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Very recently, I. Altun, M. Aslantas and H. Sahin [1] introduced the notion of $$p$$ -proximal contractions and surveyed the existence of best proximity points for such class of non-self mappings in metric spaces. In this note we show that this existence result is a straightforward consequence of the same conclusion in fixed point theory.

Sendov’s conjecture states that if all the zeroes of a complex polynomial P(z) of degree at least two lie in the unit disk, then within a unit distance of each zero lies a critical point of P(z). In a paper that appeared in 2014, Degot proved that, for each a ∈ (0, 1), there exists an integer N such that for any polynomial P(z) with degree greater than N, if P(a) = 0 and all zeroes lie inside the unit disk, the disk |z − a| ≤ 1 contains a critical point of P(z). Based on this result, we derive an explicit formula N(a) for each a ∈ (0, 1) and, consequently obtain a uniform bound N for all a ∈ [α, β] where 0 < α < β < 1. This (partially) addresses the questions posed in Degot’s paper.

Propositions 1.1–1.3 stated below contribute to results and certain problems considered in [E-S], on the behavior of products $$\Pi^n_1(1-z^{a_j}),1\leq{a_1}...\leq{a_n}$$ integers. In the discussion below, {a1,..., an} will be either a proportional subset of {1,..., n} or a set of large arithmetic diameter.

In a recent paper, (Math Notes 91(3–4): 508–516, 2012) Zarhin proved that each member of a naturally defined family of linear maps \({\mathbb {C}}^n \rightarrow {\mathbb {C}}^n\) has co-rank one. We present a direct proof of Zarhin’s result about complex polynomials with distinct roots; it is rather similar to that of Appendix by Vik.S. Kulikov to Zarhin’s paper but we give explicit constants. We also discuss the case of a polynomial with multiple roots.

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 1935, Erdos proved that all natural numbers can be written as a sum of a square of a prime and a square-free number. In 1939, Pillai derived an asymptotic formula for the number of such representations. The mathematical review of Pillai’s paper stated that the proof of the above result contained inaccuracies, thus casting a doubt on the correctness of the paper. In this paper, we re-examine Pillai’s paper and show that his argument was essentially correct. Afterwards, we improve the error term in Pillai’s theorem using the Bombieri–Vinogradov theorem.

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.

An error has been detected (and also corrected) in Theorem 2.8 of the paper entitled “Adjoining an Order Unit to a Matrix Ordered Space” (Positivity, (2005)9: 207–223; DOI 10.1007/s11117-003-2778-5). Accordingly, some of the results of the paper have been modified. Also, a notion of C*-matricially, Riesz normed spaces has been introduced.

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

The paper [3] contains an upper bound to the weighted density of a packing of circles on the unit sphere with radii from a given finite set. This bound is attained by many packings and has applications to problems of solidity. In the present note it is shown that a certain condition imposed on the set of admissible radii can be removed by modifying the original proof of the theorem.

We generalize a result of F. Luca and A. Sankaranarayanan by proving that the set of n for which ϕ(1) + + ϕ(n) is squareful is of zero density. A similar statement holds for σ (n) instead of ϕ(n) and for some other multiplicative functions.

It is proved that the set of those natural numbers which cannot be written as n-Ω(n) is of positive lower density. Here Ω(n) is the number of the prime power divisors of n. This is a refinement of a theorem of F. Luca.

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.

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.

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 \(nd(n)\).

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).

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.

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.

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.

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

We show functoriality of γα in our joint paper with A. Csaszar “Further remarks on δ- and θ-modifications”.

Professor Zbigniew Lipecki from the University of Wroclaw has told me that the definition of “small ball property” introduced in the paper “Metric spaces with the small ball property” by E. Behrends and V. Kadets (Studia Math. 148, No. 3, 275-287, 2001), has been investigated under different names earlier in the literature and that some of the results in this paper and in my paper quoted in the title are rediscoveries of known facts. In particular this applies to Proposition 2.3. I am grateful to Professor Lipecki for providing the following references to us: – Ch. Bandt, Mathematika 28 (1981), 206-210. – R. Duda and R. Telgarsky, Czechoslovak Math. J. 18(93) (1968), 66-82. – E. Szpilrajn-Marczewski, Fund. Math. 15 (1930), 126-27. – E. Szpilrajn-Marczewski, Fund. Math. 22 (1934), 303-311. – B. Tsaban, Real Analysis Exchange 27 (2001/2002), 757-764.

We note that Theorem 5.4 of the paper ``Characteristic cycles of perverse sheaves and Milnor fibers" (by P. Nang and K. Takeuchi, published online in Math. Zeitschrift, October 15, 2004) holds for "any" non-zero complex number $a$., Comment: 1page

This paper deals with Schrodinger operators as they were treated by Kato - Ikebe [3]. It is shown that every element of the domain of definition of the adjoint of such an operator has locally square integrable distributional derivatives up to the order 2. For this the potential of the Schrodinger operator must fulfil a local Stummel condition; if the potential is only locally square integrable a somewhat weaker statement is possible for three dimensions (see remark 2 at the end of this paper).

