On the minimum time map

Santo Si studiano alcune proprietà della funzione di tempo minimo per un'equazione differenziale multivoca su una varietà C. Si mette in relazione la funzione di tempo minimo per una famiglia di campi vettoriali con guella per l'equazione multivoca associata. Inoltre si prova che da una estensione alle varietà riemanniane complete della condizione classica F(t, x)(t)+v(t)x, si hanno le stesse conseguenze che in Rⁿ. Si prova, infine, sotto la stessa condizione, che la funzione che ad ogni t associa l'insieme raggiungibile al tempo t è localmente lipschitziana (rispetto alla metrica di Hausdorff).

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This work was performed under the auspices of the National Research Council of Italy (C.N.R.).     

Simultaneous approximation of algebraic irrationalities

This paper proves three theorems concerning the simultaneous approximation of numbers from a totally real algebraic number field. It is shown that for two given numbers ₁ and ₂ from a totally real algebraic number field, the constant ₁₂ can be explicitly calculated, this being the upper limit of the numbers c₁₂ such that the inequality max (q₁, q₂)(qc₁₂)^–1/2 holds for infinitely many natural numbers q; likewise for the constant a₁₂ such that the inequality q₁·q₂< a₁₂(q^logq) holds for infinitely many natural numbers q. It is shown that there exist n –1 numbers ₁, ..., _n–1 in an algebraic number field of degree n and discriminant d such that the inequality holds only for finitely many natural numbers q if 2^{ - \left[ {\tfrac{{n - 1}}{2}} \right]} \sqrt d $$ " align="middle" border="0"> . is fixed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 142–154, 1982.     

Approximating derivations on ideals ofC *-algebras

For each^*-derivation of a separableC ^*-algebraA and each >0 there is an essential idealI ofA and a self-adjoint multiplierx ofI such that (–ad(ix))|I< and x.     

Daugavet type inequalities for operators on L p–spaces

Let T be a regular operator from L ^p L ^p. Then , where T_r denotes the regular norm of T, i.e., T_r=|T| where |T| denotes the modulus operator of a regular operator T. For p=1 every bounded linear operator is regular and T=T_r, so that the above inequality generalizes the Daugavet equation for operators on L ¹–spaces. The main result of this paper (Theorem 9) is a converse of the above result. Let T be a regular linear operator on L _p and denote by T _A the operator T_A. Then for all A with (A)>0 if and only if .     

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.     

Hankel operators and problems of best approximation of unbounded functions

For each function f, f VMO, there exists a unique function f₀, analytic in the circle and such that f–f₀=_f{gVMO_A}. We define the operator of best approximation (nonlinear) A, Af=f₀, fVMO, In the paper one considers the question of the preservation of a class under the action of the operator i.e. finding the classes X, X VMO, AX X. One investigates the classes X containing unbounded functions. It is proved that if P_X is the space of the symbols of the Hankel operators from a Banach space E of functions into the Hardy space H², then AX X. For E one can take almost any space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 141, pp. 5–17, 1985.     

Jordan axioms for C*-algebras

A complex Banach spaceA which is also an associative algebra provided with a conjugate linear vector space involution * satisfying (a ²)^*=(a ^*)², aa ^* a=a³ and ab+ba2ab for alla, b inA is shown to be a C^*-algebra. The assumptions onA can be expressed in terms of the Jordan algebra obtained by symmetrization of the product ofA and are satisfied by any C^*-algebra. Thus we obtain a purely Jordan characterization of C^*-algebras.     

Problem of instability in the first approximation

Let E be a Banach space, A be a continuous linear operator such that (A) ; Re>0 Ø, and F(t, x) be a continuous function on [0, )×E satisfying the condition F(t, x) q x (q= const). An example of a system dx/dt=Ax + F(t, x) is given which has an exponentially stable zero solution for certain F(t, x) with arbitrarily small q.Translated from Matematicheskie Zametki, Vol. 23, No. 5, pp. 721–723, May, 1978.     

An asymptotic variant of the Fuglede-Putnam theorem on commutators for elements of Banach algebras

The Fuglede-Putnam theorem (in Moore's asymptotic form) on the commutators of normal operators of a Hilbert space is generalized, in particular, in the following form. Leta ₁,a ₁, b₁ and b₂ be the elements of a complex Banach algebra such that [a ₁ b₁]=[a ₂, b₂]=0 and as . Then the inequality b ₁ x–xb ₂(a ₁ x–xa ₂), where () as 0, holds uniformly in every ball xR<.Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 179–188, August, 1977.     

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.     

Symmetrization of functions in Sobolev spaces and the isoperimetric inequality

A positive measurable function f on R^d can be symmetrized to a function f^* depending only on the distance r, and with the same distribution function as f. If the distribution derivatives of f are Radon measures then we have the inequality f^*f, where f is the total mass of the gradient. This inequality is a generalisation of the classical isoperimetric inequality for sets. Furthermore, and this is important for applications, if f belongs to the Sobolev space H¹,P then f^* belongs to H¹,P and f^*_pf_p.     

