A locally non-trivial holomorphic family of Riemann surfaces over a hyperbolic Riemann surface B determines a homomorphism (monodromy) of the fundamental group of B to the modular group. We give a necessary condition for a homomorphism to be a mondromy of a holomorphic family of Riemann surfaces. And we estimate the number of holomorphic families in terms of topological information for holomorphic families over Riemann surfaces of genus 0.

A limit of rational varieties need not be rational, even when all varieties in the family are projective and have at most terminal singularities.

Given any domain Ω in ℝN we consider spaces X = X(Ω),Y = Y(Ω) which are Banach function spaces in the sense of Luxemburg [12]. From these we form what we call the abstract Sobolev space W(X, Y), which is denned to be the linear space of all ƒ ∈ X such that for i = 1, …, N, the distributional derivative ∂ƒ/∂xi belongs to Y; equipped with the norm

W(X, Y) is a Banach space. The closure of (Ω) in W(X, Y) is denoted by W0(X, Y). Let w to bea weight function on Ω, that is, a measurable function which is positive and finite almost everywhere on Ω. We say that [w, X, Y] supports the (weighted) Poincaré inequality if there is a positive constant K such that for all u ∈ W(X, Y),

analogously, [X, Y] is said to support the Friedrichs inequality if there is a positive constant K such that for all u ∈ W0(X, Y),

By the Cameron–Martin theorem, if a function f is in the Dirichlet space D, then B + f has the same a.s. properties as standard Brownian motion, B. In this paper we examine properties of B + f when fD. We start by establishing a general 0-1 law, which in particular implies that for any fixed f, the Hausdorff dimension of the image and the graph of B + f are constants a.s. (This 0-1 law applies to any Lévy process.) Then we show that if the function f is Hölder(1/2), then B + f is intersection equivalent to B. Moreover, B + f has double points a.s. in dimensions d ≤ 3, while in d ≥ 4 it does not. We also give examples of functions which are Hölder with exponent less than 1/2, that yield double points in dimensions greater than 4. Finally, we show that for d ≥ 2, the Hausdorff dimension of the image of B + f is a.s. at least the maximum of 2 and the dimension of the image of f.

Let K be a field of characteristic zero and suppose that f: → K satisfies a recurrence of the form

\[f(n) = \sum_{i=1}^d P_i(n) f(n-i),\]

Let A be a right Λ-complex and C be a left Λ-complex where Λ is a ring which is both left and right hereditary; a principal ideal domain is a special case of such a ring. This paper is concerned with the Künneth theorem expressing the homology H(A⊗C) of the product complex A⊗C (= A⊗ΛC) in terms of H(A) and H(C). Any notations we do not explain are those of (1).

Some explicit formulae for the right coaction of A* on H*BO are given. As corollaries, closed formulae for χ(ξk) and other elements in A* are given.

For a domain G in the one-point compactification $\overline{\mathbb{R}}^n = {\mathbb{R}}^n \cup \{ \infty\}$ of ${\mathbb{R}}^n, n \geqslant 2$ , we characterise the completeness of the modulus metric $\mu_G$ in terms of a potential-theoretic thickness condition of $\partial G\,,$ Martio’s M-condition [ 35 ]. Next, we prove that $\partial G$ is uniformly perfect if and only if $\mu_G$ admits a minorant in terms of a Möbius invariant metric. Several applications to quasiconformal maps are given.

As a general framework for the studies of t-structures on triangulated categories and torsion pairs in abelian categories, we introduce the notions of extriangulated categories with negative first extensions and s-torsion pairs. We define a heart of an interval in the poset of s-torsion pairs, which naturally becomes an extriangulated category with a negative first extension. This notion generalises hearts of t-structures on triangulated categories and hearts of twin torsion pairs in abelian categories. In this paper, we show that an interval in the poset of s-torsion pairs is bijectively associated with s-torsion pairs in the corresponding heart. This bijection unifies two well-known bijections: one is the bijection induced by the HRS-tilt of t-structures on triangulated categories. The other is Asai–Pfeifer’s and Tattar’s bijections for torsion pairs in an abelian category, which is related to $\tau$ -tilting reduction and brick labelling.

About twenty years ago a number of results were given expressing the sum of n terms of an ordinary hypergeometric series with unit argument in terms of an infinite series of the type 3F2. The interest in the subject was aroused by a theorem due to Ramanujan, who stated that

Let be an oriented knot in S3 and a solid torus endowed with a preferred framing which contains in its interior. By ρ and λ we denote the wrapping and winding numbers of in respectively. That is, they are the geometric and algebraic intersection numbers of and a meridian disk of . For an integer μ, let τμ be an orientation-preserving homeomorphism of satisfying τμ(m) = m and τμ(l) = l + μm in H1(∂), where (m, l) is a meridian-longitude pair of . We call τμ(), denoted by μ, the knot obtained from by μ-twisting along .

