Convex decompositions in the plane and continuous pair colorings of the irrationals

We address the structure of nonconvex closed subsets of the Euclidean plane. A closed subsetS⊆ℝ² which is not presentable as a countable union of convex sets satisfies the following dichotomy:

Monotone perturbations of the laplacian inL 1(R N )

The semilinear perturbation of Poisson’s equation (E): −Δu+β(u)∋f, where β is a maximal monotone graph inR, has been investigated by Ph. Bénilan, H. Brézis and M. Crandall forf∈L ¹(R ^N ),N≧1, under the assumptions 0∈β(0) ifN≧3 and 0∈β(0) ∩ Int β(R) ifN=1,2. We discuss in this paper the solvability and well-posedness of (E) in terms of any maximal monotone graph β. In particular, if β takes only positive values andN≧3 we prove that no solution exists; ifN=2 we give necessary and sufficient conditions on β andf for (E) to be solvable in a natural sense.     

On the Convergence in Capacity of Rational Approximants

Let {r _n } be a sequence of rational functions deg( r _n ≤ n) that converge rapidly in measure to an analytic function f on an open set in C ^N . We show that {r _n } converges rapidly in capacity to f on its natural domain of definition W _f (which, by a result of Goncar, is an open subset of C ^N ). In particular, for f meromorphic on C ^N and analytic near zero the sequence of Padé approximants {π _n (z, f, λ)} (as defined by Goncar) converges rapidly in capacity to f on C ^N . January 14, 1999. Date revised: October 7, 1999. Date accepted: November 1, 1999.     

Capturing Ridge Functions in High Dimensions from Point Queries

Constructing a good approximation to a function of many variables suffers from the “curse of dimensionality”. Namely, functions on ℝ ^N with smoothness of order s can in general be captured with accuracy at most O(n ^−s/N ) using linear spaces or nonlinear manifolds of dimension n. If N is large and s is not, then n has to be chosen inordinately large for good accuracy. The large value of N often precludes reasonable numerical procedures. On the other hand, there is the common belief that real world problems in high dimensions have as their solution, functions which are more amenable to numerical recovery. This has led to the introduction of models for these functions that do not depend on smoothness alone but also involve some form of variable reduction. In these models it is assumed that, although the function depends on N variables, only a small number of them are significant. Another variant of this principle is that the function lives on a low dimensional manifold. Since the dominant variables (respectively the manifold) are unknown, this leads to new problems of how to organize point queries to capture such functions. The present paper studies where to query the values of a ridge function f(x)=g(a⋅x) when both a∈ℝ ^N and g∈C[0,1] are unknown. We establish estimates on how well f can be approximated using these point queries under the assumptions that g∈C ^s [0,1]. We also study the role of sparsity or compressibility of a in such query problems.     

Rigidity of isometric immersions of higher codimension

Given an isometric immersionf:M ⁿ → ℝ ^N into Euclidean space, we provide sufficient conditions onf so that any 1-regular isometric immersion ofM ⁿ into ℝ ^N+1 is necessarily obtained as a composition off with a local isometric immersion ℝ ^N ⊃U → ℝ ^N+1 . This result has several applications.     

Sard’s theorem for mappings between Fréchet manifolds

We prove an infinite-dimensional version of Sard’s theorem for Fréchet manifolds. Let M (respectively, N) be a bounded Fréchet manifold with compatible metric d _M (respectively, d _N ) modeled on Fréchet spaces E (respectively, F) with standard metrics. Let f : M → N be an MC ^k -Lipschitz–Fredholm map with k > max{Ind f, 0}: Then the set of regular values of f is residual in N.     

Highly nonintegrable functions in the unit ball

In this note we construct the functionf holomorphic in the unit ballB in ℂ ^N such that for every positive-dimensional subspace Π of ℂ ^N ,f|П⋂B is notL ²-integrable. We present also some possible generalizations of this result. Partially supported by the KBN Grant 2 PO3A 060 08.     

A variational problem for pullback metrics

Let (M, g) and (N, h) be Riemannian manifolds without boundary and let f : M → N be a smooth map. Let ||f^*h||{\|f^*h\|} denote the norm of the pullback metric of h by f. In this paper, we consider the functional F(f) = ò_M ||f^*h||² dv_g{{\Phi (f) = \int_M \|f^*h\|^2 dv_g}}. We prove the existence of minimizers of the functional Φ in each 3-homotopy class of maps, where maps f ₁ and f ₂ are 3-homotopic if they are homotopic on the three dimensional skeltons of a triangulation of M. Furthermore, we give a monotonicity formula and a Bochner type formula.     

