Motion transformation in a higher order dimensional space

The objective of the work presented in this paper is an attempt at solving and transforming of the known from the classical mechanics two dimensional-plane single mass mechanical and mathematical vibration models in a higher order dimensional space with any virtual sectional curvature-positive or negative, constant or variable. A characterization of the Riemannian manifolds is performed by means of curvature operators. The computer codes Mathematica and MATLAB are used in the numerical simulation. The objects of the investigation are a sphere – with a positive constant sectional curvature, a cylinder-with a zero constant sectional curvature, helicoid-with a negative variable sectional curvature, a torus-with a variable (±) sectional curvature, any virtual surface of second order-with a variable (±) sectional curvature, pseudo-sphere – with a negative constant sectional curvature and a saddle-with a negative variable sectional curvature. The system motion is investigated in a qualitative aspect in time and frequency domain on the cited surfaces. The common algorithm derived in the paper can transform any motion from 3D space to curved manifold. We can derive the trajectory in an explicit form on the curved manifold. We can change the trajectory by a suitable variation of the curved manifold.     

Minimal surfaces: a geometric three dimensional segmentation approach

Summary. A novel geometric approach for three dimensional object segmentation is presented. The scheme is based on geometric deformable surfaces moving towards the objects to be detected. We show that this model is related to the computation of surfaces of minimal area (local minimal surfaces). The space where these surfaces are computed is induced from the three dimensional image in which the objects are to be detected. The general approach also shows the relation between classical deformable surfaces obtained via energy minimization and geometric ones derived from curvature flows in the surface evolution framework. The scheme is stable, robust, and automatically handles changes in the surface topology during the deformation. Results related to existence, uniqueness, stability, and correctness of the solution to this geometric deformable model are presented as well. Based on an efficient numerical algorithm for surface evolution, we present a number of examples of object detection in real and synthetic images. Received January 4, 1996 / Revised version received August 2, 1996     

Brownian motion in infinite dimensional manifolds

We discuss the extension to infinite dimensional Riemannian—Wiener manifolds of the transport approximation to Brownian motion, which was formulated by M. Pinsky for finite dimensional manifolds. A global representation is given for the Laplace—Beltrami operator in terms of the Riemannian spray and a homogenizing operator based upon the central hitting measure of the surface of the unit ball with respect to the Brownian motion on the model space.Research supported by NSF grant MCS8202319.     

Motion of vortex-filaments for superconductivity

In this paper, we study the asymptotic behavior of solutions to the simplified Ginzburg–Landau model for superconductivity. We prove that, asymptotically, vortex-filaments evolves according to the mean curvature flow in the sense of weak formulation. This can be seen as a first attempt to understand the nature of the motion of vortex filaments in three dimensions with magnetic field. On the other hand, this paper revisits the pioneering work of Bethuel–Orlandi–Smets [F. Bethuel, G. Orlandi, D. Smets, Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature, Ann. of Math. 163 (2006) 37–163] in a slightly relaxed setting.     

Finite dimensional semisimple Q-algebras

A Q-algebra can be represented as an operator algebra on an infinite dimensional Hilbert space. However we don’t know whether a finite n-dimensional Q-algebra can be represented on a Hilbert space of dimension n except n = 1, 2. It is known that a two dimensional Q-algebra is just a two dimensional commutative operator algebra on a two dimensional Hilbert space. In this paper we study a finite n-dimensional semisimple Q-algebra on a finite n-dimensional Hilbert space. In particular we describe a three dimensional Q-algebra of the disc algebra on a three dimensional Hilbert space. Our studies are related to the Pick interpolation problem for a uniform algebra.     

Convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature

Summary. Motion by (weighted) mean curvature is a geometric evolution law for surfaces, representing steepest descent with respect to (an)isotropic surface energy. It has been proposed that this motion could be computed by solving the analogous evolution law using a ``crystalline' approximation to the surface energy. We present the first convergence analysis for a numerical scheme of this type. Our treatment is restricted to one dimensional surfaces (curves in the plane) which are graphs. In this context, the scheme amounts to a new algorithm for solving quasilinear parabolic equations in one space dimension. Received January 28, 1993     

