On the computation of multi-dimensional solution manifolds of parametrized equations

Summary A new algorithm is presented for computing vertices of a simplicial triangulation of thep-dimensional solution manifold of a parametrized equationF(x)=0, whereF is a nonlinear mapping fromR ⁿ toR ^m ,p=n–m>1. An essential part of the method is a constructive algorithm for computing moving frames on the manifold; that is, of orthonormal bases of the tangent spaces that vary smoothly with their points of contact. The triangulation algorithm uses these bases, together with a chord form of the Gauss-Newton process as corrector, to compute the desired vertices. The Jacobian matrix of the mapping is not required at all the vertices but only at the centers of certain local triangulation patches. Several numerical examples show that the method is very efficient in computing triangulations, even around singularities such as limit points and bifurcation points. This opens up new possibilities for determining the form and special features of such solution manifolds.Dedicated to Professor Ivo Babuka on the occasion of his sixtieth birthdayThis work was supported in part by the National Science Foundation under Grant DCR-8309926, the Office of Naval Research under contract N-00014-80-C-9455, and the Air Force Office of Scientific Research under Grant 84-0131     

Transience and capacity of minimal submanifolds

We prove explicit lower bounds for the capacity of annular domains of minimal submanifolds P ^m in ambient Riemannian spaces N ⁿ with sectional curvatures bounded from above. We characterize the situations in which the lower bounds for the capacity are actually attained. Furthermore we apply these bounds to prove that Brownian motion defined on a complete minimal submanifold is transient when the ambient space is a negatively curved Hadamard-Cartan manifold. The proof stems directly from the capacity bounds and also covers the case of minimal submanifolds of dimension m > 2 in Euclidean spaces.     

A Note on the <Emphasis Type="Italic">p</Emphasis>-Parabolicity of Submanifolds

We give a geometric criterion which shows p-parabolicity of a class of submanifolds in a Riemannian manifold, with controlled second fundamental form, for p ≥ 2.     

The -equation and the Hartogs phenomenon on weakly q-pseudoconcave C R manifolds

We show that the Hartogs phenomenon holds in minimal, weakly 2-pseudoconcave generic C R submanifolds of a Stein manifold with trivial normal bundle. We also prove some results concerning the local and/or global solvability of the tangential Cauchy-Riemann equations for smooth forms and currents on weakly q-pseudoconcave C R manifolds.     

Fast diffusion flow on manifolds of nonpositive curvature

We consider the fast diffusion equation (FDE) u _t = Δu ^m (0 < m < 1) on a nonparabolic Riemannian manifold M. Existence of weak solutions holds. Then we show that the validity of Euclidean–type Sobolev inequalities implies that certain L ^p −L ^q smoothing effects of the type ∥u(t)∥ _q ≤ Ct ^−α ∥u ₀∥^γ _p , the case q = ∞ being included. The converse holds if m is sufficiently close to one. We then consider the case in which the manifold has the addition gap property min σ(−Δ) > 0. In that case solutions vanish in finite time, and we estimate from below and from above the extinction time.      

Zero sets of solutions to semilinear elliptic systems of first order

Consider a nontrivial smooth solution to a semilinear elliptic system of first order with smooth coefficients defined over an n-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution is contained in a countable union of smooth (n−2)-dimensional submanifolds. Hence it is countably (n−2)-rectifiable and its Hausdorff dimension is at most n−2. Moreover, it has locally finite (n−2)-dimensional Hausdorff measure. We show by example that every real number between 0 and n−2 actually occurs as the Hausdorff dimension (for a suitable choice of operator). We also derive results for scalar elliptic equations of second order. Oblatum 22-V-1998 & 26-III-1999 / Published online: 10 June 1999     

The approximation theory for the p-version finite element method and application to non-linear elliptic PDEs

Approximation theoretic results are obtained for approximation using continuous piecewise polynomials of degree p on meshes of triangular and quadrilateral elements. Estimates for the rate of convergence in Sobolev spaces , are given. The results are applied to estimate the rate of convergence when the p-version finite element method is used to approximate the -Laplacian. It is shown that the rate of convergence of the p-version is always at least that of the h-version (measured in terms of number of degrees of freedom used). If the solution is very smooth then the p-version attains an exponential rate of convergence. If the solution has certain types of singularity, the rate of convergence of the p-version is twice that of the h-version. The analysis generalises the work of Babuska and others to the case . In addition, the approximation theoretic results find immediate application for some types of spectral and spectral element methods. Received August 2, 1995 / Revised version received January 26, 1998     

