若干不等式.}     

Radii of convexity and close-to-convexity of certain integral representations

Strict upper bounds are determined for ¦s(z)¦, ¦Re s(z)¦, and ¦Im s(z) ¦ in the class of functions s(z)=a _nzⁿ+a_n+1zⁿ⁺¹+... (n¹) regular in ¦z¦<1 and satisfying the condition ¦u (₁) –u (₂) ¦K¦ ₁- ₂¦, where U()=Re s (eⁱ ), K>0, and ₁ and ₂ are arbitrary real numbers. These bounds are used in the determination of radii of convexity and close-to-convexity of certain integral representations.Translated from Matematicheskie Zametki, Vol. 7, No. 5, pp. 581–592, May, 1970.The author wishes to thank L. A. Aksent'ev for his guidance in this work.