A New Bound for the Fejér-Jackson Sum

We establish a new positive functional estimate for the Fejér-Jackson sum Sn() = k=1n k-1 sin k. This result enables us to give a simple proof of a result of Askey and Steinig on a monotonic sine sum associated with the sum Sn().     

On the Lower Bound of the Minimum Area of Convex Lattice Polygons

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A Fréchet-Optimal Strengthening of the Dawson-Sankoff Lower Bound

This paper proposes a lower bound for the probability that at least one out of arbitrary events occurs. The information used consists of the first- and second- degree Bonferroni summations in conjunction with and , where is the probability that exactly one event occurs and is the probability that all events occur. We prove that the proposed bound is a Fréchet optimal lower bound, which is a criterion difficult to achieve in general. The two additional non-negative terms used in the proposed bound make it at least as good as the Dawson–Sankoff lower bound, a Fréchet optimal degree two lower bound using the first- and second- degree Bonferroni summations only. A numerical example is presented to illustrate that in some cases, the improvement can be substantial. *Part of the work of this author was done as Visiting Scholar, 2004, at the University of Sydney.     

Lower Bound of the First Eigenvalue for the Laplace Operator on Compact Riemannian Manifold

Let M be a compact m-dimensional Riemannian manifold, let d denote, its diameter, -R(R>O) the lower bound of the Ricci curvature, and λ_1 the first eigerivalue for the Laplacian on M. Then there exists a constant C_m=max{2~(1/m-1),2~(1/2)}, Such thatλ_1≥π~2/d~2·1/(2-(11)/(2π~2))+11/2π~2e~cm、(?)     

A Lower Bound for the Differences of Powers of Linear Operators

Let T be a bounded linear operator in a Banach space, with σ（T）={1}. In 1983, Esterle-Berkani＇ s conjecture was proposed for the decay of differences （I - T） T^n as follows： Eitheror lim inf （n→∞（n＋1）｜｜（I-T）T^n｜｜≥1/e or T = I. We prove this claim and discuss some of its consequences.     

Lower Bound for the Poles of Igusa’s p-adic Zeta Functions

Let K be a p-adic field, R the valuation ring of K, P the maximal ideal of R and q the cardinality of the residue field R/P. Let f be a polynomial over R in n >1 variables and let χ be a character of . Let M _i (u) be the number of solutions of f = u in (R/P ⁱ ) ⁿ for and. These numbers are related with Igusa’s p-adic zeta function Z _f,χ(s) of f. We explain the connection between the M _i (u) and the smallest real part of a pole of Z _f,χ(s). We also prove that M _i (u) is divisible by , where the corners indicate that we have to round up. This will imply our main result: Z _f,χ(s) has no poles with real part less than − n/2. We will also consider arbitrary K-analytic functions f.     

Rigidity and the Lower Bound Theorem for Doubly Cohen–Macaulay Complexes

We prove that for d≥3, the 1-skeleton of any (d?1)-dimensional doubly Cohen–Macaulay (abbreviated 2-CM) complex is generically d-rigid. This implies that Barnette’s lower bound inequalities for boundary complexes of simplicial polytopes (Barnette, D. Isr. J. Math. 10:121–125, 1971; Barnette, D. Pac. J. Math. 46:349–354, 1973) hold for every 2-CM complex of dimension ≥2 (see Kalai, G. Invent. Math. 88:125–151, 1987). Moreover, the initial part (g ₀,g ₁,g ₂) of the g-vector of a 2-CM complex (of dimension ≥3) is an M-sequence. It was conjectured by Björner and Swartz (J. Comb. Theory Ser. A 113:1305–1320, 2006) that the entire g-vector of a 2-CM complex is an M-sequence.     

