A bayesian approach to the probability density estimation

Summary A Bayesian procedure for the probability density estimation is proposed. The procedure is based on the multinomial logit transformations of the parameters of a finely segmented histogram model. The smoothness of the estimated density is guaranteed by the introduction of a prior distribution of the parameters. The estimates of the parameters are defined as the mode of the posterior distribution. The prior distribution has several adjustable parameters (hyper-parameters), whose values are chosen so that ABIC (Akaike's Bayesian Information Criterion) is minimized. The basic procedure is developed under the assumption that the density is defined on a bounded interval. The handling of the general case where the support of the density function is not necessarily bounded is also discussed. The practical usefulness of the procedure is demonstrated by numerical examples. The Institute of Statistical Mathematics     

A group invariant solution for a pre-existing fluid-driven fracture in permeable rock

The propagation of a two-dimensional pre-existing fracture in permeable rock by the injection of a viscous, incompressible Newtonian fluid is considered. The fluid flow in the fracture is laminar. By the application of lubrication theory, a partial differential equation relating the half-width of the fracture to the fluid pressure and leak-off velocity is derived. The model is closed by the adoption of the PKN formulation in which the fluid pressure is proportional to the fracture half-width. The partial differential equation admits four Lie point symmetries provided the leak-off velocity satisfies a first order linear partial differential equation. The solution of this equation yields the leak-off velocity as a function of the distance along the fracture and time. The group invariant solution is derived by considering a linear combination of the Lie point symmetries. The boundary value problem is reformulated as a pair of initial value problems. The model in which the leak-off velocity is proportional to the fracture half-width is considered. The working condition of constant pressure at the fracture entry is analysed in detail.     

Elenbaas Problem of Electric Arc Discharge

The Elenbaas problem of electric discharge origination is considered. The mathematical model is an elliptic boundary-value problem with a parameter and discontinuous nonlinearity. The nontrivial solutions of the problem determine the free boundaries separating different phase states. A survey of results obtained for this problem is given. The greatest lower bound λmin of the values of the parameter λ for which the electric discharge is possible is obtained. The fact that the discharge domain appears for any λ ≥ λ_min is proved. The range of the parameter values for which the boundary of the discharge domain is of two-dimensional Lebesgue measure zero is determined. An unsolved problem is formulated.     

Invariance and asymptotic stability

The stability of the zero solution of an autonomous non-linear system is considered. The problem of finding the variables in relation to which the solution is asymptotically stable if the Lyapunov function with a sign-definite derivative is known, is formulated and solved. The maximality of the set in relation to which the solution is asymptotically stable is established. The investigation is based on the method of auxiliary functions and clarifies the relation between the properties of invariance and the asymptotic stability of dynamical systems. The constructiveness of the results obtained is demonstrated by an example.     

Laplace方程有孔椭圆区域边值问题的解析解

本文利用相似流动替换方法 ,解决了中心有圆孔的椭园形区域上 Laplace方程第一类边值问题 ;采用分区域解法 ,给出了中心有椭园孔的椭园形区域上 Laplace方程第一类边值问题的解析通解 .这一结果在许多工程领域有重要应用 ,本文给出了油藏工程实例     

Kolmogorov-Spieqel-Siveshinky方程大时间问题的Fourier拟谱逼近

该文讨论了Kolmogorov-Spieqel-Siveshinky方程的周期初值问题, 研究了半离散Fourier拟谱解的长时间行为, 证明了半离散系统的收敛性和整体吸引子的存在性. 构造了全离散的三层显式Fourier拟谱格式, 并证明了该格式的收敛性, 最后通过数值计算验证了格式的可信性. 数值结果表明: 该格式是长时间稳定并可取时间大步长. 作者模拟了方程的解在相空间的轨线, 得到了一些有意义的结论.     

