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.

Necessary and sufficient conditions for Abel's integral equation to have a solution have been given by Tamarkin [18]. The form of the solution was obtained by Abel [1]. The corresponding integral equation for an infinite range of integration was introduced by Liouville [12], who found a solution in a restricted class of cases. In the present paper, we find necessary and sufficient conditions for Liouville's equation to have a solution, and also give the form of the solution.

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

Let I be an ideal of a Noetherian ring R. The purpose of this paper is to study the relationship between the vanishing of the local cohomology modules , and the comparison of the topologies defined by the I-adic {In}n≥0, the symbolic {I(n)}n≥0 and the integral filtration

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.

We consider the spectral function

associated with the Sturm-Liouville equation

in situations where q(x) →−∞ as x → ∞ and (1.1) is in the Weyl limit-point case at ∞. As usual, q is real-valued and locally integrable in [0, ∞], and our particular concern is where q(x) has the form

where c (>0) is a parameter, s and p are non-negative on [0, ∞], p(x) → ∞ and p(x) = 0{s(x) } as x → ∞. As the boundary condition at x = 0, we take the Dirichlet condition y(0) = 0 for convenience: we can equally take the Neumann condition y′ (0) = 0 or generally a1y(0) + a2y′ (0) = 0 with real a1 and a2.

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.

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).

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.

The theorem proved in this paper is basic for a general intersection theory applicable to multigraded polynomial rings. When the polynomial ring is graded by the non-negative integers the facts, in one form or another, are well known, but on passing to more general gradings fresh complications appear. These are not wholly trivial and, as the author was unable to find an account of these matters in the literature, it may be of interest if one is given here.

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.

Laplace operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of types ${\it\delta}$ and ${\it\delta}^{\prime }$. Assuming rational independence of edge lengths, necessary and sufficient conditions for isospectrality of two Laplacians defined on the same graph are derived and scrutinized. It is proved that the spectrum of a graph Laplacian uniquely determines matching conditions for “almost all” graphs.

We find the best asymptotic lower bounds for the coefficient of the leading term of the ${L}_{1} $ norm of the two-dimensional axis-parallel discrepancy that can be obtained by Roth’s orthogonal function method among a large class of test functions. We use methods of combinatorics, probability, and complex and harmonic analysis.

We generalize a method of Conrey and Ghosh [Simple zeros of the Ramanujan $\unicode[STIX]{x1D70F}$-Dirichlet series. Invent. Math. 94(2) (1988), 403–419] to prove quantitative estimates for simple zeros of modular form $L$-functions of arbitrary conductor.

Let $S=\{q_{1},\ldots ,q_{s}\}$ be a finite, non-empty set of distinct prime numbers. For a non-zero integer $m$, write $m=q_{1}^{r_{1}}\cdots q_{s}^{r_{s}}M$, where $r_{1},\ldots ,r_{s}$ are non-negative integers and $M$ is an integer relatively prime to $q_{1}\cdots q_{s}$. We define the $S$-part $[m]_{S}$ of $m$ by $[m]_{S}:=q_{1}^{r_{1}}\cdots q_{s}^{r_{s}}$. Let $(u_{n})_{n\geqslant 0}$ be a linear recurrence sequence of integers. Under certain necessary conditions, we establish that for every $\unicode[STIX]{x1D700}>0$, there exists an integer $n_{0}$ such that $[u_{n}]_{S}\leqslant |u_{n}|^{\unicode[STIX]{x1D700}}$ holds for $n>n_{0}$. Our proof is ineffective in the sense that it does not give an explicit value for $n_{0}$. Under various assumptions on $(u_{n})_{n\geqslant 0}$, we also give effective, but weaker, upper bounds for $[u_{n}]_{S}$ of the form $|u_{n}|^{1-c}$, where $c$ is positive and depends only on $(u_{n})_{n\geqslant 0}$ and $S$.

We transpose the parametric geometry of numbers, recently created by Schmidt and Summerer, to fields of rational functions in one variable and analyze, in that context, the problem of simultaneous approximation to exponential functions.

Erdős (see [4]) asked if there are infinitely many integers k which are not a difference or a sum of two powers, i.e., if there are infinitely many positive integers k with k≠|um ± vn| for u, v, m, n ε ℤ. This is certainly a very difficult problem. For example, it is known that the Catalan equation, i.e., the equation um − vn = 1 with uv≠0 and min (m, n)≥2 has only finitely many solutions (u, v, m, n), but there is no other positive integer k≥1 for which it is known that the equation

has only finitely many solution (u, v, m, n) with min (m, n)≥2.

In this note we consider a problem which is suggested by a paper of W. Feller. A plane set A lies in a circle C of centre O and radius 1, and is such that the linear measure of the intersection of A with any straight line does not exceed 21, where 0 < l < 1. To find the upper bound of the plane measure of A. Both linear measure and plane measure are taken in the sense of Lebesgue, and they will be denoted by m and m2 respectively.

A right S-system over a monoid S is a set A on which S acts unitarily on the right. That is, there is a function A such that (φ,1)φ and (a, st)φ = ((a, s) t)φ for all a є A and for all s, t є S. We shall refer to right S-systems simply as S-systems. It is clear what is meant by S-homomorphism, S-subsystem etc.; further details of the terms used in this Introduction are given in Section 2.