Implicitly defined harmonic PHH submersions

The concept of harmonic pseudo-horizontally homothetic submersion, alias harmonic PHH submersion, expresses the more geometric notion of smooth family of minimal submanifolds indexed by a K?hler parameter manifold. We give results that allow to construct harmonic PHH submersions by solving some implicit equations and we apply this method to the case of sphere bundles. Received: 3 September 1998 / Accepted: 19 March 1999     

Inertial manifolds and linear multi-step methods

We determine the existence and C ¹ convergence of an inertial manifold for a strongly A(α) stable, pth order, p≧1, linear multi-step method approximating a sectorial evolution equation that satisfies a gap condition. This inertial manifold gives rise to a one-step method that C ¹ approximates the inertial form of the evolution equation and yields further approximation properties of the multi-step method. This revised version was published online in June 2006 with corrections to the Cover Date.     

On submanifolds of Lxf F satisfying Chen's basic equality

It is well known that the warped product Lx_f F of a line L and a Kaehler manifold F is an almost contact Riemannian manifold which is characterized by some tensor equations appeared in (1.7) and (1.8). In this paper we determine submanifolds of Lx_f F which are tangent to the structure vector field and satisfy Chen's basic equality. Also, we investigate tubular hypersurfaces of Lx_f CE ^m which satisfy Chen's basic equality where CE ^m is a complex Euclidean m-space. This revised version was published online in June 2006 with corrections to the Cover Date.     

Localization for Involutions in Floer Cohomology

We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds in a symplectic manifold M. Suppose that M carries a symplectic involution, which preserves both submanifolds. Under various topological hypotheses, we prove a localization theorem for Floer cohomology, which implies a Smith-type inequality for the Floer cohomology groups in M and its fixed point set. Two applications to symplectic Khovanov cohomology are included.     

Differentiable mappings of affine spaces into manifolds of <Emphasis Type="Italic">m</Emphasis>-planes in a multidimensional Euclidean space

In this paper we consider a differentiable mapping of a p-dimensional affine space into the differentiable manifold $ \mathfrak{M} $ \mathfrak{M} _N of all centered m-planes in an n-dimensional Euclidean space. We pay special attention to describing geometric images defined by a fundamental geometric object of the mentioned mapping.     

Riesz potentials and integral geometry in the space of rectangular matrices

Riesz potentials on the space of rectangular n×m matrices arise in diverse “higher rank” problems of harmonic analysis, representation theory, and integral geometry. In the rank-one case m=1 they coincide with the classical operators of Marcel Riesz. We develop new tools and obtain a number of new results for Riesz potentials of functions of matrix argument. The main topics are the Fourier transform technique, representation of Riesz potentials by convolutions with a positive measure supported by submanifolds of matrices of rank<m, the behavior on smooth and L^p functions. The results are applied to investigation of Radon transforms on the space of real rectangular matrices.     

Invariant submanifolds for systems of vector fields of constant rank

Given a system of vector fields on a smooth manifold that spans a plane field of constant rank, we present a systematic method and an algorithm to find submanifolds that are invariant under the flows of the vector fields. We present examples of partition into invariant submanifolds, which further gives partition into orbits. We use the method of generalized Frobenius theorem by means of exterior differential systems.     

Non-horizontal submanifolds and coarea formula

Let k be a positive integer and let m be the dimension of the horizontal subspace of a stratified group. Under the condition k ≤ m, we show that all submanifolds of codimension k are generically non-horizontal. For these submanifolds, we prove an area-type formula that allows us to compute their Q − k dimensional spherical Hausdorff measure. Finally, we observe that a.e. level set of a sufficiently regular vector-valued mapping on a stratified group is a non-horizontal submanifold. This allows us to establish a sub-Riemannian coarea formula for vector-valued Riemannian Lipschitz mappings on stratified groups.     

