Modelling and stability analysis in human population genetics with selection and mutation

Population genetics is a scientific discipline that has extensively benefitted from mathematical modelling; since the Hardy‐Weinberg law (1908) to date, many mathematical models have been designed to describe the genotype frequencies evolution in a population. Existing models differ in adopted hypothesis on evolutionary forces (such as, for example, mutation, selection, and migration) acting in the population. Mathematical analysis of population genetics models help to understand if the genetic population admits an equilibrium, ie, genotype frequencies that will not change over time. Nevertheless, the existence of an equilibrium is only an aspect of a more complex issue concerning the conditions that would allow or prevent populations to reach the equilibrium. This latter matter, much more complex, has been only partially investigated in population genetics studies. We here propose a new mathematical model to analyse the genotype frequencies distribution in a population over time and under two major evolutionary forces, namely, mutation and selection; the model allows for both infinite and finite populations. In this paper, we present our model and we analyse its convergence properties to the equilibrium genotype frequency; we also derive conditions allowing convergence. Moreover, we show that our model is a generalisation of the Hardy‐Weinberg law and of subsequent models that allow for selection or mutation. Some examples of applications are reported at the end of the paper, and the code that simulates our model is available online at https://www.ding.unisannio.it/persone/docenti/del-vecchio for free use and testing.     

Spatially almost periodic solutions for active scalar equations

We are concerned with a family of dissipative active scalar equation on . By using similar methods from the previous paper of Y. Giga et al. (see Introduction below), we construct a unique real, spatially almost periodic mild solution θ of 1.1 satisfying 1.11 . In this paper, we consider some countable sum‐closed frequency sets (see Remark 1.1 ). We show that the property of the solution is rather different from Chae et al 1 and obtain that with some initial data θ₀ for all t≥0, and 0≤α≤ω, where ω is a fixed constant. Furthermore, arranging the elements of a countable sum‐closed frequency set F_δ as in Remark 1.3 , we have for any 0≤α≤ω that belongs to , where F_δ is defined in 1.4 or 1.5 .     

Semimartingale: Itô or not ?

Itô semimartingales are the semimartingales whose characteristics are absolutely continuous with respect to Lebesgue measure. We study the importance of this assumption for statistical inference on a discretely sampled semimartingale in terms of the identifiability of its characteristics, their estimation, and propose tests of the Itô property against the non-Itô alternative when the observed semimartingale is continuous, or discontinuous with finite activity jumps, and under a number of technical assumptions.     

Conjugate processes: Theory and application to risk forecasting

Many dynamical phenomena display a cyclic behavior, in the sense that time can be partitioned into units within which distributional aspects of a process are homogeneous. In this paper, we introduce a class of models – called conjugate processes – allowing the sequence of marginal distributions of a cyclic, continuous-time process to evolve stochastically in time. The connection between the two processes is given by a fundamental compatibility equation. Key results include Laws of Large Numbers in the presented framework. We provide a constructive example which illustrates the theory, and give a statistical implementation to risk forecasting in financial data.     

埃尔米特梁单元的分块对角与高阶质量矩阵

埃尔米特梁单元常用的集中质量矩阵,是由挠度自由度对应的一致质量矩阵元素通过行求和或节点积分构造。然而,数值结果表明该集中质量矩阵在求解包含自由端的梁振动问题时,会出现频率精度掉阶现象。本文首先从保障质量矩阵最优收敛性的数值积分精度出发,分别针对三次和五次梁单元,发展了质量矩阵的梯度增强节点积分方案。利用梯度增强节点积分方案,可以得到具有分块对角形式的单元质量矩阵,而其组装的整体质量矩阵除边界节点外仍然呈现对角形式。对于两种单元,其分块对角质量矩阵分别具有4阶最优精度和6阶次优精度。再者,将标准一致质量矩阵和具有同阶精度的梯度增强节点积分质量矩阵进行优化组合,建立了具有超收敛特性的高阶质量矩阵。最后,通过数值算例系统验证了三次和五次单元的分块对角与高阶质量矩阵的频率计算精度。     