近似可展曲面的构造及应用

1引言经典微分几何中Gauss曲率为零的曲面称为可展曲面,它是一种特殊的直纹面．可展曲面有且只有三种,即锥面、柱面和切线面,它对于自由曲面造型具有重要的意义．例如,如果物体外壳是可展曲面,那么它可以没有形变地展开到平面上,从而可以用平板材料无形变地设计出来．这一性质对于造船业、航空业中的外形设计具有重要的意义．关于可展曲面的微分几何性质,可以在任何一本微分几何教材中找到,例如[3]．可展曲面可以说是微分几何中比较简单的一类曲面,但是在计算机辅助几何设计(CAGD)和计算机图形学中至今还不存在简单有效的设计方法．在[1,2,5]以及它们的参考文献中     

Threshold Dynamics for Networks with Arbitrary Surface Tensions

We present and study a new algorithm for simulating the N‐phase mean curvature motion for an arbitrary set of (isotropic) surface tensions. The departure point is the threshold dynamics algorithm of Merriman, Bence, and Osher for the two‐phase case. A new energetic interpretation of this algorithm allows us to extend it in a natural way to the case of N phases, for arbitrary surface tensions and arbitrary (isotropic) mobilities. For a large class of surface tensions, the algorithm is shown to be consistent in the sense that at every time step, it decreases an energy functional that is an approximation (in the sense of Gamma convergence) of the interfacial energy. A broad range of numerical tests shows good convergence properties. An important application is the motion of grain boundaries in polycrystalline materials: It is also established that the above‐mentioned large class of surface tensions contains the Read‐Shockley surface tensions, both in the two‐dimensional and three‐dimensional settings.© 2015 Wiley Periodicals, Inc.     

Potential theory of infinite dimensional Lévy processes

We study the potential theory of a large class of infinite dimensional Lévy processes, including Brownian motion on abstract Wiener spaces. The key result is the construction of compact Lyapunov functions, i.e., excessive functions with compact level sets. Then many techniques from classical potential theory carry over to this infinite dimensional setting. Thus a number of potential theoretic properties and principles can be proved, answering long standing open problems even for the Brownian motion on abstract Wiener space, as, e.g., formulated by R. Carmona in 1980. In particular, we prove the analog of the known result, that the Cameron-Martin space is polar, in the Lévy case and apply the technique of controlled convergence to solve the Dirichlet problem with general (not necessarily continuous) boundary data.     

Left invariant metrics and curvatures on simply connected three‐dimensional Lie groups

For each simply connected three‐dimensional Lie group we determine the automorphism group, classify the left invariant Riemannian metrics up to automorphism, and study the extent to which curvature can be altered by a change of metric. Thereby we obtain the principal Ricci curvatures, the scalar curvature and the sectional curvatures as functions of left invariant metrics on the three‐dimensional Lie groups. Our results improve a bit of Milnor's results of [7] in the three‐dimensional case, and Kowalski and Nikv?cevi?'s results [6, Theorems 3.1 and 4.1] (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global geometry and topology of spacelike stationary surfaces in the 4-dimensional Lorentz space

For spacelike stationary (i.e. zero mean curvature) surfaces in 4-dimensional Lorentz space, one can naturally introduce two Gauss maps and a Weierstrass-type representation. In this paper we investigate the global geometry of such surfaces systematically. The total Gaussian curvature is related with the surface topology as well as the indices of the so-called good singular ends by a Gauss–Bonnet type formula. On the other hand, as shown by a family of counterexamples to Osserman?s theorem, finite total curvature no longer implies that Gauss maps extend to the ends. Interesting examples include the deformations of the classical catenoid, the helicoid, the Enneper surface, and Jorge–Meeks? k-noids. Each family of these generalizations includes embedded examples in the 4-dimensional Lorentz space, showing a sharp contrast with the 3-dimensional case.     

Three dimensional distribution of Brownian motion extrema

This paper describes the joint distributions of minima, maxima and endpoint values for a three dimensional (3D) Wiener process. In particular, the results provide the joint cumulative distributions for the maxima and/or minima of the components of the process. The method of images is used to derive explicit expressions for the densities; the analysis can only be carried out for special correlation structures and requires a detailed study of partitions of the sphere by means of spherical triangles. The joint densities obtained can be used in several applied fields such as financial mathematics to obtain analytical expressions for prices of options for the 3D geometric Brownian motion process.     

