The results developed by Watson [1] are interpreted to indicate how the slow viscous flow due to the rotation of a small circular cylinder in the presence of a stationary cylinder can be calculated. It is shown how the stream function is given as a combination of the force-free representations corresponding to a line rotlet and a line stokeslet outside the stationary body, plus the streaming flow past the body. The coefficients which multiply these representations are calculated by techniques already described by Watson.

We recall that if S is a d - simplex then each facet and each vertex figure of S is a (d − 1)-simplex and S is a self-dual. We introduce a d-polytope P, called a d-multiplex, with the property that each facet and each vertex figure of P is a (d − 1)-multiplex and P is self-dual.

A tiling of a convex m-gon by a finite number r of convex n-gons is said to be of type <m, n, r>. The Main Theorem of this paper gives necessary and sufficient conditions on m, n and r for a tiling of type <m, n, r> to exist.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{F}_q$ be the finite field of $q$ elements. An analogue of the regular continued fraction expansion for an element $\alpha $ in the field of formal Laurent series over $\mathbb{F}_q$ is given uniquely by

$$\begin{equation*} \alpha = A_0(\alpha )+\cfrac {1}{A_1(\alpha )+\cfrac {1}{A_2(\alpha )+\ddots }}, \end{equation*}$$

$(A_n(\alpha ))_{n=0}^\infty $

$\mathbb{F}_q$

$\deg (A_n(\alpha ))\ge 1$

$n\ge 1.$

$(p_n)_{n=1}^\infty $

$$\begin{equation*} \lim _{n\to \infty }\frac {1}{n}\sum _{j=1}^n \deg (A_{p_j}(\alpha )) = \frac {q}{q-1} \end{equation*}$$

$\alpha $

$(p_n)_{n=1}^\infty $

$(n)_{n=1}^\infty ,$

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 $d\geqslant 10$.

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].

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).

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.

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.

In this paper we prove two analogues of the following theorem:

If is a collection of distinct subsets of {1, …, m} such that , i ≠ j ⇒ Ai ∩ Aj ≠ ∅, Ai ∪ Aj ≠ {1, …, m} then .

Let a1 …, ar be elements of a commutative ring R with identity. When the R-length of

is finite and R is Noetherian the multiplicity e(a1 …, ar\R) can be defined in one of three ways. Starting either from the earliest definition by means of Hilbert functions (see Samuel [4]) or from the latest by induction on r(see Wright [5]) it can be shown that e(a1 …, ar\R) depends only on the ideal

and the number r of generators. The other definition due to Auslander and Buchsbaum [1] is as the Euler-Poincare characteristic of a Koszul complex. The main purpose of this note is to investigate the extent to which the separate homology modules and the Koszul complex itself are independent of the sequence a1 …, ar. First it is shown that when A = (a1, …, ar) and B = (b1 …, br) are related by a reversible R-linear transformation the two Koszul complexes are isomorphic. A special case of the final theorem states that if A and B are minimal generating sets of the same ideal and R is a local ring then again the Koszul complexes are isomorphic.

In the present note we show that the elements of GF(q2) (q = 2n) can be represented in “polar form” in such a way that GF(q2) acts like an “Argand diagram” over its “real subfield” GF(q). From this polar representation it is easy to develop a trigonometry of the plane GF(q2), including definitions of circles and orthogonality. As an application of these ideas we show, in §4, that the circles and lines orthogonal to a given circle yield a new model satisfying Graves' axioms for finite homogeneous hyperbolic planes.

Minkowski's fundamental theorem and the Minkowski-Hlawka theorem play basic complementary roles in the Geometry of Numbers. Blichfeldt showed essentially that Minkowski's fundamental theorem was a simple consequence of a more general theorem, in which the convex body was replaced by any measurable set and the lattice was replaced by a discrete set of points having a definite asymptotic density. Hlawka himself showed that the Minkowski-Hlawka theorem could be proved in a slightly modified form, when the star body was replaced by any measurable set, but he did not replace the lattice by a more general set of points.