On a Cochrane Sum and Its Hybrid Mean Value Formula

The main purpose of this paper is to use a mean value theorem of Dirichlet L-functions to study the asymptotic property of a hybrid mean value of a Cochrane sum and to give an interesting mean value formula.     

An identity involving Dedekind sums and generalized Kloosterman sums

The various properties of classical Dedekind sums S(h, q) have been investi-gated by many authors. For example, Yanni Liu and Wenpeng Zhang: A hybrid mean value related to the Dedekind sums and Kloosterman sums, Acta Mathematica Sinica, 27 (2011), 435–440 studied the hybrid mean value properties involving Dedekind sums and generalized Kloosterman sums K(m, n, r; q). The main purpose of this paper, is using the analytic methods and the properties of character sums, to study the computational problem of one kind of hybrid mean value involving Dedekind sums and generalized Kloosterman sums, and give an interesting identity.     

A test for superadditivity of the mean value function of a non-homogeneous Poisson process

Let N(t) be a non-homogeneous Poisson process with mean value function Λ(t) and rate of occurrence λ(t). We propose a conditional test of the hypothesis that the process is homogeneous, versus alternatives for which the mean value function is superadditive. Specific models leading to superadditivity are presented, and the superadditive test is compared, on the basis of consistency and asymptotic relative efficiency, with the Cox-Lewis test, the latter being directed to alternatives where λ(t) is increasing.     

Mittelwerte von Hecke-Polynomen

In generalization of a classical result in number theory, we derive an estimate for the mean value of the Hecke polynomial $$S(t, \lambda ) = \sum {a(\mathfrak{a})} \lambda (\mathfrak{a})N\mathfrak{a}^{ - it} .$$ Here the sum runs over the integral ideals a of an algebraic number fieldK with normsNa≤X, and the mean value is defined via integration of |S(t, λ)|² with respect tot∈[?T,T], and summation of the result over the Hecke Groessencharacters λ modulo some ideal f ofK having exponents <T. From this we derive an upper bound for the mean value of the fourth power of Hecke's zeta functions on the half-line Res=1/2.     

On the mean value of L-functions with the weight of character sums

The main purpose of this paper is using the estimate for character sums and the analytic method to study the mean value of the Dirichlet L-functions with the weight of character sums, and give an interesting mean value theorem.     

Interpolation error estimates for mean value coordinates over convex polygons

In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson coordinates in Gillette et al. (Adv Comput Math 37(3), 417–439, 2012), we prove interpolation error estimates for the mean value coordinates on convex polygons suitable for standard finite element analysis. Our analysis is based on providing a uniform bound on the gradient of the mean value functions for all convex polygons of diameter one satisfying certain simple geometric restrictions. This work makes rigorous an observed practical advantage of the mean value coordinates: unlike Wachspress coordinates, the gradients of the mean value coordinates do not become large as interior angles of the polygon approach π.     

The prescribed mean curvature equation for nonparametric surfaces

We study the prescribed mean curvature equation for nonparametric surfaces, obtaining existence and uniqueness results in the Sobolev space W^2,p. We also prove that under appropriate conditions the set of surfaces of mean curvature H is a connected subset of W^2,p. Moreover, we obtain existence results for a boundary value problem which generalizes the one-dimensional periodic problem for the mean curvature equation.     

On an optimal harvesting problem

Kaplan [1972] treated a harvesting problem as a discrete time stochastic control model with independent disturbances and with decreasing mean value increase depending either on the value or on the age of the asset. We consider the more general model where the mean value increase depends on the value and also on the age. As main result we obtain the existence of optimal control-limit policies with respect to three different natural orders in state space. A basic role play additional convexity and boundedness assumptions. Our findings extend and correct the main result of Kaplan. The paper contains further detailed information about the solution.     

Optimization designs and performance comparison of two CUSUM schemes for monitoring process shifts in mean and variance

In statistical process control (SPC), when dealing with a quality characteristic x that is a variable, it is usually necessary to monitor both the mean value and variability. This article proposes an optimization algorithm (called the holistic algorithm) to design the CUSUM charts for this purpose. It facilitates the determination of the charting parameters of the CUSUM charts and considerably or significantly increases their overall detection effectiveness. A single CUSUM chart (called the ABS CUSUM chart) has been developed by the holistic algorithm and fully investigated. This chart is able to detect two-sided mean shifts and increasing variance shifts by inspecting the absolute value of sample mean shift. The results of performance studies show that the overall performance of the ABS CUSUM chart is nearly as good as an optimal 3-CUSUM scheme (a scheme incorporating three individual CUSUM charts). However, since the ABS CUSUM chart is easier for implementation and design, it may be more suitable for many SPC applications in which both mean and variance of a variable have to be monitored.     

