We prove that the eigenvalues of a certain highly non-self-adjoint operator that arises in fluid mechanics correspond, up to scaling by a positive constant, to those of a self-adjoint operator with compact resolvent; hence there are infinitely many real eigenvalues which accumulate only at ±∞. We use this result to determine the asymptotic distribution of the eigenvalues.

We raise and investigate the following problem which one can regard as a very close relative of the densest sphere packing problem. If the Euclidean 3-space is partitioned into convex cells each containing a unit ball, how should the shapes of the cells be designed to minimize the average surface area of the cells? In particular, we prove that the average surface area in question is always at least

Let $\mathbb{A}=(A,+)$ be a (possibly non-commutative) semigroup. For $Z\subseteq A$, we define $Z^{\times }:=Z\cap \mathbb{A}^{\times }$, where $\mathbb{A}^{\times }$ is the set of the units of $\mathbb{A}$ and

$$\begin{eqnarray}{\it\gamma}(Z):=\sup _{z_{0}\in Z^{\times }}\inf _{z_{0}\neq z\in Z}\text{\text{ord}}(z-z_{0}).\end{eqnarray}$$

${\it\gamma}(\cdot )$

$\mathbb{A}$

$X,Y\subseteq A$

$$\begin{eqnarray}|X+Y|\geqslant \min ({\it\gamma}(X+Y),|X|+|Y|-1).\end{eqnarray}$$

$\mathbb{A}$

${\it\gamma}(X+Y)$

$|S|$

$S$

$\mathbb{A}$

For every integer $k\geqslant 3$ we construct a $k$-gonal curve $C$ along with a very ample divisor of degree $2g+k-1$ (where $g$ is the genus of $C$) to which the vanishing statement from the Green–Lazarsfeld gonality conjecture does not apply.

Given a set D of positive real numbers, let Xn(D) denote the graph with ℝn as the vertex set such that two points are joined if their distance is in D. Bukh conjectured in [Measurable sets with excluded distances. Geom. Funct. Anal.18 (2008), 668–697] that if D is algebraically independent, then Chr(Xn(D)), the chromatic number of Xn(D), is finite. Here we prove that Chr(Xn(D)) is countable and that, if n=2 , even the coloring number is countable. Furthermore, we prove that Chr (Y ) is countable, where Y is the following graph on ℂn: let 𝔽 be a countable subfield of ℂ and let D⊆ℂ be algebraically independent over 𝔽; join a,b∈ℂn if there is some p(x,y)∈𝔽[x,y] such that p(x,x) is identically zero and p(a,b)≠0 is algebraic over some d∈𝔽∪D.

In this paper, we study a finite connected graph which admits a quasi-monomorphism to hyperbolic spaces and give a geometric bound for the Cheeger constants in terms of the volume, an upper bound of the degree, and the quasi-monomorphism.

We give a simple graph-theoretic proof of a classical result due to Nash-Williams on covering graphs by forests. Moreover, we derive a slight generalization of this statement where some edges are preassigned to distinct forests.

1. Certain geometric properties of the valuation theory were considered by O. Zariski in [7]. We have proved some related results in [1] and we consider further similar problems in this paper.

Let V be an irreducible algebraic primal situated in Sd, where d≥3. Throughout the ground field is the field K of complex numbers. For simplicity we assume that V lies in an affine space Ad with coordinates x1,…,xd. Let O be a point on V not at infinity and we take it to be the origin of Ad. Apply a monoidal transformation to V with O as the basis; We obtain thereby a (d−l)-fold V1 lying on a non-singular d-fold U1 situated in an affine space of dimension N1 Since V and V1 are birationally equivalent, we may identify their function fields and thus we denote their common function field by Σ.

Let τ(n; k, h) denote the number of divisors of n which are congruent to h (mod k), and τ(n; k) the number of divisors of n prime to k, so that

Let

Erdős [1] proved that, if ε and η are fixed arbitrary positive numbers, then for almost all integers n ≤ x, we have

provided

Hall and Sudbery [2] showed that it is sufficient that

and apart from the ε, this upper bound for k is best possible, for it is clear that k must not exceed the normal order of τ(n). For n ≤ x, Hardy and Ramanujan [3] showed that this normal order is (log x)log 2 = 2 log log x.

§0. We begin by stating a theorem of Jarník [1]. Let m be a positive integer, 0 < β < m, and Ψ(q) a positive function of integers q > 0. Let E be the set of points (x1, …, xm) for which the inequality

admits infinitely many solutions q ≥ 1. Then, supposing

Jarník proves that E has infinite Hausdorff β-measure.

