Harmonic maps with fixed singular sets

Suppose Ω is a smooth domain in R^m,N is a compact smooth Riemannian manifold, andZ is a fixed compact subset of Ω having finite (m − 3)-dimensional Minkowski content (e.g.,Z ism − 3 rectifiable). We consider various spaces of harmonic mapsu: Ω →N that have a singular setZ and controlled behavior nearZ. We study the structure of such spacesH and questions of existence, uniqueness, stability, and minimality under perturbation. In caseZ = 0,H is a Banach manifold locally diffeomorphic to a submanifold of the product of the boundary data space with a finite-dimensional space of Jacobi fields with controlled singular behavior. In this smooth case, the projection ofu εH tou |ϖΩ is Fredholm of index 0. R. H.’s research partially supported by the National Science Foundation.     

Nonexistence Results of Minimal Immersions

Let M be a Riemannian m-dimensional manifold with m ≥ 3, endowed with non zero parallel p-form. We prove that there is no minimal isometric immersions of M in a Riemannian manifold N with constant strictly negative sectional curvature. Next we show that, under the conform flatness of the manifold N and some assumptions on the Ricci curvature of N, there is no α-pluriharmonic isometric immersion.     

Intrinsic normalizations of a nonholonomic hypersurface with <Emphasis Type="Italic">m</Emphasis>-dimensional generators in affine space <Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

We consider some intrinsic normalizations of a nonholonomic hypersurface with m-dimensional generators in the n-dimensional affine spaced. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 532–562, October–December, 2007.     

A bilinear algorithm for sparse representations

We consider the following sparse representation problem: represent a given matrix X∈ℝ ^m×N as a multiplication X=AS of two matrices A∈ℝ ^m×n (m≤n<N) and S∈ℝ ^n×N , under requirements that all m×m submatrices of A are nonsingular, and S is sparse in sense that each column of S has at least n−m+1 zero elements. It is known that under some mild additional assumptions, such representation is unique, up to scaling and permutation of the rows of S. We show that finding A (which is the most difficult part of such representation) can be reduced to a hyperplane clustering problem. We present a bilinear algorithm for such clustering, which is robust to outliers. A computer simulation example is presented showing the robustness of our algorithm.     

Geometry of distributions of flags on Grassmann manifolds of a projective space. I

We consider intrinsic normalizations of distributions of flags of type (m, m+1) on the Grassmann manifold ofm-planes of a (2m+2)-dimensional projective space. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 214–227, April–June, 2000. Translated by V. Mackevičius     

Geometry of distributions of flags on Grassmann manifolds of a projective space. II

We consider intrinsic normalizations of distributions of flags of type (m, m+1) on the Grassmann manifold ofm-planes of a (2m+2)-dimensional projective space. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 335–349, July–September, 2000. Translated by V. Mackevičius     

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.     

Weak interaction limit for nuclear matter and the time-dependent Hartree-Fock equation

We consider an effective model of nuclear matter including spin and isospin degrees of freedom, described by an N-body Hamiltonian with suitably renormalized two-body and three-body interaction potentials. We show that the corresponding mean-field theory (the time-dependent Hartree-Fock approximation) is “exact” as N tends to infinity.     

Subspaces ofl p N of small codimension

In this paper the structure of subspaces and quotients ofl _p ^N of dimension very close toN is studied, for 1≤p≤∞. In particular, the maximal dimensionk=k(p, m, N) so that an arbitrarym-dimensional subspaceX ofl _p ^N contains a good copy ofl _p ^k , is investigated form=N−o(N). In several cases the obtained results are sharp.     

Sets of Periods for Expanding Maps on Flat Manifolds

 It is proven that the sets of periods for expanding maps on n-dimensional flat manifolds are uniformly cofinite, i.e. there is a positive integer m ₀, which depends only on n, such that for any integer , for any n-dimensional flat manifold ℳ and for any expanding map F on ℳ, there exists a periodic point of F whose least period is exactly m.     

Large deviations for the symmetric simple exclusion process in dimensions d≥ 3

We consider symmetric simple exclusion processes with L=&ρmacr;N ^d particles in a periodic d-dimensional lattice of width N. We perform the diffusive hydrodynamic scaling of space and time. The initial condition is arbitrary and is typically far away form equilibrium. It specifies in the scaling limit a density profile on the d-dimensional torus. We are interested in the large deviations of the empirical process, N ^{− d} [∑ ^L ₁δ _{xi (·)}] as random variables taking values in the space of measures on D[0.1]. We prove a large deviation principle, with a rate function that is more or less universal, involving explicity besides the initial profile, only such canonical objects as bulk and self diffusion coefficients. Received: 7 September 1997 / Revised version: 15 May 1998     

Determinants of the Hypergeometric Period Matrices of an Arrangement and Its Dual

We fix three natural numbers k, n, N, such that n+k+1 = N, and introduce the notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement of N hyperplanes in a k-dimensional affine space, the other is an arrangement of N hyperplanes in an n-dimensional affine space. We assign weights α ₁, . . . , α _N to the hyperplanes of the arrangements and for each of the arrangements consider the associated period matrices. The first is a matrix of k-dimensional hypergeometric integrals and the second is a matrix of n-dimensional hypergeometric integrals. The size of each matrix is equal to the number of bounded domains of the corresponding arrangement. We show that the dual arrangements have the same number of bounded domains and the product of the determinants of the period matrices is equal to an alternating product of certain values of Euler’s gamma function multiplied by a product of exponentials of the weights. Supported in part by NSF grant DMS-0244579.     