Über Folgen,die BezÜglich Eines Mittels Rekurrent Sind

Let X be a convex subset of a finite-dimensional real vector space. A function M: X ^k → X is called a strict mean value, if M(x₁,…, x_k) lies in the convex hull of x₁,…, x_k), but does not coincide with one of its vertices. A sequence (x_n)_{n∈ ?} in X is called M-recursive if x_n+k = M(x_n, x_n+1,…, x_n+k?1) for all n. We prove that for a continuous strict mean value M every M-recursive sequence is convergent. We give a necessary and sufficient condition for a convergent sequence in X to be M-recursive for some continuous strict mean value M, and we characterize its limit by a functional equation. 39 B 72, 39 B 52, 40 A 05.     

Second moments of Dirichlet L-functions weighted by Kloosterman sums

For the general modulo q ? 3 and a general multiplicative character χ modulo q, the upper bound estimate of |S(m, n, 1, χ, q)| is a very complex and difficult problem. In most cases, the Weil type bound for |S(m, n, 1, χ, q)| is valid, but there are some counterexamples. Although the value distribution of |S(m, n, 1, χ, q)| is very complicated, it also exhibits many good distribution properties in some number theory problems. The main purpose of this paper is using the estimate for k-th Kloosterman sums and analytic method to study the asymptotic properties of the mean square value of Dirichlet L-functions weighted by Kloosterman sums, and give an interesting mean value formula for it, which extends the result in reference of W. Zhang, Y.Yi, X.He: On the 2k-th power mean of Dirichlet L-functions with the weight of general Kloosterman sums, Journal of Number Theory, 84 (2000), 199–213.     

On the mean square of the error term for the asymmetric two-dimensional divisor problems(II)

Let Δ(a, b; x) denote the error term of the asymmetric two-dimensional divisor problem. In this paper we shall study the relation between the discrete mean value ${\sum_{n\leq T} \Delta^2(a,b;n)}$ and the continuous mean value ${\int_1^T\Delta^2(a,b;x)dx}$ .     

Some new results on waiting time and busy time in M/G/1 queue

This paper considers an M/G/1 queue with Poisson rate λ > 0 and service time distribution G(t) which is supposed to have finite mean 1/μ. The following questions are first studied: (a) The closed bounds of the probability that waiting time is more than a fixed value; (b)The total busy time of the server, which including the distribution, probability that are more than a fixed value during a given time interval (0, t], and the expected value. Some new and important results are obtained by theories of the classes of life distributions and renewal process.     

Integral mean values of modular L-functions

We consider the mean squares of L-functions associated to modular forms with respect to Hecke congruence subgroups, expressing the mean value as an inner product. This avoids the discussion of generalized additive divisor problems. As applications, we obtain asymptotic formulas for both weighted and unweighted mean squares.     

A note on the upper bound estimate of high-dimensional Cochrane sums

An upper bound estimate of high-dimensional Cochrane sums is given in this note using Weinstein's version of Deligne's estimate for the hyper-Kloosterman sum and a mean value theorem of Dirichlet L-functions.     

The mean value of the squared path-difference distance for rooted phylogenetic trees

The path-difference metric is one of the oldest distances for the comparison of fully resolved phylogenetic trees, but its statistical properties are still quite unknown. In this paper we compute the mean value of the square of the path-difference metric between two fully resolved rooted phylogenetic trees with n leaves, under the uniform distribution. This complements previous work by Steel and Penny, who computed this mean value for fully resolved unrooted phylogenetic trees.     

On the average orders of the error term in the Dirichlet divisor problem

Let Δ(x) be the error term in the Dirichlet divisor problem. The purpose of this paper is to study the difference between two kinds of mean value formulas of Δ(x), that is, the mean value formulas and ∑_n?xΔ(n)^k with a natural number k. In particular we study the case k=2 and 3 in detail.     

On the mean value of dedekind sum weighted by the quadratic gauss sum

Various properties of classical Dedekind sums S(h, q) have been investigated by many authors. For example, Wenpeng Zhang, On the mean values of Dedekind sums, J. Théor. Nombres Bordx, 8 (1996), 429–442, studied the asymptotic behavior of the mean value of Dedekind sums, and H.Rademacher and E.Grosswald, Dedekind Sums, The Carus Mathematical Monographs No. 16, The Mathematical Association of America, Washington, D.C., 1972, studied the related properties. In this paper, we use the algebraic method to study the computational problem of one kind of mean value involving the classical Dedekind sum and the quadratic Gauss sum, and give several exact computational formulae for it.     

The Cauchy problem for the Liouville equation and Bryant surfaces

We give a construction that connects the Cauchy problem for the 2-dimensional elliptic Liouville equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3-space H³, and solve both of them. We construct the unique mean curvature one surface in H³ that passes through a given curve with a given unit normal along it, and provide diverse applications. In particular, topics such as period problems, symmetries, finite total curvature, planar geodesics, rigidity, etc. are treated for these surfaces.