关于Euler e-函数和n的指数互素因子

对于任意正整数n,设n=pα11pα22…pαrr为n的标准素因数分解式,如果对于de n且de=pβ11pβ22…pβrr有(βi,αi)=1(i=1,2,…,k),则称de为n的指数互素因子.本文利用初等及解析方法研究了正整数n的所有de因子的求和及求积的计算问题,获得了两个有趣的计算公式;同时还研究了n的所有de因子个数函数,即Eu ler e函-数φe(n)的均值性质,并给出了一个较强的渐近公式.     

Complexity of inverting the Euler function

Given an integer , how hard is it to find the set of all integers such that , where is the Euler totient function? We present a certain basic algorithm which, given the prime number factorization of , in polynomial time ``on average' (that is, ), finds the set of all such solutions . In fact, in the worst case this set of solutions is exponential in , and so cannot be constructed by a polynomial time algorithm. In the opposite direction, we show, under a widely accepted number theoretic conjecture, that the PARTITION PROBLEM, an NP-complete problem, can be reduced in polynomial (in the input size) time to the problem of deciding whether has a solution, for polynomially (in the input size of the PARTITION PROBLEM) many values of (where the prime factorizations of these are given). What this means is that the problem of deciding whether there even exists a solution to , let alone finding any or all such solutions, is very likely to be intractable. Finally, we establish close links between the problem of inverting the Euler function and the integer factorization problem.

     

Oscillations of the remainder term related to the Euler totient function

We split the remainder term in the asymptotic formula for the mean of the Euler phi function into two summands called the arithmetic and the analytic part respectively. We show that the arithmetic part can be studied with a mild use of the complex analytic tools, whereas the study of the analytic part heavily depends on the properties of the Riemann zeta function and on the distribution of its non-trivial zeros in particular.     

Hilbert重级数定理的一个新推广

通过引入一个适当的对数函数建立了一种新的Hilbert型不等式.利用Euler-Maclaurin求和公式对权函数进行了估计.证明了常数因子π2r＋1Er是最佳的,其中Er是Euler数.作为应用,给出了一些互相等价的不等式.     

WEAK APPROXIMATION OF OBLIQUELY REFLECTED DIFFUSIONS IN TIME-DEPENDENT DOMAINS

In an earlier paper, we proved the existence of solutions to the Skorohod problem with oblique reflection in time-dependent domains and, subsequently, applied this result to the problem of constructing solutions, in time-dependent domains, to stochastic differential equations with oblique reflection. In this paper we use these results to construct weak approximations of solutions to stochastic differential equations with oblique reflection, in time-dependent domains in Rd, by means of a projected Euler scheme. We prove that the constructed method has, as is the case for normal reflection and time-independent domains, an order of convergence equal to 1/2 and we evaluate the method empirically by means of two numerical examples. Furthermore, using a well-known extension of the Feynman-Kac formula, to stochastic differential equations with reflection, our method gives, in addition, a Monte Carlo method for solving second order parabolic partial differential equations with Robin boundary conditions in time-dependent domains.     

Controlled differential equations as Young integrals: A simple approach

The theory of rough paths allows one to define controlled differential equations driven by a path which is irregular. The most simple case is the one where the driving path has finite p-variations with 1?p<2, in which case the integrals are interpreted as Young integrals. The prototypal example is given by stochastic differential equations driven by fractional Brownian motion with Hurst index greater than 1/2. Using simple computations, we give the main results regarding this theory - existence, uniqueness, convergence of the Euler scheme, flow property … - which are spread out among several articles.     

Method of relaxed streamline upwinding for hyperbolic conservation laws

In this work a new finite element based Method of Relaxed Streamline Upwinding is proposed to solve hyperbolic conservation laws. Formulation of the proposed scheme is based on relaxation system which replaces hyperbolic conservation laws by semi-linear system with stiff source term also called as relaxation term. The advantage of the semi-linear system is that the nonlinearity in the convection term is pushed towards the source term on right hand side which can be handled with ease. Six symmetric discrete velocity models are introduced in two dimensions which symmetrically spread foot of the characteristics in all four quadrants thereby taking information symmetrically from all directions. Proposed scheme gives exact diffusion vectors which are very simple. Moreover, the formulation is easily extendable from scalar to vector conservation laws. Various test cases are solved for Burgers equation (with convex and non-convex flux functions), Euler equations and shallow water equations in one and two dimensions which demonstrate the robustness and accuracy of the proposed scheme. New test cases are proposed for Burgers equation, Euler and shallow water equations. Exact solution is given for two-dimensional Burgers test case which involves normal discontinuity and series of oblique discontinuities. Error analysis of the proposed scheme shows optimal convergence rate. Moreover, spectral stability analysis gives implicit expression of critical time step.     

An Asymptotic Preserving scheme for the Euler equations in a strong magnetic field

This paper is concerned with the numerical approximation of the isothermal Euler equations for charged particles subject to the Lorentz force (the ‘Euler–Lorentz’ system). When the magnetic field is large, or equivalently, when the parameter ε$\epsilon$ representing the non-dimensional ion cyclotron frequency tends to zero, the so-called drift-fluid (or gyro-fluid) approximation is obtained. In this limit, the parallel motion relative to the magnetic field direction splits from perpendicular motion and is given implicitly by the constraint of zero total force along the magnetic field lines. In this paper, we provide a well-posed elliptic equation for the parallel velocity which in turn allows us to construct an Asymptotic-Preserving (AP) scheme for the Euler–Lorentz system. This scheme gives rise to both a consistent approximation of the Euler–Lorentz model when ε$\epsilon$ is finite and a consistent approximation of the drift limit when ε→0$\epsilon \to 0$. Above all, it does not require any constraint on the space and time-steps related to the small value of ε$\epsilon$. Numerical results are presented, which confirm the AP character of the scheme and its Asymptotic Stability.     

Analysis of supersonic flow around two bodies of revolution near a surface

The results of experimental and numerical investigations of the peculiarities of flow around two identical cylindrical bodies of revolution of diameter D = 50 mm and the body aspect ratio λ = 5 with conical forebodies whose apex angles are θ = 40° and 60°, which are located above a horizontal surface in parallel with one another and with the flow, are presented for the Mach numbers M_∞ = 4.03, Reynolds numbers Re₁ ≈ 55·10⁶ m⁻¹, fixed distance from the surface Y = Δy/D = 0.96, and the gaps between their axes Z = Δz/D = 1.06−2.4. The peculiarities of three-dimensional turbulent separated flows realizing on the bodies and on the plate as well as the possibilities of predicting the aerodynamic forces and moments acting on the bodies on the basis of numerical computations within the framework of the Euler equations are considered.     

Some problems associated with the control of distributed structures

Control of structures can be carried out conveniently by modal control, whereby the structure is controlled by controlling its modes. Modal control requires the estimation of the modal states for feedback, which can present a problem. One approach that does not require modal state estimation is direct feedback control, which implies collocated sensors and actuators. This paper examines some problems encountered in direct feedback control of distributed structures in conjunction with pole placement. A perturbation technique permits the computation of control gains for multi-input systems. The paper demonstrates that the difficulties experienced in using direct feedback in conjunction with pole placement are endemic to the approach.This research was sponsored by the Air Force Office of Scientific Research Grant No. AFOSR-83-0017, monitored by Dr. A. K. Amos, whose support is fully appreciated. This paper was presented at the Meeting on Optimal Control and Calculus of Variations, Oberwolfach, West Germany, June 15–21, 1986.