Finite-dimensional discrete linear stochastic accelerated-time systems and their application to quadratic stochastic dynamical systems

The limiting behavior of the trajectories {x ⁽ⁿ⁾ } of linear discrete stochastic systems of the form (K, P ^an+b ) _nN , whereK is the standard simplex in ^N ,P: ^{N N} is a linear operator,PK K,a ft,b ,a+b>0, is described. An application to a class of quadratic stochastic dynamical systems is considered.Translated fromMatematicheskie Zametki, Vol. 59, No. 5, pp. 709–718, May, 1996.     

Approximate first-order and second-order directional derivatives of a marginal function in convex optimization

Given a convex functionf: ^p × ^q (–, +], the marginal function is defined on ^p by (x)=inf{f(x, y)|y ^q }. Our purpose in this paper is to express the approximate first-order and second-order directional derivatives of atx ₀ in terms of those off at (x ₀,y ₀), wherey ₀ is any element for which (x ₀)=f(x ₀,y ₀).The author is indebted to one referee for pointing out an inaccuracy in an earlier version of Theorem 4.1.     

Two questions in diophantine approximation

The first question is about a possible variation on Dirichlet's approximation theorem for linear forms x+y+z, wherex,y are restricted topositive integers. The second question, which turns out to be related to the first, is about approximation to elements in a power series fieldk((t ^–1)) by solutions of first order linear differential equationsx+y+z=0, wherex,y,z are polynomials int. Dedicated to Prof. Dr. E. Hlawka on the occasion of his 60th birthday     

Symmetrische Permutationsmengen

A permutation set (M, I) consisting of a setM and a set of permutations ofM, is calledsymmetric, if for any two permutations, the existence of anx M with (x) (x) and ^–1 (x) = ^–1 (x) implies ^–1 = ^–1 , andsharply 3-transitive, if for any two triples (x ₁,x ₂,x ₃), (y ₁,y ₂,y ₃) M ³ with|{x ₁,x ₂,x ₃ }| = |{y ₁,y ₂,y ₃ }| = 3 there is exactly one permutation with(x ₁) =y ₁,(x ₂) =y ₂,(x ₃) =y ₃. The following theorem will be proved.THEOREM.Let (M, ) be a sharply 3-transitive symmetric permutation set with |M|3, such that contains the identity. Then is a group and there is a commutative field K such that and the projective linear group PGL(2, K) are isomorphic.     

Some Landau's type inequalities for infinitesimal generators

Summary Lett T(t) be a strongly continuous contraction semigroup on a complex Banach space and letA be its infinitesimal generator. We prove that, forx D(A ³), the following inequalities hold true: Ax³ 243/8 x²A ³ x, A ² x 24 xA ³ x². Ift T(t) is a contraction group (resp. cosine function) we get the analogous but better inequalities with constants 9/8 and 3 (resp. 81/40 and 72/25) instead of 243/8 and 24. We consider also uniformly bounded semigroups, groups and cosine functions.     

Ein neues Surjektivitätskriterium im Hilbertraum

LetA, B andC be linearm-accretive operators in a Hilbert space. Suppose further thatC is bounded, thatb:=inf {Re (C y, y)| y=1}>0, thatA ^–1 exists as a bounded operator and that Re (B ^* x, A ^–1 x)+a x²0 holds for allxD (B ^*) and a constanta with 0a<b. ThenCA+B is surjective, (CA+B)^–1 exists and C Ax+Bx (b–a) A x holds for allxD (A) D (B). This criterion can be applied to evolution equations of the formdu/dt+C(t)A(t)u=f(t) whereB:=d/dt.     

