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 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.

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.

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.

Sub-Differential Characterizations of Non-Smooth Lower Semi-Continuous Pseudo-Convex Functions on Real Banach Spaces

In this paper, we characterize lower semi-continuous pseudo-convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the pseudo-monotonicity of its Clarke-Rockafellar Sub-differential. We extend the results on the characterizations of non-smooth convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the monotonicity of its sub-differentials to the lower semi-continuous pseudo-convex functions on real Banach spaces.

AN APPROXIMATE EXPRESSION RELATED WITH RSA FIXED POINTS

Let T=T(n,e,α) be the number of fixed points o f RSA(n,e) that a re co prime with n=pq,and A,B be sets of prime numbers in (1,x) and (1,y) respectively. An estimation on the mean value M(A,B,e,α)=1 (#A)(#B)∑p∈A,q∈B,(p,q)=1logT(pq,e,α) is given.

Meridional Distributions of Historical Zonal Averages and Their Use to Quantify the Global and Spheroidal Mean Near-Surface Temperature of the Terrestrial Atmosphere

The zonal averages of temperature (the so-called normal temperatures) for numerous parallels of latitude published between 1852 and 1913 by Dove, Forbes, Ferrel, Spitaler, Batchelder, Arrhenius, von Bezold, Hopfner, von Hann, and Börnstein were used to quantify the global (spherical) and spheroidal mean near-surface temperature of the terrestrial atmosphere. Only the datasets of Dove and Forbes published in the 1850s provided global averages below 〈T〉=14°C, mainly due to the poor coverage of the Southern Hemisphere by observations during that time. The global averages derived from the distributions of normal temperatures published between 1877 and 1913 ranged from 〈T〉=14.0°C (Batchelder) to 〈T〉=15.1°C (Ferrel). The differences between the global and the spheroidal mean near-surface air temperature are marginal. To examine the uncertainty due to interannual variability and different years considered in the historic zonal mean temperature distributions, the historical normal temperatures were perturbed within ±2σ to obtain ensembles of 50 realizations for each dataset. Numerical integrations of the perturbed distributions indicate uncertainties in the global averages in the range of ±0.3°C to ±0.6°C and depended on the number of available normal temperatures. Compared to our results, the global mean temperature of 〈T〉=15.0°C published by von Hann in 1897 and von Bezold in 1901 and 1906 is notably too high, while 〈T〉=14.4°C published by von Hann in 1908 seems to be more adequate within the range of uncertainty. The HadCRUT4 record provided 〈T〉≌?13.7°C for 1851-1880 and 〈T〉=13.6°C for 1881-1910. The Berkeley record provided 〈T〉=13.6°C and 〈T〉≌?13.5°C for these periods, respectively. The NASA GISS record yielded 〈T〉=13.6°C for 1881-1910 as well. These results are notably lower than those based on the historic zonal means. For 1991-2018, the HadCRUT4, Berkeley, and NASA GISS records provided 〈T〉=14.4°C, 〈T〉=14.5°C, and 〈T〉=14.5°C, respectively. The comparison of the 1991-2018 globally averaged near-surface temperature with those derived from distributions of zonal temperature averages for numerous parallels of latitude suggests no change for the past 100 years.

CHARACTERISTICS OF SUBDIFFERENTIALS OF FUNCTIONS

In the presem paper, some important characteristics of Fenchel-, Frechet-,Hademard-, and Gateaux-Subdifferentials are showed up, and properties of functions, especially. convexity of functions, are described by subdifferentials.

SOLUTIONS TO A CLASS OF PARABOLIC INHOMOGENEOUS NORMALIZED p-LAPLACE EQUATIONS

In this article, we prove that viscosity solutions of the parabolic inhomogeneous equationsn＋p/put-△p^Nu=f（x,t）can be characterized using asymptotic mean value properties for all p ≥ 1, including p = 1 and p = ∞. We also obtain a comparison principle for the non-negative or non-positive inhomogeneous term f for the corresponding initial-boundary value problem and this in turn implies the uniqueness of solutions to such a problem.