The unregularized gradient flow of the symplectic action

The symplectic action can be defined on the space of smooth paths in a symplectic manifold P which join two Lagrangian submanifolds of P. To pursue a new approach to the variational theory of this function, we define on a subset of the path space the flow whose trajectories are given by the solutions of the Cauchy-Riemann equation with respect to a suitable almost complex structure on P. In particular, we prove compactness and transversality results for the set of bounded trajectories.     

Study of linear information for classes of polynomial equations

Summary We study linear sequential (adaptive) information for approximating zeros of polynomials of unbounded degree and establish a theorem on constrained approximation of smooth functions by polynomials.For a positive we seek a pointx ^* such that|x ^* – _p | , where _p is a zero of a real polynomialp in the interval [a, b]. We assume thatp belongs to the classF ₁ of polynomials of bounded arbitrary seminorm and having a root in [a, b] or to the classF ₂ of polynomials which are nonpositive ata, nonnegative atb and have exactly one simple zero in [a, b]. The information onp consists ofn sequential (adaptive) evaluations of arbitrary linear functionals. The pointx ^* is constructed by means of an algorithm which is an arbitrary mapping depending on the information onp. We show that there exists no information and no algorithm for computingx ^* for everyp fromF ₁, no matter how large the value ofn is. This is a stronger result than that obtained by us for smooth functions.For the classF ₂ we can find a pointx ^* for arbitraryp and. Anoptimal algorithm, i.e., an algorithm with the smallest error, is thebisection of the smallest known interval containing the root ofp. We also exhibitoptimal information operators, i.e., the linear functionals for which the error of an optimal algorithm that uses them is minimal. It turns out that in the class of nonsequential (parallel) information, i.e., when the functionals are given simultaneously, optimal information consists of the evaluations of a polynomial atn-equidistant points in [a, b]. In the class of sequential continuous information, optimal information consists of evaluations of a polynomial atn points generated by thebisection method. To prove this result we establish a theorem on constrained approximation of smooth functions by polynomials. More precisely, we prove that a smooth function can be arbitrarily well uniformly approximated by a polynomial which satisfies constrains given byn arbitrary continuous linear functionals.Our results indicate that the problem of finding an -approximation to a real zero of a real polynomial (of unknown degree) is essentially of the same difficulty as the problem of finding an -approximation to a zero of an infinitely differentiable function.     

Mean Location and Sample Mean Location on Manifolds: Asymptotics, Tests, Confidence Regions

In a previous investigation we studied some asymptotic properties of the sample mean location on submanifolds of Euclidean space. The sample mean location generalizes least squares statistics to smooth compact submanifolds of Euclidean space. In this paper these properties are put into use. Tests for hypotheses about mean location are constructed and confidence regions for mean location are indicated. We study the asymptotic distribution of the test statistic. The problem of comparing mean locations for two samples is analyzed. Special attention is paid to observations on Stiefel manifolds including the orthogonal groupO(p) and spheresS^k−1, and special orthogonal groupsSO(p). The results also are illustrated with our experience with simulations.     

Existence of a somewhere injective pseudo-holomorphic disc

Given a smooth totally real submanifold L {\cal L} in an almost complex manifold (M,J) and a J-holomorphic disc with boundary in L {\cal L} , by restriction of the initial disc and factorization, one gets a smooth simple J-holomorphic curve still with boundary in L {\cal L} . As a consequence one gets a proof of the Arnold-Givental conjecture for a class of Lagrangian submanifolds in a symplectic manifold.     

Fast conformal mapping of multiply connected regions

An iterative method is presented which constructs for an unbounded region G with m holes and sufficiently smooth boundary a circular region H and a conformal mapping Φ from H to G. With the usual normalization both H and Φ are uniquely determined by G. With a few modifications the method can also be applied to a bounded region G with m holes. The canonical region H is then the unit disc with m circular holes. The proposed method also determines the centers and radii of the boundary circles of H and requires, at each iterative step, the solution of a Riemann–Hilbert (RH) problem, which has a unique solution. Numerically, the RH problem can be treated efficiently by the method of successive conjugation using the fast Fourier transform (FFT). The iteration for the solution of the RH problem converges linearly. The conformal mapping method converges quadratically. The results of some test calculations exemplify the performance of the method.