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).     

Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

We study the regularity of the minimizer u for the functional F (u,f)=|u|² + |u–f{² over all maps uH ¹(, S ²). We prove that for some suitable functions f every minimizer u is smooth in if ₀ and for the same functions f, u has singularities when is large enough.

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Résumé On étudie la régularité des minimiseurs u du problème de minimisation min_ueH ¹(,S²)(|u|² + |u–f{². On montre que pour certaines fonctions f, u est régulière lorsque ₀ et pour les mêmes f, si est assez grand, alors u possède des singularités.

     

Quasimonotonicity,regularity and duality for nonlinear systems of partial differential equations

Summary We prove partial regularity for the vector-valued differential forms solving the system (A(x, ))=0, d=0, and for the gradient of the vector-valued functions solving the system div A(x, Du)=B(x, u, Du). Here the mapping A, with A(x, w) (1+ + ¦¦²)^{(p – 2)/2} (p2), satisfies a quasimonotonicity condition which, when applied to the gradient A(x, )=Df(x, ) of a real-valued functionf, is analogous to but stronger than quasiconvexity for f. The case 1相似文献   

A sandwich theorem and Hyers—Ulam stability of affine functions

It is shown that two real functionsf andg, defined on a real intervalI, satisfy the inequalitiesf(x + (1 – )y) g(x) + (1 – )g(y) andg(x + (1 – )y) f(x) + (1 – )f(y) for allx, y I and [0, 1], iff there exists an affine functionh: I such thatf h g. As a consequence we obtain a stability result of Hyers—Ulam type for affine functions.     

Algebraic systems of quadratic forms of number fields and function fields

In this paper we examine for which Witt classes ,..., _n over a number field or a function fieldF there exist a finite extensionL/F and ₂,..., _n L^* such thatT _L/F ()=1 andTr _L/F (_i)=_i fori=2,...n.     

Sukzessive Minima mit und ohne Nebenbedingungen

Given semi-normsf andg on ⁿ and a real number >0. Then the successive minima off under the constraintg are defined by _j : = inf {: there existj linear independent vectors inZ ⁿ withf andg}. The main theorem of this paper (Lagrange multiplier theorem) states that the successive minima of a certainnorm h on ⁿ (without constraints) coincide with the _j 's up to bounded factors. Moreover, this norm is constructed explicitly. Using Minkowski's wellknown theorem on successive minima and our result certain inequalities on simultaneous Diophantine approximations are derived.     

An interior point multiplicative method for optimization under positivity constraints

We analyze an algorithm for the problem minf(x) s.t.x 0 suggested, without convergence proof, by Eggermont. The iterative step is given by x _j ^k+1 =x _j ^k (1-_kf(x ^k)_j) with _k > 0 determined through a line search. This method can be seen as a natural extension of the steepest descent method for unconstrained optimization, and we establish convergence properties similar to those known for steepest descent, namely weak convergence to a KKT point for a generalf, weak convergence to a solution for convexf and full convergence to the solution for strictly convexf. Applying this method to a maximum likelihood estimation problem, we obtain an additively overrelaxed version of the EM Algorithm. We extend the full convergence results known for EM to this overrelaxed version by establishing local Fejér monotonicity to the solution set.Research for this paper was partially supported by CNPq grant No 301280/86.     

On the typicality of linear Hamiltonian systems with exponential separateness

Linear Hamiltonian systems of ordinary differential equations,are considered, where f ^t is a continuous action of the group R on a complete metric space , A is an element of the space S _H ,consisting of the continuous bounded mappings of into the set of all pseudosymmetric matrices and endowed with the uniform convergence metric. By ₁ (A, x) _2n (A, x) we denote the Lyapunov exponents of such systems. The typicality (in the Baire sense) in the space S _H × is proved for those pairs (A, x) for which one has the alternative: either _k (A, x) = _k+1 (A, x) or the linear subspace of the solutions of the corresponding system with exponents less than _k (A, x) is exponentially separated from any of its algebraic complements in the space of all the solutions of the system. From here, in particular, there follows the typicality of the formulated alternative for linear Hamiltonian systems with continuous quasiperiodic coefficients (with the same frequency module) and also for linear Hamiltonian systems with continuous almost periodic coefficients. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 16, pp. 137–181, 1992.     

Topological classification of linear endomorphisms

Letf=f ₊f _–f f ₀ be a linear direct sum decomposition of a real endomorphismf End ⁿ , where the components correspond respectively to absolute values of eigenvalues ||: 0<||<1(f ₊); ||>1(f _–); ||=0(f ); and ||=1(f ₀). We conjecture that a complete set of invariants with respect to topological equivalence consists of the dimensions and orientations off ₊ and off _–, together with the linear isomorphy classes off andf ₀. We prove that this is true in case ⁿ contains nof-periodic points of periods 5 or 7. The conjecture is also true in case it is true for all periodic rotations. In §9 we make some comments on this unsolved case. The main interest of the paper is in §6 and 7.Research supported by the National Science Foundation (Contract number 144-B 695), the Wisconsin Alumni Research Foundation (project number 120432), and l'Institut des Hautes Etudes Scientifiques.     

Shift invariant spaces inL 2

If is a complex, separable Hilbert space, letL ² () denote theL ²-space of functions defined on the unit circle and having values in . The bilateral shift onL ²() is the operator (U f)()=f(). A Hilbert spaceH iscontractively contained in the Hilbert spaceK ifHK and the inclusion mapH–K is a contraction. We describe the structure of those Hilbert spaces, contractively contained inL ²(), that are carried into themselves contractively byU . We also do this for the subcase of those spaces which are carried into themselves unitarily byU .     

