Surfaces In Rn Research Articles

Controllability for the Nonlinear Fuzzy Control System with Nonlocal Initial Condition in EnN

This paper we study the exact controllability for the nonlinear fuzzy control system with nonlocal initial condition in Eⁿ N by using the concept of fuzzy number of dimension n whose values are normal, convex, upper semicontinuous and compactly supported surface in Rⁿ. Eⁿ N be the set of all fuzzy numbers in Rⁿ with edges having bases parallel to axis X₁, X₂,… ,X n .

Fourth order partial differential equations on general geometries

We extend a recently introduced method for numerically solving partial differential equations on implicit surfaces [M. Bertalmío, L.T. Cheng, S. Osher, G. Sapiro. Variational problems and partial differential equations on implicit surfaces, J. Comput. Phys. 174 (2) (2001) 759–780] to fourth order PDEs including the Cahn–Hilliard equation and a lubrication model for curved surfaces. By representing a surface in RN as the level set of a smooth function, ϕ, we compute the PDE using only finite differences on a standard Cartesian mesh in RN. The higher order equations introduce a number of challenges that are of less concern when applying this method to first and second order PDEs. Many of these problems, such as time-stepping restrictions and large stencil sizes, are shared by standard fourth order equations in Euclidean domains, but others are caused by the extreme degeneracy of the PDEs that result from this method and the general geometry. We approach these difficulties by applying convexity splitting methods, ADI schemes, and iterative solvers. We discuss in detail the differences between computing these fourth order equations and computing the first and second order PDEs considered in earlier work. We explicitly derive schemes for the linear fourth order diffusion, the Cahn–Hilliard equation for phase transition in a binary alloy, and surface tension driven flows on complex geometries. Numerical examples validating our methods are presented for these flows for data on general surfaces.

Controllability of the nonlinear Fuzzy Integro-Differential Equations on EnN

In this paper we study the controllability for the nonlinear fuzzy integro-differential equations on Eⁿ N by using the concept of fuzzy number for dimension n whose values are normal, convex, semicontinuous and compactly supported surface in Rⁿ. Eⁿ N be the set of all fuzzy numbers in Rⁿ with edges having bases parallel to axis X₁, X₂, X₃…X n .

Boundary flow for the minimal surfaces in Rn with Plateau boundary condition

We introduce the notion of the boundary flow for minimal surfaces in Rn with Plateau boundary condition and establish the global existence and uniqueness of the flow as well as the continuous dependence on the initial data.

Boundary flow for the minimal surfaces in Rn with Plateau boundary condition

We introduce the notion of the boundary flow for minimal surfaces in Rn with Plateau boundary condition and establish the global existence and uniqueness of the flow as well as the continuous dependence on the initial data.

The Barrier Principle for Minimal Submanifolds of Arbitrary Codimension

We establish a barrier principle for minimal submanifolds of a Riemannianmanifold of arbitrary codimension. We construct examples of barriers fortwo-dimensional minimal surfaces in ℝn, n ≥ 4, and apply these to deduceexistence as well as nonexistence theorems for Plateau's problem.

Weighted singular integral operators in Clifford analysis

AbstractWe consider the Cauchy‐type integral and singular integral operator having a weighted Cauchy kernel, both over a domain bounded by an n‐dimensional surface in ℝn+1, n⩾2. The aim of this paper is to study the behaviour of the weighted Cauchy singular integral operators near the integration boundary as well as to establish the basic relation among them. In certain sense, this approach has the advantage of being better related with boundary value problems than what is concerned in the setting of smooth surfaces. Copyright © 2002 John Wiley & Sons, Ltd.

Long time behavior of Riemannian mean curvature flow of graphs

In this paper we consider long time behavior of a mean curvature flow of nonparametric surface in Rn, with respect to a conformal Riemannian metric. We impose zero boundary value, and we prove that the solution tends to 0 exponentially fast as t→∞. Its normalization u/supu tends to the first eigenfunction of the associated linearized problem.

