A Bombieri-type mean-value theorem concerning exponential sums over primes

In several classical problems in number theory,like the Goldbach problem or the twinprime problem,one has to treat the exponential

The Mean-value of Meromorphic Functions with Respect to a Generalized Boolean Transformation

A formula for the mean-value distribution of certain meromorphic functions on a vertical line s = σ +iR under a generalized Boolean transformation, called rational Boolean transformation from R into itself, is derived using Birkhoff 's ergodic theorem. This formula is represented as a computable integral. Using the Cauchy's integral theorem, values of this integral corresponding to various possible cases are explicitly computed.

Damage alarming of long-span suspension bridge based on GPS-RTK monitoring

Structure damage identification and alarming of long-span bridge were conducted with three-dimensional dynamic displacement data collected by GPS subsystem of health monitoring system on Runyang Suspension Bridge.First,the effects of temperature on the main girder spatial position coordinates were analyzed from the transverse,longitudinal and vertical directions of bridge,and the correlation regression models were built between temperature and the position coordinates of main girder in the longitudinal and vertical directions;then the alarming indices of coordinate residuals were conducted,and the mean-value control chart was applied to making statistical pattern identification for abnormal changes of girder dynamic coordinates;and finally,the structural damage alarming method of main girder was established.Analysis results show that temperature has remarkable correlation with position coordinates in the longitudinal and vertical directions of bridge,and has weak correlation with the transverse coordinates.The 3%abnormal change of the longitudinal coordinates and 5%abnormal change of the vertical ones caused by structural damage are respectively identified by the mean-value control chart method based on GPS dynamic monitoring data and hence the structural abnormalities state identification and damage alarming for main girder of long-span suspension bridge can be realized in multiple directions.

Common Properties of Riemann Zeta Function, Bessel Functions and Gauss Function Concerning Their Zeros

The behavior of the zeros in finite Taylor series approximations of the Riemann Xi function (to the zeta function), of modified Bessel functions and of the Gaussian (bell) function is investigated and illustrated in the complex domain by pictures. It can be seen how the zeros in finite approximations approach to the genuine zeros in the transition to higher-order approximation and in case of the Gaussian (bell) function that they go with great uniformity to infinity in the complex plane. A limiting transition from the modified Bessel functions to a Gaussian function is discussed and represented in pictures. In an Appendix a new building stone to a full proof of the Riemann hypothesis using the Second mean-value theorem is presented.

Star-shaped Differentiable Functions and Star-shaped Differentials

Based on the isomorphism between the space of star-shaped sets and the space of continuous positively homogeneous real-valued functions, the star-shaped differential of a directionally differentiable function is defined. Formulas for star-shaped differential of a pointwise maximum and a pointwise minimum of a finite number of directionally differentiable functions, and a composite of two directionaUy differentiable functions are derived. Furthermore, the mean-value theorem for a directionaUy differentiable function is demonstrated.

Approach to Riemann Hypothesis by Combined Commensurable Step Function Approximation with Bonnet Method

To the Riemann hypothesis, we investigate first the approximation by step-wise Omega functions Ω(u) with commensurable step lengths u0 concerning their zeros in corresponding Xi functions Ξ(z). They are periodically on the y-axis with period proportional to inverse step length u0. It is found that they possess additional zeros off the imaginary y-axis and additionally on this axis and vanish in the limiting case u0 → 0 in complex infinity. There remain then only the “genuine” zeros for Xi functions to continuous Omega functions which we call “analytic zeros” and which lie on the imaginary axis. After a short repetition of the Second mean-value (or Bonnet) approach to the problem and the derivation of operational identities for Trigonometric functions we give in Section 8 a proof for the position of these genuine “analytic” zeros on the imaginary axis by construction of a contradiction for the case off the imaginary axis. In Section 10, we show by a few examples that monotonically decreasing of the Omega functions is only a sufficient condition for the mentioned property of the positions of zeros on the imaginary axis but not a necessary one.

On the mean-square of the error term related ton ∑_（n≤x）λ~2（nj）

We prove a non-trivial upper bound for the quantity ∫X2|X ∑_(n≤x)λ~2(nj)-c(j-1)x| 2dx for j=2, 3, 4.

On Odd Goldbach Problem Under General Riemann Hypothesis

This paper proves that every odd integer N≥exp(114) can be written as a sum of three primes under GRH.

Harmonic coordinates for real-time image cloning

Traditional gradient domain seamless image cloning is a time consuming task,requiring the solving of Poisson's equations whenever the shape or position of the cloned region changes.Recently,a more efficient alternative,the mean-value coordinates(MVCs) based approach,was proposed to interpolate interior pixels by a weighted combination of values along the boundary.However,this approach cannot faithfully preserve the gradient in the cloning region.In this paper,we introduce harmonic cloning,which uses harmonic coordinates(HCs) instead of MVCs in image cloning.Benefiting from the non-negativity and interior locality of HCs,our interpolation generates a more accurate harmonic field across the cloned region,to preserve the results with as high a quality as with Poisson cloning.Furthermore,with optimizations and implementation on a graphic processing unit(GPU),we demonstrate that,compared with the method using MVCs,our harmonic cloning gains better quality while retaining real-time performance.

SOME ALMOST ALL RESULTS CONCERNINGTHE PJATECKII-SAPIRO PRIME NUMBERS

The distribution of the Pjateckii-Sapiro prime numbers in arithmetic progressions is investigated, and a Bombier i- Vinogr adov typ e mean- value t heorem and anot her almost all resultconcerning this problem are established.

General divisor functions in arithmetic progressions to large moduli

We prove a result on the distribution of the general divisor functions in arithmetic progressions to smooth moduli which exceed the square root of the length.

GOLDBACH CONJECTURE AND PARITY PROBLEM

Let N be a large positive even number and p, q, P1,... primes.

DISTRIBUTION OF k-FULL INTEGERS

In this paper we discuss the error function △_k(x) in the problem of distribution of k-fullintegers. Under the Riemann hypothesis, an asymptotic formula for the mean-value of △_k(x)is given.

Model Transduction for Triangle Meshes

This paper proposes a novel method, called model transduction, to directly transfer pose between different meshes, without the need of building the skeleton configurations for meshes. Different from previous retargetting methods, such as deformation transfer, model transduction does not require a reference source mesh to obtain the source deformation, thus effectively avoids unsatisfying results when the source and target have different reference poses. Moreover, we show other two applications of the model transduction method： pose correction after various mesh editing operations, and skeleton-free deformation animation based on 3D Mocap （Motion capture） data. Model transduction is based on two ingredients： model deformation and model correspondence. Specifically, based on the mean-value manifold operator, our mesh deformation method produces visually pleasing deformation results under large angle rotations or big-scale translations of handles. Then we propose a novel scheme for shape-preserving correspondence between manifold meshes. Our method fits nicely in a unified framework, where the similar type of operator is applied in all phases. The resulting quadratic formulation can be efficiently minimized by fast solving the sparse linear system. Experimental results show that model transduction can successfully transfer both complex skeletal structures and subtle skin deformations.

THE GOLDBACH-VINOGRDOV THEOREM WITH THREE PRIMES IN A THIN SUBSET

It is proved constructively that there exists a thin subset S of primes, satisfying for some absolute constant c＞0, such that every sufficiently large odd integer N can beLet r be prime, and hi positive integers with (bj, r) = 1,j = 1, 2, 3. It is also proved that, for almost all prime moduli r＜ log- N, every sufficiently large odd integer N = b1 + b2 +ba(modr) can be represented as where c ＞ 0 is an absolute constant.