Iterative Pexider equation modulo a subset

Summary LetX, Y, Z be arbitrary nonempty sets,E be a nonempty subset ofZ ^z andK be a groupoid. Assume that {F _t} _{t K} Z ^X, {G _t} _{t K} Y ^X, {H _t} _{t K} Z ^Y are families of functions satisfying the functional equationF _st = k_(s,t) H_s G_t for (s, t) D(K), whereD(K) stands for the domain of the binary operation on the groupoidK andk _(s,t) E for (s, t) D(K). Conditions are established under which the equation can be reduced to the corresponding Cauchy equation. This paper generalizes some results from [4] and [1].     

On a generalization of the Cauchy functional equation

Summary While looking for solutions of some functional equations and systems of functional equations introduced by S. Midura and their generalizations, we came across the problem of solving the equationg(ax + by) = Ag(x) + Bg(y) + L(x, y) (1) in the class of functions mapping a non-empty subsetP of a linear spaceX over a commutative fieldK, satisfying the conditionaP + bP P, into a linear spaceY over a commutative fieldF, whereL: X × X Y is biadditive,a, b K\{0}, andA, B F\{0}. Theorem.Suppose that K is either R or C, F is of characteristic zero, there exist A ₁,A ₂,B ₁,B ₂, F\ {0}with L(ax, y) = A ₁ L(x, y), L(x, ay) = A ₂ L(x, y), L(bx, y) = B ₁ L(x, y), and L(x, by) = B ₂ L(x, y) for x, y X, and P has a non-empty convex and algebraically open subset. Then the functional equation (1)has a solution in the class of functions g: P Y iff the following two conditions hold: L(x, y) = L(y, x) for x, y X, (2)if L 0, then A ₁ =A ₂,B ₁ =B ₂,A = A ₁ ² ,and B = B ₁ ² . (3) Furthermore, if conditions (2)and (3)are valid, then a function g: P Y satisfies the equation (1)iff there exist a y ₀ Y and an additive function h: X Y such that if A + B 1, then y ₀ = 0;h(ax) = Ah(x), h(bx) =Bh(x) for x X; g(x) = h(x) + y ₀ + 1/2A ₁ ^-1 B ₁ ^-1 L(x, x)for x P.     

A structure theorem for sum form functional equations

Summary It is proved that, iff _ij:]0, 1[ C (i = 1, ,k;j = 1, ,l) are measurable, satisfy the equation (1) (with some functionsg _it, h_jt:]0, 1[ C), then eachf _ij is in a linear space (called Euler space) spanned by the functionsx x ^j(logx) ^k (x ]0, 1[;j = 1, ,M;k = 0, ,m _j – 1), where ₁, , _M are distinct complex numbers andm ₁, , m_M natural numbers. The dimension of this linear space is bounded by a linear function ofN.     

A conditional dilatation equation

Summary The following theorem holds true. Theorem. Let X be a normed real vector space of dimension 3 and let k > 0 be a fixed real number. Suppose that f: X X and g: X × X are functions satisfying x – y = k f(x) – f(y) = g(x, y)(x – y) for all x, y X. Then there exist elements and t X such that f(x) = x + t for all x X and such that g(x, y) = for all x, y X with x – y = k.     

On weak convergence of functionals of stochastic processes with continuously differentiable paths

We obtain a criterion for weak convergence of a sequence of stochastic processes _n(t), t [0, 1],n N, _n(t) R ^m in the spaceC _m ^k [0, 1] of continuously differentiable functions. We consider several examples of weakly convergent sequences of stochastic processes inC _m ^k [0, 1] and several integer functionals defined on these random variables.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 85–90, 1987.     

Moser-type inequalities in Lorentz spaces

Moser-type estimates for functions whose gradient is in the Lorentz space L(n, q), 1q, are given. Similar results are obtained for solutions uH _inf0 ^sup1 of Au=(f _i ) _{x i} , where A is a linear elliptic second order differential operator and |f|L(n, q), 2q.Work partially supported by MURST (40%).     

Convergence des solutions formelles de certaines équations fonctionnelles

Résumé Soitq un nombre algébrique de module 1, qui ne soit pas une racine de l'unité, etP [X, Y ₀,Y ₁] un polynôme non nul. Dans cet article, nous montrons que toute solution de l'équation fonctionnelleP(z, (z), (qz))=0, qui est une série formelle (z) dansQ[[z]], a un rayon de convergence non nul.

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Summary Letq Q be an algebraic number of modulus one that is not a root of unity. LetP Q[X, Y ₀,Y ₁] be a non zero polynomial. In this paper, we show that every formal power series,(z) Q[[z]], solution of the functional equationP(z), (z), (qz)) = 0 has a non zero radius of convergence.

     

On functions having Cauchy differences in some prescribed sets

Summary AssumeE is a real topological vector space,F is a real Banach space,K is a discrete subgroup ofF andC is a symmetric, convex and compact subset ofF such thatK (6C) = {0}. If a functionh:E F is continuous at at least one point andh(x + y) – h(x) – h(y) K + C for allx, y E, then there exists a continuous linear functiona:E F such thath(x) – a(x) K + C for everyx E.     