Obstructions to fibering a manifold

Given a map f : M → N of closed topological manifolds we define torsion obstructions whose vanishing is a necessary condition for f being homotopy equivalent to a projection of a locally trivial fiber bundle. If N = S ¹, these torsion obstructions are identified with the ones due to Farrell (Indiana Univ Math J 21:315–346, 1971/1972).     

On minimax identification of nonparametric autoregressive models

We consider the problem of nonparametric identification for a multi-dimensional functional autoregression y _t = f(y _{t −1}, …,y _t−d ) + e _t on the basis of N observations of y _t . In the case when the unknown nonlinear function f belongs to the Barron class, we propose an estimation algorithm which provides approximations of f with expected L ₂ accuracy O(N ^1/4ln^1/4 N). We also show that this approximation rate cannot be significantly improved. The proposed algorithms are “computationally efficient”– the total number of elementary computations necessary to complete the estimate grows polynomially with N. Received: 23 September 1997 / Revised version: 28 January 1999     

Functions of normal operators under perturbations

In Peller (1980) [27], Peller (1985) [28], Aleksandrov and Peller (2009) [2], Aleksandrov and Peller (2010) [3], and Aleksandrov and Peller (2010) [4] sharp estimates for f(A)−f(B) were obtained for self-adjoint operators A and B and for various classes of functions f on the real line R. In this paper we extend those results to the case of functions of normal operators. We show that if a function f belongs to the Hölder class Λα(R²), 0<α<1, of functions of two variables, and N₁ and N₂ are normal operators, then ‖f(N₁)−f(N₂)‖?const‖fΛα_‖‖N₁−N₂α^‖. We obtain a more general result for functions in the space for an arbitrary modulus of continuity ω. We prove that if f belongs to the Besov class , then it is operator Lipschitz, i.e., . We also study properties of f(N₁)−f(N₂) in the case when f∈Λα(R²) and N₁−N₂ belongs to the Schatten–von Neumann class Sp.     

Approximation of Functions of Few Variables in?High?Dimensions

Let f be a continuous function defined on Ω:=[0,1] ^N which depends on only ℓ coordinate variables, f(x₁,?,x_N)=g(x_i1,?,x_il)f(x_{1},\ldots,x_{N})=g(x_{i_{1}},\ldots,x_{i_{\ell}}). We assume that we are given m and are allowed to ask for the values of f at m points in Ω. If g is in Lip1 and the coordinates i ₁,…,i _ℓ are known to us, then by asking for the values of f at m=L ^ℓ uniformly spaced points, we could recover f to the accuracy |g|_Lip1 L ⁻¹ in the norm of C(Ω). This paper studies whether we can obtain similar results when the coordinates i ₁,…,i _ℓ are not known to us. A prototypical result of this paper is that by asking for C(ℓ)L ^ℓ (log ₂ N) adaptively chosen point values of f, we can recover f in the uniform norm to accuracy |g|_Lip1 L ⁻¹ when g∈Lip1. Similar results are proven for more general smoothness conditions on g. Results are also proven under the assumption that f can be approximated to some tolerance ε (which is not known) by functions of ℓ variables.     

The first negative coefficients of symmetric square <Emphasis Type="Italic">L</Emphasis>-functions

Let n_sym²fn_{\mathrm{sym}^{2}f} be the greatest integer such that l_sym²f(n) 3 0\lambda_{\mathrm{sym}^{2}f}(n)\ge0 for all n < n_sym²fnn,N)=1, where l_sym²f(n)\lambda_{\mathrm{sym}^{2}f}(n) is the nth coefficient of the Dirichlet series representation of the symmetric square L-function L(s,sym² f) associated to a primitive form f of level N and of weight k. In this paper, we establish the subconvexity bound: n_sym²f << (k²N²)^40/113n_{\mathrm{sym}^{2}f}\ll(k^{2}N^{2})^{40/113} where the implied constant is absolute.     

Counting the Faces of Randomly-Projected Hypercubes and Orthants, with Applications

