Mean Value Formula Research Articles

A sum analogous to the high-dimensional Kloosterman sums and its upper bound estimate

The main purpose of this paper is, using the properties of Gauss sums and the estimate for the generalized exponential sums, to study the upper bound estimate problem of one kind sums analogous to the high-dimensional Kloosterman sums and to give some interesting mean value formula and an upper bound estimate for it. MSC:11L05.

Asymptotic mean value formula for sub-p-harmonic functions on the Heisenberg group

In this paper, we prove an asymptotic mean value formula of sub-p-harmonic functions in the viscosity sense on the Heisenberg group. As an application, we give a new proof of the Harnack inequality for sub-p-harmonic functions on the Heisenberg group.

On the Value Function of the M/G/1 FCFS and LCFS Queues

We consider a single-server queue with Poisson input operating under first-come–first-served (FCFS) or last-come–first-served (LCFS) disciplines. The service times of the customers are independent and obey a general distribution. The system is subject to costs for holding a customer per unit of time, which can be customer specific or customer class specific. We give general expressions for the corresponding value functions, which have elementary compact forms, similar to the Pollaczek–Khinchine mean value formulae. The results generalize earlier work where similar expressions have been obtained for specific service time distributions. The obtained value functions can be readily applied to develop nearly optimal dispatching policies for a broad range of systems with parallel queues, including multiclass scenarios and the cases where service time estimates are available.

On the Value Function of the M/G/1 FCFS and LCFS Queues

We consider a single-server queue with Poisson input operating under first-come–first-served (FCFS) or last-come–first-served (LCFS) disciplines. The service times of the customers are independent and obey a general distribution. The system is subject to costs for holding a customer per unit of time, which can be customer specific or customer class specific. We give general expressions for the corresponding value functions, which have elementary compact forms, similar to the Pollaczek–Khinchine mean value formulae. The results generalize earlier work where similar expressions have been obtained for specific service time distributions. The obtained value functions can be readily applied to develop nearly optimal dispatching policies for a broad range of systems with parallel queues, including multiclass scenarios and the cases where service time estimates are available.

Density in mandibular trabeculae of beta thalassemia major patients at the age of 11-13 years

Introduction: Density of mandible trabeculae is defined as density of trabeculae at posterior region of mandibulae. Systemic diseases which can affect bone density are osteoporosis and many types of anemia, including which has significant influence that is beta thalassemia major. The objective of this investigation was to obtain the data of density in mandible trabeculae of beta thalassemia major patients at the age of 11-13 years. Methods: The descriptive method was used in this study and the samples were selected using the total sampling technique. 15 periapical photos were obtained using the total sampling echnique. The periapical photos are administered using 3D-Doctor software. Data were analyzed using simple mean value formula. Results: Statistical count resulted in the minimum score of sample which was 98, 039 grayscale, the maximum score of sample which was 231, 34 grayscale, and the range score of sample which was 126.503 grayscale Conclusion: Density of mandible trabeculae of thalassemia beta major patients at the age of 11-13 is 126.503 grayscale.

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 derivation of new mean-value formulas for linear differential equations with constant coefficients

In the framework of a symbolic approach to mean-value formulas, we suggest a method for the derivation of new mean-value formulas for some classes of partial differential equations.

A hybrid mean value related to certain hardy sums and Kloosterman sums

The main purpose of this paper is using the mean value formula of Dirichlet L-functions and the analytic methods to study a hybrid mean value problem related to certain Hardy sums and Kloosterman sums, and give some interesting mean value formulae and identities for it.

Mean values of L-functions and relative class numbers of cyclotomic fields

Using formulas for quadratic mean values of L-functions at s = 1, we recover previously known explicit upper bounds on relative class numbers of cyclotomic elds. We also obtain new better bounds.

Intrinsic volumes of the maximal polytope process in higher dimensional STIT tessellations

Stationary and isotropic iteration stable random tessellations are considered, which are constructed by a random process of iterative cell division. The collection of maximal polytopes at a fixed time t within a convex window W ⊂ R d is regarded and formulas for mean values, variances and a characterization of certain covariance measures are proved. The focus is on the case d ≥ 3 , which is different from the planar one, treated separately in Schreiber and Thäle (2010) [12]. Moreover, a limit theorem for suitably rescaled intrinsic volumes is established, leading — in sharp contrast to the situation in the plane — to a non-Gaussian limit.