An analogue of minimal surface theory in 𝑆𝐿(𝑛,𝐂)/𝐒𝐔(𝐧)

We shall discuss the class of surfaces with holomorphic right Gauss maps in non-compact duals of compact semi-simple Lie groups (e.g. SL ⁡ ( n , C ) / SU ⁡ ( n ) \operatorname {SL}(n,\mathbf {C})/\operatorname {SU}(n) ), which contains minimal surfaces in R n \mathbf {R}^n and constant mean curvature 1 1 surfaces in H 3 \mathcal {H}^3 . A Weierstrass type representation formula and a Chern-Osserman type inequality for such surfaces are given.

Fourier Analysis on Starlike Lipschitz Surfaces

A theory of a class of singular integrals on starlike Lipschitz surfaces in Rn is established. The class of singular integrals forms an operator algebra identical to the class of bounded holomorphic Fourier multipliers, as well as to the Cauchy–Dunford bounded holomorphic functional calculus of the spherical Dirac operator. The study proposes a new method inducing Clifford holomorphic functions from holomorphic functions of one complex variable, by means of which problems on the sphere are reduced to those on the unit circle.

Isothermic Surfaces in Spaces of Arbitrary Dimension: Integrability, Discretization, and Bäcklund Transformations—A Discrete Calapso Equation

A vector analog of the classical Calapso equation governing isothermic surfaces in Rn+2 is introduced. It is shown that this vector Calapso system admits a nonlocal) scalar Lax pair based on the classical Moutard equation. The analog of Darboux's Bäcklund transformation for isothermic surfaces in R3 is derived in a systematic manner and shown that it may be formulated in terms of the classical Moutard transformation acting on the scalar Lax pair. A permutability theorem for isothermic surfaces is set down that manifests itself in an explicit superposition principle for the vector Calapso system. This superposition principle in vectorial form is shown to constitute an integrable discretization of the vector Calapso system and, therefore, defines discrete isothermic surfaces in Rn+2. The discrete Calapso equation is related to the discrete Korteweg–de Vries equation and discrete holomorphic functions. A matrix Lax pair based on Clifford algebras and a scalar Lax pair are derived for the discrete Calapso equation. A discrete Moutard‐type transformation for the discrete Calapso equation is obtained, and it is shown that the discrete Calapso equation may be specialized to an integrable discrete version of the O(n+2) nonlinear σ‐model.

Moebius geometry of submanifolds in 𝕊 n

In this paper we define a Moebius invariant metric and a Moebius invariant second fundamental form for submanifolds in ?n and show that in case of a hypersurface with n≥ 4 they determine the hypersurface up to Moebius transformations. Using these Moebius invariants we calculate the first variation of the moebius volume functional. We show that any minimal surface in ?n is also Moebius minimal and that the image in ?n of any minimal surface in ℝn unter the inverse of a stereographic projection is also Moebius minimal. Finally we use the relations between Moebius invariants to classify all surfaces in ?3 with vanishing Moebius form.

The classification of complete orientable 2-dimensional area-minimizing mod 2 surfaces in ℝn

Let S ⊂ ℝn be a complete 2-dimensional areaminimizing mod 2 surface. Then S = x1 (M1) ∪ … ∪ xr (Mr) where each Mj is connected, xj: Mj → Vj is a classical minimal immersion into an affine subspace Vj of ℝn, and the subspaces V1,…, Vr are pairwise orthogonal. Here we prove that if Mj is orientable, then xj (Mj) is either aflat plane or, in suitable coordinates, a generalized complex hyperbola.

Approximation by spherical waves inL p -spaces

The paper is devoted to the following problem. Consider the set of all radial functions with centers at the points of a closed surface inRn. Are such functions complete in the spaceLq(Rn)? It is shown that the answer is positive if and only ifq is not less than 2n/(n + 1). A similar question is also answered for Riemannian symmetric spaces of rank 1. Relations of this problem with the wave and heat equations are also discussed.

