We give a short proof of the uniqueness of finite spherical buildings of rank at least 3 in terms of the structure of the rank 2 residues and use this result to prove a result making it possible to identify an arbitrary finite group of Lie type from knowledge of its “parabolic structure” alone. Our proof also involves a connection between loops, “Latin chamber systems” and buildings.

The aim of this paper is to prove that the inverse of the normalization functor in the Dold–Kan correspondence $D:{\sf Ch}({\sf Ab})\rightarrow {\sf sAb}$ is an $E_{\infty}$-monoidal functor. This proves that generalized Eilenberg–MacLane spectra on differential graded commutative algebras are $E_{\infty}$-monoids in the category of $H{\bb Z}$-module spectra.

We give a locally minimal, but not globally minimal, bridge position of a knot, that is, an unstabilized, nonminimal bridge position of a knot. It implies that a bridge position cannot always be simplified so that the bridge number monotonically decreases to the minimal.

In (2) Lennox and Stonehewer proved:Let G be a group with subnormal subgroups H and K, and let J = 〈H, K〉. Suppose that J/J′ has finite rank. Then

(i) given positive integers a and b, there exists c such that γc(J) ≤ γa(H). γb(K) and

(ii) J is subnormal in G.

We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups: ${\mathfrak{A}}_5$ , ${\text{PSL}}_2(\textbf{F}_7)$ , ${\mathfrak{A}}_6$ , ${\text{SL}}_2(\textbf{F}_8)$ , ${\mathfrak{A}}_7$ , ${\text{PSp}}_4(\textbf{F}_3)$ , ${\text{SL}}_2(\textbf{F}_{7})$ , $2.{\mathfrak{A}}_5$ , $2.{\mathfrak{A}}_6$ , $3.{\mathfrak{A}}_6$ or $6.{\mathfrak{A}}_6$ . All of these groups with a possible exception of $2.{\mathfrak{A}}_6$ and $6.{\mathfrak{A}}_6$ indeed act on some rationally connected threefolds.

The ordinary one-dimensional wave equation

has special integrals of the form

which satisfy the first-order equations

respectively, and are often called progressive waves, or progressive integrals, of (1·1). The straight lines

in an xt-plane are the characteristics of (1·1). It follows from (1·2) that progressive integrals of (1·1) are constant on some particular characteristic, and are characterized by this property.

The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.

A set is called n-generic if it is Cohen generic for n-quantifier arithmetic. A (Turing) degree is n-generic if it contains an n-generic set. Our interest in this paper is the relationship between n-generic (indeed 1-generic) degrees and minimal degrees, i.e. degrees which are non-recursive and which bound no degrees intermediate between them and the recursive degree. It is known that n-generic degrees and minimal degrees have a complex relationship since Cohen forcing and Sacks forcing are mutually incompatible. The goal of this paper is to show.

This paper is concerned with the problem of the flow of an incompressible electrically conducting fluid along a rectangular duet under a transverse magnetic field. The case in which the walls perpendicular to the field are perfectly conducting and those parallel to the field are non-conducting has been considered by Hunt (1), who derives the solution in two ways; as the limiting cases of the flows with (a) non-conducting walls parallel and thin walls of arbitrary conductivity perpendicular to the field, and (b) thin walls of arbitrary conductivity parallel and perfectly conducting walls perpendicular to the field. We show that these two limiting solutions derived by Hunt are in fact equivalent. In addition, we extend the solution of case (b) above by removing the thin wall restriction.

We study the coefficients of Kac and Wakimoto's character formulas for the affine Lie superalgebras sℓ(n+1|1)∧. The coefficients of these characters are the weight multiplicities of irreducible modules of the Lie superalgebras, and their asymptotic study begins with Kac and Peterson's earlier use of modular forms to understand the coefficients of characters for affine Lie algebras. In the affine Lie superalgebra setting, the characters are products of weakly holomorphic modular forms and Appell-type sums, which have recently been studied using developments in the theory of mock modular forms and harmonic Maass forms. Using our previously developed extension of the Circle Method for products of mock modular forms along with the Saddle Point Method, we find asymptotic series expansions for the coefficients of the characters with polynomial error.

