Geometrical Characterization of Strict Suns in ℓ∞(n)

A subset M of a normed linear space X is called a strict sun if, for any x X\M, the set of its nearest points from M is nonempty and for any point y M which is nearest to x, the point y is a nearest point from M to any point of the ray {x + (1 - )y | > 0\}. We give an intrinsic geometrical characterization of strict suns in the space (n).     

The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations

Summary The Skorohod oblique reflection problem for (D, , w) (D a general domain in ^d , (x),xD, a convex cone of directions of reflection,w a function inD(⁺, ^d )) is considered. It is first proved, under a condition on (D, ), corresponding to (x) not being simultaneously too large and too much skewed with respect to D, that given a sequence {w ⁿ} of functions converging in the Skorohod topology tow, any sequence {(x ⁿ, ⁿ)} of solutions to the Skorohod problem for (D, , w ⁿ) is relatively compact and any of its limit points is a solution to the Skorohod problem for (D, , w). Next it is shown that if (D, ) satisfies the uniform exterior sphere condition and another requirement, then solutions to the Skorohod problem for (D, , w) exist for everywD(⁺, ^d ) with small enough jump size. The requirement is met in the case when D is piecewiseC _b ¹ , is generated by continuous vector fields on the faces ofD and (x) makes and angle (in a suitable sense) of less than /2 with the cone of inward normals atD, for everyxD. Existence of obliquely reflecting Brownian motion and of weak solutions to stochastic differential equations with oblique reflection boundary conditions is derived.     

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.     

On a characterization of polynomials by divided differences

Summary We consider the functional equationf[x ₁,x ₂,, x _n ] =h(x ₁ + +x _n ) (x ₁,,x _n K, x _j x _k forj k), (D) wheref[x ₁,x ₂,,x _n ] denotes the (n – 1)-st divided difference off and prove Theorem. Let n be an integer, n 2, let K be a field, char(K) 2, with # K 8(n – 2) + 2. Let, furthermore, f, h: K K be functions. Then we have that f, h fulfil (D) if, and only if, there are constants a_j K, 0 j n (a := a_n, b := a_{n – 1}) such thatf = ax ⁿ +bx ^{n – 1} + +a ₀ and h = ax + b.     

A Class of Iterative Algorithms and Solvability of Nonlinear Variational Inequalities Involving Multivalued Mappings

The solvability of the following class of nonlinear variational inequality (NVI) problems based on a class of iterative procedures, which possess an equivalence to a class of projection formulas, is presented.Determine an element x ^* K and u ^* T(x ^*) such that u ^*, x – x ^* 0 for all x K where T: K P(H) is a multivalued mapping from a real Hilbert space H into P(H), the power set of H, and K is a nonempty closed convex subset of H. The iterative procedure adopted here is represented by a nonlinear variational inequality: for arbitrarily chosen initial points x ⁰, y ⁰ K, u ⁰ T(y ⁰) and v ⁰ T(x ⁰), we have u ^k + x ^k+1 – y ^k , x – x ^k+1 0, x K, for u ^k T(y ^k ) and for k 0where v ^k + y ^k – x ^k , x – y ^k 0, x K and for v ^k T(x ^k ).     

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.     

Theorem on a convergence condition in the spaces

For a given -function (u), a condition on a -function (u) is found such that it is necessary and sufficient for the following to hold: if f_n(x) f(x) and f _n (x)M (n=1, 2, ...) where M>0 is an absolute constant, then f _n (x)–f(x)0(n). An analogous condition for convergence in Orlicz spaces is obtained as a corollary.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 615–626, May, 1977.The author thanks V. A. Skvortsov for his constant attention and guidance on this paper.     

The Busy Period of the M/GI/∞ Queue

An equation for the distribution Z() of the duration T of the busy period in a stationary M/GI/ service system is constructed from first principles. Two scenarios are examined, being distinguished by the half-plane Re()>₀ for some ₀0 in which the generic service time random variable S, always assumed to have a finite mean E(S), has an analytic Laplace–Stieltjes transform E(e^–S ). If ₀<0 then E(e^–T ) is analytic in a half-plane (₁,), where ₀₁<0 and ₁ is determined by the distribution of S; then for any 0<s<|₁|.When ₀=0, E(e^–T ) is analytic in (0,), and now more is known about T. Inequalities on the tail () are used to show that for any 1, E(T ) is finite if and only if E(S ) is finite. It follows that the point process consisting of the starting epochs of busy periods is long range dependent if and only if E(S ²)=, in which case it has Hurst index equal to [frac12](3–), where is the moment index of S.If also the tail (x)=Pr{Sx} of the service time distribution satisfies the subexponential density condition ₀ ^x (x–u) (u)du/ (x)2E(S) as x, then (x)/ (x)e^E(S), where is the arrival rate.     

Kleine Nullstellen homogener quadratischer Gleichungen

Let 0 be a real quadratic form inn variables, which takes on integral values on ⁿ . Denote by the largest coefficient of in absolute value. Suppose vanishes on ad-dimensional rational subspace. It is shown that has a zero (x ₁,...,x _n ⁿ \{(0,...,0)} with max |x _i ^(n-d/2d).     

Sur les équations fonctionnelles auxq-différences

Summary In this paper, we study the convergence of formal power series solutions of functional equations of the formP _k(x)(^[k](x))=(x), where ^[k] (x) denotes thek-th iterate of the function.We obtain results similar to the results of Malgrange and Ramis for formal solutions of differential equations: if(0) = 0, and(0) =q is a nonzero complex number with absolute value less than one then, if(x)=a(n)x ⁿ is a divergent solution, there exists a positive real numbers such that the power seriesa(n)q ^sn(n+1)2 x ⁿ has a finite and nonzero radius of convergence.