Conditions for separately subharmonic functions to be subharmonic

Letu be a function on ^m × ⁿ , wherem2 andn2, such thatu(x, .) is subharmonic on ⁿ for each fixedx in ^m andu(.,y) is subharmonic on ^m for each fixedy in ⁿ . We give a local integrability condition which ensures the subharmonicity ofu on ^m × ⁿ , and we show that this condition is close to being sharp. In particular, the local integrability of (log⁺ u ⁺) ^m+n–2+ is enough to secure the subharmonicity ofu if >0, but not if <0.     

The Real Period Function of A3 Singularity and Perturbations of the Spherical Pendulum

We prove that the Hessian matrix of the real period function () associated with the real versal deformation f (x)=±x ⁴+₂ x ²+₁ x+₀ of a singularity of type A ₃, is nondegenerate, provided that ³ does not belong to the discriminant set of the singularity. We explain the relation between this result and the perturbations of the spherical pendulum.     

On discrete and continuous norms in Paley-Wiener spaces and consequences for exponential frames

It is well known that for certain sequences {t_n}_n the usual L^p norm ·_p in the Paley-Wiener space PW ^p is equivalent to the discrete norm f_p,{tn}:=( _n=– |f(t_n)|^p)^1/p for 1 p = < and f_,{tn}:=sup _n|f(t_n| for p=). We estimate f_p from above by Cf_p, _n and give an explicit value for C depending only on p, , and characteristic parameters of the sequence {t_n}_n. This includes an explicit lower frame bound in a famous theorem of Duffin and Schaeffer.     

A geometric property of extremal surfaces

Let the surface R³ be defined by the equation z = f(x, y), where f(x, y) is a function 3 times continuously differentiable in R². It is proved that if the total (Gaussian) curvature of the surface is nonzero almost everywhere on in the sense of Lebesgue measure in R²), then is extremal, i.e., for almost all (x,y) R² the inequality max (||qx||, ||qy, qf (x, y)) > q^–1/s–. holds for all integral q qo (f), where x is the distance from the real number x to the nearest integer and > 0 is arbitrarily small.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 177–181, February, 1978.In conclusion, the author thanks V. G. Sprindzhuk for suggesting the problem.     

The Discrete Spectrum Asymptotics with Respect to a Large Coupling Constant under Strong Nonnegative Perturbations

Let A be a self-adjoint operator, let (, ) be an inner gap in the spectrum of the operator A, and let B(t) = A + tW ^* W, where the operator W(A – iI)^-1 is not necessarily bounded. Conditions are obtained under which the spectrum of B(t) in (, ) is discrete. Let N(, A, W, ), (, ), > 0, be the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to . The asymptotics of N(, A, W, ) as + is obtained in terms of the spectral asymptotics of a certain self-adjoint compact operator. Bibliography: 5 titles.     

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 .     

On the equation of similar profiles

Summary Considerf+ ff+ (1–f²)+ f=0 together with the boundary conditionsf(0)=f(0)=0,f ()=1. If=–1,>0, arbitrary there is at least one solution which satisfies 0<f<1 on (0, ). By the additional conditionf>0 on (0, ) or, alternately 0<1, the uniqueness of the solution is demonstrated.If=1,<0, arbitrary the existence of solutions for which –1<f<0 in some initial interval (0,t) and satisfying generallyf>1 is established. In both problems, bounds forf (0) and qualitative behavior of the solutions are shown.

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Sommario Si consideri il problema definito dall'equazionef+ f f+ (1–f²)+ f=0 e dalle condizioni al contornof(0)=f (0)=0,f()=1. Assumendo=–1,>0, arbitrario si dimostra che esiste almeno una soluzione che soddisfa 0<f<1 nell'intervallo (0, ). Se in aggiunta si ipotizzaf>0 in (0, ), oppure 0<=1, l'unicità délia soluzione è assicurata.Successivamente si considéra il problema di valori al contorno con=1,<0, arbitrario. In questo caso esiste un'intera classe di soluzioni che soddisfano –1<f<0 in un intorno dell'origine e tali chef>1, in generale.Di detti problemi viene studiato il comportamento délle soluzioni e vengono determinate dalle maggiorazioni e minorazioni del valoref(0).

     

Weak Mixing of Random Walks on Groups

Let G be a locally compact -compact group with right Haar measure m and a regular probability measure on G. We say that is weakly mixing if for all gL (G) and all fL ¹(G) with fdm=0 we have n ^{–1 n} _k=1| ^k *f,g|0. We show that is weakly mixing if and only if is ergodic and strictly aperiodic. To prove this we use and prove some results about unimodular eigenvalues for general Markov operators.     

Posterior Sensitivity to the Sampling Distribution and the Prior: More than One Observation

Sensitivity of a posterior quantity (f, P) to the choice of the sampling distribution f and prior P is considered. Sensitivity is measured by the range of (f, P) when f and P vary in nonparametric classes ^f and ^P respectively. Direct and iterative methods are described which obtain the range of (f, P) over f ^f when prior P is fixed, and also the overall range over f ^f and P ^P . When multiple i.i.d. observations X ₁,...,X _k are observed from f, the posterior quantity (f, P) is not a ratio-linear function of f. A method of steepest descent is proposed to obtain the range of (f, P). Several examples illustrate applications of these methods.     

Absolute values of the coefficients of the polynomials in Weierstrass's approximation theorem

The following problem, bound up with Weierstrass's classical approximation theorem, is solved definitively: to determine the sequence of positive numbersM _k such that, for anyf(z)c[0,1] and > 0 there exists the polynomial that f–P< and _k <M _k ,k=1, ...,n.Translated from Matematicheskii Zametki, Vol. 22, No. 2, pp. 269–276, August, 1977.