For knots in S3 we obtain new criteria for periodicity. We show that the Casson–Gordon invariants of a periodic knot are preserved under the periodic action lifted to the cyclic covers. As an application, we consider a family of knots with the Seifert form of a period 3 knot, and using Casson–Gordon invariants show that knots in this family do not have period 3. We also obtain periodicity criteria in terms of the homology groups of cyclic branched covers of S3.

A development of the “echo” method of observation of the height of the Kennelly-Heaviside layer is described. Short wave-trains of radiation lasting sec. are transmitted at regular sec. intervals, and observed stroboscopically, together with their echoes, by means of a cathode ray oscillograph. Details of the transmitting and receiving apparatus are given. The rapid fluctuation in amplitude of the echoes is easily studied by this method.

The process of Fronsdal inducing is discussed as a method of quantizing physical systems, and is seen to reduce to a technique for obtaining representations of *-algebras.

It is asked whether this above process is at all successful, and whether it is at all related to the representation inducing methods of Fell. The two inducing techniques are shown to be equivalent for algebras satisfying three specified properties, and it is also shown that all the strictly irreducible representations of such algebras can be found by these methods.

Examples are produced whereby it is shown that algebras of the type required by the theory are of physical interest.

A solution which is exact within the framework of the classical theory of elasticity is obtained for the title problem assuming that the half space is homogeneous and isotropic, and that the die indents at a constant rate. If the shape of the die and the elastic medium are specified, the rate of indentation uniquely determines the outward speed of the edge of the expanding contact zone. The magnitude of this speed, relative to the speeds of the dilatational, rotational and Rayleigh waves in the elastic medium, determines which of four possible characteristic transient stress states will occur. Each of the four ranges of contact speed is solved by the method of rotational superposition of self-similar potentials which is briefly described in the Appendix.

The Jacobi identity for Whitehead products,

where α ɛ πl(X), β ɛ πm(M), γ ɛ πn (X), l, m, n ≥ 2 is now well known. In this note we extend the identity to the case l, m, n > 1; the extension is easy and we are merely filling a small gap in the literature. However, the extension does enable us to improve a result in (2).

The formulae of this paper are chiefly concerned with multiple secants and tangents to surfaces in spaces of dimension four to seven; they are obtained by the functional method of Cayley and Severi, which was first applied to scrolls by C. G. F. James. The majority of the results have already been given for scrolls in James's third paper; but whereas the functional equations in this case involve only two variables, those of the present paper contain four, and the work is considerably more complicated. It is also to be noted that, though the formulae for scrolls might perhaps have been expected to follow as particular examples from the corresponding formulae for surfaces, this is not always the case. The reason for the discrepancy is not always clear, although in some instances theoretical reasons for disagreement can be assigned.

Various authors, e.g. Sommerfeld (8), Bouwkamp(i), Groschwitz and Hönl(2), Müller and Westpfahl(7), have considered approximate solutions to the problem of the diffraction of a plane wave by a narrow perfectly conducting strip. If the electric vector is polarized parallel to the strip they find that the scattering coefficient, for normal incidence, is given by, where k is the wave number, 2b the width of the strip and γ(= 0·5572…) being Euler's constant. Any attempt to discuss the convergence of such a pseudo-power series requires a knowledge of the higher order terms and naturally leads to the question of what is the proper expansion parameter.

When the approximate methods of Bragg and Williams or Bethe are used in theoretical work on cooperative phenomena it is customary to suppose that only the configurational energy of the assembly depends on the molecular arrangement; the contribution which a molecule makes to the partition function, apart from its share in this configurational energy, is supposed to be independent of the positions of the other molecules.

We give non-torsion counterexamples to the integral Tate conjecture over finite fields. We extend the result due to Pirutka and Yagita for prime numbers 2, 3, 5 to all prime numbers.