     

Une caracterisation du type de la loi de Cauchy-conforme sur ℝ n

Résumé La loi de Cauchy-conforme est la mesure de probabilité sur ⁿ de densitéC/(1+X²)ⁿ. Le type d'une mesure sur ⁿ étant l'ensemble des mesures images de par les similitudes-translations de ⁿ et étant une mesure de probabilité sans atome, on démontre que le type de est invariant par les inversions de ⁿ si et seulement si est du type de la loi de Cauchy-conforme.

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The conformal Cauchy law is the probability on ⁿ with densityC/(1+X²)ⁿ. It is shown that for a non-atomic measure on ⁿ the following is true: its type is invariant under inversions of ⁿ if and only if it is the type of a conformal Cauchy law. (The type of a measure is defined as the set of its images under similarities and translations.)

     

Diagonals of positive semigroups

LetE be a Dedekind complete complex Banach lattice and letD denote the diagonal projection from the spaceL _r (E) onto the centerZ(E) ofE. Let {T(t)} _t0 be a positive strongly continuous semigroup of linear operators with generatorA. The first main result is that if the spectral bounds(A) equals to zero, then the functionD(T(t)) is a center valuedp-function. The second main result is that if for >0 the diagonalD(R(, A)) of the resolvent operatorR(, A) is strictly positive, then (D(R(, A))) ^–1 is a center valued Bernstein function. As an application of these results it follows that the order limit lim₀D(R(,A)) exists inZ(E) and equals the order limit lim _m D((R(, A)) ^m ) for any >0.     

Fundamental equation of information revisited

Summary This paper presents a new, shorter and more direct proof of the following result of J. Aczél and C. T. Ng: IfM: J R (J =]0, 1[ ^k ) is both multiplicative and additive, then the general solution: J R of(x) + M(1 – x)(y/1 – x) = (y) + M(1 – y)(x/1 – y) (x, y, x + y J) is given by(x) = ifM = 0,(x) = M(x)[L(x) + ] + M(1 – x)L(1 – x) ifM 0,where is an arbitrary constant andL: J R is an arbitrary solution of the logarithmic functional equationL(xy) = L(x) + L(y) (x, y J). Also, some extensions of this result to fields more general than the reals are given.     

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {X _k , 1 k n} be n independent and real-valued random variables with common subexponential distribution function, and let {k, 1 k n} be other n random variables independent of {X _k , 1 k n} and satisfying a _k b for some 0 < a b < for all 1 k n. This paper proves that the asymptotic relations P (max_{1 m n k=1} ^m _k X _k > x) P (sum _k=1 ⁿ _k X _k > x) sum _k=1 ⁿ P ( _k X _k > x) hold as x . In doing so, no any assumption is made on the dependence structure of the sequence { _k , 1 k n}. An application to ruin theory is proposed.     

Weak Sequential Completeness of β-duals of Double Sequence Spaces

We consider dual pairs E,E ⁽⁾ of double sequence spaces E and E ⁽⁾, where E ⁽⁾ is the -dual space of E with respect to the -convergence of double sequences for = p (Pringsheim convergence), bp (bounded p-convergence) and r (regular convergence). Motivated by Boos, Fleming and Leiger [3], we introduce two oscillating properties (signed P_OSCP(k), k {1,2}) for a double sequence space E such that the signed P_OSCP(1) guarantees the (E ^(p), E)-sequential completeness of E ^(p), whereas the signed P_OSCP(2) implies the equalities E ^(r) = E ^(bp) = E ^(p) and the (E ⁽⁾, E)-sequentialcompleteness of E ⁽⁾ for = bp and r.     

On harmonic renewal measures

Summary Let be a probability and the corresponding harmonic renewal measure. Complementing earlier results where is concentrated on a halfline we investigate the behaviour ofv _h ([x, x + 1]) and the harmonic renewal functionG(x) =v _h((–,x])asx ifm ₁=x(dx)>0. We also consider the casem ₁=0.     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

Singular parts of contractive analytic operator functions

One considers the class _G of holomorphic functions in a domain G, whose values are contractions in a separable Hilbert space. It is proved that if T(·) _G , T(z₀) is a weak contraction, its singular part T^s(z₀) is complete, and the increments T(z)–T(z₀) are not too large (for example, finite-dimensional), then the operator T^s(z₀) is complete for almost all zG. If, however, T(z₀) is, in addition, completely nonunitary and satisfies definite smoothness conditions, then in the nontrivial case the spectrum [z] of the contraction T^s(z) (zG) is a thin set: The proof of the mentioned results is based on the investigation of the formulas obtained in the paper, connecting the characteristic functions of the contractions T(z) for different values of zG.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 30–44, 1987.     

An implicit function theorem

Suppose thatF:DR ⁿ×R^mRⁿ, withF(x ⁰,y ⁰)=0. The classical implicit function theorem requires thatF is differentiable with respect tox and moreover that ₁ F(x ⁰,y ⁰) is nonsingular. We strengthen this theorem by removing the nonsingularity and differentiability requirements and by replacing them with a one-to-one condition onF as a function ofx.     

Ouverts d"harmonicité pour les fonctions séparément harmoniques

Let D ^N , G ^M be two open sets, E D and F G two compact sets which satisfy the condition (H) (that is a harmonic condition similar to Leja"s condition). We find an open set ^N+M such that each separately harmonic function f : X : = (D× F) (E × G) (i.e.: for all x in E, f(x,.) is harmonic on G; for all y in F, f(., y) is harmonic on D) extends to a harmonic function on .