Abschnittspositive Matrizen und Positivitätsfaktoren in der Limitierungstheorie

Let A and B be normal matrices. In :={x=(x_k) ¦ x_k} we define the order relation _A by x_A0:<=> _k=0 ⁿ a_nkx_k0 (n ). Let T be a row-finite matrix. A is called T-section-positive, if _kt_mkx_ke^(k) _A0 (m ) for x_A0 (see [5]). We study the relation between T-sectional positivity and T-sectional boundedness. An (A,B)-summability factor sequence =(_k) is called positive, if (_kx_k)_B0 for each xc_A with x_A0. For B-section-positive matrices A we give a functional analytic characterization of positive (A,B)-summability factor sequences.

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Die Arbeit entstand während eines vom DAAD unterstützten Forschungsaufenthalts an der Fernuniversität-Gesamthochschule Hagen     

Applications of the zero–one law to problems connected with Cauchy's functional equation

Summary Marek Kuczma's book, entitled An Introduction To The Theory Of Functional Equations And Inequalities, mentions a certain setV ₀ in several places and presents references as to where this set is discussed in the literature. The main result of this paper is a proof of the fact that the setA _M (V ₀)={xV ₀ f(x)>M} is saturated non-measurable for each additive discontinuous functionf and each real numberM. Other results aboutV ₀ are also presented. Connections between measure and category are stressed. The main tool in our proofs is a certain so-called zero–one law and its topological analogue. In addition it is shown that the zero–one law is equivalent to Smital's lemma.     

Analysis of a class of fractional programming problems

We propose a solution strategy for fractional programming problems of the form max _xx g(x)/ (u(x)), where the function satisfies certain convexity conditions. It is shown that subject to these conditions optimal solutions to this problem can be obtained from the solution of the problem max _xx g(x) + u(x), where is an exogenous parameter. The proposed strategy combines fractional programming andc-programming techniques. A maximal mean-standard deviation ratio problem is solved to illustrate the strategy in action.     

Geometry and the Jones projection of a state

LetA be a von Neumann algebra and a faithful normal state. ThenO = { ºAd(g ¹) :g G _A }andU = { ºAd(u ^*) :u U _A are homogeneous reductive spaces. IfA is aC ^* algebra,e the Jones projection of the faithful state viewed as a conditional expectation, then we prove that the similarity orbit ofe by invertible elements ofA can be imbedded inAA in such a way thate is carried to 1 1 and the orbit ofe to a homogeneous reductive space and an analytic submanifold ofAA.     

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.     

Efficiency of Proximal Bundle Methods

We give efficiency estimates for proximal bundle methods for finding f_*min_Xf, where f and X are convex. We show that, for any accuracy <0, these methods find a point x^kX such that f(x^k)–f^* after at most k=O(1/³) objective and subgradient evaluations.     

Some propositions related to a dilation theorem of W. Benz

Summary This paper begins with another proof of a theorem of W. Benz [2] concerning dilations in normed linear spaces. Our proof motivates several questions which are addressed thereafter. For instance it is shown that, ifI is an open interval in ,: I ⁿ , is continuously differentiable and there exista ₁,...,a _n I such that {(a ₁,...,(a _n )} is linearly independent, then {(t): t I} contains a Hamel basis for ⁿ over .     

On convergence subsystems of orthonormal systems

It is proved that for any sequence {R _k} _k=1 of real numbers satisfyingR _kk (k1) andR _k=o(k log₂ k),k, there exists an orthonormal system {n _k(x)} _n=1 ,x (0;1), such that none of its subsystems {n _k(x)} _k=1 withn _kR_k (k1) is a convergence subsystem.     

On a Property of Subexponential Distributions

Let F be a distribution function (d.f.) on [0, ) with finite first moment m >0. We define the integrated tail distribution function F ₁ of F by F ₁(t)=m^-1 ₀ ^t (1- F(u))du, t0. In this paper, we obtain sufficient conditions under which implications FSF ₁S and F ₁S FS hold, where S is the class of subexponential distributions.     

Aufeinanderfolgende Elemente in multiplikativen Zahlenmengen

LetM be a multiplicative set with 1M andmnM if and only ifmM,nM for (m,n)=1. It is shown by elementary means that there exists the asymptotic density of the setM(M–1) for every multiplicative setM. The density is positive if and only ifM possesses a positive density and 2M for some . This result is slightly generalized to sums over multiplicative functionsf with |f|1.     

The lyapunov equation and the problem of stability for linear bounded discrete-time systems in Hilbert space

This paper is devoted to a study of the properties of the equationA ^*FA–F=–G, where FL() is unknown, AL(), GL() is positive and is a Hilbert space. It is shown that necessary and sufficient (in some sense) conditions for the existence of positive definite solutions of this equation are directly connected with the stability of infinite dimensional linear systemx _k+1=Ax _k . The relationships between stability of such a system and stability of a continuous-time system generated by a strongly continuous semigroup are given also. As an example the case of the delayed system in Rⁿ is considered.This work was supported in part by the Polish Academy of Sciences under the contract Problem Miedzyresortowy I.1, Grupa Tematyczna 3 This paper was written while the author was with the Instytut Automatyki, the same university.     

The caratheodory number for thek-core

Thek-core of the setS ⁿ is the intersection of the convex hull of all setsA S with ¦SA¦<-k. The Caratheodory number of thek-core is the smallest integerf (d,k) with the property thatx core _{kS, S} ⁿ implies the existence of a subsetT S such thatx core_kT and ¦T¦f (d, k). In this paper various properties off(d, k) are established.Research of this author was partially supported by Hungarian National Science Foundation grant no. 1812.