Following an example discovered by John Berge [2], we show that there is a 4-component link L ⊂ (S1 × S2)#(S1 × S2) so that, generically, the result of Dehn surgery on L is a 3-manifold with two inequivalent genus 2 Heegaard splittings, and each of these Heegaard splittings is of Hempel distance 3.

The potential barrier governed by a second-order ordinary differential equation has been studied for decades, both exactly and approximately. These concepts are generalized so as to be applicable to differential equations of arbitrary even order. The barrier is defined, and the approximate solutions both inside and outside are derived. Transmission through this barrier is investigated by deriving connexion formulae through the two transition points of arbitrary odd order. The transmitted solution is linked through the barrier to the incident and reflected solutions in a unique manner, enabling an approximate transmission coefficient to be calculated and its structure to be ascertained. Every concept in the investigation is a suitable generalization of the elementary case, the generalization possible being a product of the approximate processes adopted and how they limit the nature of the generalization. An inspection of the elementary case and its transmission coefficient gives no hint as to the nature of the generalization, the limitations necessarily imposed, and the structure of the final transmission coefficient, since the elementary case has special features not present in the generalization.

This paper contains an attempt, begun several years ago and only partially successful, to do for the canonical curve of genus five something similar to what W. P. Milne had done for that of genus four. A fundamental feature of Milne's work was the use of a rational normal curve of order three drawn through the six points of contact of a quadric with the sextic curve; here it is not in general possible to pass a rational curve of order four through the eight points of contact. (Exceptionally it may be possible to do so and then the development follows much the same lines as in Milne's paper: only the general case is discussed in what follows.)

Let N ⊆ M be a pair of closed smooth manifolds and L an algebraic model for the submanifold N. In this paper, we will give an obstruction to finding an algebraic model X of M so that the submanifold N corresponds in X to an algebraic subvariety isomorphic to L.

1. Introduction. In this note, we consider some estimates for the modulus of infinite products of the form

where the lv are near to an arithmetic progression of common difference unity. In a subsequent note, an application will be given of these estimates to the study of a class of Cauchy exponential series which includes, as a special case, the series of Korous(l).

For any topological space X let C(X) be the realization of the singular cubical set of X; let [midast ] be the topological space consisting of one point. In [1] Antolini proves, as a corollary to a general theorem about cubical sets, that C(X) and X×C([midast ]) are homotopy equivalent, provided X is a CW-complex. In this note we give a short geometric proof that for any topological space X there is a natural weak homotopy equivalence between C(X) and X×C([midast ]).

The Gliding of a Plate on a Stream of Finite Depth*

Page 597. The line after equation (44) should read

Proc. Cambridge Phil. Soc. 31 (1935), 589–603.

Let $t:{\mathbb F_p} \to C$ be a complex valued function on ${\mathbb F_p}$. A classical problem in analytic number theory is bounding the maximum

$$M(t): = \mathop {\max }\limits_{0 \le H < p} \left| {{1 \over {\sqrt p }}\sum\limits_{0 \le n < H} {t(n)} } \right|$$

$(1/\sqrt p )\sum\nolimits_{0 \le n < H} {t(n)} $

$$M(t) \le {\left\| {\hat t} \right\|_\infty }\log 3p,$$

$\hat t:{\mathbb F_p} \to \mathbb C$

$\varepsilon > 0$

$a \in \mathbb F_p^ \times $

$${\rm{k}}{1_{a,1,p}}:x \mapsto e((ax + \bar x)/p)$$

$$M({\rm{k}}{1_{a,1,p}}) \ge \left( {{{1 - \varepsilon } \over {\sqrt 2 \pi }} + o(1)} \right)\log \log p,$$

$p \to \infty $

${\{ M({\rm{k}}{1_{a,1,p}})\} _{a \in \mathbb F_p^ \times }}$

Let us say that a set points in a Euclidean space has property (2) if no 2 points X, Y ∈ S have X Y > 1. Then an easy observation is the following:

The maximum area (Lebesgue measure) of a plane set S with property(2) is ¼π, attained only when S is (essentially) a disc.

