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A classical result of K. L. Chung and W. Feller deals with the partial sums S k S_k arising in a fair coin-tossing game. If N n N_n is the number of “positive” terms among S 1 S_1 , S 2 S_2 , …, S n S_n then the quantity P ( N 2 n = 2 r ) P(N_{2n} = 2r) takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for P ( N 2 n + 1 = r ) P(N_{2n+1} = r) , r = 0 r = 0 , 1 1 , 2 2 , …, 2 n + 1 2n+1 . We get to this ansatz by adaptating the Feynman–Kac methodology.

We introduce a graph Ramsey game called Ramsey, Paper, Scissors. This game has two players, Proposer and Decider. Starting from an empty graph on $n$ vertices, on each turn Proposer proposes a potential edge and Decider simultaneously decides (without knowing Proposer's choice) whether to add it to the graph. Proposer cannot propose an edge which would create a triangle in the graph. The game ends when Proposer has no legal moves remaining, and Proposer wins if the final graph has independence number at least $s$. We prove a threshold phenomenon exists for this game by exhibiting randomized strategies for both players that are optimal up to constants. Namely, there exist constants $0B\sqrt{n}\log{n}$. This is a factor of $\Theta(\sqrt{\log{n}})$ larger than the lower bound coming from the off-diagonal Ramsey number $r(3,s)$.

We complete a gap in the proof that exceptional polynomials are complete orthogonal systems in the associated Hilbert spaces.

We indicate how recent results of Grafakos and Teschl (J Fourier Anal Appl 19:167–179, 2013), and Estrada (J Fourier Anal Appl 20:301–320, 2014), relating the Fourier transform of a radial function in $$\mathbb R^n$$ and the Fourier transform of the same function in $$\mathbb R^{n+2}$$ and $$\mathbb R^{n+1}$$ , respectively, are located within known results on transplantation for Hankel transforms.

In the present paper, necessary and sufficient conditions are given for the equality of the power rezidue symbols $$ {\left( {\frac{\alpha }{a}} \right)_n} $$ and $$ {\left( {\frac{\alpha }{a}} \right)_n} $$ in the cyclotomic field ℚ(ζ n ), 2 ∤ n, for a ∈ ℤ, (a, n) = 1. This result is a generalization of the classical Eisenstein reciprocity law and its continuation in a Hasse’s paper. Bibliography: 3 titles.

Congruences mod p for a prime p and partial products of the numbers 1,…, p − 1 are obtained. Bibliography: 2 titles.

Kolmogorov-Arnold-Moser (or KAM) theory was developed for conservative dynamical systems that are nearly integrable. Integrable systems in their phase space usually contain lots of invariant tori, and KAM theory establishes persistence results for such tori, which carry quasi-periodic motions. We sketch this theory, which begins with Kolmogorov's pioneering work.

It is shown that an approach earlier used by the author for classification of closed simply connected 6-manifolds (reduction to the problem of calculating certain bordism groups) can also be applied for easily obtaining the results by Barden (1965) on classification of closed simply connected 5-manifolds. Bibliography: 11 titles.

A remarkable contribution to the theory of differential equations was made by Adolph Steen in 1874. His results were presented in Danish in a journal containing few articles on mathematics and were wholly ignored by the mathematical community for more than a century. During this time much of his work was rediscovered and extended by numerous authors; in fact the relevant literature embraces five languages in addition to the original Danish and is scattered over correspondingly many countries. In Part I we outline the historical development of Steen's ideas and in Part II we reproduce his pioneering article in an English translation. Though correction of a historical injustice is the primary goal, not everything in his paper has been rediscovered by others. Hence some of the results retain, even at this late date, more than merely historical interest.

The Award Committee – Peter Kritzer, Johannes Kepler University Linz, Austria and Jan Vybiral, Charles University, Czech Republic – determined that the following paper exhibits exceptional merit and therefore awarded the prize to: Bernd Carl, Aicke Hinrichs, and Philipp Rudolph for their paper ‘‘Entropy numbers of convex hulls in Banach spaces and applications’’, which appeared in October, 2014. Vol. 30, pp. 555–587. The $3000 prize will be divided between the winners. Each author will also receive a plaque.