GENERAL SOLUTION OF A 2-D WEAK SINGULAR INTEGRAL EQUATION WITH CONSTRAINT AND ITS APPLICATIONS

In this paper, the solution, more general than [1], of a weak singular integral equation integral(0)(pi)integral(-infinity)(infinity) p(s,psi)d sk(psi)d psi=F(r,theta), (r,theta)epsilon (Q) over bar=Q+partial derivative Q subject to constraint p(s,psi)=0, for (s,psi)=(r,theta)is not an element of Q={r,theta)/F(r,theta)>c*} is found p=2/pi[root w g'(0)+integral(0)(w) root w-u g '(u)du] where k and F are given continuous functions; (s,psi) is a local polar coordinating with origin at M(r,theta); (r,theta) is the global polar coordinating with origin at O(0,0) F(r,theta)=c* (const.) is the boundary contour partial derivative Q of the considered range Q; g(w)=F(r,theta)/[pi k(psi(0))]; g'=dg/dw; w=N-r(2)sin(2)(theta+psi(0)); psi(0) and N are mean values. The solution shown in type (2.19) of [1] is a special case of the above solution and only suits F(r,theta)=w. The solution of a rigid cone contact with elastic half space, more simple and clear than Love's (1939), is given as an example of application.

Four-Step Iteration Scheme to Approximate Fixed Point for Weak Contractions

Fixed point theory is one of the most important subjects in the setting of metric spaces since fixed point theorems can be used to determine the existence and the uniqueness of solutions of such mathematical problems.It is known that many problems in applied sciences and engineering can be formulated as functional equations.Such equations can be transferred to fixed point theorems in an easy manner.Moreover,we use the fixed point theory to prove the existence and uniqueness of solutions of such integral and differential equations.Let X be a non-empty set.A fixed point for a self-mapping T on X is a point𝑒𝑒∈𝑋𝑋that satisfying T e=e.One of the most challenging problems in mathematics is to construct some iterations to faster the calculation or approximation of the fixed point of such problems.Some mathematicians constructed and generated some new iteration schemes to calculate or approximate the fixed point of such problems such as Mann et al.[Mann(1953);Ishikawa(1974);Sintunavarat and Pitea(2016);Berinde(2004b);Agarwal,O’Regan and Sahu(2007)].The main purpose of the present paper is to introduce and construct a new iteration scheme to calculate or approximate the fixed point within a fewer number of steps as much as we can.We prove that our iteration scheme is faster than the iteration schemes given by Sintunavarat et al.[Sintunavarat and Pitea(2016);Agarwal,O’Regan and Sahu(2007);Mann(1953);Ishikawa(1974)].We give some numerical examples by using MATLAB to compare the efficiency and effectiveness of our iterations scheme with the efficiency of Mann et al.[Mann(1953);Ishikawa(1974);Sintunavarat and Pitea(2016);Abbas and Nazir(2014);Agarwal,O’Regan and Sahu(2007)]schemes.Moreover,we introduce a problem raised from Newton’s law of cooling as an application of our new iteration scheme.Also,we support our application with a numerical example and figures to illustrate the validity of our iterative scheme.

关于Hardy和的均值（英文）

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.

An Adaptive Lasso Grey Model for Regional FDI Statistics Prediction

To overcome the deficiency of traditional mathematical statistics methods,an adaptive Lasso grey model algorithm for regional FDI(foreign direct investment)prediction is proposed in this paper,and its validity is analyzed.Firstly,the characteristics of the FDI data in six provinces of Central China are generalized,and the mixture model’s constituent variables of the Lasso grey problem as well as the grey model are defined.Next,based on the influencing factors of regional FDI statistics(mean values of regional FDI and median values of regional FDI),an adaptive Lasso grey model algorithm for regional FDI was established.Then,an application test in Central China is taken as a case study to illustrate the feasibility of the adaptive Lasso grey model algorithm in regional FDI prediction.We also select RMSE(root mean square error)and MAE(mean absolute error)to demonstrate the convergence and the validity of the algorithm.Finally,we train this proposedal gorithm according to the regional FDI statistical data in six provinces in Central China from 2006 to 2018.We then use it to predict the regional FDI statistical data from 2019 to 2023 and show its changing tendency.The extended work for the adaptive Lasso grey model algorithm and its procedure to other regional economic fields is also discussed.

Exploring the Three-dimensional Framework of Knowledge Service in the field of Library and Information Science(LIS)

The evolution of the society and economy has stimulated the development of Knowledge Service(KS), making it an indispensable solution to address future challenges facing libraries and information institutions. However at present, academic research on knowledge service is falling short and its definition is far from clear and complete. As such,this article proposes the Three-dimensional Framework Knowledge Service(TdFKS) for libraries and information institutions based on the knowledge value chain model. By making reliability analysis and mean value analysis of a questionnaire survey result, the article clarifies the structure of the three-dimensional framework and verifies the rationality of the TdFKS.

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.

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.

On the Mean Value of the Complete Trigonometric Sums with Dirichlet Characters

The main purpose of this paper is, using the analytic method, to study the mean value properties of the complete trigonometric sums with Dirichlet characters, and give an exact calculating formula for its fourth power mean.