The interplay between symmetries, conservation laws, and variational principles is a rich and varied one and extends well beyond the classical Noether's theorem. Recall that Noether's first theorem asserts that to every r dimensional Lie algebra of (generalized) symmetries of a variational problem there are r conserved quantities for the corresponding Euler-Lagrange equations. Noether's second theorem asserts that infinite dimensional symmetry algebras (depending upon arbitrary functions of all the independent variables) lead to differential identities for the Euler-Lagrange equations.

1. The velocity distribution in fluid of small viscosity, or at large Reynolds number, in the laminar boundary layer associated with a thin flat plate along the main direction of flow, has been worked out by Blasius on the basis of the Prandtl theory. These original investigations covered the region from the forward edge of the plate to the downstream edge. In a recent paper Goldstein investigated the velocity distribution in the wake immediately behind the downstream edge over a region within about 0·5l from it, l being the length of the plate. Later

Tollmien obtained a first approximation to the flow at a great distance down-stream. Goldstein § extended Tollmien's solution by finding a second approximation and by joining it up with his previous work on the flow very near the plate.

The aim of this note is to answer in the negative a question of W. -D. Geyer, asked at the 1983 Group Theory Meeting in Oberwolfach: Is a maximal abelian subgroup A of a free profinite group F necessarily isomorphic to , the profinite completion of

Cluster categories and cluster algebras encode two dimensional structures. For instance, the Auslander–Reiten quiver of a cluster category can be drawn on a surface, and there is a class of cluster algebras determined by surfaces with marked points.

Cluster characters are maps from cluster categories (and more general triangulated categories) to cluster algebras. They have a tropical shadow in the form of so-called tropical friezes, which are maps from cluster categories (and more general triangulated categories) to the integers.

This paper will define higher dimensional tropical friezes. One of the motivations is the higher dimensional cluster categories of Oppermann and Thomas, which encode (d + 1)-dimensional structures for an integer d ⩾ 1. They are (d + 2)-angulated categories, which belong to the subject of higher homological algebra.

We will define higher dimensional tropical friezes as maps from higher cluster categories (and more general (d + 2)-angulated categories) to the integers. Following Palu, we will define a notion of (d + 2)-angulated index, establish some of its properties, and use it to construct higher dimensional tropical friezes.

One of the classical models of applied probability is that which may be described as the covering of a line with a random collection of intervals of random length. It appears as such in ((9), pp. 23–25), but also as the problem of Type II counters with random dead time (1, 4, 10), as a model for a pedestrian crossing a busy road(5), and as the queue M/G/∞. If an excuse is required for returning to a problem which has been the object of so much research in the past, it is to be found in the fact, noted by Kendall (5), that the model gives rise to the general stable infinitely divisible p-function, and that the theory of the semigroup P of standard p-functions (6) requires deeper information than previous, practically motivated, studies have provided.

This note is concerned with the problem of the slow motions of an inviscid, incompressible rotating fluid, and in particular with the motion of a sphere along the axis of rotation. This problem was studied recently by Stewartson (2), who overcomes the principal mathematical difficulty, viz. that of formulating the problem in a coordinate system in which the appropriate differential equation can be solved simply and in which the sphere is a coordinate surface, by a very elegant transformation of independent variables. Stewartson, however, uses inappropriate initial conditions. It is the purpose of this note to discuss the question of initial conditions in the light of results previously obtained by the writer (1).

The class of Toeplitz operators has attracted the attention of several mathematicians and plays an important part in operator theory and related fields. Here we have a special interest in connection with the theory of shift operators, Toeplitz operators, and Hardy classes of vector and operator valued functions as in [12].

We establish a connection between gaps problems in Diophantine approximation and the frequency spectrum of patches in cut and project sets with special windows. Our theorems provide bounds for the number of distinct frequencies of patches of size r, which depend on the precise cut and project sets being used, and which are almost always less than a power of log r. Furthermore, for a substantial collection of cut and project sets we show that the number of frequencies of patches of size r remains bounded as r tends to infinity. The latter result applies to a collection of cut and project sets of full Hausdorff dimension.

The free product A* B of groups A and B can be described in two ways.