Let Σan be a given infinite series with the sequence of partial sums {Sn}. Let the sequence be defined by

where is given by

Let [Fscr ] be the category of functors from the category of finite-dimensional [ ]2-vector spaces to [ ]2-vector spaces. The concept of &∇tilde;-nilpotence in the category [Fscr ] is used to define a ‘dimension’ for the category of analytic functors which has good properties. In particular, the paper shows that the tensor product F [otimes ] G of analytic functors which are respectively &∇tilde;s and &∇tilde;t nilpotent is &∇tilde;s+t − 1-nilpotent.

The notion of &∇tilde;-nilpotence is extended to define a dimension in the category of unstable modules over the mod 2 Steenrod algebra, which is shown to coincide with the transcendence degree of an unstable Noetherian algebra over the Steenrod algebra.

In recent years Cantor manifolds have assumed great importance in topology. This is due chiefly to the work of Urysohn and to the later work of Alexandroff. Urysohn, in the paper in which he introduced the theory of dimensions, showed that a 2-dimensional closed surface is a 2-dimensional Cantor manifold. Alexandroff has not only extended this result to n dimensions but has also established the existence on the cutting set Z of the surface F of generalized cycles homologous to zero on the various parts of F − Z. These results show that the deepest structural characteristics of the closed surface come to light when one dissects it.

KMS states of a twisted convolution algebra of Schwartz functions on a vector group are classified and related to KMS states of twisted L1-algebras for certain subquotients. The KMS states for the subquotient algebras are also related to Fock states of vector groups. In the particular case of the subquotient Tn × ℤn of ℚ2n this links the Fock space construction of the theta functions with their appearance in KMS states of loop groups and in the Kac character formula.

The separation dimension of a graph G is the minimum positive integer d for which there is an embedding of G into ℝd, such that every pair of disjoint edges are separated by some axis-parallel hyperplane. We prove a conjecture of Alon et al. [SIAM J. Discrete Math. 2015] by showing that every graph with maximum degree Δ has separation dimension less than 20Δ, which is best possible up to a constant factor. We also prove that graphs with separation dimension 3 have bounded average degree and bounded chromatic number, partially resolving an open problem by Alon et al. [J. Graph Theory 2018].

A group K is said to be a B-group if every permutation group containing K as a regular subgroup is either imprimitive or 2-transitive. In the second edition of his influential textbook on finite groups, Burnside published a proof that cyclic groups of composite prime-power degree are B-groups. Ten years later, in 1921, he published a proof that every abelian group of composite degree is a B-group. Both proofs are character-theoretic and both have serious flaws. Indeed, the second result is false. In this paper we explain these flaws and prove that every cyclic group of composite order is a B-group, using only Burnside’s character-theoretic methods. We also survey the related literature, prove some new results on B-groups of prime-power order, state two related open problems and present some new computational data.

We obtain upper and lower bounds of the same order of magnitude for the error between the distribution of a sum of independent and identically distributed random variables, and a normal approximation by a portion of a Chebychev-Cramér series. Our results are sufficiently general to contain the familiar characterizations by Ibragimov(4), Heyde and Leslie (3) and Lifshits(5), and complement some of those obtained earlier by the author (2).

It was proved by Littlewood that, for every large positive T, ζ (s) has a zero β + iγ satisfying

where A is an absolute constant.

We establish the existence and uniqueness of a factorization for geometric morphisms that generalizes the pure, complete spread factorization for geometric morphisms with a locally connected domain. A complete spread with locally connected domain over a topos is a geometric counterpart of a Lawvere distribution on the topos, and the factorization itself is of the comprehensive type. The new factorization removes the topologically restrictive local connectedness requirement by working with quasicomponents in topos theory. In the special case when the codomain topos of the geometric morphism coincides with the base topos, the factorization gives the locale of quasicomponents of the domain topos, or its ‘0-dimensional’ reflection.

The commutators of the infinitesimal operators of the n-dimensional rotation group Rn with vector operators under Rn are expressed in a vectorial notation. The infinitesimal operators for the representations (l 0…0) are treated in detail. Shift operators for l are constructed and are used to derive the branching rule for these representations. The energy levels and degeneracy of bound states of a particle under an inverse square force in n dimensions are found by wave mechanics and by expressing the Hamiltonian in terms of Casimir's operator for Rn+1. Differential operators which transform one radial wave function into another are obtained.