弹簧地基上大直径单桩海上风机的一阶自振频率

自振频率是海上风机前期动力设计的主要难点之一,计算精度要求极高,而桩-土相互作用对自振频率影响显著。目前,针对桩-土相互作用普遍采用弹簧地基简化模型,为对比分析不同弹簧地基简化模型的准确性和精度,本文采用回传射线矩阵法,建立固定端、单弹簧、双弹簧和三弹簧这4种地基简化模型,基于Bernoulli-Euler梁理论和Timoshenko梁理论,求解风机系统的自振频率,并与风机基频实测值作对比,进一步分析三弹簧简化地基模型中地基旋转刚度、地基耦合刚度和地基水平刚度对风机系统基频的影响。研究结果表明,桩-土相互作用不可忽略,三弹簧地基简化模型计算风机系统基频精度最高;地基刚度对风机系统基频的敏感性次序为,地基耦合刚度>地基旋转刚度>地基水平刚度;当地基旋转(水平)刚度较小时,地基耦合刚度与地基旋转(水平)刚度的耦合效应对风机系统的基频影响较大;地基耦合刚度与地基旋转刚度的耦合效应对风机系统基频的影响程度大于地基耦合刚度与地基水平刚度的耦合效应。     

An Adaptive Nonmonotone Projected Barzilai-Borwein Gradient Method with Active Set Prediction for Nonnegative Matrix Factorization

In this paper, we first present an adaptive nonmonotone term to improve the efficiency of nonmonotone line search, and then an active set identification technique is suggested to get more efficient descent direction such that it improves the local convergence behavior of algorithm and decreases the computation cost. By means of the adaptive nonmonotone line search and the active set identification technique, we put forward a global convergent gradient-based method to solve the nonnegative matrix factorization (NMF) based on the alternating nonnegative least squares framework, in which we introduce a modified Barzilai-Borwein (BB) step size. The new modified BB step size and the larger step size strategy are exploited to accelerate convergence. Finally, the results of extensive numerical experiments using both synthetic and image datasets show that our proposed method is efficient in terms of computational speed.     

求解低秩矩阵填充的改进的交替最速下降法

矩阵填充是指利用矩阵的低秩特性而由部分观测元素恢复出原矩阵,在推荐系统、信号处理、医学成像、机器学习等领域有着广泛的应用。采用精确线搜索的交替最速下降法由于每次迭代计算量小因而对大规模问题的求解非常有效。本文在其基础上采用分离地精确线搜索,可使得每次迭代下降更多但计算量相同,从而可望进一步提高计算效率。本文分析了新算法的收敛性。数值结果也表明所提出的算法更加有效。     

Optimal Decisions in a Time Priority Queue

We show how the position of a limit order (LO) in the queue influences the decision of whether to cancel the order or let it rest. Using ultra-high-frequency data from the Nasdaq exchange, we perform empirical analysis on various LO book events and propose novel ways for modelling some of these events, including cancellation of LOs in various positions and size of market orders. Based on our empirical findings, we develop a queuing model that captures stylized facts on the data. This model includes a distinct feature which allows for a potentially random effect due to the agent’s impulse control. We apply the queuing model in an algorithmic trading setting by considering an agent maximizing her expected utility through placing and cancelling of LOs. The agent’s optimal strategy is presented after calibrating the model to real data. A simulation study shows that for the same level of standard deviation of terminal wealth, the optimal strategy has a 2.5% higher mean compared to a strategy which ignores the effect of position, or an 8.8% lower standard deviation for the same level of mean. This extra gain stems from posting an LO during adverse conditions and obtaining a good queue position before conditions become favourable.     

On the Global Convergence of a Projective Trust Region Algorithm for Nonlinear Equality Constrained Optimization

A trust-region sequential quadratic programming (SQP) method is developed and analyzed for the solution of smooth equality constrained optimization problems. The trust-region SQP algorithm is based on filter line search technique and a composite-step approach, which decomposes the overall step as sum of a vertical step and a horizontal step. The algorithm includes critical modifications of horizontal step computation. One orthogonal projective matrix of the Jacobian of constraint functions is employed in trust-region subproblems. The orthogonal projection gives the null space of the transposition of the Jacobian of the constraint function. Theoretical analysis shows that the new algorithm retains the global convergence to the first-order critical points under rather general conditions. The preliminary numerical results are reported.