§1. Preliminaries. A Cauchy process in d-dimensional Euclidean space, Rd, is a stochastic process, Xt(ω), with stationary independent increments and with a continuous transition density, p(t, y − x) defined by

and

where m, the isotropic measure, is a probability measure on Sd, the unit sphere in Rd, such that when d > 1 the support of m is not contained in any d − 1 dimensional subspace. In (2) w is given by

where . It follows that for each t > 0 and y we have p(t, y) > 0 and that for each t > 0 p(t, y) is a bounded and continuous function of y. Xt(ω) can be considered as being a standard Markov process (for a full description of the definition of such a process see Chapter 1 of [1]) and in particular we can assume that the sample functions of Xt(ω) are right continuous and have left limits. We can also assume that Xt(ω) enjoys the strong Markov property. We write Px and Ex for probabilities and expectations conditional on X0(ω) = x, and we write P for P0.

In this note we shall show that the assumptions made by Murray [1] on the initial state of the gas and the initial magnetic field are inconsistent with the steady state momentum equation and the energy balance equation (20). The form of second term in equation (20) requires

Let M be a finite extension field of the rational numbers Q, and let CM denote the ideal class group of M. Let l be a rational prime, and let AM denote the l-class group of M (i.e., the Sylow l-subgroup of CM). If G is any finite abelian l-group, we define

where ℤl is the ring of l-adic integers and Fl is the finite field of l elements. One of the classical results of algebraic number theory is the specification of rank AM when M is a quadratic extension of Q and l = 2. This result was obtained by means of Gauss's theory of genera. A generalization of this result can be found in [1], where A. Fröhlich has obtained upper and lower bounds for rank AM when M is a cyclic extension of Q of degree l. His methods also show how to compute rank AM exactly when l = 3. In [4], G. Gras has described a procedure for analyzing the l-class groups of relatively cyclic extensions of degree l. However when l > 3, the computations can be very difficult.

Let K be an algebraic number field of degree n = rl + 2r2 (in the usual notation) over the rationals with discriminant d. Let ZK denote the ring of integers in K. It is usual to speak of an integer Πi ∈ Zk as an almost-prime of order l, if the principal ideal (Πi) has at most l prime ideal factors, counted according to multiplicity. Let P1, …, Pn be positive real numbers with Pk = Pk+r2, k = r1 + l, …, r1 + r2 and P = P1 … Pn ≥ 1.

The authors regret that a misprint has occurred in the main theorem of their recent paper with the above title [Mathematika, 23 (1976), 220–226]. The congruence condition which p1 should satisfy modulo if p1 has odd order.

Let K0(x) be a simple transcendental extension of a field K0, υ0 be a valuation of K0 with value group G0 and residue field K0. Suppose is an inclusion of totally ordered abelian groups with [G1: G0] < ∞ such that G is the direct sum of G1 and an infinite cyclic group. It is proved that there exists an (explicitly constructible) valuation υ of K0(x) extending υ0 such that the value group of υ is G and its residue field is k, where k is a given finite extension of k0. This is analogous to a result of Matignon and Ohm [2, Corollary 3.2] for residually non-algebraic prolongations of υ0 to K0(x).

Given a sequence a(l), a(2), a(3), … of complex numbers such that a(n) ≤ 0(nc) for some c > 0, we define, for Im(z) > 0,

where q(λ) = exp (2πiz/λ), λ > 0 and a is a real number. Throughout this paper, for complex numbers x, w with x ≠ 0, xw = exp (w log x) and the principal branch is taken for the logarithm. Then it is easily verified that the infinite product converges absolutely and uniformly in every compact subset of the upper half plane H. Hence f(z) is holomorphic in H. The aim of this paper is to determine holomorphic functions in H defined by (1) which satisfy the special transformation formula

for some real number k under certain assumptions.

In this note we present a simple way of obtaining the universal field of fractions of certain free rings as a subfield of an ultrapower of a (by no means unique) skew field. This method of embedding was discovered by Amitsur in [1]; our presentation uses Cohn's specialization lemma and the embedding is constructed in terms of full matrices over the rings in question (Theorem 3.2). In particular, if k is an infinite commutative field, the universal field of fractions of a free k-algebra can be realized as a subfield of an ultrapower of any skew extension of k, with centre k, which is infinite dimensional over k. Thus many problems concerning the universal field of fractions of a free k-algebra can be settled by studying skew extensions of k of relatively simple structure. More precisely and more generally, let E be a skew field with centre k and denote by R the free E-ring on X over k. Write U for the universal field of fractions of R and assume that U embeds in an ultrapower of a skew field D. Then by Łos' theorem, U inherits the first-order properties of D which can be expressed by universal sentences.

An expression is found here for the small transverse displacement of a thin elliptic plate due to a force applied at an arbitrary point of the plate. The plate is in the form of a complete ellipse and is clamped along the boundary. The displacement is expressed in terms of infinite series in §§2–4. The convergence of the series is rapid unless the eccentricity of the ellipse is nearly unity. The simplest case in which the force is applied at the centre of the plate is considered in §5; the displacement of the centre due to this force is compared in §6 with the corresponding displacements of a circular plate and of an infinite strip.

We consider the first boundary value problem for a second-order parabolic equation with variable coefficients in the domain $K\times \mathbb{R}^{n-m}$, where $K$ is an $m$-dimensional cone. The main results of the paper are pointwise estimates of the Green’s function.