Hyperbolic geometry and reflection groups

The n-dimensional pseudospheres are the surfaces in Rn+l given by the equations x12+x22+...+xk2-xk+12-...-xn+12=1(1 ??? k ??? n+1). The cases k=l, n+1 give, respectively a pair of hyperboloids, and the ordinary n-sphere. In the first chapter we consider the pseudospheres as surfaces h En+1,k, where Em,k=Rk x (iR)m-k, and investigate their geometry in terms of the linear algebra of these spaces. The main objects of investigation are finite sequences of hyperplanes in a pseudosphere. To each such sequence we associate a square symmetric matrix, the Gram matrix, which gives information about angle and incidence properties of the hyperplanes. We find when a given matrix is the Gram matrix of some sequence of hyperplanes, and when a sequence is determined up to isometry by its Gram matrix. We also consider subspaces of pseudospheres and projections onto them. This leads to an n-dimensional cosine rule for spherical and hyperbolic simplices. In the second chapter we derive integral formulae for the volume of an n-dimensional spherical or hyperbolic simplex, both in terms of its dihedral angles and its edge lengths. For the regular simplex with common edge length ?? we then derive power series for the volume, both in u = sin??/2, and in ?? itself, and discuss some of the properties of the coefficients. In obtaining these series we encounter an interesting family of entire functions, Rn(p) (n a nonnegative integer and p??C). We derive a functional equation relating Rn(p) and Rn-1(p). Finally we classify, up to isometry, all tetrahedra with one or more vertices truncated, for which the dihedral angles along the edges formed by the truncatons. are all ??/2, and the remaining dihedral angles are all sub-multiples of ??. We show how to find the volumes of these polyhedra, and find presentations and small generating sets for the orientation-preserving subgroups of their reflection groups. For particular families of these groups, we find low index torsion free subgroups, and construct associated manifolds and manifolds with boundary In particular, we find a sequence of manifolds with totally geodesic boundary of genus, g???2, which we conjecture to be of least volume among such manifolds.

Self-intersection points in classical area-minimizing surfaces

In this paper we study the structure at a point of tangential self-intersection in an area-minimizing classical minimal surface inR n. The result is that the lowest order homogeneous term in the splitting function, i.e. the difference of the height functions from the tangent plane, has the form $$p(z) = az^k + \bar a\bar z^k , with a \cdot a = 0.$$

The inversion of fractional integrals on a sphere

The purpose of the paper is to invert Riesz potentials and some other fractional integrals on then-dimensional spherical surface in ℝn+1 in the closed form. New descriptions of spaces of the fractional smoothness on a sphere are obtained in terms of spherical hypersingular integrals. It is shown that Riesz potentials of the ordersn,n + 2,n + 4, ... on a sphere are Noether operators and theird-characteristic depends on the radius of the sphere.

Calibrations and the size of Grassmann faces

In the past fifteen years or so, convex geometry and the theory of calibrations have provided a deeper understanding of the behavior and singular structure ofm-dimensional area-minimizing surfaces inR n . Calibrations correspond to faces of the GrassmannianG(m,R n ) of orientedm-planes inR n , viewed as a compact submanifold of the exterior algebra Λ m R n . Large faces typically provide many examples of area-minimizing surfaces. This paper studies the sizes of such faces. It also considers integrands Φ more general than area. One result implies that form-dimensional surfaces inR n , with 2 ⩽m ⩽n − 2, for any integrand Φ, there are Φ-minimizing surfaces with interior singularities.

Asymptotic expansions associated with the variance estimator of the normal observation error in nonlinear regression

Asymptotic expansions are derived for the least-squares variance estimator of the normal observation error in nonlinear regression and for its distribution function. The first three terms of the asymptotic expansion are given. The third expansion term is interpreted geometrically as the invariant of some surface in Rn.

Regularity of solutions and interfaces of a generalized porous medium equation inR N

We consider the Cauchy problem for the generalized porous medium equation ut=Δϕ(u) where u=u(x, t), xeRn and t>0, and the initial datum u(x, 0) is assumed to be nonnegative, integrable mid to nave compact support. The nonlinearity ϕ(u) is a C1 function defined for u≥O which grows like a power of u. Our assumptions generalize the porous medium case, ϕ(u)=um, m>1, and also include the equation of the Marshak waves. This problem has finite speed of propagation. We estimate the rate of growth of the support of the solution with precise estimates for t→ 0 and t→∞. Our main result deals with the regularity of the solutions. We show that after a certain time t0 the pressure, defined by v=ψ(u), with ψ′(u)=ϕ(u)/u and ψ(0)=0, is a Lipschitz-continuous function of x and t and the interface is a Lipschitz-continuous surface in RN+1; the solution u is Holder continuous for all times t> 0.