Diffraction of a plane wave by a narrow cone

The problem of diffraction of a plane scalar wave by a narrow cone is considered. The shape of the cone is arbitrary. The boundary condition is the Dirichlet or Neumann one. The wave scattered by the cone vertex arises as a result of the diffraction process. The subject of this paper is to calculate the wave amplitude. If the cone is narrow, it is possible to obtain simpler approximate formulas in comparison with Smyshlayev's one. The exactness of the approximate formulas is checked numerically. The etalon is a solution in explicit form in the axially symmetric case. The calculation shows that our formula is more exact in the case of the Dirichlet boundary condition than Felsen's formula. The approximate formula is a generalization of Felsen's one for circular cone to an arbitrary narrow cone in the case of the Neumann boundary condition. Bibliography: 6 titles. Dedicated to N. N. Uraltseva on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 67–74. Translated by D. B. Dement'ev.     

基于Fuzzy等价关系的矩形聚类方法

证明了Boole矩阵与模糊矩阵等价的充分必要条件,总结出Fuzzy聚类的矩形方法.针对《中国大学评价》中的6所林业院校,应用矩形方法,直接从模糊关系矩阵逐步降低阈值,得到分类的矩形表.此方法操作简单,形象直观,分类合理.     

Decision-making in a fuzzy environment with fuzzy termination time

The multistage control of a deterministic and stochastic system in a fuzzy environment is considered. The fuzzy environment is meant as fuzzy constraints imposed on subsequent controls and a fuzzy goal to be attained. The fuzzy decision is assumed to be the intersection of fuzzy constraints and a fuzzy goal. The problem is to find a maximizing decision. The termination time is given as a specified fuzzy set in the space of control stages. For solving the problem, the dynamic programming is applied.     

Exact solution of Marangoni convection in a binary fluid with throughflow and Soret effect

The onset of Marangoni convection in a non-reactive binary fluid layer in the presence of throughflow and Soret effect is determined. The bottom boundary of the fluid layer is assumed to be either conducting or insulating to temperature and solute concentration perturbations while the top boundary is free and insulating. The linear stability analysis is followed and an exact solution is obtained for the corresponding eigenvalue problem by assuming that stationary convection is exhibited at the neutral state. The contribution from the Soret effect is seen only when the throughflow is weak, but however for a wider range of upward throughflow when the bottom boundary is conducting. The instability gets advanced/delayed when the Soret parameter assumes negative/positive values. The results agree well with the existing results in the literature for some particular cases.     

The linearization method in the inverse problem of determination of low-permeability zones in an oil-bearing layer

The possibility of determination of low-permeability zones in an oil-bearing layer from bore-pressure measurements is studied. The formulation of the problem on the pressure distribution in the layer is considered in the case when the pressure is invariable across the layer. The problem is reduced to an integral equation. The linearization of the inverse problem of determination of low-permeability zones is considered. An iterative technique is proposed for solution of this problem. The technique involves reduction of the inverse problem to an operator equation.     

Numerical solution of positive control problem via linear programming

The reachability problem for linear time-invariant discrete-time control systems with sign-restricted input is considered. The time-optimal control is constructed by an iterative procedure. Each step of the iteration is defined as a linear programming problem. This problem is solved by the simplex algorithm. The initial feasible solution for the simplex algorithm is provided by the preceding step of the iteration. The inversion of the basis matrix is reduced to a bordering procedure. The structural stability of the solution is investigated.     

Maximum Likelihood Estimation of Hidden Markov Multivariate Normal Distribution Parameters

Hidden Markov model is widely used in statistical modeling of time, space and state transition data. The definition of hidden Markov multivariate normal distribution is given. The principle of using cluster analysis to determine the hidden state of observed variables is introduced. The maximum likelihood estimator of the unknown parameters in the model is derived. The simulated observation data set is used to test the estimation effect and stability of the method. The characteristic is simple classical statistical inference such as cluster analysis and maximum likelihood estimation. The method solves the parameter estimation problem of complex statistical models.     