On the Harris Recurrence of Iterated Random Lipschitz Functions and Related Convergence Rate Results

A result by Elton⁽⁶⁾ states that an iterated function system

Epsilon efficiency

This paper considers the extension of -optimality for scalar problems to vector maximization problems, or efficiency problems, which havem objective functions defined on a set .It is shown that the natural extension of the scalar -optimality concepts [viz, given >0, given a solution setS, ifxS there exists an efficient solutiony with f(x)–f(y), and given an efficient solutiony, there exists anxS with f(x)–f(y)] do not hold for some methods used. Six concepts of -efficient sets are introduced and examined, to a very limited extent, in the context of five methods used for generating efficient points or near efficient points.In doing so, a distinction is drawn between methods in which the surrogate optimizations are carried out exactly, and those where terminal -optimal solutions are obtained.The author would like to thank the referees whose thoroughness was extremely helpful for the revised paper.     

Best approximation and best unilateral approximation of the kernel of a biharmonic equation and optimal renewal of the values of operators

For the classB _p , 0 < 1, 1p , of 2-periodic functions of the form f(t)=u(,t), whereu (,t) is a biharmonic function in the unit disk, we obtain the exact values of the best approximation and best unilateral approximation of the kernel K(t) of the convolution f= K *g, gl, with respect to the metric of L₁. We also consider the problem of renewal of the values of the convolution operator by using the information about the values of the boundary functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 11, pp. 1549–1557, November, 1995.     

Lipschitz and darbo conditions for the superposition operator in ideal spaces

Summary In this paper we give necessary and sufficient conditions for the superposition operator Fx(s)=f(s, x(s)) to satisfy a Lipschitz condition Fx₁ - Fx₂kx₁ - x₂ or a Darbo condition (FN)k(N) in ideal spaces of measurable functions, where is the Hausdorff measure of noncompactness. Moreover, we characterize a large class of spaces in which the above mentioned two conditions are equivalent.

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Sunto In questo lavoro diamo delle condizioni necessarie e sufficienti perchè l'operatore di sovrapposizione Fx(s)=f (s, x(s)) soddisfi alla condizione di Lipschitz Fx₁–Fx₂ kx₁–x₂ o quella di Darbo (FN)k(N) in spazi ideali di funzioni misurabili, ove è la misura di non compattezza di Hausdorff. Inoltre, caratterizziamo un'ampia classe di spazi in cui le suddette due condizioni sono equivalenti.

     

Isometries of symmetric operator spaces associated with AFD factors of typeII and symmetric vector-valued spaces

If is a surjective isometry of the separable symmetric operator spaceE(M, ) associated with the approximately finite-dimensional semifinite factorM and if · _E(M,) is not proportional to · _{L 2}, then there exist a unitary operatorUM and a Jordan automorphismJ ofM such that(x)=UJ(x) for allxME(M, ). We characterize also surjective isometries of vector-valued symmetric spacesF((0, 1), E(M, )).Research supported by the Australian Research Council     

Über die Existenz gemeinsamer Elemente bester Approximation bezüglich zweier Normen II

The present paper deals with the possibility of existence of best approximation elements, simultaneously with respect to two norms ·_i,i=1,2, for all the elements of a class of subspaces. In case this class in any of the following: (a) All n-dimensional subspaces, (b) All ·₁-or ·||₂-closed, n-codimensional subspaces, (c) All ·₁-or ·₂-closed subspaces with infinite dimension and codimension, we prove that the two norms differ at most by a constant factor.     

On a condition on the radical of a Banach algebra ensuring strong decomposability

The main result is the following theorem. Let be a commutative Banach algebra with radical R, where the factor algebra is isomorphic to the algebra of all continuous functions on a totally disconnected compact space. If rⁿ¹ /n 0 as n uniformly for r R, rl, then the algebra is strongly decomposable, i.e., there exists a closed subalgebra B isomorphic to such that =BR.This is a strengthening of the theorem of A. Ya. Khelemskii, who assumed . There are 4 references.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 589–592, December, 1967.     

Sufficient conditions for bang-bang control in Hilbert space

Sufficient conditions for bang-bang and singular optimal control are established in the case of linear operator equations with cost functionals which are the sum of linear and quadratic terms, that is,Ax=u,J(u)=(r,x)+(x,x), >0. For example, ifA is a bounded operator with a bounded inverse from a Hilbert spaceH into itself and the control setU is the unit ball inH, then an optimal control is bang-bang (has norm l) if 0<1/2;A ^–1*r·A ^–1–2, but is singular (an interior point ofU) if >1/2A ^–1*r·A².This work was supported by NRC Grant No. A-4047 and NSF Grant No. GP-7445.     

Rate of convergence in the central limit theorem of the UH-statistics

The problem of estimating the rate of convergence in the central limit theorem for the U-statistics with values in the Hilbert space is considered. Bounds of the order o(n^–1/2) if the kernel satisfies the nondegenerate condition and E³ < .Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 8, pp. 1043–1050, August, 1990.