The distribution of values of trigonometric sums with linearly independent frequencies: Kac's problem revisited

The average frequency with which a trigonometric sum having linearly independent frequencies achieves a given value in a specified time interval was solved by Kac in the form of a double integral. In the present paper, Kac's solution is generalized by allowing the amplitudes of the sinusoidal terms to be statistically independent random variables possessing compact probability density functions. The average frequency is expressed as a Fourier cosine series. Representative numerical calculations are shown. The asymptotic form of the average frequency is also discussed.     

关于αn~k模1的分布

讨论了αｎ￣ｋ模１的分布，证明了对于有无穷多个自然数ｎ满足     

Average size of -Selmer groups of elliptic curves, I

In this paper, we study a class of elliptic curves over with -torsion group , and prove that the average order of the -Selmer groups is bounded.

     

On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces

Let be a real reflexive Banach space with dual and open and bounded and such that  Let be maximal monotone with and and with and A general and more unified eigenvalue theory is developed for the pair of operators  Further conditions are given for the existence of a pair such that

The ``implicit" eigenvalue problem, with in place of is also considered.  The existence of continuous branches of eigenvectors of infinite length is investigated, and a Fredholm alternative in the spirit of Necas is given for a pair of homogeneous operators No compactness assumptions have been made in most of the results.  The degree theories of Browder and Skrypnik are used, as well as the degree theories of the authors involving densely defined perturbations of maximal monotone operators.  Applications to nonlinear partial differential equations are included.

     

On the mean value of

Let χ be the Dirichlet character modulo q3 and L(s,χ) denote the corresponding Dirichlet L-function. The mean value of is studied and a few asymptotic formulae are given. Hybrid mean value of , general Kloosterman sums and general quadratic Gauss sums are considered.     

渐进非对称陀螺分子16O3及同位素16O18O16O的配分函数研究

采用乘积近似法计算了臭氧分子16O3及其同位素16O18O16O在20-6000 K温度范围内的总配分函数。其中转动配分函数用Watson的刚性转子模型,振动配分函数用谐振子模型．总的温度范围被划分为五个温度段。计算的配分函数在这五个温度段分别被拟合到一个温度T的五阶多项式,从而在每个温度段均得到六个拟合系数。由这些拟合系数就可以快速、准确的获得分子在所研究温度范围内任意温度的总配分函数。研究结果为目标特性识别、气动物理的高超声速技术等领域的研究提供了重要的参考信息。     

On the Hybrid Power Mean Involving the Two-Term Exponential Sums and Polynomial Character Sums

The main purpose of this paper is using the analytic method and the properties of trigonometric sums and character sums to study the computational problem of one kind hybrid power mean involving two-term exponential sums and polynomial character sums. Then the authors give some interesting calculating formulae for them.     

Recursions associated to trapezoid,symmetric and rotation symmetric functions over Galois fields

Rotation symmetric Boolean functions are invariant under circular translation of indices. These functions have very rich cryptographic properties and have been used in different cryptosystems. Recently, Thomas Cusick proved that exponential sums of rotation symmetric Boolean functions satisfy homogeneous linear recurrences with integer coefficients. In this work, a generalization of this result is proved over any Galois field. That is, exponential sums over Galois fields of some rotation symmetric polynomials satisfy linear recurrences with integer coefficients. In the particular case of ${\mathbb{F}}_2$, an elementary method is used to obtain explicit recurrences for exponential sums of some of these functions. The concept of trapezoid Boolean function is also introduced and it is showed that the linear recurrences that exponential sums of trapezoid Boolean functions satisfy are the same as the ones satisfied by exponential sums of the corresponding rotations symmetric Boolean functions. Finally, it is proved that exponential sums of trapezoid and symmetric polynomials also satisfy linear recurrences with integer coefficients over any Galois field ${\mathbb{F}}_q$. Moreover, the Discrete Fourier Transform matrix and some Complex Hadamard matrices appear as examples in some of our explicit formulas of these recurrences.     

Lattice points in bodies of revolution II

In [3] it was shown that when a three-dimensional smooth convex body has rotational symmetry around a coordinate axis one can find better bounds for the lattice point discrepancy than what is known for more general convex bodies. To accomplish this, however, it was necessary to assume a non-vanishing condition on the third derivative of the generatrix. In this article we drop this condition, showing that the aforementioned bound holds for a wider family of revolution bodies, which includes those with analytic boundary. A novelty in our approach is that, besides the usual analytic methods, it requires studying some Diophantine properties of the Taylor coefficients of the phase on the Fourier transform side.