Distance-Dominating Cycles in Quasi Claw-Free Graphs

We say that a graph G is quasi claw-free if every pair (a ₁, a ₂) of vertices at distance 2 satisfies {u∈N (a ₁)∩N (a ₂) | N[u]⊆N[a ₁]∪N [a ₂]}≠∅. A cycle C is m-dominating if every vertex of G is of distance at most m from C. We prove that if G is a κ-connected (κ≥2) quasi claw-free graph then either G has an m-dominating cycle or G has a set of at least κ+1 vertices such that the distance between every pair of them is at least 2m+3. Received: June 12, 1996 Revised: November 9, 1998     

Vertex Numbers of Weighted Faces in Poisson Hyperplane Mosaics

In the random mosaic generated by a stationary Poisson hyperplane process in ℝ ^d , we consider the typical k-face weighted by the j-dimensional volume of the j-skeleton (0≤j≤k≤d). We prove sharp lower and upper bounds for the expected number of its vertices.     

A splitting theorem for the weighted measure

Let M be an n-dimensional complete noncompact Riemannian manifold, h be a smooth function on M and dμ = e ^h dV be the weighted measure. In this article, we prove that when the spectrum of the weighted Laplacian \triangle_m{\triangle_{\mu}} has a positive lower bound λ₁(M) > 0 and the m(m > n)-dimensional Bakry-émery curvature is bounded from below by -\fracm-1m-2l₁(M){-\frac{m-1}{m-2}\lambda_1(M)}, then M splits isometrically as R × N whenever it has two ends with infinite weighted volume, here N is an (n − 1)-dimensional compact manifold.     

Sets of Periods for Expanding Maps on Flat Manifolds

 It is proven that the sets of periods for expanding maps on n-dimensional flat manifolds are uniformly cofinite, i.e. there is a positive integer m ₀, which depends only on n, such that for any integer , for any n-dimensional flat manifold ℳ and for any expanding map F on ℳ, there exists a periodic point of F whose least period is exactly m. (Received 10 April 1998; in revised form 20 January 1999)     

Submanifolds with Parallel Second Fundamental Form Studied via the Gauss Map

For an arbitrary n-dimensional Riemannian manifold N and an integer m ∈ {1,…,n−1} a covariant derivative on the Grassmann bundle ^ := G_m(T N) is introduced which has the property that an m-dimensional submanifold M ⊂ N has parallel second fundamental form if and only if its Gauss map M → ^ is affine. (For N Rⁿ this result was already obtained by J. Vilms in 1972.) By means of this relation a generalization of Cartan's theorem on the total geodesy of a geodesic umbrella can be derived: Suppose, initial data (p,W,b) prescribing a tangent space W ∈ G_m(T_pN) and a second fundamental form b at p ∈ N are given; for these data we construct an m-dimensional ‘umbrella’ M = M(p,W,b) ⊂ N the rays of which are helical arcs of N; moreover, we present tensorial conditions (not involving ) which guarantee that the umbrella M has parallel second fundamental form. These conditions are as well necessary, and locally every submanifold with parallel second fundamental form can be obtained in this way. Mathematics Subject Classifications (2000): 53B25, 53B20, 53B21.     

On the global theory of projective mappings

We consider the theory of constant rank projective mappings of compact Riemannian manifolds from the global point of view. We study projective immersions and submersions. As an example of the results, letf:(M, g) → (N, g′) be a projective submersion of anm-dimensional Riemannian manifold (M, g) onto an (m−1)-dimensional Riemannian manifold (N, g′). Then (M, g) is locally the Riemannian product of the sheets of two integrable distributions Kerf _* and (Kerf _*)^⊥ whenever (M, g) is one of the two following types: (a) a complete manifold with Ric ≥ 0; (b) a compact oriented manifold with Ric ≤ 0. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 111–118, July, 1995. This work was partially supported by the Russian Foundation for Basic Research grant No. 94-01-0195.     

The element number of the convex regular polytopes

Let Σ be a set of n-dimensional polytopes. A set Ω of n-dimensional polytopes is said to be an element set for Σ if each polytope in Σ is the union of a finite number of polytopes in Ω identified along (n − 1)-dimensional faces. In this paper, we consider the n-dimensional polytopes in general, and extend the notion of element sets to higher dimensions. In particular, we will show that in the 4-space, the element number of the six convex regular polychora is at least 2, and in the n-space (n ≥ 5), the element number is 3, unless n + 1 is a square number.     

Diffusion approximation for a heavily loaded multi-user wireless communication system with cooperation

A cellular wireless communication system in which data is transmitted to multiple users over a common channel is considered. When the base stations in this system can cooperate with each other, the link from the base stations to the users can be considered a multi-user multiple-input multiple-output (MIMO) downlink system. For such a system, it is known from information theory that the total rate of transmission can be enhanced by cooperation. The channel is assumed to be fixed for all transmissions over the period of interest and the ratio of anticipated average arrival rates for the users, also known as the relative traffic rate, is fixed. A packet-based model is considered where data for each user is queued at the transmit end. We consider a simple policy which, under Markovian assumptions, is known to be throughput-optimal for this coupled queueing system. Since an exact expression for the performance of this policy is not available, as a measure of performance, we establish a heavy traffic diffusion approximation. To arrive at this diffusion approximation, we use two key properties of the policy; we posit the first property as a reasonable manifestation of cooperation, and the second property follows from coordinate convexity of the capacity region. The diffusion process is a semimartingale reflecting Brownian motion (SRBM) living in the positive orthant of N-dimensional space (where N is the number of users). This SRBM has one direction of reflection associated with each of the 2 ^N −1 boundary faces, but show that, in fact, only those directions associated with the (N−1)-dimensional boundary faces matter for the heavy traffic limit. The latter is likely of independent theoretical interest.