This paper concerns the interaction between group theory and classical mechanics in general and with the application of this theory to plasma physics in particular.

Mechanics has had a profound role in the historical development of mathematics. Leading original thinkers in pure mathematics such as Newton, Euler, Lagrange, Jacobi, Laplace, Cauchy, Gauss, Riemann, Poincaré, Hilbert, Birkhofï, Smale and Arnold were also great original thinkers in various facets of mechanics. The interaction between mechanics and pure mathematics remains one of the most active and flourishing areas of current research. It is this flavor which I hope to convey in this lecture.

Dickson polynomials over finite fields are familiar examples of permutation polynomials, i.e. of polynomials for which the corresponding polynomial mapping is a permutation of the finite field. We prove that a Dickson polynomial can be a complete mapping polynomial only in some special cases. Complete mapping polynomials are of interest in combinatorics and are defined as polynomials f(x) over a finite field for which both f(x) and f(x) + x are permutation polynomials. Our result also verifies a special case of a conjecture of Chowla and Zassenhaus on permutation polynomials.

There has been a great deal of work recently concerning the relationship between the commutativity of a ring JR and the existence of certain specified derivations of R. Bell, Herstein, Procesei, Schacher, Ligh, Martindale, Putcha, Wilson, and Yaqub [1, 2, 6, 8, 9, 10, 11, 12, 14] have studied conditions on commutators which imply the commutativity of rings.

Let A be an n × n complex matrix. The singular values of A are the non-negative square-roots of the eigenvalues of A*A. G. N. De Oliviera [4] gave a necessary condition for the existence of a matrix A with a1..., an as diagonal elements and α1,..., αn as singular values. We shall give another necessary condition which implies the above author’s condition and we show that this is also a sufficient condition for the case n =2.

The purpose of this expository note is to establish the fact mentioned in the title. While this is not difficult and requires no new ideas, it seems worth doing, as such a simple characterization does not appear explicitly in a recent treatise on the subject of flat modules [Bourbaki XXVII, Chapter 1].

In this paper, we consider the oscillatory behavior of the second order neutral delay differential equation

where t ≥ t0,T and σ are positive constants, a,p, q € C(t0, ∞), R),f ∊ C[R, R]. Some sufficient conditions are established such that the above equation is oscillatory. The obtained oscillation criteria generalize and improve a number of known results about both neutral and delay differential equations.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, of conductor $N$ and without complex multiplication. For any positive integer $l$, let ${{\phi }_{1}}$ be the Galois representation associated to the $l$-division points of $E$. From a celebrated 1972 result of Serre we know that ${{\phi }_{1}}$ is surjective for any sufficiently large prime $l$. In this paper we find conditional and unconditional upper bounds in terms of $N$ for the primes $l$ for which ${{\phi }_{1}}$ is not surjective.

In Riemannian geometry the autoparallels associated with the affine connexion coincide with the geodesies which arise from the metric. This is not the case in a modification of Riemannian geometry suggested by Lyra. A sufficient condition that the two classes of curves coincide is obtained.

Recent work of Ding and Huang shows that if we perturb a bounded operator (between Hilbert spaces) which has closed range, then the perturbed operator again has closed range. We extend this result by introducing a weaker perturbation condition, and our result is then used to prove a theorem about the stability of frames for a subspace.

We study the total graph of a finite commutative ring. We calculate its metric dimension in the case when the Jacobson radical of the ring is nontrivial, and we examine the metric dimension of the total graph of a product of at most two fields, obtaining either exact values in some cases or bounds in other, depending on the number of elements in the respective fields.

The purpose of this note is to generalize a result of Gulliksen, Ribenboim and Viswanathan which characterized local group rings when both the ring and the group are commutative.

We assume throughout that all rings are associative with identity. If R is a ring we call R local if R/J(R) is a division ring where J(R) denotes the Jacobson radical of R. It is well known that R is local if and only if each element of R\J(R) is a unit. We need the following.

Consider the delay differential equation

where α(t) and β(t) are positive, periodic, and continuous functions with period w > 0, and m is a nonnegative integer. We show that this equation has a positive periodic solution x*(t) with period w. We also establish a necessary and sufficient condition for every solution of the equation to oscillate about x*(t) and a sufficient condition for x*(t) to be a global attractor of all solutions of the equation.

We consider a class of strongly $q$-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly $q$-log-convex. We also prove that the Bessel transformation preserves log-convexity.

Simply connected conformally flat Riemannian manifolds are characterized as hypersurfaces in the light cone of a standard flat Lorentzian space, transversal to its generators. Some applications of this fact are given.

We prove that for a contact process restricted to the cube [1,n]d and initially fully occupied, the time to die out, when it is suitably normalized, converges to an exponential random variable as n tends to infinity.

Some structural results about quotients and tensor products of hereditary respectively quasi-hereditary algebras are presented. They are related to properties of stratifications of derived module categories. The concept of derived-simplicity for an algebra is introduced and studied.

In this paper the following result is proved.

Let n > 0 and k ≥ 0 be integers with n — 2k ≥ 1. Given any n — k integers

there is a non-empty subset of indices I ⊂ {1, 2,…, n — k} such that the sum Σi∊I ≡ 0(mod n) if at most n — 2k of the integers (1) lie in the same residue class modulo n.

The result is best possible if n ≥ 3k — 2 in the sense that if "at most n — 2k" is replaced by "at most n — 2k + 1" the result becomes false. This can be seen by taking aj = 1 for 1 ≤ j ≤ n — 2k + 1 and aj = 2 for n —2k + 2 ≤ j ≤ n — k, noting that the number of 2's here is n — k — (n — 2k + 1)= k — 1 ≤ n — 2k + 1.

A well known theorem of van der Waerden [1] states that given any two positive integers k and t, there exists a positive integer m such that in every distribution of the numbers 1,2, …, m into k classes, at least one class contains an arithmetic progression of t + 1 terms. Other proofs and generalizations of this theorem have been given by Griinwald [2], Witt [3] and Lukomskaya [4]. The last mentioned proof appears in the booklet of Khinchin “Three pearls of number theory” in which van der Waerden's theorem plays the role of the first pearl.

Fan ([2, Theorem 2]) has proved the following theorem:

Let K be a nonempty compact convex set in a normed linear space X. For any continuous map f from K into X, there exists a point u∈K such that

In this note, we prove that the above theorem is true for a continuous condensing map defined on a closed ball in a Banach space. We also prove that it is true for a continuous condensing map defined on a closed convex bounded subset of a Hilbert space.

Let R be an integral domain. An element u of the quotient field of R is said to be pseudo-integral over R if uIv ⊆ Iv for some nonzero finitely generated ideal I of R. The set of all pseudo-integral elements forms an integrally closed (but not necessarily pseudo-integrally closed) overling R ofR. It is shown that , where X is a family of indeterminates; pseudo-integrality is analyzed in rings of the form D + M; and an example is given to show that pseudo-integrality does not behave well with respect to localization.