Bootstrap of the offspring mean in the critical process with a non-stationary immigration

In applications of branching processes, usually it is hard to obtain samples of a large size. Therefore, a bootstrap procedure allowing inference based on a small sample size is very useful. Unfortunately, in the critical branching process with stationary immigration the standard parametric bootstrap is invalid. In this paper, we consider a process with non-stationary immigration, whose mean and variance vary regularly with nonnegative exponents α and β, respectively. We prove that 1+2α is the threshold for the validity of the bootstrap in this model. If β<1+2α, the standard bootstrap is valid and if β>1+2α it is invalid. In the case β=1+2α, the validity of the bootstrap depends on the slowly varying parts of the immigration mean and variance. These results allow us to develop statistical inferences about the parameters of the process in its early stages.     

On a dual model with a dividend threshold

In insurance mathematics, a compound Poisson model is often used to describe the aggregate claims of the surplus process. In this paper, we consider the dual of the compound Poisson model under a threshold dividend strategy. We derive a set of two integro-differential equations satisfied by the expected total discounted dividends until ruin and show how the equations can be solved by using only one of the two integro-differential equations. The cases where profits follow an exponential or a mixture of exponential distributions are then solved and the discussion for the case of a general profit distribution follows by the use of Laplace transforms. We illustrate how the optimal threshold level that maximizes the expected total discounted dividends until ruin can be obtained, and finally we generalize the results to the case where the surplus process is a more general skip-free downwards Lévy process.     

On proper secrets, (<Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">k</Emphasis>)-bases and linear codes

This paper contains three parts where each part triggered and motivated the subsequent one. In the first part (Proper Secrets) we study the Shamir’s “k-out-of-n” threshold secret sharing scheme. In that scheme, the dealer generates a random polynomial of degree k−1 whose free coefficient is the secret and the private shares are point values of that polynomial. We show that the secret may, equivalently, be chosen as any other point value of the polynomial (including the point at infinity), but, on the other hand, setting the secret to be any other linear combination of the polynomial coefficients may result in an imperfect scheme. In the second part ((t, k)-bases) we define, for every pair of integers t and k such that 1 ≤ t ≤ k−1, the concepts of (t, k)-spanning sets, (t, k)-independent sets and (t, k)-bases as generalizations of the usual concepts of spanning sets, independent sets and bases in a finite-dimensional vector space. We study the relations between those notions and derive upper and lower bounds for the size of such sets. In the third part (Linear Codes) we show the relations between those notions and linear codes. Our main notion of a (t, k)-base bridges between two well-known structures: (1, k)-bases are just projective geometries, while (k−1, k)-bases correspond to maximal MDS-codes. We show how the properties of (t, k)-independence and (t, k)-spanning relate to the notions of minimum distance and covering radius of linear codes and how our results regarding the size of such sets relate to known bounds in coding theory. We conclude by comparing between the notions that we introduce here and some well known objects from projective geometry.      

阈值电路特性对腔衰荡光谱测量影响的实验研究

腔衰荡光谱技术(cavity ring-down spectroscopy, CRDS)是一种高灵敏的激光吸收光谱技术,被广泛应用于镜片反射率测量、光谱分析以及痕量气体检测等研究领域。首先从理论上描述了CRDS技术的实验原理,然后基于555定时器自行设计了一种可应用于CRDS技术的阈值电路,并通过实验进行了验证。为了优化阈值电路的性能,研究了不同输入阻抗及容抗下阈值电路对衰荡事件测量的影响。通过直接接入不同值的电容与电阻,发现输入电容越大,输入阻抗越小,对衰荡事件的测量影响越大,尤其当阈值电路输入电阻小于50 Ω时,衰荡事件的线型严重扭曲。对不同输入电阻与输入电容时C₂H₂气体在6 531.780 5 cm-1处的吸收信号进行了采集与分析,发现输入电容与输入电阻的改变对测量结果有很大的影响,得到的吸收信号会失真,甚至无法进行拟合。最终给出了阈值电路的最佳设计方案,在阈值电路的设计中选取的输入阻抗越大越好,最小要在100 Ω以上,选取的容抗越小越好,最好低于1 nF,同时要保证阈值电路的时间常数远小于衰荡时间。该研究对于CRDS技术应用过程中阈值电路的设计具有重要的参考价值。     

