An asymptotic estimate related to Selberg's sieve

Suppose 1 ≤ z₁ ≤ z₂ ≤ N, and let $\text{Λ}{}_{\mathrm{i}}\text{(d) = μ(d)}\text{max(log}\text{(}\text{z}{}_{\mathrm{i}}\text{d}\text{), 0)}$ for i = 1, 2. We show that

$\text{∑}\text{N?n}\text{(}\text{∑}\text{d|n}\text{Λ}{}_1\text{(d))(}\text{∑}\text{e|n}\text{Λ}{}_1\text{(e)) = N}\text{log z}{}_1\text{+ O(N).}$

We then use this to improve a result of Barban-Vehov which has applications to zero-density theorems.     

Some applications of the double large sieve

We sharpen a procedure of Cao and Zhai (J Théorie Nombres Bordeaux,11: 407–423, 1999) to estimate the sum $$\begin{aligned} \sum _{m\sim M} \sum _{n\sim N} a_m b_n \, e\left(\frac{F m^\alpha n^\beta }{M^\alpha N^\beta }\right) \end{aligned}$$ with $|a_m|,\ |b_n| \le 1$ . We apply this to give bounds for the discrepancy (mod 1) of the sequence $\{p^c: p\le X\}$ where $p$ is a prime variable, in the range $\frac{130}{79}\le c \le \frac{11}{5}$ . An alternative strategy is used for the range $1.48 \le c \le \frac{130}{79}$ . We use further exponential sum estimates to show that for large $R>0$ , and a small constant $\eta >0$ , the inequality $$\begin{aligned} \left| p_1^c+p_2^c+p_3^c+p_4^c+p_5^c - R\right| < R^{-\eta } \end{aligned}$$ holds for many prime tuples, provided $2<c\le 2.041$ . This improves work of Cao and Zhai (Monatsh Math, 150:173–179, 2007) and a theorem claimed by Shi and Liu (Monatsh Math, published online, 2012).     

Some comments on Rota's Umbral Calculus

Rota's Umbral Calculus is put in the context of general Fourier analysis. Also, some shortcuts in the proofs are illustrated and a new characterization of sequences of binomial type is given. Finally it is shown that there are few (classical) orthogonal polynomials of binomial type.     

On some applications of Wald's identity to dams

It is known that the main difficulty in applying the Markovian analogue of Wald's Identity is the presence, in the Identity, of the last state variable before the random walk is terminated. In this paper we show that this difficulty can be overcome if the underlying Markov chain has a finite state space. The absorption probabilities thus obtained are used, by employing a duality argument, to derive time-dependent and limiting probabilities for the depletion process of a dam with Markovian inputs.The second problem that is considered here is that of a non-homogeneous but cyclic Markov chain. An analogue of Wald's Identity is obtained for this case, and is used to derive time- dependent and limiting probabilities for the depletion process with inputs forming a non- homogeneous (cyclic) Markov chain.     

Some hypergraphs and packing problems associated with matrices of 0's and 1's

We define various classes of hypergraphs associated with m × n matrices of 0's and 1's and determine for which classes the maximum cardinality of a set of pairwise disjoint edges equals the minimum cardinality of a set of nodes that cover all edges independently of the matrix.     

The Chinese concept of Cavalieri's principle and its applications

Bonaventura Cavalieri (1598–1647) was noted for his method of indivisibles which led to the principle which bears his name. In the third century, while attempting to derive the volume of a sphere, Liu Hui applied a similar principle to determine the ratio of the volumes of a sphere and a solid circumscribing the sphere. This solid is formed by the intersection of two perpendicular cylinders circumscribing the sphere and is called mou he fang gai. Liu Hui left unresolved the problem of finding the volume of the mou he fang gai. In the fifth century Zu Geng, also applying Cavalieri's principle, solved this problem and thus derived the volume of a sphere. The influence of Zu Geng's method on later mathematicians is discussed in the latter part of the article.     