Upper Bound for the Bethe–Sommerfeld Threshold and the Spectrum of the Poisson Random Hamiltonian in Two Dimensions

We consider the Schrödinger operator on ${\mathbb{R}^2}$ with a locally square-integrable periodic potential V and give an upper bound for the Bethe–Sommerfeld threshold (the minimal energy above which no spectral gaps occur) with respect to the square-integrable norm of V on a fundamental domain, provided that V is small. As an application, we prove the spectrum of the two-dimensional Schrödinger operator with the Poisson type random potential almost surely equals the positive real axis or the whole real axis, according as the negative part of the single-site potential equals zero or not. The latter result completes the missing part of the result by Ando et al. (Ann Henri Poincaré 7:145–160, 2006).     

A Berry-Esséen Bound for the Product-Limit Estimator under Left-Truncation and Right-Censoring

For left-truncated and right-censored data, the product-limit estimator F?_xis a well-known nonparametric estimator for the distribution function F_x of the target variable X such as the survival time. Since F?_xas a very complicated product form we establish first the Berry-Esseen bound for the cumulative hazard estimator of F_x The cumulative hazard estimator can be represented as a U-statistic. By using the result in Helmers and van Zwet [6], we derive the Berry-esséen bound for this U-statistic. Then Berry-Esseen bounds for the distribution of the cumulative hazard estimator and the normal distribution and the distribution of the product-limit estimator and the normal distribution are obtained.     

A Lower Bound on Unknotting Number

In this paper the authors use a modified Wirtinger presentation to give a lower bound on the unknotting number of a knot in S3.     

On a Lower Bound of the Uniform Distance in the Central Limit Theorem for ϕ-Mixing Random Variables

We derive a lower bound of the uniform distance in the central limit theorem for real -mixing random variables under the finiteness of the eighth moments of summands. The main result of the present paper generalizes the corresponding author"s result obtained in 1997 for m-dependent random variables to the case of -mixing random variables.     

Virtual Bound Levels in a Gap of the Essential Spectrum of the Weakly Perturbed Periodic Schrödinger Operator

In the space \({L_{2}(\mathbf{R}^{d}) (d \le 3)}\) we consider the Schrödinger operator \({H_{\gamma}=-{\Delta}+ V(\mathbf{x})\cdot+\gamma W(\mathbf{x})\cdot}\), where \({V(\mathbf{x})=V(x_{1}, x_{2}, \dots, x_{d})}\) is a periodic function with respect to all the variables, \({\gamma}\) is a small real coupling constant and the perturbation \({W(\mathbf{x})}\) tends to zero sufficiently fast as \({|\mathbf{x}|\rightarrow\infty}\). We study so called virtual bound levels of the operator \({H_\gamma}\), i.e., those eigenvalues of \({H_\gamma}\) which are born at the moment \({\gamma=0}\) in a gap \({(\lambda_-,\,\lambda_+)}\) of the spectrum of the unperturbed operator \({H_0=-\Delta+ V(\mathbf{x})\cdot}\) from an edge of this gap while \({\gamma}\) increases or decreases. We assume that the dispersion function of H₀, branching from an edge of \({(\lambda_-,\lambda_+)}\), is non-degenerate in the Morse sense at its extremal set. For a definite perturbation \({(W(\mathbf{x})\ge 0)}\) we show that if d ≤ 2, then in the gap there exist virtual eigenvalues which are born from this edge. We investigate their number and an asymptotic behavior of them and of the corresponding eigenfunctions as \({\gamma\rightarrow 0}\). For an indefinite perturbation we estimate the multiplicity of virtual bound levels. In particular, we show that if d = 3 and both edges of the gap \({(\lambda_-,\,\lambda_+)}\) are non-degenerate, then under additional conditions there is a threshold for the birth of the impurity spectrum in the gap, i.e., \({\sigma(H_\gamma)\cap(\lambda_-,\,\lambda_+)=\emptyset}\) for a small enough \({|\gamma|}\).     

