On the optimality of a characterization theorem for the gamma function using the multiplication formula

Summary The following Artin type characterization of : _{+ +} is proved: Assume thatf: _{+ +} satisfies the Gauss multiplication formula for some fixedp 2,f is absolutely continuous on [l/p, 1 + ] for some > 0 and lim _{x 0} xf(x) = 1. Thenf(x) = (x) forx > 0.The optimality of this result is checked by means of counterexamples. For instance, it is shown that the result is no longer true, if f is absolutely continuous is replaced by f is continuous and of finite variation.     

On the boundedness of Weyl sums

Leta be irrational and letf:[0,1] be Riemann-integrable with integral zero. Letf _n (x) denote the Weyl sumf _n (x):= _k=0 ^n–1 f({x k>}),x/[0,1[,n. We prove criteria for the boundedness of the sequence (f _n ) _n1 and discuss the relation of this question to irregularities of the distribution of sequences.     

On solvability of linear difference equations in smooth and real analytic vector functions of several variables

We investigate the multidimensional equations _j=1 ^q A_j(x)y(x+e _j )=f(x),e _j ⁿ wherex ⁿ andA _j : ⁿ Hom( ^p , ^m ),f : ^{n m} are given maps. Sufficient conditions for smooth and analytic solvability for anyf C ^k ,k are found.Research partially supported by the Israel Ministry of ScienceAMS classification 39B Functional equations     

On some functional equations of Goląb-Schinzel type

Summary LetE be a real Hausdorff topological vector space. We consider the following binary law * on ·E:(, ) * (, ) = (, ^k + ) for(, ), (, ) × E where is a nonnegative real number,k andl are integers.In order to find all subgroupoids of ( ·E, *) which depend faithfully on a set of parameters, we have to solve the following functional equation:f(f(y) ^k x +f(x) ^l y) =f(x)f(y) (x, y E). (1)In this paper, all solutionsf: of (1) which are in the Baire class I and have the Darboux property are obtained. We obtain also all continuous solutionsf: E of (1). The subgroupoids of (^* ·E, *) which dapend faithfully and continuously on a set of parameters are then determined in different cases. We also deduce from this that the only subsemigroup ofL _n ¹ of the form {(F(x ₂,x ₃, ,x _n ),x ₂,x ₃, ,x _n ); (x ₂, ,x _n ) ^{n – 1} }, where the mappingF: ^{n – 1 *} has some regularity property, is {1} × ^{n – 1} .We may noitice that the Gob-Schinzel functional equation is a particular case of equation (1)(k = 0, l = 1, = 1). So we can say that (1) is of Gob—Schinzel type. More generally, whenE is a real algebra, we shall say that a functional equation is of Gob—Schinzel type if it is of the form:f(f(y) ^k x +f(x) ^l y) =F(x,y,f(x),f(y),f(xy)) wherek andl are integers andF is a given function in five variables. In this category of functional equations, we study here the equation:f(f(y) ^k x +f(x) ^l y) =f(xy) (x, y f: ). (4)This paper extends the results obtained by N. Brillouët and J. Dhombres in [3] and completes some results obtained by P. Urban in his Ph.D. thesis [11] (this work has not yet been published).Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth     

C 1 changes of variable: Beurling-Helson type theorem and Hörmander conjecture on Fourier multipliers

We prove that if aC ¹ smooth change of variable : generates a bounded composition operatorff° in the spaceA _p()=L ^p ,p2, then is linear (affine).We also prove that for a nonlinearC ¹ mapping , the norms of exponentialse ⁱ as Fourier multipliers inL ^p () tend to infinity (,||). In both results the condition C ¹ is sharp, it cannot be replaced by the Lipschitz condition.     

Asymptotic behaviour of Gaussian random fields

Summary Let X={X(t), t ^N} be a centred Gaussian random field with covariance X(t)X(s)=r(t–s) continuous on ^N×^N and r(0)=1. Let (t,s)=((X(t)–X(s)) ²)^1/2; (t,s) is a pseudometric on ^N. Assume X is -separable. Let D ₁ be the unit cube in ^N and for 0<k, D _k= {x^N: k ^–1 xD₁}, Z(k)=sup{X(t),tD _k}. If X is sample continuous and ¦r(t)¦ =o(1/log¦t¦) as ¦t¦8 then Z(k)-(2Nlogk) ^1/20 as k a.s.     

A functional inequality characterizing convex functions,conjugacy and a generalization of Hölder's and Minkowski's inequalities

Summary The main result says that, iff: _{+ +} satisfies the functional inequalityaf(s) + bf(t) f (as + bt) (s,t 0) for somea, b such that 0 <a < 1 <a + b, thenf(t) = f(1)t, (t 0). A relevant result for the reverse inequality is also discussed. Applying these results we determine the form of all functionsf: _k ⁺ ₊ satisying the above inequalities. This leads to a characterization of both concave and convex functions defined on ₊ ^k–1 , to a notion of conjugate functions and to a general inequality which contains Hölder's and Minkowski's inequalities as very special cases.     

Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh

Summary We study here the discretisation of the nonlinear hyperbolic equationu _t +div(vf(u))=0 in ³ × ₊, with given initial conditionu(.,0)=u ₀(.) in ², wherev is a function from ² × ₊ to ² such that divv=0 andf is a given nondecreasing function from to . An explicit Euler scheme is used for the time discretisation of the equation, and a triangular mesh for the spatial discretisation. Under a usual stability condition, we prove the convergence of the solution given by an upstream finite volume scheme towards the unique entropy weak solution to the equation.     