Let {Sn} be the sequence of partial sums of the infinite seriesΣαn. Let {pn} be a sequence of constants real or complex and let us set

The sequence {tn} of Nörlund means (5) or simply (N, pn) means of the sequence {Sn} generated by the sequence of coefficients {pn} is defined by the following sequence -to-sequence transformation

The series ∑αn or the sequence {Sn} is said to be summable (N, pn) to the sum S, if

and is said to be absolutely summable (N, pn) or summable |N, pn|, if the sequence {tn} is of bounded variation, that is, the series ∑|tn − tn−1| is convergent (2).

The present note was inspired by the desire to see whether the exposition of the theory of the Segre cubic primal of ten nodes was simplified by using only five coordinates instead of the six redundant coordinates introduced by Stéphanos and Castelnuovo. But the simple remark that the ten nodes may be separated into two simplexes which are polars of one another in regard to a quadric—which may or may not be novel—suggests the comparison of the theory of five associated lines in space of four dimensions with familiar properties of eight associated points in three dimensions; especially as it appears (§ 8) that the ten nodes do not form an associated set. With the repetition, for the sake of clearness, of several results which are familiar in general terms, there seems enough novelty to make the note of some utility. The form found for the equation of the Segre primal in five coordinates (§ 5) seems also noticeable.

A repetition has been made, with the introduction of electrical counting methods, of the experiment of Bothe and Geiger on the simultaneous production of a scattered quantum and of a recoil electron in the scattering of X-rays. Evidence has been obtained that such coincidences do actually occur, in agreement with the results of the original experiment, and with the Compton theory of scattering.

Let Qn be a real indefinite quadratic form in n variables x1, x2,…, xn, of determinant D ≠ 0 and of type (r, s), 0 < r < n, n = r + s. Let σ denote the signature of Qn so that σ = r − s. It is known (see e.g. Blaney(4)) that, given any real numbers c1 c2, …, cn, there exists a constant C depending upon n and σ only such that the inequality

has a solution in integers x1, x2, …, xn. Let Cr, s denote the infimum of all such constants. Clearly Cr, s = Cs, r, so we need consider non-negative signatures only. For n = 2, C1, 1 = ¼ follows from a classical result of Minkowski on the product of two linear forms. When n = 3, Davenport (5) proved that C2, 1 = 27/100. For all n and σ = 0, Birch (3) proved that Cr, r = ¼. In 1962, Watson(18) determined the values of Cr, s for all n ≥ 21 and for all signatures σ. He proved that

Watson also conjectured that (1·2) holds for all n ≥ 4. Dumir(6) proved Watson's conjecture for n = 4. For n = 5, it was proved by Hans-Gill and Madhu Raka(7, 8). The author (12) has proved the conjecture for σ = 1 and all n. In the preceding paper (13) we proved that C5, 1 = 1. In this paper we prove Watson's conjecture for σ = 2, 3 and 4.

The main result of this paper is an oscillation theorem on linear self-adjoint differential systems and a corresponding eigenvalue problem. It establishes a formula between the number of focal points of a so-called conjoined basis of the differential system on a given compact interval and the number of eigenvalues which are less than the given eigenvalue parameter. It extends an earlier result of the author and generalizes an oscillation theorem of M. Morse. Among others the proof of the theorem requires a formula on the index of the difference of symmetric solutions of a corresponding Riccati matrix differential equation. This index formula is the other new result presented.

The basic equations of non-relativistic magnetohydrodynamics are briefly reviewed. Beginning with the energy-momentum tensor for the coupled electromagnetic and fluid fields relativistic energy and momentum equations are formulated. The effect of heat conduction is included and relativistic constitutive equations are discussed. A relativistic first law of thermodynamics is developed in terms of internal energy and entropy. Finally, the second law and entropy production rate are derived in a covariant fashion.

1. The object of this paper is to prove some formulae of Jacobi polynomials including a generating function. The results (2·l)–(2·4), (2·6)–(2·9), (3·l)–(3·4), and (4·1) are believed to be new.

It is well known that a path-connected Hausdorff space is arc-connected. Indeed, given a path P, that is, a continuous map. f of the unit interval I0(0 ≤ t ≤ 1) into the Hausdorff space X, there is an arc joining the points f(0), f(1) (supposed distinct) which is obtained by ‘cutting out loops in P’. More precisely, there exist a continuous increasing map α of I0 onto itself and a homeomorphism φ of I0 onto φ(I0) ⊂ X, such that if [t0, t1] is any maximal interval of constancy of α (including the case t0 = t1) then f(t0) = f(t1) = φοα(t0). The function φ, α, can be defined by a constructive process.

Preface. In order to make this paper intelligible to the reader, I repeat the following definitions and results from my paper ‘On lattice points in star domains’, which is to appear in the Proceedings of the London Mathematical Society.