§1. Introduction. Let two probability spaces (X, , μ,) and (Y, ℬ, ν) be given. For a subset D of X × Y and a real number d ≥ 0 we consider the following problem

(MP) Does there exist a measure » on X × Y having μ and ν as marginals and such that λ (D) ≥ 1 − d?

This problem comes from Strassen's paper [12], where Borel probabilities on Polish spaces were treated. Further, it was investigated by many authors in more general settings (cf. [2], [4]-[7], [11]-[13]).

Let G be a graph with b+v vertices, each of the b vertices P1, …, Pb having valency k and each of the remaining v vertices Q1 …, Qv having valency r, and each edge joining a vertex Pm to a vertex Qn. Suppose also that b≥v; then r≥k since bk = vr.

This paper is concerned with diffusion into a turbulent atmosphere from an infinite ground level line source at right angles to the direction of the mean wind velocity. A solution is obtained for a mechanism which takes into account diffusion in the direction of the velocity, and the predictions of the solution are found to be in good agreement with experimental data in adiabatic atmospheric conditions.

Barban and Vehov posed in [1] the problem of minimizing the quadratic form

under the conditions

where 1 ≤ z1 < z2 and μ(n) is the Möbius function. They also commented on the connections of this problem with zero-density estimates of L-functions near the lines δ = 1 and ½, and with Linnik's prime number theorem. This program was carried out some years later by other authors, in the first place by Selberg [10], and later by Motohashi ([7]–[9]), Graham [2] and the present author ([4], [5]).

We study a kernel function of the twisted symmetric square $L$-function of elliptic modular forms. As an application, several exact special values of the $L$-function are computed.

In 1940 Paul Erdős made a conjecture about the distribution of reduced residues. Here we study the distribution of $k$-tuples of reduced residues.

We investigate the problem of the decomposition of balls into a finite number of congruent pieces in dimension $d=2k$. In addition, we prove that the $d$-dimensional unit ball $B_{d}$ can be divided into a finite number of congruent pieces if $d=4$ or $d\geqslant 6$. We show that the minimal number of required pieces is less than $20d$, if $d\geqslant 10$.

This article considers a family of functionals J to be maximized over the planar convex sets K for which the perimeter and Steiner point have been fixed. Assuming that J is the integral of a positive quadratic expression in the support function h and its derivative, the maximizer is always either a triangle or a line segment (which can be considered as a collapsed triangle). Among the concrete consequences of the main theorem is the fact that, given any convex body K1 of finite perimeter, the set in this class that is farthest away in the sense of the L2 distance is always a line segment. The same property is proved for the Hausdorff distance.

We study the number of non-trivial simple zeros of the Dedekind zeta-function of a quadratic number field in the rectangle $\{\unicode[STIX]{x1D70E}+\text{i}t:0<\unicode[STIX]{x1D70E}<1,0<t<T\}$. We prove that such a number exceeds $T^{6/7-\unicode[STIX]{x1D700}}$ if $T$ is sufficiently large. This improves upon the classical lower bound $T^{6/11}$ established by Conrey et al [Simple zeros of the zeta function of a quadratic number field. I. Invent. Math.86 (1986), 563–576].

We show that if a permutation $\unicode[STIX]{x1D70B}$ contains two intervals of length 2, where one interval is an ascent and the other a descent, then the Möbius function $\unicode[STIX]{x1D707}[1,\unicode[STIX]{x1D70B}]$ of the interval $[1,\unicode[STIX]{x1D70B}]$ is zero. As a consequence, we prove that the proportion of permutations of length $n$ with principal Möbius function equal to zero is asymptotically bounded below by $(1-1/e)^{2}\geqslant 0.3995$. This is the first result determining the value of $\unicode[STIX]{x1D707}[1,\unicode[STIX]{x1D70B}]$ for an asymptotically positive proportion of permutations $\unicode[STIX]{x1D70B}$. We further establish other general conditions on a permutation $\unicode[STIX]{x1D70B}$ that ensure $\unicode[STIX]{x1D707}[1,\unicode[STIX]{x1D70B}]=0$, including the occurrence in $\unicode[STIX]{x1D70B}$ of any interval of the form $\unicode[STIX]{x1D6FC}\oplus 1\oplus \unicode[STIX]{x1D6FD}$.