In a recent paper, Chao-Ping Chen and Feng Qi (2005) established sharp upper and lower bounds for the sequence P n := 1.3 … ( 2 n − 1 ) 2.4 … 2 n P_{n}:=\frac {1.3\ldots (2n-1)}{2.4\ldots 2n} . We show that their result follows easily from a theorem of G. N Watson published in 1959. We also show that the main result of Chen and Qi’s paper is a special case of a more general inequality which admits a very short proof.

Zannier proved that for a Laurent series f(x) satisfying the functional equation of type f(x m ) = P(x, f(x)), where \( P(x, y) \in {\Bbb C}(x)[y] \), if f(x) is not rational the set of such m consists of the powers of a single integer. He mentioned that the case f(x) = P(x, f(x m )) should be proved in a similar way. In this paper we first verify this statement and second we show a theorem which is useful for proving the transcendence of a Laurent series satisfying a certain functional equation. This theorem is a generalization of the result that a Laurent series which satisfiesf(x m ) = P(x, f(x)), where\( P(x, y) \in {\Bbb C}(x, y),\,m \geq 2 \)cannot represent an algebraic function unless it is rational (Ke. Nishioka [1], Zannier [8], Section 3).

This is an attachment to a letter of the author to Alexander Zvonkin (dated November 1, 1995). It is reproduced here with the kind permission of the author.

Concerns have been voiced about the correctness of certain technical points in DiPerna’s paper (Comm. Math. Phys. 91 (1983), 1–30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods. In [Di], DiPerna found a global entropy solution of the isentropic Euler equations for the following exponents in the equation of state for the pressure: γ = 1 + 2/(2m+ 1), m ≥ 2 integer. (1) He divided his arguments into the following two steps. 1. Compactness framework Assume that a sequence of approximate solutions (ρ (x, t),m (x, t)), 0 ≤ t ≤ T , satisfies: (i). There exists a constant C(T ) > 0, independent of > 0, such that 0 ≤ ρ (x, t) ≤ C, |m (x, t)/ρ (x, t)| ≤ C; (ii). For all weak entropy pairs (η, q) of the isentropic Euler equations, the measure sequence η(ρ ,m )t + q(ρ ,m )x is contained in a compact subset of H −1 loc (R× [0, T ]). If γ satisfies (1), then the sequence (ρ (x, t),m (x, t)) is compact in Lloc(R× [0, T ]). The reason for the restriction on the number γ is that, in such a case, any weak entropy function is a polynomial function of the Riemann invariants (w, z). This is the key step in DiPerna’s arguments and is also his main contribution to the compensated compactness method in this aspect. Received by the editors May 16, 1996. 1991 Mathematics Subject Classification. Primary 35K55, 35L65; Secondary 76N15, 35L60, 65M06.

The Award Committee – Michael Gnewuch, University of Kaiserslautern, Germany and Friedrich Pillichshammer, Johannes Kepler University, Austria – determined that the following two papers exhibited exceptional merit and therefore awarded the prize to: ShuTezuka for his paper ‘‘On the discrepancy of generalizedNiederreiter sequences’’, which appeared in June–August, 2013. Vol. 29, pp. 240–247. Joos Heintz, Bart Kuijpers, and Andres Rojas Paredes for their paper ‘‘Software engineering and complexity in effective Algebraic Geometry’’, which appeared in February, 2013. Vol. 29, pp. 92–138. The $3000 prize will be divided between the winners. Each author will also receive a plaque.

Here we give a result on the postulation (i.e. the 2-normality) of nonlinearly normal embeddings of Abelian varieties. This result improves some of the results proved in a recent paper by Ch. Birkenhake.

