On the Mean Value of High-Powers of a Special Character Sum Modulo a Prime

In this paper,we use the elementary methods,the properties of Dirichlet character sums and the classical Gauss sums to study the estimation of the mean value of high-powers for a special character sum modulo a prime,and derive an exact computational formula.It can be conveniently programmed by the“Mathematica”software,by which we can get the exact results easily.

THE SCHUR CONVEXITY OF GINI MEAN VALUES IN THE SENSE OF HARMONIC MEAN

We prove that the Gini mean values S（a,b; x,y） are Schur harmonic convex with respect to （x,y）∈（0,∞）×（0,∞） if and only if （a, b） ∈{（a, b）：a≥0,a ≥ b,a＋b＋1≥0}∪{（a,b）：b≥0,b≥a,a＋b＋1≥0} and Schur harmonic concave with respect to （x,y） ∈ （0,∞）×（0,∞） if and only if （a,b）∈{（a,b）：a≤0,b≤0,a｜b｜1≤0}.

The Asymptotic Property of Generalized Taylor Remainder "Mean Value Point"

This paper discussed asymptotic property of Taylor remainder 'mean value point' in normed Linear space. The asymptotic progerty of 'mean value point' is solved when f(n+i)(x0)h(n+i) = 0(i = 1, 2,..., p - 1) and f(n+p)(x0)h(h+p) don't exist. Meanwhile, achieve more general asymptotic estimation formula. Make many former results are just because of special case of the pager.

Some Hybrid Mean Values of a New Number Theoretic Function

Let p and q be two distinct primes, epq（n） denotes the largest exponent of power pq which divides n. In this paper, we study the mean value properties of function epq（n）, and give some hybrid mean value formulas for epq（n） and Dirichlet divisor function d（n）. Key words： largest exponent; asymptotic formula; hybrid mean value; Dirichlet divisor function d（n）

THE MEAN VALUE THEOREM AND CONVERSE THEOREM OF ONE CLASS THE FOURTH-ORDER PARTIAL DIFFERENTIAL EQUATIONS

For the formal presentation about the definite problems of ultra-hyperbolic equations, the famous Asgeirsson mean value theorem has answered that Cauchy problems are ill-posed to ultra-hyperbolic partial differential equations of the second-order. So it is important to develop Asgeirsson mean value theorem. The mean value of solution for the higher order equation hay been discussed primarily and has no exact result at present. The mean value theorem for the higher order equation can be deduced and satisfied generalized biaxial symmetry potential equation by using the result of Asgeirsson mean value theorem and the properties of derivation and integration. Moreover, the mean value formula can be obtained by using the regular solutions of potential equation and the special properties of Jacobi polynomials. Its converse theorem is also proved. The obtained results make it possible to discuss on continuation of the solutions and well posed problem.

On the Lucas Base and Computation of Counting Function Mean Value

In this paper, we introduce a new counting function a(m) related to the Lucas number, then use conjecture and induction methods to give an exact formula Ar(N)=α(n), (r=1,2,3) and prove them.

Hybrid Mean Value of the Hyper Cochrane Sums

The main purpose of this paper is to use the analytic methods to study the hybrid mean value involving the hyper Cochrane sums, and give several sharp asymptotic formulae.

On the Mean Value of the Infinite Order Legendre Series

In this paper, we studies the relations between the mean value and the maximun norm of the infinite order entire functions which defined by legendre series. We obtained that if f(z) is an infinite order entire function with a positive exponenatial lower order. then loaM (α) ～logMδ(α) ～ logMδ(α) (α→+∞).

A New Compound Function and Its Mean Value

The main purpose of this paper is using the analytic method to study the mean value properties of the arithmetical functions δk（（m, n））, δk（[m,n]/m）,and give several interesting asymptotic formulae for them.

Mean Values of the Hardy Sum

Let p≥5 be a prime. For any integer h, the Hardy sum is defined by H(h,p)=sum((-1)^(j+1+[(hj)/p]))from (p-1) to (j=1) which is related to the classical Dedekind sum. The mean values of the Hardy sum in short intervals are studied by using the mean value theorems of Dirichlet L-functions.