The problem of the contact between a linear elastic body and elastic and rigid bodies (a variational approach)

The problem of the contact between a linear elastic body and a rigid body is formulated as a one-sided problem. The solution is determined from the variational inequality, equivalent to the problem of minimizing the energy functional in a set of allowable displacements. The regularity of the solution is established down to internal points of the contact boundary. A measure is constructed in the subsets of the contact boundary that enables the effect of a stamp on an elastic body to be characterized. The absolute continuity of this measure is proved at the internal point. The problem of the contact of two elastic bodies is examined in a similar formulation. The regularity of the solution is established and the nature of the effect of one body on the other is clarified.     

B-Spline Collocation Method for Nonlinear Singularly-Perturbed Two-Point Boundary-Value Problems

A B-spline collocation method is presented for nonlinear singularly-perturbed boundary-value problems with mixed boundary conditions. The quasilinearization technique is used to linearize the original nonlinear singular perturbation problem into a sequence of linear singular perturbation problems. The B-spline collocation method on piecewise uniform mesh is derived for the linear case and is used to solve each linear singular perturbation problem obtained through quasilinearization. The fitted mesh technique is employed to generate a piecewise uniform mesh, condensed in the neighborhood of the boundary layers. The convergence analysis is given and the method is shown to have second-order uniform convergence. The stability of the B-spline collocation system is discussed. Numerical experiments are conducted to demonstrate the efficiency of the method.     

线性方程组的多维投影算法

本文分析了求解线性方程组的一维投影算法即最小剩余法。定义了长轴陷阱及陷阱深度，用它们刻划了该算法迭代过程中锯齿现象的几何特征。本文给出了基于残差序列的避开长轴陷阱的扰动技巧，即多维投影算法。数值试验表明，投影算法要优于现在流行的主要算法。     

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??Hidden Markov model is widely used in statistical modeling of time, space and state transition data. The definition of hidden Markov multivariate normal distribution is given. The principle of using cluster analysis to determine the hidden state of observed variables is introduced. The maximum likelihood estimator of the unknown parameters in the model is derived. The simulated observation data set is used to test the estimation effect and stability of the method. The characteristic is simple classical statistical inference such as cluster analysis and maximum likelihood estimation. The method solves the parameter estimation problem of complex statistical models.     

证券市场上含有多个基金时的最优投资策略

本文研究了证券市场中包含多个基金和股票时的均值-方差最优投资决策模型,得到了最优投资组合的解析表达形式,以及对应的投资有效前沿,证明了两基金分离问题,由于最优解是不唯一的,进而讨论了最优解集合的结构,并对实例进行计算与分析。     

Inventory Lot Size Models with Vanishing Market

The paper is concerned with equipment which is being phased out because it is superseded by a technologically more advanced product. It nevertheless still requires efficient handling for the rest of its life span.An inventory lot size model without backlogs for a single commodity with a vanishing market is used. The vanishing market is represented by- a demand function that decreases with time and eventually reaches zero, implying that the planning horizon is finite and known. It is assumed that there is no lead time for replenishments. The problem is to devise an optimal management scheme for this inventory system, namely, to determine the number of replenishments and their schedule over the predicted survival time horizon of the product.The problem is completely solved for the class of demand functions which can be approximated to by positive powers of time. The solution is given in a nearly closed form and is proven to be unique.     

An estimate of the attraction domain with a specified exponential stability index in a wheeled robot control problem

The problem of synthesizing a law for the control of the plane motion of a wheeled robot is investigated. The rear wheels are the drive wheels and the front wheels are responsible for the turning of the platform. The aim of the control is to steer a target point to a specified trajectory and to stabilize the motion along it. The trajectory is assumed to be specified by a smooth curve. The actual curvature of the trajectory of the target point, which is related to the angle of rotation of the front wheels by a simple algebraic relation, is considered as the control. The control is subjected to bilateral constraints by virtue of the fact that the angle of rotation of the front wheels is limited. The attraction domain in the distance to trajectory - orientation space, is investigated for the proposed control law. Arrival at a trajectory with a specified exponential stability index is guaranteed in the case of initial conditions belonging to the given domain. An estimate of the attraction domain in the form of an ellipse is given.