Some results on Hu's conjecture concerning binary trees

Given n weights, w₁, w₂,…, w_n, such that 0?w₁?w₂???w₁, we examine a property of permutation π¹, where π¹=(w₁, w_n, w₂, w_n?1,…), concerning alphabetical binary trees.For each permutation π of these n weights, there is an optimal alphabetical binary tree corresponding to π, we denote it's cost by V(π). There is also an optimal almost uniform alphabetical binary tree, corresponding to π, we denote it's cost by V_u(π).This paper asserts that V_u(π¹)?V_u(π)?V(π) for all π. This is a preliminary result concerning the conjecture of T.C. Hu. Hu's conjecture is V(π¹)?V(π) for all π.     

Some basic observations on Kelly's conjecture for graphs

P.J. Kelly first mentioned the possibility of determining a graph from subgraphs obtained by deleting several points. While such problems have received a great deal of attention in the case of deletions of single points, the problem for several points is virtually untouched. This paper contains some basic results on that problem, including the negative observation that for every k, there exist two non-isomorphic graphs with the same collection of k-point subgraphs.     

A generalization of Sylvester's identity on determinants and some applications

A generalization of Sylvester's identity on determinants is proved by elimination techniques. As an application some inequalities for determinants of totally positive or positive definite matrices are derived.     

On a modification of Marquardt's compromise: Rationale and applications

The method of Marquardt's compromise for handling the problem of near-singularity in nonlinear regression is modified by (a) utilizing methods from ridge regression for computing the Marquardt parameter k, and (b) utilizing a search method of interpolation-extrapolation for determining the step size. The performance of these modifications is evaluated on a number of standard test problems from the literature. It is shown that the Lawless-Wang estimate of k without the benefit of interpolation- extrapolation competes favorably with standard algorithms implemented in packaged programs. An application to a model of plasma insulin dynamics is also given. Comparison of the modified procedure with the Nelder-Mead simplex method is given.     

Kuratowski's and Wagner's theorems for matroids

In an earlier paper we proved the following theorem, which provides a strengthening of Tutte's well-known characterization of regular (totally unimodular) matroids: A binary matroid is regular if it does not have the Fano matroid or its dual as a series-minor (parallel-minor). In this paper we prove two theorems (Theorems 5.1 and 6.1) which provide the same kind of strengthening for Tutte's characterization of the graphic matroids (i.e., bond-matroids). One interesting aspect of these theorems is the introduction of the matroids of “type R”. It turns out that these matroids are, in at least two different senses, the smallest regular matroids which are neither graphic nor cographic (Theorems 6.2 and 6.3).     

Some remarks on Kuzmin's theorem for F-expansions

In this paper a general Kuzmin theorem for a class of multidimensional F-expansions is given. This result gives the uniform rate at which a certain sequence of approximates converges to the density of the invariant measure associated with an F-expansion. Some metric theorems are also given. This work extends and corrects some earlier results. The Jacobi algorithm is included as an example.     

Mersenne's solution for Galileo's problem of the rotating earth

More than twenty years before Huygens and Newton developed formulas for centrifugal acceleration, Mersenne contrived a statisfactory solution for Galileo's problem of the extrusion of bodies from the earth as a result of its daily rotation. Mersenne was able to overcome an error in Galileo's approach without the use either of an explicit notion of infinitesimals or of any clear concept of force. His solution depends on comparing the lengths of two lines, a technique that several historians have claimed to be inadequate for this problem.     

Some geometric applications of Dilworth’s theorem

A geometric graph is a graph drawn in the plane such that its edges are closed line segments and no three vertices are collinear. We settle an old question of Avital, Hanani, Erdős, Kupitz, and Perles by showing that every geometric graph withn vertices andm>k ⁴ n edges containsk+1 pairwise disjoint edges. We also prove that, given a set of pointsV and a set of axis-parallel rectangles in the plane, then either there arek+1 rectangles such that no point ofV belongs to more than one of them, or we can find an at most 2·10⁵ k ⁸ element subset ofV meeting all rectangles. This improves a result of Ding, Seymour, and Winkler. Both proofs are based on Dilworth’s theorem on partially ordered sets. The research by János Pach was supported by Hungarian National Foundation for Scientific Research Grant OTKA-4269 and NSF Grant CCR-91-22103.