On local oscillatory integrals with variable Calderón-Zygmund kernels, II

Let , be a real analytic function or a real-C function on ⁿ andk be a variable Calderón-Zygmund kernel. Define the oscillatory singular integral operatorT by

Some remarks on Turán's inequality. III: The completion

. 0pq, 1–1/p+1/p0. f(x) — n, [–1,1],

On the computation of modules of long quadrilaterals

We study quadrilateralsQ which are given by two intervals on {:Im = 0} and {:Im = 1}, and two Jordan arcs ₁, ₂, in {:0 Im 1} connecting these two intervals. Many practical problems require the determination of the modulem(Q) ofQ, but ifQ is long, i.e., if

Circularity of Finite Groups without Fixed Points

Let áA, B | A^m=1, Bⁿ=A^t, BAB^-1=A^r?\langle A, B\,\vert\, A^\mu=1,\, B^\nu=A^t,\, BAB^{-1}=A^\rho\rangle where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

Some notes about affine diameters of convex figures

Given a pointx in a convex figureM, let(x) denote the number of all affine diameters ofM passing throughx. It is shown that, for a convex figureM, the following conditions are equivalent.

The GHS inequality and the Riemann hypothesis

LetV(t) be the even function on (–, ) which is related to the Riemann xi-function by (x/2)=4 _– exp(ixt–V(t))dt. In a proof of certain moment inequalities which are necessary for the validity of the Riemann Hypothesis, it was previously shown thatV'(t)/t is increasing on (0, ). We prove a stronger property which is related to the GHS inequality of statistical mechanics, namely thatV' is convex on [0, ). The possible relevance of the convexity ofV' to the Riemann Hypothesis is discussed.Communicated by Richard Varga.     

Spectral synthesis for Euler type operators

A=(a _ij) _i ^j=1 — k-o ,a _ij . :

The geodesic “straightest” lines of a nonlinear nonholonomic manifold in E4

The following properties of the geodesic straightest lines of a system of two Monge equations inE ₄ are established:

Investigations on slow versus fast growing: How to majorize slow growing functions nontrivially by fast growing ones

Let T() be the ordinal notation system from Buchholz-Schütte (1988). [The order type of the countable segmentT()₀ is — by Rathjen (1988) — the proof-theoretic ordinal the proof-theoretic ordinal ofACA ₀ + ( ₁ ^l –TR).] In particular let _a denote the enumeration function of the infinite cardinals and leta ₀ a denote the partial collapsing operation on T() which maps ordinals of T() into the countable segment T ₀ of T(). Assume that the (fast growing) extended Grzegorczyk hierarchy and the slow growing hierarchy are defined with respect to the natural system of distinguished fundamental sequences of Buchholz and Schütte (1988) in the following way:

Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents

We are concerned with the semilinear polyharmonic model problem (–)K v = v +v|v| ^s–1 inB,D v|B = 0 for ¦|<-K – 1. HereK ,B is the unit ball in ⁿ,n >2K, is the critical Sobolev exponent. Let ₁ denote the first Dirichlet eigenvalue of (-)^K inB. The existence of a positive radial solutionv is shown for

Some nodes matrices appearing in the numerical analysis for singular integral equations

Let {p _m (w)} be the sequence of Jacobi polynomials corresponding to the weightw(x)=(1–x)(1+x), 0, <1. denote=">x _k (w)=cos _m,k (w),k=1,...,m, the zeros ofp _m (w). If +=0, then the estimates

On the convergence of multivariable Lagrange interpolants

LetA _dj be a triangular array in a compact setXC ⁿ . Forf analytic in a neighborhood ofX, letL _d (f) denote the Lagrange interpolant tof at staged of the array. In the caseX is locally regular, we construct a continuous function satisfying the complex Monge-Ampère equation onC ⁿ –X, such that iff is analytic onR forR>1 then, for someB>0, we have L _d (f)–f _x B exp(–d logR. In particular, since1 onX, iff is analytic on1, then lim _d L _d (f)–f _x =0.Communicated by Edward B. Saff.     

The existence of non-trivial hyperfactorizations ofK 2n

A -hyperfactorization ofK _2n is a collection of 1-factors ofK _2n for which each pair of disjoint edges appears in precisely of the 1-factors. We call a -hyperfactorizationtrivial if it contains each 1-factor ofK _2n with the same multiplicity (then =(2n–5)!!). A -hyperfactorization is calledsimple if each 1-factor ofK _2n appears at most once. Prior to this paper, the only known non-trivial -hyperfactorizations had one of the following parameters (or were multipliers of such an example)

Comparative cooperative game theory

Given two side-payment gamesv andw, both defined for the same finite player-setN, the following three welfare criteria are characterized in terms of the datav andw: (A) For everyy C(w) there existsx C(v) such thatyx; (A) For everyxC(v) there existsyC(w) such thatyx; and (B) There existyC(w) andxC(v) such thatyx. (HereC(v) denotes the core ofv.) Given two non-side-payment gamesv andw, sufficient conditions for the criteria (A) and (B) are established, by observing that an ordinal convex game has a large core.In memory of my teacher in Japan, Professor Ryuichi Watanabe, 1928–1986.     

On multivariate polynomial interpolation

We provide a map which associates each finite set in complexs-space with a polynomial space from which interpolation to arbitrary data given at the points in is possible and uniquely so. Among all polynomial spacesQ from which interpolation at is uniquely possible, our is of smallest degree. It is alsoD- and scale-invariant. Our map is monotone, thus providing a Newton form for the resulting interpolant. Our map is also continuous within reason, allowing us to interpret certain cases of coalescence as Hermite interpolation. In fact, our map can be extended to the case where, with eachgq, there is associated a polynomial space P, and, for given smoothf, a polynomialqQ is sought for which

A test of convergence of Fourier series with respect to multiplicative systems,analogous to the Jordan test

, {p _n} _n=0 (p₀=1, _n2 n2). : x f(t) V(G)