A backward Euler alternating direction implicit difference scheme for the three‐dimensional fractional evolution equation

A backward Euler alternating direction implicit (ADI) difference scheme is formulated and analyzed for the three‐dimensional fractional evolution equation. In our method, the Riemann‐Liouville fractional integral term is treated by means of first order convolution quadrature suggested by Lubich. Meanwhile, an ADI technique is adopted to reduce the multidimensional problem to a series of one‐dimensional problems. A fully discrete difference scheme is constructed with space discretization by finite difference method. Two new inner products and corresponding norms are defined to analyze the scheme. The verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature. Numerical experiments are reported to demonstrate the efficiency of our scheme.     

Three dimensional line stochastic matrices and extreme points

Several properties of the extreme points of the convex set of three dimensional line stochastic matrices of order n are presented. The existence of many different classes of extremal configurations is established. These extremal matrices exhibit a large variety of patterns with some unexpected configurations. Latin squares of special types are used in some of the existence results. Furthermore, three questions raised by Brualdi and Csima are answered concerning the extreme points of three dimensional plane stochastic matrices of order n.     

Space‐time spectral method for two‐dimensional semilinear parabolic equations

In this paper, a high‐order accurate numerical method for two‐dimensional semilinear parabolic equations is presented. We apply a Galerkin–Legendre spectral method for discretizing spatial derivatives and a spectral collocation method for the time integration of the resulting nonlinear system of ordinary differential equations. Our formulation can be made arbitrarily high‐order accurate in both space and time. Optimal a priori error bound is derived in the L²‐norm for the semidiscrete formulation. Extensive numerical results are presented to demonstrate the convergence property of the method, show our formulation have spectrally accurate in both space and time. John Wiley & Sons, Ltd.     

A splitting technique of higher order for the Navier–Stokes equations

This article presents a splitting technique for solving the time dependent incompressible Navier–Stokes equations. Using nested finite element spaces which can be interpreted as a postprocessing step the splitting method is of more than second order accuracy in time. The integration of adaptive methods in space and time in the splitting are discussed. In this algorithm, a gradient recovery technique is used to compute boundary conditions for the pressure and to achieve a higher convergence order for the gradient at different points of the algorithm. Results on the ‘Flow around a cylinder’s- and the ‘Driven Cavity’s-problem are presented.     

Diffusion-Generated Motion by Mean Curvature for Filaments

Summary. Diffusion-generated motion by mean curvature is a simple algorithm for producing motion by mean curvature of a surface, in which the motion is generated by alternately diffusing and renormalizing a characteristic function. In this paper, we generalize diffusion-generated motion to a procedure that can be applied to the curvature motion of filaments, i.e., curves in R ^3, that may initially consist of a complex configuration of links. The method consists of applying diffusion to a complex-valued function whose values wind around the filament, followed by normalization. We motivate this approach by considering the essential features of the complex Ginzburg-Landau equation, which is a reaction-diffusion PDE that describes the formation and propagation of filamentary structures. The new algorithm naturally captures topological merging and breaking of filaments without fattening curves. We justify the new algorithm with asymptotic analysis and numerical experiments. Received October 30, 2000; accepted September 28, 2001 Online publication December 5, 2001     

Rigidity in the class of orientable compact surfaces of minimal total absolute curvature

Consider an orientable compact surface in three-dimensional Euclidean space with minimum total absolute curvature. If the Gaussian curvature changes sign to finite order and satisfies a nondegeneracy condition along closed asymptotic curves, we show that any other isometric surface differs by at most a Euclidean motion.     

Harmonic Maps in Connection of Phase Transitions with Higher Dimensional Potential Wells

This is in the sequel of authors' paper [Lin, F. H., Pan, X. B. and Wang, C. Y.,Phase transition for potentials of high dimensional wells, Comm. Pure Appl. Math., 65(6),2012, 833–888] in which the authors had set up a program to verify rigorously some formal statements associated with the multiple component phase transitions with higher dimensional wells. The main goal here is to establish a regularity theory for minimizing maps with a rather non-standard boundary condition at the sharp interface of the transition.The authors also present a proof, under simplified geometric assumptions, of existence of local smooth gradient flows under such constraints on interfaces which are in the motion by the mean-curvature. In a forthcoming paper, a general theory for such gradient flows and its relation to Keller-Rubinstein-Sternberg's work(in 1989) on the fast reaction, slow diffusion and motion by the mean curvature would be addressed.