On the Singularity of Random Bernoulli Matrices— Novel Integer Partitions and Lower Bound Expansions

We prove a lower bound expansion on the probability that a random ±1 matrix is singular, and conjecture that such expansions govern the actual probability of singularity. These expansions are based on naming the most likely, second most likely, and so on, ways that a Bernoulli matrix can be singular; the most likely way is to have a null vector of the form e _i ±e _j , which corresponds to the integer partition 11, with two parts of size 1. The second most likely way is to have a null vector of the form e _i ±e _j ±e _k ±e _? , which corresponds to the partition 1111. The fifth most likely way corresponds to the partition 21111. We define and characterize the “novel partitions” which show up in this series. As a family, novel partitions suffice to detect singularity, i.e., any singular Bernoulli matrix has a left null vector whose underlying integer partition is novel. And, with respect to this property, the family of novel partitions is minimal. We prove that the only novel partitions with six or fewer parts are 11, 1111, 21111, 111111, 221111, 311111, and 322111. We prove that there are fourteen novel partitions having seven parts. We formulate a conjecture about which partitions are “first place and runners up” in relation to the Erd?s-Littlewood-Offord bound. We prove some bounds on the interaction between left and right null vectors.     

Newton’s Method for Computing the Nearest Correlation Matrix with a Simple Upper Bound

The standard nearest correlation matrix can be efficiently computed by exploiting a recent development of Newton’s method (Qi and Sun in SIAM J. Matrix Anal. Appl. 28:360–385, 2006). Two key mathematical properties, that ensure the efficiency of the method, are the strong semismoothness of the projection operator onto the positive semidefinite cone and constraint nondegeneracy at every feasible point. In the case where a simple upper bound is enforced in the nearest correlation matrix in order to improve its condition number, it is shown, among other things, that constraint nondegeneracy does not always hold, meaning Newton’s method may lose its quadratic convergence. Despite this, the numerical results show that Newton’s method is still extremely efficient even for large scale problems. Through regularization, the developed method is applied to semidefinite programming problems with simple bounds.     

Bound states and the Szeg? condition for Jacobi matrices and Schrödinger operators

For Jacobi matrices with a_n=1+(−1)ⁿαn^−γ, b_n=(−1)ⁿβn^−γ, we study bound states and the Szeg? condition. We provide a new proof of Nevai's result that if , the Szeg? condition holds, which works also if one replaces (−1)ⁿ by . We show that if α=0, β≠0, and , the Szeg? condition fails. We also show that if γ=1, α and β are small enough ( will do), then the Jacobi matrix has finitely many bound states (for α=0, β large, it has infinitely many).     

Two-phases Method and Branch and Bound Procedures to Solve the Bi–objective Knapsack Problem

The classical 0–1 knapsack problem is considered with two objectives. Two methods of the two–phases type are developed to generate the set of efficient solutions. In the first phase, the set of supported efficient solutions is determined by optimizing a parameterized single-objective knapsack problem. Two versions are proposed for a second phase, determining the non-supported efficient solutions: both versions are Branch and Bound approaches, but one is breadth first, while the other is depth first. Extensive numerical experiments have been realized to compare the results of both methods.     

A Berry–Esseen Bound for U-Statistics in the Non-I.I.D. Case

Let X ₁,..., X _n be independent, not necessarily identically distributed random variables. An optimal Berry–Esseen bound is derived for U-statistics of order 2, that is, statistics of the form T=_1i<jn g _ij(X _i, X _j), where the g _ij are measurable functions such that |g _ij(X _i, X _j)|<. An application is given concerning Wilcoxon's rank-sum test.     

A Berry-Esseen Bound for k-sample Symmetric Statistics

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Global Complexity Bound Analysis of the Levenberg–Marquardt Method for Nonsmooth Equations and Its Application to the Nonlinear Complementarity Problem

We investigate a global complexity bound of the Levenberg–Marquardt Method (LMM) for nonsmooth equations. The global complexity bound is an upper bound to the number of iterations required to get an approximate solution that satisfies a certain condition. We give sufficient conditions under which the bound of the LMM for nonsmooth equations is the same as smooth cases. We also show that it can be reduced under some regularity assumption. Furthermore, by applying these results to nonsmooth equations equivalent to the nonlinear complementarity problem (NCP), we get global complexity bounds for the NCP. In particular, we give a reasonable bound when the mapping involved in the NCP is a uniformly P-function.