Patterns in stock market movements tested as random number generators

This paper shows that tests of Random Number Generators (RNGs) may be used to test the Efficient Market Hypothesis (EMH). It uses the Overlapping Serial Test (OST), a standard test in RNG research, to detect anomalous patterns in the distribution of sequences of stock market movements up and down. Our results show that most stock markets exhibit idiosyncratic recurrent patterns, contrary to the efficient market hypothesis; also that OST detects a different kind of non-randomness to standard econometric long- and short-memory tests. Exposure of these anomalies should contribute to making markets more efficient.     

Random Numbers from a Delay Equation

Delay differential equations can have “chaotic” solutions that can be used to mimic Brownian motion. Since a Brownian motion is random in its velocity, it is reasonable to think that a random number generator might be constructed from such a model. In this preliminary study, we consider one specific example of this and show that it satisfies criteria commonly employed in the testing of random number generators (from TestU01’s very stringent “Big Crush” battery of tests). A technique termed digit discarding, commonly used in both this generator and physical RNGs using laser feedback systems, is discussed with regard to the maximal Lyapunov exponent. Also, we benchmark the generator to a contemporary common method: the multiple recursive generator, MRG32k3a. Although our method is about 7 times slower than MRG32k3a, there is in principle no apparent limit on the number of possible values that can be generated from the scheme we present here.     

Strengthening the security of authenticated key exchange against bad randomness

Recent history has revealed that many random number generators (RNGs) used in cryptographic algorithms and protocols were not providing appropriate randomness, either by accident or on purpose. Subsequently, researchers have proposed new algorithms and protocols that are less dependent on the RNG. One exception is that all prominent authenticated key exchange (AKE) protocols are insecure given bad randomness, even when using good long-term keying material. We analyse the security of AKE protocols in the presence of adversaries that can perform attacks based on chosen randomness, i.e., attacks in which the adversary controls the randomness used in protocol sessions. We propose novel stateful protocols, which modify memory shared among a user’s sessions, and show in what sense they are secure against this worst case randomness failure. We develop a stronger security notion for AKE protocols that captures the security that we can achieve under such failures, and prove that our main protocol is correct in this model. Our protocols make substantially weaker assumptions on the RNG than existing protocols.     

Product numerical range in a space with tensor product structure

We study operators acting on a tensor product Hilbert space and investigate their product numerical range, product numerical radius and separable numerical range. Concrete bounds for the product numerical range for Hermitian operators are derived. Product numerical range of a non-Hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum. While the latter set is convex, the product range needs not to be convex nor simply connected. The product numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors.     

带漂移的单侧正规化回火分数阶扩散方程的Crank-Nicolson拟紧格式

基于已有的针对单侧正规化回火分数阶扩散方程的三阶拟紧算法,将该算法的思想应用于带漂移的单侧正规化回火分数阶扩散方程的数值模拟,并结合Crank-Nicolson方法导出数值格式.证明了数值格式的稳定性与收敛性,且数值格式的时间收敛阶和空间收敛阶分别是二阶和三阶.通过数值试验验证了数值格式的有效性和理论结果.     

Error analysis of the numerical solution of split differential equations

The operator splitting method is a widely used approach for solving partial differential equations describing physical processes. Its application usually requires the use of certain numerical methods in order to solve the different split sub-problems. The error analysis of such a numerical approach is a complex task. In the present paper we show that an interaction error appears in the numerical solution when an operator splitting procedure is applied together with a lower-order numerical method. The effect of the interaction error is investigated by an analytical study and by numerical experiments made for a test problem.     

基于人工鱼群算法求任意函数的数值积分

将不等距离分割方法与人工鱼群算法相结合,提出一种基于人工鱼群算法求任意函数数值积分的方法,该方法除能计算通常意义下任意函数的定积分外,还能计算奇异函数积分、振荡函数积分以及原函数不易求得的被积函数的积分.最后给出几个数值积分算例,并与传统数值积分方法作了比较,仿真结果分析表明,该算法十分有效,能够快速有效地获得任意函数的数值积分值.     

A numerical approach for solving an extended Fisher-Kolomogrov-Petrovskii-Piskunov equation

In the present paper a numerical method, based on finite differences and spline collocation, is presented for the numerical solution of a generalized Fisher integro-differential equation. A composite weighted trapezoidal rule is manipulated to handle the numerical integrations which results in a closed-form difference scheme. A number of test examples are solved to assess the accuracy of the method. The numerical solutions obtained, indicate that the approach is reliable and yields results compatible with the exact solutions and consistent with other existing numerical methods. Convergence and stability of the scheme have also been discussed.     

A numerical approximation method for solving a three-dimensional space Galilei invariant fractional advection-diffusion equation

In this paper, a numerical approximation method for solving a three-dimensional space Galilei invariant fractional advection diffusion equation is presented. The convergence and stability of the numerical approximation method are discussed by a new technique of Fourier analysis. The solvability of the numerical approximation method also is analyzed. Finally, applying Richardson extrapolation technique, a high-accuracy algorithm is structured and the numerical example demonstrated the theoretical results.     