This paper presents a review of the elementary theories on the bending of straight and curved beams, on plates and shells, using asymptotic approximations of the basic linearized equations of elasticity in three dimensions. The maximun norm has been chosen to specify the orders of magnitude of the quantities involved. The expansions are given as usual in terms of the small geometrical parameter characterizing the thinness of the structure. Most of the ideas and results are well known. Nevertheless, in the cases where more than one small parameter may be involved, such as small curvatures (shallow structures) or the small loading parameter used to linearize the equations of elasticity, the discussion on the limits of validity of the different theories lead to some interesting newer aspects. Moreover, the main ideas presented in this paper concerning multiple parameter expansions may be applied to discuss the behaviour of the structures and to obtain valuable analytical results in more complicated situations such as moderate and strong anisotropy, dynamic problems, stability etc.

Consider the first order differential equation (1) , where pi, and τi, for i = 1,…,n, are positive constants. To find necessary and sufficient conditions, in terms of the coefficients and the delays only, under which all solutions of (1) oscillate, is a problem of great importence. In a recent paper, Bowcock and Yu claimed that is a necessary and sufficient condition for all solutions of (1) to be oscillatory. In this paper a counterexample shows that the above result is not valid and the error in this paper is indicated.

I regret to announce that there is a serious error in my paper in these Transactions, volume 84, pp. 430-443. The error was discovered by Louis Kokoris who found that on line 8 of page 434 the expression given as 4[g(bz)](az) should have been 4[g(az)](bz). As a consequence the computation of P(z, g, az, b) yields nothing, the proof of formula (30) is not valid, and the important Lemma 9 is not proved. Thus the paper does not give a proof of its major result stated as Theorem 1. Nevertheless, the theorems of the paper are all correct and we shall provide a revision of the proof here. This revised proof has been checked by Louis Kokoris to whom the author wishes to express his great thanks. We observe first that the equation

Introduction. In a recent paper [2] R. H. Bruck has introduced the concept of right neoring and discussed some properties of these systems. In particular, he has considered analogues of certain properties of the ring of integers. This paper is essentially a commentary on Bruck's paper and we generalize some of his results as follows. The construction of the universal right neoring in [2] is applied to the free monogenic $3-loop in any subvariety $3 of the variety of loops and a complete analogue of Theorem 4.1 of [2] is obtained for any one of these subvarieties. Then, using a result similar to those obtained in [5], it is shown that this construction yields uncountably many right neorings with an identity which generates the additive loop of the right neoring. Conversely, every right neoring with an identity which generates its additive loop can be obtained from a free monogenic $3-loop by the above construction. Each of these right neorings has some properties resembling those of the ring of integers. One possible answer is given to the question raised by Bruck concerning the existence of universal right neorings with free additive loop of arbitrary rank. A brief proof is given, using the results of [4; 5], of the cancellation properties of the monogenic universal right neoring. Finally, we discuss briefly the relationship between right neorings and the logarithmetics of Etherington.

The purpose of this note is to present a new proof of a generalized form of Day's fixed point theorem. The proof we give is suggested by the work of T. Mitchell in his paper, Function algebras, means, and fixed points, [2]. The version of Day's theorem which we present here has not appeared explicitly in the literature before, and seems especially well suited for application to questions concerning fixed point properties of topological semigroups. 1. Preliminaries. We adopt the terminology and notation of [2] except where otherwise specified. New terminology will be introduced as needed. Let y be a convex compactum (compact convex set in a real locally convex linear topological space E), and let A( Y) denote the Banach space of all (real) continuous affine functions on Y under the supremum norm. Observe that A(Y) contains every function of the form h=f\Y + r where fe E* and r is real; thus A(Y) separates points of Y.

The study of the dynamics of the size of a population via mathematical modelling is a problem of interest and widely studied. Traditionally, continuous deterministic methods based on differential equations have been used to deal with this problem. However discrete versions of some models are also available and sometimes more adequate. In this paper, we randomize the Pielou logistic equation in order to include the inherent uncertainty in modelling. Taking advantage of the method of transformation of random variables, we provide a full probabilistic description to the randomized Pielou logistic model via the computation of the probability density functions of the solution stochastic process, the steady state and the time until a certain level of population is reached. The theoretical results are illustrated by means of two examples, the first one consists of a numerical experiment and the second one shows an application to study the diffusion of a technology using real data.

It is shown that a result by Moeckel holds not only for admissible subgroups of SL (2, ℤ), but also for arbitrary subgroups of finite index.