Remarks on some zero-sum problems

Let Z denote the ring of integers and for a prime p and positive integers r and d, let f_r(P, d) denote the smallest positive integer such that given any sequence of f_r(p, d) elements in (Z/pZ(^d, there exists a subsequence of (rp) elements whose sum is zero in (Z/pZ(^d. That f₁(p, 1) = 2p − 1, is a classical result due to Erdős, Ginzburg and Ziv. Whereas the determination of the exact value of f₁(p, 2) has resisted the attacks of many well known mathematicians, we shall see that exact values of f_r(p, 1) for r ≥ 1 can be easily obtained from the above mentioned theorem of Erdős, Ginzburg and Ziv and those of f_r(p, 2) for r ≥ 2 can be established by the existing techniques developed by Alon, Dubiner and Rónyai in connection with obtaining good upper bounds for f₁(p, 2). We shall also take this opportunity to describe some of the early results in the introduction.     

Remarks on some existence theorems for optimal control

First, a remark is made that a growth condition contained in previous papers by Cesari concerning existence theorems for optimal controls can be replaced by a slightly more general condition. In this more general condition, a constantM 0 is replaced by any functionM (t)0 which is assumed to beL-integrable in every finite interval.Then, the remark is made that the same condition, which is usually required to be satisfied by the functionsf ₀(t, x, u),f(t, x, u) characterizing the control, can be required to be satisfied only by the admissible pairsx(t),u(t) of the class in which the optimum is being sought. This generalization requires a subtle argument. The new condition parallels now the usual conditions of the type |x| ^p dtM, which are required to be satisfied by the admissible pairs of the class .The work of L. Cesari was partially supported by the Office of Scientific Research, United States Air Force, Grant No. 942-65. The work of T. Nishiura was partially supported by the National Science Foundation, Grant No. GP-7676.     

Remarks on some recent metrical common fixed point theorems

In an interesting article, Bouhadjera and Godet-Thobie (2009) [6] introduced notions of subcompatibility and subsequential continuity, and utilized them to prove several common fixed point theorems. The results of the aforementioned article contain flaws, and they are not correct in their present form. But these results can be recovered in two ways by strengthening either of the newly introduced definitions. The results, in their corrected form, still bring about noted improvements over a multitude of relevant fixed point theorems of the existing literature, substantiating the utility of these (two) new notions.     

Remarks on sphere-type theorems

We prove if is a complete Riemannian manifold with an embedded totally geodesic compact hypersurface such that has nonnegative sectional curvature, and the sectional curvature of is strictly positive in a neighborhood of , then the pair is diffeomorphic to the pair . This result gives an affirmative answer to a question of H. Wu in the case when is compact and simply connected.

     

Remarks on multigrid convergence theorems

Multigrid has become an important iterative method for the solution of discrete elliptic equations. However, there is much to be done in the theory of convergence proofs. At the present time there are two general two-level methods for general convergence proofs: an algebraic approach and a duality approach. While these theories do not give sharp estimates, they provide good, general, rigorous convergence theorems. In this note we study the relationship between these theories. While the approach and thought process leading to these theories are different, the results are essentially the same. Indeed, the basic estimates required by these theories are the same.     

Remarks on two theorems of Qi-Keng Lu

Two alternate arguments in the framework of intrinsic metrics and measures respectively of part of the proof of a famous theorem due to Qi-Keng Lu on Bergman metric with constant negative holomorphic sectional curvature are presented.A relationship between the Lu constant and the holo- morphic sectional curvature of the Bergman metric is given.Some recent progress of the Yau's porblem on the characterization of domain of holomorphy on which the Bergman metric is K(?)hler-Einstein is described.     

On some theorems on circulant matrices

In [2] some theorems on n × n circulant matrices were introduced under the hypothesis n a prime number. We extend these theorems to the case n =2?r where r is a prime greater then 2.     

Remarks on some completely monotonic functions

Applying the Euler-Maclaurin summation formula, we obtain upper and lower polynomial bounds for the function , x>0, with coefficients the Bernoulli numbers Bk. This enables us to give simpler proofs of some results of H. Alzer and F. Qi et al., concerning complete monotonicity of certain functions involving the gamma function Γ(x), the psi function ψ(x) and the polygamma functions ψ(n)(x), n=1,2,… .