三维数值流形方法的理论研究

在二维数值流形方法的基础上,对三维数值流形进行了理论研究.研究了三维覆盖位移函数,进行了三维数值流形的力学分析,给出了三维流形单元的刚度矩阵,详细推导了三维数值流形的Hammer积分及剖分规则,系统地研究三维数值流形的理论体系与数值实现方法.作为数值算例,给出了相应的悬臂梁的计算结果,计算结果表明算法的精度和计算效益较高.     

Smooth factorizations of matrix valued functions and their derivatives

Summary In this paper we study the numerical factorization of matrix valued functions in order to apply them in the numerical solution of differential algebraic equations with time varying coefficients. The main difficulty is to obtain smoothness of the factors and a numerically accessible form of their derivatives. We show how this can be achieved without numerical differentiation if the derivative of the given matrix valued function is known. These results are then applied in the numerical solution of differential algebraic Riccati equations. For this a numerical algorithm is given and its properties are demonstrated by a numerical example.     

一类强色散DGH方程的多辛普雷斯曼格式

DGH方程作为一类重要的非线性水波方程有着许多广泛的应用前景.基于Hamilton系统的多辛理论研究了一类强色散DGH方程的数值解法,利用多辛普雷斯曼方法构造了一种典型的半隐式的多辛格式.分析了该格式的局部能量和动量守恒律误差,并给出了数值算例.数值算例结果表明该多辛离散格式具有较好的长时间数值稳定性.     

Numerical analysis for a variable-order nonlinear cable equation

In this paper, a variable-order nonlinear cable equation is considered. A numerical method with first-order temporal accuracy and fourth-order spatial accuracy is proposed. The convergence and stability of the numerical method are analyzed by Fourier analysis. We also propose an improved numerical method with second-order temporal accuracy and fourth-order spatial accuracy. Finally, the results of a numerical example support the theoretical analysis.     

土壤水流与溶质耦合运移问题的混合元-迎风广义差分法研究

本文研究了一维非饱和土壤水流与溶质耦合运移问题的数学模型, 建立了求其数值解的守恒混合元-迎风广义差分格式. 对非线性土壤水分入渗方程, 采用守恒混合元法进行离散模拟, 同时得到了土壤含水量和水分通量; 而对对流-扩散形式的溶质运移方程, 利用迎风的广义差分法离散求解. 且分析了解的存在唯一性, 并讨论了误差估计. 最后给出数值算例, 模拟结果表明利用本文格式来求解非饱和土壤水流与溶质耦合运移问题是可靠的, 且该格式具有稳定性和可实用性.     

The numerical method of successive interpolations for Fredholm functional integral equations

A new numerical method for Fredholm functional integral equations is proposed. The method combines the fixed point technique with numerical integration and cubic spline interpolation. The convergence and the numerical stability of the method are proved and tested on some numerical examples.     

一类非线性延迟微分方程数值解的振动性分析

本文考虑一类非线性延迟微分方程-带有单调造血率的造血模型数值解的振动性.通过研究特征方程根的情况得到数值解振动的条件并且讨论了非振动的数值解的一些性质.为了更有力的说明我们的结果,最后给出了相应的算例.     

Recovering the local volatility in Black–Scholes model by numerical differentiation

In this article, a numerical method for recovering the local volatility in Black–Scholes model is proposed based on the Dupire formula in which the numerical derivatives are used. By Tikhonov regularization, a new numerical differentiation method in two-dimensional (2-D) case is presented. The convergent analysis and numerical examples are also given. It shows that our method is efficient and stable.     

A Fourier method and an extrapolation technique for Stokes’ first problem for a heated generalized second grade fluid with fractional derivative

In this article, we consider Stokes’ first problem for a heated generalized second grade fluid with fractional derivative (SFP-HGSGF). Implicit and explicit numerical approximation schemes for the SFP-HGSGF are presented. The stability and convergence of the numerical schemes are discussed using a Fourier method. In addition, the solvability of the implicit numerical approximation scheme is also analyzed. A Richardson extrapolation technique for improving the order of convergence of the implicit scheme is proposed. Finally, a numerical test is given. The numerical results demonstrate the good performance of our theoretical analysis.     

Numerical pseudodifferential operator and Fourier regularization

The concept of numerical pseudodifferential operator, which is an extension of numerical differentiation, is suggested. Numerical pseudodifferential operator just is calculating the value of the pseudodifferential operator with unbounded symbol. Many ill-posed problems can lead to numerical pseudodifferential operators. Fourier regularization is a very simple and effective method for recovering the stability of numerical pseudodifferential operators. A systematically theoretical analysis and some concrete examples are provided.     

Differential quadrature method based on the highest derivative and its applications

The numerical differentiation is often used when dealing with the differential equations. Using the numerical differentiation, the differential equations can be transformed into algebraic equations. Then we can get the numerical solution from the algebraic equations. But the numerical differentiation process is very sensitive to even a small level of errors. In contrast, it is expected that on average the numerical integration process is much less sensitive to errors. In this paper, we provide a new method using the DQ method based on the interpolation of the highest derivative (DQIHD) for the differential equations. The original function is then obtained by integration. In this paper, the DQIHD method was applied to the buckling analysis of thin isotropic plates and Winkler plates, the numerical results agree well with the analytic solutions, and the results show that our method is of high accuracy, of good convergence with little computational efforts. And it is easy to deal with the boundary conditions.