A functional inequality for real-analytic functions

Summary Forf ( C ⁿ() and 0 t x letJ _n (f, t, x) = (–1)ⁿ f(–x)f ⁽ⁿ⁾(t) +f(x)f ⁽ⁿ⁾ (–t). We prove that the only real-analytic functions satisfyingJ _n (f, t, x) 0 for alln = 0, 1, 2, are the exponential functionsf(x) = c e ^x,c, . Further we present a nontrivial class of real-analytic functions satisfying the inequalitiesJ ₀ (f, x, x) 0 and ₀ ^x (x – t)^{n – 1}J_n(f, t, x)dt 0 (n 1).     

Weak solutions of the Stokes problem in weighted Sobolev spaces

For n2 we consider the Stokes problem in ⁿ, -u + p=f, -divu=g, in weighted Soboiev spaces H ₆ ^m,r , where the weights are proportional to (1+|x|). We prove the existence of weak solutions for any K, whereK is a discrete set of critical values. Furthermore, we characterize the solutions of the homogeneous problem.This research was supported by the DFG research group Equations of Hydrodynamics, Universities of Bayreuth and Paderborn.     

Orderh 2 method for a singular two-point boundary value problem

A method is described based on auniform mesh for the singular two-point boundary value problem:y+(/x)y+f(x, y)=0, 0<x1,y(0)=0,y(1)=A, and it is shown to be orderh ² convergent forall 1.     

On the Uniqueness ofL 2-solutions in half-space of certain differential equations

Square integrable solutions to the equation{– ²/y² + P(D_x)+b(y)–}u(x, y) = f(x, y) are considered in the half-spacey>0, x ⁿ , whereP(D _x) is a constant coefficient operator. Under suitable conditions on lim_y0u(x, y), b(y), f(x, y) and , it is shown that suppu = suppf. This generalizes a result due to Walter Littman.Research partially supported by USNSF Grant 79-02538-A02.     

Numerical stability of the Chebyshev method for the solution of large linear systems

Summary This paper contains the rounding error analysis for the Chebyshev method for the solution of large linear systemsAx+g=0 whereA=A ^* is positive definite. We prove that the Chebyshev method in floating point arithmetic is numerically stable, which means that the computed sequence {x _k} approximates the solution such that x _k – is of order AA ^–1 where is the relative computer precision.We also point out that in general the Chebyshev method is not well-behaved, which means that the computed residualsr _k=Ax _k+g are of order A²A ^–1.This work was supported in part by the Office of Naval Research under Contract N0014-67-0314-0010, NR 044-422 and by the National Science Foundation under Grant GJ32111     

Primes in short intervals

Let (x) stand for the number of primes not exceedingx. In the present work it is shown that if 23/421,yx andx>x() then (x)–(x–y)>y/(100 logx). This implies for the difference between consecutive primes the inequalityp _n+1–p _n p _n ^23/42 .     

D'Alembert's functional equation on restricted domains

Summary In the class of functionalsf:X , whereX is an inner product space with dimX 3, we study the D'Alembert functional equationf(x + y) + f(x – y) = 2f(x)f(y) (1) on the restricted domainsX ₁ = {(x, y) X ²/x, y = 0} andX ₂ = {(x, y) X ²/x = y}. In this paper we prove that the equation (1) restricted toX ₁ is not equivalent to (1) on the whole spaceX. We also succeed in characterizing all common solutions if we add the conditionf(2x) = 2f²(x) – 1. Using this result, we prove the equivalence between (1) restricted toX ₂ and (1) on the whole spaceX. This research follows similar previous studies concerning the additive, exponential and quadratic functional equations.     

On the semimartingale representation of reflecting Brownian motion in a cusp

Summary LetC be the symmetric cusp {(x, y)²:–x yx ,x0} where >1. In this paper we decide whether or not reflecting Brownian motion inC has a semimartingale representation. Here the reflecting Brownian motion has directions of reflection that make constant angles with the unit inward normals to the boundary. Our results carry through for a wide class of asymmetric cusps too.     

