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A new property of a complete intersection in a graded Noetherian ring is proved; this result, in conjunction with the Jacobian criterion, is then applied to the description of the loci of singularities in a family of affine varieties introduced in a well-known paper on forms of many variables.

In this note, we give a counterexample for Theorem 2.3 of the above mentioned paper that is a generalization of the Grothendieck non-vanishing theorem to a class of modules larger than finitely generated modules.

We considered in Example 3.1 of the paper [1] an S-structure on R2n+s . We concluded that when s > 1 this manifold cannot be of constant φ-sectional curvature. Unfortunately this result is wrong. In fact, essentially due to a sign mistake in defining the φ-structure and a consequent transposition of the elements of the φ-basis (3.2), some of the Christoffel’s symbols were incorrect. In the present rectification, using a more slendler tecnique, we prove that our manifold is of constant φ-sectional curvature −3s and then it is η-Einstein.

I give a simple analysis of the game that I previously published in Scientific American which shows the paradoxical behavior whereby two losing games randomly combine to form a winning game. The game, modeled on a random walk, requires only two states and is described by a first-order Markov process.

This paper presents a sufficient condition for a one-dimensional Dirac operator with a potential tending to infinity at infinity to have no eigenvalues. It also provides a quick proof (and suggests variations) of a related criterion given by Evans and Harris.

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.

Let S be a surface obtained from a plane polygon by identifying infinitely many pairs of segments along its boundary. A condition is given under which the complex structure in the interior of the polygon extends uniquely across the quotient of its boundary to make S into a closed Riemann surface. When this condition holds, a modulus of continuity is obtained for a uniformizing map on S., 36 pages, 11 figures

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.

We give the asymptotics of the sum \(\sum\nolimits_{x \leqslant n \leqslant x + h} {f(n)}\), where \(h \gg x^{7/12 + \varepsilon }\), for the multiplicative functions \(f(n) = z^{\omega (n)} , z^{\omega (n)} |\mu (n)|, 1/d_k (n)\), and \(nd(n)\).

We correct the proof of Theorem 3.8 in [GP2].

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).

The size Ramsey number ˆ(F1 ,F 2) is the smallest number of edges that an (F1 ,F 2)-arrowing graph can have. Let Si,n be obtained from the star K1,n by subdividing one edge by new i − 1 vertices. Bielak [Periodica Math. Hung. 18

In [1] Kohli and Vashistha gave an analogue of probabilistic version of Pant‘s Theorem ([2], Theorem 1). We note that mappings defined in Examples 3.6 to 3.8 of [1] are not self maps as claimed in the Definitions 3.1 and 3.2. In this context, we provide some relevant examples to complete the interesting results.

In this note two theorems obtained by S. Zahid Ali Zenei in [11] are refined.

We study spectral multipliers for a distinguished Laplacian on certain groups of exponential growth. We obtain a stronger optimal version of the results proved in (CGHM) and (A).

It was proved by Ibukiyama and Skoruppa that the spaces J1,m(l) of Jacobi forms of weight 1, index m and level l vanish if gcd (l,m)=1. We will show that the result holds without the gcd hypothesis.

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.

In this paper, it is shown that von Neumann continuous geometry may be regarded as the first attempt towards formulating a general quantum spacetime geometry akin to that of Cantorian spacetime E(∞) and noncommutative geometry.

Fine and Wilf's theorem has recently been extended to words having three periods. Following the method of the authors we extend it to an arbitrary number of periods and deduce from that a characterization of generalized Arnoux-Rauzy sequences or episturmian infinite words.

We consider the matrix for the Satake isomorphism with respect to natural bases. We give a simple proof in the case of Chevalley groups that the matrix coefficients which are not obviously zero are in fact positive numbers. We also relate the matrix coefficients to Kazhdan–Lusztig polynomials and to Bernstein functions.

This note concerns the paper "On the controllability of a coupled system of two Korteweg–de Vries equations" by Micu et al. [2]. They study a nonlinear coupled system of two Korteweg–de Vries equations and prove that the system is controllable by using four boundary controls. Here, we prove that in some cases it is possible to get the controllablity of the system by using only two controls. This can be done depending on both the spatial domain and the control time.

Using Hodge theory and L2-cohomology we study the singularities and topology of configuration and moduli spaces of polygonal linkages in the 2-sphere. As a consequence we describe the local deformation space of a folded paper cone in R3.

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.

We fill a gap in the proof of one of the central theorems in Epp's paper, concerning $p$-cyclic extensions of complete discrete valuation rings., Comment: 3 pages

International audience; In a previous paper, we have described the construction of an automaton from a rational expression which has the property that the automaton built from an expression which is itself computed from a co-deterministic automaton by the state elimination method is co-deterministic. It turned out that the definition on which the construction is based was inappropriate, and thus the proof of the property was flawed. We give here the correct definition of the broken derived terms of an expression which allow to define the automaton and the detailed full proof of the property.

(1996). A Parenthetical Note (to a Paper of Guy) Mathematics Magazine: Vol. 69, No. 4, pp. 257-260.

We correct an inaccuracy in the papers of Le, Mignotte, and Bugeaud on the generalized Ramanujan-Nagell equation.