For a probability measure μ on a real separable Hilbert space H, we are interested in “volume-based” approximations of the d-dimensional least squares error of μ, i.e., least squares error with respect to a best fit d-dimensional affine subspace. Such approximations are given by averaging real-valued multivariate functions which are typically scalings of squared (d+1)-volumes of (d+1)-simplices in H. Specifically, we show that such averages are comparable to the square of the d-dimensional least squares error of μ, where the comparison depends on a simple quantitative geometric property of μ. This result is a higher dimensional generalization of the elementary fact that the double integral of the squared distances between points is proportional to the variance of μ. We relate our work to two recent algorithms, one for clustering affine subspaces and the other for Monte-Carlo singular value decomposition based on volume sampling.

We obtain explicit inversion formulas for the Radon-like transform that assigns to a function on the unit sphere the integrals of that function over hemispheres lying in lower-dimensional central cross-sections. The results are applied to the determination of star bodies from the volumes of their central half-sections.

Let Q be an infinite set of positive integers. Denote by Wτ,n(Q) (respectively, Wτ,n) the set of points in dimension n≥1 that are simultaneously τ-approximable by infinitely many rationals with denominators in Q (respectively, in ℕ*). When n≥2 and τ>1+1/(n−1) , a non-trivial lower bound for the Hausdorff dimension of the liminf set Wτ,n ∖Wτ,n (Q) is established in the case where the set Q satisfies some divisibility properties. The computation of the actual value of this Hausdorff dimension and the one-dimensional analogue of the problem are also discussed.

In this note it is shown that

(i) there is a plane convex set of minimal width λ which does not contain any convex set whose width is constant and greater than λ/(3—31/2),

(ii) every plane convex set of minimal width λ contains a convex set whose width is constant and is equal to λ/(3—31/2).

In recent papers the present authors considered the effects of small cross-flow on the evolution of two unequal oblique waves. In these studies the relative size of the crossflow meant that a diffusion (or buffer) layer was required around the critical layer to smooth out the algebraic growth in the mean-flow distortion generated by the nonlinear critical-layer interactions. The present analysis increases the cross-flow to an order of magnitude such that the buffer and critical layers coalesce. In this instance the nonlinear critical layer contains viscous as well as nonequilibrium effects. The resulting amplitude equations are solved for perturbations initiated at a fixed station in the flow.

An internal wave motion, below a layer of uniform fluid, induces a weak current on the free surface in the form of a long wave with phase velocity cI. A uniform progressive train of surface waves, whose wave-length is much shorter than that of the current is incident on it from infinity and undergoes modification. In particular, when the group velocity cg of the progressive wave is equal to cI, the resonance takes place and then, even though the amplitude of the current is small, the interaction builds up near a number of its wavelengths until the train of surface waves is significantly modified. The equations governing the modifications are derived, using the method of multiple scales, and the roles of the Döppler shift and the radiation stress in resonant situations are elucidated. Three-dimensional interactions are discussed and an analogy is drawn between the fundamental equation describing the interactions and Schrödinger's equation.

Let s1, s2, … denote the squarefree numbers in ascending order. In [1], Erdős showed that, if 0 ≤ γ ≤ 2, then

where B(γ) is a function only of γ. In 1973 Hooley [4] improved the range of validity of this result to 0 ≤ γ ≤ 3, and then later gained a further slight improvement by a method he outlined at the International Number Theory Symposium at Stillwater, Oklahoma in 1984. We have, however, independently obtained the better improvement that (1) holds for

in contrast to the range

derived by Hooley. The main purpose of this paper is to substantiate our new result. Professor Hooley has informed me that there are similarities between our methods as well as significant differences.

A linear second order partial differential equation with variable coefficients is considered. The equation is relevant in a number of physical situations. Simple general solutions are obtained subject to the coefficients satisfying certain constraints.

The equations governing the evolution of the modulated amplitude of a pointcentred disturbance to a slightly supercritical flow are shown to have solutions which become infinite at a finite time and at a single point. The amplitude develops a sharp peak and the structure of this peak is found, for real and complex coefficients in the governing equations. Such a solution can only occur if the coefficients satisfy certain conditions

In the study of heat transfer to fluids flowing in pipes or channels, the inversion of a Fourier transform requires consideration of the zeros of a certain function. The principal contribution here is a proof that these zeros are all real and simple, properties assumed by previous authors.