基于提升小波变换的阈值改进去噪算法在紫外可见光谱中的研究

在紫外可见光谱定量分析中,由于分光光度计内部的光学系统、光源、检测器、电子元器件,电路设计以及外部环境干扰等因素产生的随机噪声,严重影响光谱定量分析结果的准确性,为提高紫外可见光谱分析精度,需要对光谱数据进行去噪预处理。由于小波分析具有多分辨率,低熵性、去相关性等特点,基于小波分析的去噪算法优于传统的去噪算法,目前基于小波去噪的方法主要有模极大值去噪算法,系数相关去噪算法,阈值去噪算法,工程实际应用以Donoho的阈值去噪法最为常用。根据Donoho阈值消噪原理,提出一种基于提升小波变换的阈值改进算法,一方面使用提升小波变换,提升小波变换是第二代小波变换,继承了小波的多分辨率特性,并且不需要进行傅里叶变换,从而具有算法简单,速度快,实现简单的优点;另一方面提出了一种新的阈值函数,克服了硬阈值函数在阈值处不连续以及软阈值函数存在恒定偏差的问题,同时对阈值估计进行了调整,有利于信号小波系数的保留和噪声小波系数的剔除。对三组多金属离子混合溶液的实测紫外可见光谱信号,添加随机噪声后使用该方法进行去噪处理,并使用信噪比(SNR)和均方根误差(RMSE)进行去噪性能评价。试验结果表明,提出的算法优于Donoho的软硬阈值去噪算法,能够有效提高光谱信噪比和降低均方根误差,从而更好地消除光谱信号中的噪声和保留光谱信号中一些重要的细节特征,比较适合用于紫外可见光谱数据建模之前的去噪预处理,在紫外可见光谱信号分析中具有较好的应用前景。     

Effect of quadratic pressure gradient term on a one-dimensional moving boundary problem based on modified Darcy’s law

A relatively high formation pressure gradient can exist in seepage flow in low-permeable porous media with a threshold pressure gradient, and a significant error may then be caused in the model computation by neglecting the quadratic pressure gradient term in the governing equations. Based on these concerns, in consideration of the quadratic pressure gradient term, a basic moving boundary model is constructed for a one-dimensional seepage flow problem with a threshold pressure gradient. Owing to a strong nonlinearity and the existing moving boundary in the mathematical model, a corresponding numerical solution method is presented. First, a spatial coordinate transformation method is adopted in order to transform the system of partial differential equations with moving boundary conditions into a closed system with fixed boundary conditions; then the solution can be stably numerically obtained by a fully implicit finite-difference method. The validity of the numerical method is verified by a published exact analytical solution. Furthermore, to compare with Darcy’s flow problem, the exact analytical solution for the case of Darcy’s flow considering the quadratic pressure gradient term is also derived by an inverse Laplace transform. A comparison of these model solutions leads to the conclusion that such moving boundary problems must incorporate the quadratic pressure gradient term in their governing equations; the sensitive effects of the quadratic pressure gradient term tend to diminish, with the dimensionless threshold pressure gradient increasing for the one-dimensional problem.     

Modified Maxwell–Wagner model analysis on charging and discharging in TIPS-pentacene diode by optical second harmonic generation measurement

We used time-resolved electric field induced optical second harmonic generation (TR-EFISHG) measurement and Maxwell–Wagner (MW) equivalent circuit model analysis to study the charging and discharging processes of ITO/Polyimide/6,13-Bis(triisopropylsilylethynyl)-pentacene/Au (ITO/PI/TIPS-pentacene/Au) diodes. For fresh diodes, electrode charging was followed successively by interfacial charging, whereas interfacial discharging occurred very quickly in combination with electrode discharging. In contrast, the order of charging and discharging was reversed for aged diodes. A modified MW equivalent circuit well accounts for these results, where TIPS-pentacene functions as a dielectric material before hole injection, but as a conductor after hole injection. As a result, the accumulated interfacial charge can escape very quickly from the TIPS-pentacene together with electrode discharging. These results illustrate the non-reversal behavior of the fresh TIPS-pentacene diodes.