We can construct the set of reduced words in A and B. Define a binary operation on by concatenating two words and performing as many reductions as possible. Prove that is a group; the difficult step is the proof of associativity. Define A * B = .

We use Young's raising operators to introduce and study double theta polynomials, which specialize to both the theta polynomials of Buch, Kresch, and Tamvakis, and to double (or factorial) Schur S-polynomials and Q-polynomials. These double theta polynomials give Giambelli formulas which represent the equivariant Schubert classes in the torus-equivariant cohomology ring of symplectic Grassmannians, and we employ them to obtain a new presentation of this ring in terms of intrinsic generators and relations.

A JB-algebra A is a real Jordan algebra, which is also a Banach space, the norm in which satisfies the conditions that

and

for all elements a and b in A. It follows from (1.1) and (l.2) that

for all elements a and b in A. When the JB-algebra A possesses an identity element then A is said to be a unital JB-algebra and (1.2) is equivalent to the condition that

for all elements a and b in A. For the general theory of JB-algebras the reader is referred to (2), (3), (7) and (10).

A method of deriving power series expansions for the correlation functions of lattice systems is described. The concept of lattice constants for rooted graphs is introduced and it is shown how the correlation functions can be expanded in terms of the lattice constants for connected graphs only. The weight functions in the expansions depend on correlation functions for finite systems and may, therefore, be determined by computer methods. This work is an extension of a method previously developed for the free energy. Applications to spin systems, percolation problems and the lattice gas are considered and sum rules for the weight functions are given.

In this note (cf. sections 3, 4) I shall give an axiomatization of those fields (of characteristic ≠ 2) which have a theory of quadratic forms like the -adic numbers or like the real numbers. This leads then, for instance, to a generalization of the well-known theorems on -adic forms to a wider class of fields, including non-local ones. The main purpose of the exercise is, however, to separate out the roles of the arithmetic in the underlying field, on the one hand, which solely enters into the verification of the axioms, and of the ordinary algebra of quadratic forms on the other hand. The resulting clarification of the structure of the theory is of interest even in the known -adic case.

An integral transformation is denned over a finite interval of the time domain. When the Laplace transform exists, the finite transform yields identical results. However, the finite transform is found to be considerably more general than the Laplace transform. It permits consideration of functions which are not of exponential order, leads to a simple scheme to determine system response, and is applicable to boundary-value problems.

The quantum theory of the heavy electron field is applied to calculate approximately the cross-sections for “photoelectric” absorption of heavy electrons by a bound nuclear particle and for emission of a heavy electron by a free nuclear particle on collision with a nucleus. The absorption cross-section per nuclear particle is of the order 10−26 cm.2 for heavy electrons with energies up to 108 e.V., but probably decreases rapidly at higher energies. Although the energy loss of a proton, of energy 108 e.V. or higher, due to heavy electron emission is found to be greater than that due to radiation, the cross-section is very small (⋍ 10−29 cm.2) and the phenomenon is unlikely to be observed in a cloud chamber. In the present state of theory it is impossible to decide whether radiative emission of heavy electrons by nuclear particles is capable of contributing appreciably to the heavy electrons observed at sea-level but, if so, the cross-section for the process must be very much greater at high energies than in the non-relativistic energy region which we have investigated.

1. The main object of this note is to evaluate two definite integrals involving the product of Meijer's G-function with Gauss's hypergeometric function. The first result established in this paper is the extension of the results recently given by Saxena ((3), page 490) in these proceedings and includes both of his results given there as particular cases.

Throughout this paper, the concept of a ‘freely generated algebra’ will be meant in the sense of (3), Definition 2·1. It is understood that all conventions concerning terminology and notation introduced in (3) hold in this paper. In particular, it is understood that the sign denotes a freely generated algebra, and that the definition-set and the free basis of are denoted by C and D respectively. The letter q will denote a cardinal number.

1. Let G be a compact metric 0-dimensional Abelian group. Its dual or character group Γ is a discrete countable torsion group. We denote elements of G by x, of Γ by y, the value of the character y at x by (x, y), and the Fourier transform of f by

We present a constructive proof of Gelfand duality for C*-algebras by reducing the problem to Gelfand duality for real C*-algebras.