Invariants of four- and three-dimensional singularities of integrable systems

A relationship between invariants of four-dimensional singularities of integrable Hamiltonian systems (with two degrees of freedom) and invariants of two-dimensional foliations on three-dimensional manifolds being the “boundaries” of these four-dimensional singularities is discovered. Nonequivalent singularities which, nevertheless, have equal three-dimensional invariants are found.     

A technology for 3D grid generation

New results concerning the technology for adaptive 3D grid generation are obtained. This technology is based on the numerical solution of the inverted one-, two-, and three-dimensional Beltrami and diffusion equations with respect to the monitor metric. The one- and two-dimensional equations are used to generate grids on the edges and the faces of the domain, respectively. The three-dimensional equations are used to generate a grid in the interior of the domain. Examples of adaptive 3D hexahedral and prismatic grids are demonstrated.     

Unique Determination of Locally Convex Surfaces with Boundary and Positive Curvature of Genus p ≥ 0

We prove the next result. If two isometric regular surfaces with regular boundaries, of an arbitrary finite genus, and positive Gaussian curvature in the three-dimensional Euclidean space, consist of two congruent arcs corresponding under the isometry (lying on the boundaries of these surfaces or inside these surfaces) then these surfaces are congruent.

     

Perpetual American Double Lookback Options on Drawdowns and Drawups with Floating Strikes

We present closed-form solutions to the problems of pricing of the perpetual American double lookback put and call options on the maximum drawdown and the maximum drawup with floating strikes in the Black-Merton-Scholes model. It is shown that the optimal exercise times are the first times at which the underlying risky asset price process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum as well as the maximum drawdown or maximum drawup. The proof is based on the reduction of the original double optimal stopping problems to the appropriate sequences of single optimal stopping problems for the three-dimensional continuous Markov processes. The latter problems are solved as the equivalent free-boundary problems by means of the smooth-fit and normal-reflection conditions for the value functions at the optimal stopping boundaries and the edges of the three-dimensional state spaces. We show that the optimal exercise boundaries are determined as either the unique solutions of the associated systems of arithmetic equations or the minimal and maximal solutions of the appropriate first-order nonlinear ordinary differential equations.

     

A simple and effective mesh adaptation

Simulation accuracy is greatly influenced by grid quality. Here, mesh quality indicates orthogonality of grid lines at the boundaries and quasi-orthogonality within critical regions, smoothness, solution adaptive behaviour and bounded aspect ratios. A simple, effective and computationally efficient approach for adapting quadrilateral grids to a given adaptive functional is presented. Several numerical examples are explored for supporting our claim.     

Continuous theory of dislocations and disclinations in a two-dimensional medium

A system of equations describing mobile defects in a two-dimensional Cosser at continuum, i.e. in a medium whose motion is determined by the displacement field and rotation field independent of it, is obtained.

The basic equations of the static theory /1–5/ and dynamic continuous theory /6–12/ of defects (dislocations and disclinations) are known for a three-dimensional medium, obtained by a variety of methods. A dislocation model of the misalignment surfaces used in describing the Martensitic transformations /2, 13/ is proposed. The dislocation representations were used in /14–16/ to describe the grain boundaries, and the difference dislocations within the boundaries of separation were studied in /17, 18/. The dislocation structure of internal boundaries of separation was described in /19, 20/ using the differential geometry characteristics (torsion and curvature tensors, non-holonomic object) of three-dimensional media. Surface dislocations and disclinations of the separate Volterra distortions-type were studied in /21/, with liquid crystals and various biological objects indicated as the suitable areas of application of these concepts.     

Neural network-based state fusion and adaptive tracking for maneuvering targets

An adaptive algorithm for tracking maneuvering targets is proposed. This algorithm is implemented with two filters and a multilayer feedforward neural network using state fusion, together with the current statistic model and adaptive filtering. The neural network fuses automatically all the state information of the two filters and tunes adaptively the system variance for one of the two filters to adapt to different target maneuvers when the two filters track the same maneuvering target in parallel. Simulation results show that the adaptive algorithm tracks very well maneuvering targets over a wide range of maneuvers with high precision, in both one and three-dimensional cases.     

Hiding in plain sight: criminal network analysis

The United States is faced with an increasingly complex criminal enterprise. Advances in technology, communications, transport, and economies enable a highly adaptive criminal element to hide in plain site. These advances provide criminal organizations with the same global boundaries and opportunities as legitimate organizations.     

正变坐标系下完全深度平均湍流方程组

本文针对宽浅型水域，对三维湍流时均方程组逐项进行深度平均，推导出包含自由水面和地形影响的深度平均流动控制方程组．本文还同时获得了深度平均形式的ｋ－ε湍流模型方程组．因计入了水流的三维效应，该模型称为完全深度平均模型．考虑到天然水域几何边界复杂，本文运用较简便的方法，将上述模型方程组交换至正交坐标系下．所得控制方程组可以直接运用于对实际问题的数值模拟．     

Fourier spectral method with an adaptive time strategy for nonlinear fractional Schrödinger equation

In this paper, a Fourier spectral method with an adaptive time step strategy is proposed to solve the fractional nonlinear Schrödinger (FNLS) equation with periodic initial value problem. First, we prove the conservation law of the mass and the energy for the semi-discrete Fourier spectral scheme. Second, the error estimation of the semi-discrete scheme is given in the relevant fractional Sobolev space. Then, an adaptive time-step strategy is designed to reduce central processing unit (CPU) time. Finally, the numerical experiments for the one-, two- and three-dimensional FNLSs, show that the adaptive strategy, compared to the constant time step, can reduce the CPU-time by almost half.     