Algorithmes de calcul du maximum des formes quadratiques sur la boule unité de la norme du maximum

Résumé Certaines méthodes directes et indirectes pour le calcul de Max {x ^t Ax, (x)1} sont étudiées.Les méthodes directes sont basées sur les propriétés particulières des normes ₁, ₂ et . Ces méthodes sont très simples mais ne s'appliquent qu'à certaines familles de matrices.La méthode indirecte est la méthode autoduale introduite dans [25, 26] avec = ₁. Dans ce cas, le choix du vecteur initial pour qu'il y ait convergence vers une solution optimale est largement discuté.

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Some methods for computing the maximum of quadratic from on the unit ball of the maximum norm

Summary Some direct and indirect methods are studied for computing Max {x ^t Ax, (x)1} whereA is symmetric definite positive.Direct methods are constructed using particular properties of ₁, ₂, norms. These methods are very simple, but uniquely suitable to certains families of matrices.The indirect method is the autodual method, introduced in [25, 26, 29] with = ₁. In this case the problem of choosing an initial vector so that convergence of the iterative sequence occurs to an optimal solution is largely discussed.

     

Non polynomial splines and weakly singular two-point boundary value problems

In the present paper we shall consider an application of simple non-polynomial splines to a numerical solution of a weakly singular two-point boundary value problem:x ^– (x y)=f(x,y), (0<x1) subject toy(0)=0,y(1)=c ₁(1) ory(0)=c ₂,y(1)=c ₃(0<<1). Our collocation method gives a continuously differentiable approximation and isO(h ²)-convergent.     

Monotonicity properties of zeros of generalized Airy functions

Summary We show, among other things, that the positive zeros of a solution ofy +x y=0,y(0)=0 decrease to 1 as increases, 0.

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Sommario Si dimostra, tra l'altro, che gli zeri positivi d'una soiuzione diy +x y=0,y(0)=0 decrescono al limite 1, quando cresce, 0.

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To the memory of Milo Háik

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This research was supported by grants from the Natural Sciences and Engineering Research Council (Canada) and Consiglio Nazionale delle Ricerche (Italy). Some of the work was done while the second-named author was visiting the Department of Mathematics, University of Torino.     

Présentations des groupes de Fischer (II)

In this paper we give Coxeter presentation (X, ) for the three Fischer groupsG=Fi₂₂, Fi₂₃, Fi₂₄; we apply methods exposed in the first part. Each of these groups is generated by a class of 3-transpositions (named here a Fischer class) in which elements ofX are chosen. A subset of is the set of all the relations (xy) ^m(x,y)=1, wherex andy are inX and wherem(x,y) means the order ofxy inG. We obtainG as a specified quotient of the Coxeter group (X, ) with the appropriate diagram .     

Representations of integers as sums of powers with increasing exponents

Let(n) be the least integer such thatn may be represented in the formn=x ₁ ² +x ₂ ³ +...+x _(n) ⁽ⁿ⁾⁺¹ wherex ₁,x ₂, ...,x _(n) are natural numbers. We computed(n) forn 250 000 and found that(n) 5 for all thesen exceptn=56, 160 for which(n)=6. Also(n) 4 for 41542<n<=250 000.     

A Local Approach to 1-Homogeneous Graphs

Let bea distance-regular graph with diameter d. For vertices x and y of at distancei, 1 i d, we define the setsC _i(x,y) = i–1(x) (y), A _i (x,y) = _i (x) (y) and B _i (x,y) = _i+1(x) (y).Then we say has the CAB_j property,if the partition CAB _i (x,y) = {C _i (x,y),A _i (x,y),B _i (x,y)}of the local graph of y is equitable for each pairof vertices x and y of at distance i j. We show that in with the CAB_j property then the parameters ofthe equitable partitions CAB i(x,y) do not dependon the choice of vertices x and y atdistance i for all i j. The graph has the CAB property if it has the CAB _d property. We show the equivalence of the CAB property and the1-homogeneous property in a distance-regular graph with a ₁ 0. Finally, we classify the 1-homogeneous Terwilligergraphs with c ₂ 2.