A hybrid mean value related to the Dedekind sums and Kloosterman sums

The main purpose of this paper is to use the properties of character sum and the analytic method to study a hybrid mean value problem related to the Dedekind sums and Kloosterman sums, and give some interesting mean value formulae and identities for it.

Measuring the level sets of anisotropic homogeneous functions

In this note we investigate some basic properties of the level sets of functions which are homogeneous with respect to nonisotropic dilations. In particular we obtain a formula for the volume of the level sets in terms of the area on the level surfaces. We relate the results to some well known mean value formulas for solutions of PDE’s.

A hybird mean value involving gauss sums and character sums

Let χ be a Dirichlet character modulo q > 2, and L(s, χ) denotes the Dirichlet L-function corresponding to χ. The main purpose of this paper is using the estimate for character sums and the analytic method to study the mean value properties of \( \frac{{L'}} {L}(1,\chi ) \) with the weight of Gauss sums and character sums, and give an interesting mean value formula for it.

A hybrid mean value related to the Dedekind sums and Kloosterman sums

The main purpose of this paper is using the properties of character sum and the analytic method to study a hybrid mean value problem related to the Dedekind sums and Kloosterman sums, and give some interesting mean value formulae and identities for it.

ON THE γ-TH HYPER-KLOOSTERMAN SUMS AND A PROBLEM OF D. H. LEHMER

For any integer k <TEX>$\geq$</TEX> 2, let P(c, k + 1;q) be the number of all k+1-tuples with positive integer coordinates (<TEX>$a_1,a_2,...,a_{k+1}$</TEX>) such that <TEX>$1{\leq}a_i{\leq}q$</TEX>, (<TEX>$a_i,q$</TEX>) = 1, <TEX>$a_1a_2...a_{k+1}{\equiv}$</TEX> c (mod q) and 2 <TEX>$\nmid$</TEX> (<TEX>$a_1+a_2+...+a_{k+1}$</TEX>), and E(c, k+1; q) = P(c, k+1;q) - <TEX>$\frac{{\phi}^k(q)}{2}$</TEX>. The main purpose of this paper is using the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions to study the hybrid mean value of the r-th hyper-Kloosterman sums Kl(h,k+1,r;q) and E(c,k+1;q), and give an interesting mean value formula.

Mean value properties of solutions of linear partial differential equations

The present paper suggests a uniform viewpoint to mean value theorems for linear elliptic and hyperbolic partial differential equations that, in a certain cases, allows one to obtain new mean value formulas. Also, the authors consider a method for obtaining mean value formulas for elliptic equations from those for hyperbolic equations.

MEAN VALUE OF THE CHARACTER SUMS OVER INTERVAL

Let $q\gt 8$ be an odd integer and $p$ a prime with $p$ a prime with $p\lt q$ and $p\nmid q$. The main purpose of this paper is to study the mean value properties of the character sums over interval $\left[1, \frac{q}{8}\right)$ by using the mean value theorems of the Dirichlet L-functions,and give some interesting mean value formulae.

MEAN VALUE OF THE CHARACTER SUMS OVER INTERVAL

Let $q>8$ be an odd integer and $p$ a prime with $p$ a prime with $p<q$ and $p\nmid q$. The main purpose of this paper is to study the mean value properties of the character sums over interval $\left[1, \frac{q}{8}\right)$ by using the mean value theorems of the Dirichlet L-functions, and give some interesting mean value formulae.

On character sums over a short interval

The main purpose of this paper is using the analytic methods to study the hybrid mean value involving the character sums, general quadratic Gauss sums and general Kloosterman sums, and give several interesting mean value formulae.

Hybrid mean value on the difference between an integer and its inverse modulo q

The main purpose of this paper is to study the asymptotic property of one class of number-theoretic functions and give a sharp hybrid mean-value formula by using the generalized Bernoulli numbers, Gauss sums and the mean-value theorems of Dirichlet L-functions.