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.

Why do I think that Z.v.G.II was an epoch-making paper? (1) It was the paradigm for subsequent efforts to classify the possible structures for any mathematical object. Hawkins [15] documents the fact that Killing’s paper was the immediate inspiration for the work of Cartan, Molien, and Maschke on the structure of linearassociative algebras which culminated in Wedderburn’s theorems. Killing’s success was certainly an example which gave Richard Brauer the will to persist in the attempt to classify simple groups. (2) Weyl’s theory of the representation of semi-simple Lie groups would have been impossible without ideas, results, and methods originated by Killing in Z.v.G.II. Weyl’s fusion of global and local analysis laid the basis for the work of Harish-Chandra and the flowering of abstract harmonic analysis. (3) The whole industry of root systems evinced in the writings of I. Macdonald, V. Kac, R. Moody, and others started with Killing. For the latest see [21]. (4) The Weyl group and the Coxeter transformation are in Z.v.G.II. There they are realized not as orthogonal motions of Euclidean space but as permutations of the roots. In my view, this is the proper way to think of them for general Kac-Moody algebras. Further, the conditions for symmetrisability which play a key role in Kac’s book [17] are given on p. 21 of Z.v.G.II. (5) It was Killing who discovered the exceptional Lie algebra E8, which apparently is the main hope for saving Super-String Theory—not that I expect it to be saved! (6) Roughly one third of the extraordinary work of Elie Cartan was based more or less directly on Z.v.G.II.

Cooperative games form an important class of problems in game theory, where a key goal is to distribute a value among a set of players who are allowed to cooperate by forming coalitions. An outcome of the game is given by an allocation vector that assigns a value share to each player. A crucial aspect of such games is submodularity (or convexity). Indeed, convex instances of cooperative games exhibit several nice properties, e.g. regarding the existence and computation of allocations realizing some of the most important solution concepts proposed in the literature. For this reason, a relevant question is whether one can give a polynomial-time characterization of submodular instances, for prominent cooperative games that are in general non-convex. In this paper, we focus on a fundamental and widely studied cooperative game, namely the spanning tree game. An efficient recognition of submodular instances of this game was not known so far, and explicitly mentioned as an open question in the literature. We here settle this open problem by giving a polynomial-time characterization of submodular spanning tree games.

In a paper which appeared a few years ago the authors investigated the exponential representation of functions analytic in the upper half-plane with positive imaginary part there [1]. We refer to that paper in the sequel as A-D. One of the principal results of A-D, there called Theorem A, can be extended to a considerably more general result, the proof of which is perhaps simpler than that given in A-D. We give the extended version of Theorem A here. We will use the notations and results of A-D without further explanations. Before we present the extension and its proof we would like to add some information that by oversight was omitted from the list of fundamental properties of the functions in the class P given in Section 1 of A-D. In such a comprehensive review one should mention that the classical theorem on representation of a positive harmonic function in a circle by a Poisson-Stieltjes integral is due to G. Herglotz [2]. The following results of L. H. Loomis [3] were not given: XVII. For all ~ for which /t[~] = O, the limits lira Im[~b(~ + iq)] ~l~ O and lira hOg(A) exist and are f ini te simultaneously and are equal. Their ~ 0 common value is the symmetric derivative of #(A) at 2 = ~ multiplied by ~r. XVIII. I f for two values of 0 in the interval 0 < 0 < ~

In a recent paper M.C. Cheng proposed new criteria for the simplex algorithm which guarantee that (i) a nonbasic variable of a basic feasible solution will remain nonbasic in an optimal basic solution, (ii) a basic variable of a basic solution will remain basic in an optimal basic solution. This comment gives (i) a slight generalization of the first result and (ii) a counterexample to the second proposition.

It is pointed out that theL1 convergence classes of Fourier series of even functions considered in [2] and [3] are subclasses of theL1 convergence classes of Fomin [7]. Moreover, showing the simple fact that for eachp>1 the classesCp andVp considered in [2] coincide, whereCp={(ak):ak=o (1) and\(n^{p - 1} \sum\limits_{k = n}^\infty {\left| {\Delta _{ak} } \right|^p = o} \left( 1 \right)\)}, it follows that the main result in [2] reduces to an earlier theorem of Fomin [5]. Other equivalent descriptions of these classes are also given.

It is shown that unbranched proper holomorphic maps between pseudoconvex domains with smooth C∞ boundaries, one of which satisfies subelliptic estimates for the\(\bar \partial\)-Neumann problem on (0,1)-forms, extend to unbranched C∞-coverings between the closures of the domains.