A Note on the Mean Value Theorems

The mean value theorem for derivatives says that for a given function over a closed and bounded interval, there is a point P on the graph such that the tangent at P is parallel to the secant through the two endpoints. The mean value theorem for definite integrals says that the area under the function is equal to the area of a rectangle whose base is the length of the interval and height of some point Q on the graph. These two theorems have been studied and utilized extensively and they form the backbone of many important theorems in different branches of mathematics. In this note, we pose the question: for what functions do the two points P and Q always coincide? We find that the only analytic functions satisfying this condition are linear or exponential functions.

Image Feature Computation in Encrypted Domain Based on Mean Value

In smart environments,more and more teaching data sources are uploaded to remote cloud centers which promote the development of the smart campus.The outsourcing of massive teaching data can reduce storage burden and computational cost,but causes some privacy concerns because those teaching data(especially personal image data)may contain personal private information.In this paper,a privacy-preserving image feature extraction algorithm is proposed by using mean value features.Clients use block scrambling and chaotic map to encrypt original images before uploading to the remote servers.Cloud servers can directly extract image mean value features from encrypted images.Experiments show the effectiveness and security of our algorithm.It can achieve information search over the encrypted images on the smart campus.

A Number Theoretic Function and Its Mean Value Computation

Let p be a prime, n be any positiv e integer, α(n,p) denotes the power of p in the factorization of n! . In this paper, we give an exact computing formula of the mean value ∑ n<Nα(n,p).

A Mean Value of Cochrane Sum

Let q 〉 4 be an integer. The main purpose of this paper is to study the mean value of Cochrane sum C（a, q） in quarter intervals, and obtain a sharp asymptotic formula for it.

Solution of an open problem for Schur convexity or concavity of the Gini mean values

The Schur convexity or concavity problem of the Gini mean values S(a, b; x, y) with respect to (x, y) ∈ (0, ∞) × (0, ∞) for fixed (a, b) ∈ ? × ? is still open. In this paper, we prove that S(a, b; x, y) is Schur convex with respect to (x, y) ∈ (0, ∞) × (0, ∞) if and only if (a, b) ∈ {(a, b): a ? 0, b ? 0, a + b ? 1}, and Schur concave with respect to (x, y) ∈ (0, ∞) × (0, ∞) if and only if (a, b) ∈ {(a, b): b ? 0, b ? a, a + b ? 1} ∩ {(a, b): a ? 0, a ? b, a + b ? 1}.

The mean value involving Dedekind sums and two-term exponential sums

In this paper,we use the analytic methods to study the mean value properties involving the classical Dedekind sums and two-term exponential sums,and give two sharper asymptotic formulae for it.

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.

Mesh Deformation Method Based on Mean Value Coordinates Interpolation

Mesh deformation technique is widely used in many application fields, and has re- ceived a lot of attentions in recent years. This paper focuses on the methodology and algorithm of algebraic type mesh deformation for unstructured mesh in numerical discretization. To preserve mesh quality effectively, an algebraic approach for two and three dimensional unstructured mesh is developed based on mean value coordinates interpolation combined with node visibility analysis. The proposed approach firstly performs node visibility analysis to find out the visible boundary for each grid point to be moved, then evaluates the mean value coordinates of each grid point with respect to all vertices on its visible boundary. Thus the displacements of grid points can be calculated by interpolating the boundary movement by the mean value coordinates. Compared with other methods, the proposed method has good deformation capability and predictable com- putational cost, with no need to select parameters or functions. Applications of mesh deformation in different fields are presented to demonstrate the effectiveness of the proposed approach. The results of numerical experiments exhibit not only superior deformation capability of the method in traditional applications of fluid dynamic grid, but also great potential in modeling for large deformation analysis and inverse design problems.

Trigonometric series with piecewise mean value bounded variation coefficients

In the present paper,we generalize some classical results in convergence and integrability for trigonometric series with varying coefficients(may change signs),by introducing a new ultimate condition upon coefficient sequences.This is a comprehensive systematic work on the topic.

Observer design for induction motor： an approach based on the mean value theorem

In this paper, observer design for an induction motor has been investigated. The peculiarity of this paper is the synthesis of a mono-Luenberger observer for highly coupled system. To transform the nonlinear error dynamics for the induction motor into the linear parametric varying （LPV） system, the differential mean value theorem combined with the sector nonlinearity transformation has been used. Stability conditions based on the Lyapunov function lead to solvability of a set of linear matrix inequalities. The proposed observer guarantees the global exponential convergence to zero of the estimation error. Finally, the simulation results are given to show the performance of the observer design.