On a problem of Erdős and Lovász: Random lines in a projective plane

Letn(k) be the least size of an intersecting family ofk-sets with cover numberk, and let _k denote any projective plane of orderk–1.Theorem There is a constant A such that ifH is a random set ofm Aklogk lines from _k then P_r(H<)0(k).Corollary If there exists a _k thenn(k)=O(klogk). These statements were conjectured by P. Erds and L. Lovász in 1973.Supported in part by NSF-DMS87-83558 and AFOSR grants 89-0066, 89-0512 and 90-0008     

Regularized spectral distributions

For C a bounded, injective operator with dense image, we define a C-regularized spectral distribution. This produces a functional calculus, f f(B), from C() into the space of closed densely defined operators, such that f(B)C is bounded when f has compact support. As an analogue of Stone's theorem, we characterize certain regularized spectral distributions as corresponding to generators of polynomially bounded C-regularized groups. We represent the regularized spectral distribution in terms of the regularized group and in terms of the C-resolvent. Applications include the Schrödinger equation with potential, and symmetric hyperbolic systems, all on L^p(ⁿ) (1p<), C _o(ⁿ), BUC(ⁿ), or any space of functions where translation is a bounded strongly continuous group.     

Real Milnor Fibrations and (C)-Regularity

In this paper we study Milnor fibrations associated to real isolated singularities defined by map-germs f: (^m,0)(²,0). The main result relates the existence of the Milnor fibration with the (C)-regularity of the family of hypersurfaces with isolated singularity obtained by projecting f into the family L_– of all lines through the origin in the plane ².     

Wiener-Ditkin sets for certain beurling algebras

It is proved that closed subgroups of ⁿ are Wiener-Ditkin sets for the Beurling algebrasL ¹ ( ⁿ ), <1.     

Approximation of the efficient point set by perturbation of the ordering cone

A vector optimization problem is given by a feasible setZ ⁿ , a vector-valued objective functionf: ^{n l} , and an ordering coneC ^l . We perturb the ordering cone in such a way that the weakly efficient points of the perturbed vector optimization problem given byZ, f, and the perturbed cone are efficient points of the original problem. Especially this means that scalarization methods, which compute in general only weakly efficient points, determine efficient points of the original problem, when they were applied to the perturbed problem.It turns out that the efficient points are the limits of weakly efficient points of the perturbed problems, letting the perturbation tend to zero. On the basis of this, a reference point algorithm is formulated. Finally, we apply this algorithm to a structural optimization problem.     

Note on initial topologies on rational vector spaces induced by realvalued linear mappings

LetG be a vector space over the field of rational numbers andf, g:G -linear mappings. equipped with the usual norm topology. Denote by _f , _g the initial topologies onG induced byf respectivelyg.Then the following result holds: If there is a nonvoid open setU whose complement contains at least one inner point such thatf ^–1 U _g , then there is ac withf=cg. In particular, iff0, the topologies coincide.Furthermore, a -linear mappingh: (G, _f )(G, _g ) is continuous if and only if there is a real constantc withg ^o h=cf.Dedicated to Professor János Aczél on his 60th birthday     

On local oscillatory integrals with variable Calderón-Zygmund kernels

Letp(1, ). In this paper, the authors investigate the uniformL ^p ( ⁿ ) in of the oscillatory singular integral operatorT defined by

Tessellations generated by hyperplanes

Let be a locally finite system of hyperplanes in ^d with the property that the cells of the induced cell complex decomposition of ^d have uniformly bounded diameters. If is simple and the density of the vertices in exists, then the density of thek-cells in exists and can be given explicitly (k = 1, ...,d). Also, the mean number ofj-faces of thek-cells in exists and can be calculated. For certain nonsimple systems , corresponding inequalities are obtained.     

Rotationally symmetric harmonic maps from a ball into a warped product manifold

This paper deals with the existence problem for rotationally symmetric harmonic maps from an Euclidean unit ball B ⁿ or ⁿ into a warped product manifold N_f=[0, r₀)x_fS^n–1.     

Implicit elliptic differential equations

Let be a bounded domain in ⁿ (n 3) having a smooth boundary, letY be a closed, connected and locally connected subset of ^h , letf be a real-valued function defined on × ^h × ^nh ×Y, and letL be a linear, second-order elliptic operator. In this paper, the existence of strong solutionsu W ^2,p (, ^h ) W ₀ ^1,p (, ^h ) (n<p<+) to the implicit elliptic equationf(x, u, Du, Lu)=0, whereu=(u ₁,u ₂, ...,u _h ),Du=(Du ₁,Du ₂, ...,Du _h ) andLu=(Lu ₁,Lu ₂, ...,Lu _h ), is established. The abstract framework where the equation is studied is that of set-valued analysis.Dedicated to Professor G. Pulvirenti on the occasion of his sixtieth birthday     

Zum Einbettungsproblem für Markoff-Prozesse

Every elementary Markov process with a polish state space and with a discrete set of time parameter dense in ₊, whose finite dimensional distributions are derived from a semigroup (K _t) of Markov kernels continuous in 0 ₊ and whose initial distribution satisfies K _t, can be imbedded in an elementary Markov process with the same state space and with parameter set ₊ so that the corresponding finite dimensional distributions are equal.     

Embedding a non-embeddable stable plane

In [4], K. Strambach describes a 2-dimensional stable plane admitting =SL₂ as a group of automorphisms such that there exists no -equivarient embedding into a 2-dimensional projective plane. R. Löwen [3] has given a 4-dimensional analogue , admitting =SL₂. He posed the question whether there are embeddings of Strambach's plane into . We show that such embeddings exist, in fact we determine all -equivariant embeddings of 2-dimensional stable planes admitting as atransitive group of automorphisms.