Let A be an n×N real-valued matrix with n<N; we count the number of k-faces f _k (AQ) when Q is either the standard N-dimensional hypercube I ^N or else the positive orthant ℝ₊ ^N . To state results simply, consider a proportional-growth asymptotic, where for fixed δ,ρ in (0,1), we have a sequence of matrices A_n,NnA_{n,N_{n}} and of integers k _n with n/N _n →δ and k _n /n→ρ as n→∞. If each matrix A_n,NnA_{n,N_{n}} has its columns in general position, then f _k (AI ^N )/f _k (I ^N ) tends to zero or one depending on whether ρ>min (0,2−δ ⁻¹) or ρ<min (0,2−δ ⁻¹). Also, if each A_n,NnA_{n,N_{n}} is a random draw from a distribution which is invariant under right multiplication by signed permutations, then f _k (Aℝ₊ ^N )/f _k (ℝ₊ ^N ) tends almost surely to zero or one depending on whether ρ>min (0,2−δ ⁻¹) or ρ<min (0,2−δ ⁻¹). We make a variety of contrasts to related work on projections of the simplex and/or cross-polytope. These geometric face-counting results have implications for signal processing, information theory, inverse problems, and optimization. Indeed, face counting is related to conditions for uniqueness of solutions of underdetermined systems of linear equations. Below, let A be a fixed n×N matrix, n<N, with columns in general position.

The Newton polygon of a product of power series

Summary LetF be a field with a non-trivial valuation υ:F→ℝ∪{+∞}. To any power series in one variable overF one can associate a Newton polygon with respect to this valuation. LetN ₁ andN ₂ be polygons which arise as Newton polygons of power series overF. We determine the set of polygonsN with the property that there exist power seriesf _i with respective Newton polygonN _i ,i=1,2, such that the productf ₁ f ₂ has Newton polygonN. Supported by a grant from the DFG This article was processed by the author using the LATEX style filepljour1 from Springer-Verlag.     

Minimal helix surfaces in <Emphasis Type="Italic">N</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>×ℝ

An immersed surface M in N ⁿ ×ℝ is a helix if its tangent planes make constant angle with ∂ _t . We prove that a minimal helix surface M, of arbitrary codimension is flat. If the codimension is one, it is totally geodesic. If the sectional curvature of N is positive, a minimal helix surfaces in N ⁿ ×ℝ is not necessarily totally geodesic. When the sectional curvature of N is nonpositive, then M is totally geodesic. In particular, minimal helix surfaces in Euclidean n-space are planes. We also investigate the case when M has parallel mean curvature vector: A complete helix surface with parallel mean curvature vector in Euclidean n-space is a plane or a cylinder of revolution. Finally, we use Eikonal f functions to construct locally any helix surface. In particular every minimal one can be constructed taking f with zero Hessian.     

On the central value of symmetric square L-functions

Let S _k (N, χ) be the space of cusp forms of weight k, level N and character χ. For let L(s, sym² f) be the symmetric square L-function and be the Rankin–Selberg square attached to f. For fixed k ≥ 2, N prime, and real primitive χ, asymptotic formulas for the first and second moment of the central value of L(s, sym² f) and over a basis of S _k (N, χ) are given as N → ∞. As an application it is shown that a positive proportion of the central values L(1/2, sym² f) does not vanish. The author was supported by NSERC grant 311664-05.     

Generalization of one Poletskii lemma to classes of space mappings

The paper is devoted to investigations in the field of space mappings. We prove that open discrete mappings f ∈ W ^1,n _loc such that their outer dilatation K _O (x, f) belongs to L ⁿ⁻¹ _loc and the measure of the set B _f of branching points of f is equal to zero have finite length distortion. In other words, the images of almost all curves γ in the domain D under the considered mappings f : D → ℝ ⁿ , n ≥ 2, are locally rectifiable, f possesses the (N)-property with respect to length on γ, and, furthermore, the (N)-property also holds in the inverse direction for liftings of curves. The results obtained generalize the well-known Poletskii lemma proved for quasiregular mappings.     

On the existence of convex decompositions of partially separable functions

The concept of a partially separable functionf developed in [4] is generalized to include all functionsf that can be expressed as a finite sum of element functionsf _i whose Hessians have nontrivial nullspacesN _i , Such functions can be efficiently minimized by the partitioned variable metric methods described in [5], provided that each element functionf _i is convex. If this condition is not satisfied, we attempt toconvexify the given decomposition by shifting quadratic terms among the originalf _i such that the resulting modified element functions are at least locally convex. To avoid tests on the numerical value of the Hessian, we study the totally convex case where all locally convexf with the separability structureN _i ¹ have a convex decomposition. It is shown that total convexity only depends on the associated linear conditions on the Hessian matrix. In the sparse case, when eachN _i is spanned by Cartesian basis vectors, it is shown that a sparsity pattern corresponds to a totally convex structure if and only if it allows a (permuted) LDL^T factorization without fill-in.     

Some results on higher order immersions of manifolds

Letf:M ⁿ →N ^v(n,p)+m be a map. Suppose thatm=n-1 orn<4. We obtain the necessary and sufflcient conditions forf to be homotopic to apth order immersion. Some concretepth order immersion results ofRP ⁿ are also proved. Project supported by the National Natural Science Foundation of China