Adaptive FE analysis of plates by shear-flexible quadrilateral Reissner-Mindlin elements

h-version adaptive finite element analysis of plates using QUAD4 R-M elements is discussed. The necessity of using shear flexible formulations for plate elements in adaptive FEA is explained. Two shear flexible QUAD4 plate elements formulated by reconstituted shear strain fields are selected from literature and are used to solve both thin and thick plates with various geometries and boundary conditions. The effect of boundary layers in plates with soft or free boundaries is discussed and is shown in various examples. A powerful and versatile quadrilateral automatic mesh generator (MSD) is used for the discretization of the plate domain. The error norms are computed by the Z² as well as the superconvergence theory. The global convergence rates of the adaptive solutions with respect to the energy norms are demonstrated and it is seen that the theoretical rates of convergence are exceeded in several cases.     

A streamline tracking algorithm for semi-Lagrangian advection schemes based on the analytic integration of the velocity field

A new scheme for the construction on an unstructured grid of the streamlines of the three-dimensional shallow water equations is presented. The qualitative advantages of the scheme, notably closed streamlines and realistic treatment of closed boundaries, are derived and the spatial accuracy is demonstrated.Semi-Lagrangian advection schemes offer the computational cost advantage of being explicit but also unconditionally stable with respect to time step. However, semi-Lagrangian methods based on the numerical integration of the discretised velocity field frequently have difficulty in meeting physically significant criteria such as the closure of streamlines and the inviolability of closed boundaries. Here a streamline tracking scheme based on the analytic integration of the discretised velocity field is presented.     

矩形截面梁中波传播的精确分析

将傅立叶级数方法推广应用于矩形截面梁中波传播的精确分析．不仅试着直接从三维弹性动力学方程出发,导出了矩形截面梁中波传播的一般解析解,而且给出了弹性波在自由矩形截面梁中的传播特性．波传播精确模型的提出,为实现梁波的耦合控制奠定了坚实基础．     

Conduction and convection phenomena through a slab with thermal heterogeneities

This paper addresses the computation of the three-dimensional transient heat transfer through a layered solid and/or fluid formation containing irregular inclusions. The use of appropriate Green’s functions for a flat layer formation in a boundary element method formulation avoids the discretization of the layer interface boundaries.     

Generation of 3D grids by approximate factorization

An efficient procedure is proposed for generating three-dimensional numerical grids inside domains with curvilinear boundaries. Systems of partial differential equations of elliptical type are used as the basic equations for grid generation. An approximate factorization method is proposed for numerical solution of the resulting boundary-value problem. Calculation results are reported for a prototype domain. Translated from Prikladnaya Matematika i Informatika, No. 2, pp. 99–105, 1999.     

Convergence analysis of an adaptive edge element method for Maxwell's equations

An efficient and reliable a posteriori error estimate is derived for solving three-dimensional static Maxwell's equations by using the edge elements of first family. Based on the a posteriori error estimates, an adaptive finite element method is constructed and its convergence is established. Compared with the existing results, an important advantage of the new theory lies in its feature that the usual marking of elements based on the oscillation is not needed in our adaptive algorithm, while the linear convergence of the algorithm can be still demonstrated in terms of the reduction of the energy-norm error and the oscillation. Numerical examples are provided which demonstrate the effectiveness and robustness of the adaptive methods.     

Application of the spherical metric tensor to grid adaptation and the solution of applied problems

New results concerning the construction and application of adaptive numerical grids for solving applied problems are presented. The grid generation technique is based on the numerical solution of inverted Beltrami and diffusion equations for a monitor metric. The capabilities of the spherical metric tensor as applied to adaptive grid generation are examined in detail. Adaptive hexahedral grids are used to numerically solve a boundary value problem for the three-dimensional heat equation with a moving boundary in a continuous medium with discontinuous thermophysical parameters; this problem models the interaction of a thermal wave with a thermocouple embedded in the solid.     

Clifford algebra valued boundary integral equations for three-dimensional elasticity

Applications of Clifford analysis to three-dimensional elasticity are addressed in the present paper. The governing equation for the displacement field is formulated in terms of the Dirac operator and Clifford algebra valued functions so that a general solution is obtained analytically in terms of one monogenic function and one multiple-component spatial harmonic function together with its derivative. In order to solve numerically the three-dimensional problems of elasticity for an arbitrary domain with complicated boundary conditions, Clifford algebra valued boundary integral equations (BIEs) for multiple-component spatial harmonic functions at an observation point, either inside the domain, on the boundary, or outside the domain, are constructed. Both smooth and non-smooth boundaries are considered in the construction. Moreover, the singularities of the integrals are evaluated exactly so that in the end singularity-free BIEs for the observation point on the boundary taking values on Clifford numbers can be obtained. A Clifford algebra valued boundary element method (BEM) based on the singularity-free BIEs is then developed for solving three-dimensional problems of elasticity. The accuracy of the Clifford algebra valued BEM is demonstrated numerically.     

Analysis of an optimal control problem for the three-dimensional coupled modified Navier-Stokes and Maxwell equations

The mathematical formulation and analysis of an optimal control problem associated with a viscous, incompressible, electrically conducting fluid in a bounded three-dimensional domain with fixed perfectly conducting boundaries is considered. The objective of control is the matching of the velocity and magnetic fields to given target fields; control is effected through distributed mechanical force and current controls. The existence of optimal solutions is shown, the Gâteaux differentiability for the magnetohydrodynamic system with respect to controls is proved, and the optimality system is obtained.