Efficient algorithms for robust generalized cross-validation spline smoothing

Generalized cross-validation (GCV) is a widely used parameter selection criterion for spline smoothing, but it can give poor results if the sample size n is not sufficiently large. An effective way to overcome this is to use the more stable criterion called robust GCV (RGCV). The main computational effort for the evaluation of the GCV score is the trace of the smoothing matrix, , while the RGCV score requires both and . Since 1985, there has been an efficient O(n) algorithm to compute . This paper develops two pairs of new O(n) algorithms to compute and , which allow the RGCV score to be calculated efficiently. The algorithms involve the differentiation of certain matrix functionals using banded Cholesky decomposition.     

Remarks on Hardy spaces defined by non-smooth approximate identity

We study in this paper some relations between Hardy spaces which are defined by non-smooth approximate identity ?(x), and the end-point Triebel-Lizorkin spaces (1?q?∞). First, we prove that for compact ? which satisfies a slightly weaker condition than Fefferman and Stein's condition. Then we prove that non-trivial Hardy space defined by approximate identity ? must contain Besov space . Thirdly, we construct certain functions and a function such that Daubechies wavelet function but .     

Markov interlacing property for exponential polynomials

Let Un be an extended Tchebycheff system on the real line. Given a point , where x₁<?<xn, we denote by the polynomial from Un, which has zeros x₁,…,xn. (It is uniquely determined up to multiplication by a constant.) The system Un has the Markov interlacing property (M) if the assumption that and interlace implies that the zeros of and interlace strictly, unless . We formulate a general condition which ensures the validity of the property (M) for polynomials from Un. We also prove that the condition is satisfied for some known systems, including exponential polynomials and . As a corollary we obtain that property (M) holds true for Müntz polynomials , too.     

Singular-value-like decomposition for complex matrix triples

The classical singular value decomposition for a matrix A∈C^m×n is a canonical form for A that also displays the eigenvalues of the Hermitian matrices AA^∗ and A^∗A. In this paper, we develop a corresponding decomposition for A that provides the Jordan canonical forms for the complex symmetric matrices and . More generally, we consider the matrix triple , where are invertible and either complex symmetric or complex skew-symmetric, and we provide a canonical form under transformations of the form , where X,Y are nonsingular.     

A priori estimate for a family of semi-linear elliptic equations with critical nonlinearity

We consider positive solutions of on B₁ (n?5) where μ and K>0 are smooth functions on B₁. If K is very sub-harmonic at each critical point of K in B2/3 and the maximum of u in is comparable to its maximum over , then all positive solutions are uniformly bounded on . As an application, a priori estimate for solutions of equations defined on Sn is derived.     

Strong convergence of the empirical distribution of eigenvalues of sample covariance matrices with a perturbation matrix

Let , where is a random symmetric matrix, a random symmetric matrix, and with being independent real random variables. Suppose that , and are independent. It is proved that the empirical spectral distribution of the eigenvalues of random symmetric matrices converges almost surely to a non-random distribution.     

Compactifications of the homeomorphism group of a graph

Let Γ be a countable locally finite graph and let H(Γ) and H₊(Γ) denote the homeomorphism group of Γ with the compact-open topology and its identity component. These groups can be embedded into the space of all closed sets of Γ×Γ with the Fell topology, which is compact. Taking closure, we have natural compactifications and . In this paper, we completely determine the topological type of the pair and give a necessary and sufficient condition for this pair to be a (Q,s)-manifold. The pair is also considered for simple examples, and in particular, we find that has homotopy type of RP³. In this investigation we point out a certain inaccuracy in Sakai-Uehara's preceding results on for finite graphs Γ.     

Fonction asymptotique de Samuel des sections hyperplanes et multiplicité

Let (A,m_A,k) be a local noetherian ring and I an m_A-primary ideal. The asymptotic Samuel function (with respect to I) : A?R∪{+∞} is defined by , ∀x∈A. Similarly, one defines, for another ideal J, as the minimum of as x varies in J. Of special interest is the rational number . We study the behavior of the asymptotic Samuel function (with respect to I) when passing to hyperplane sections of A as one does for the theory of mixed multiplicities.     

On Γ-convergence of pairs of dual functionals

The paper considers a slightly modified notion of the Γ-convergence of convex functionals in uniformly convex Banach spaces and establishes that under standard coercitivity and growth conditions the Γ-convergence of a sequence of functionals {Fj} to implies that the corresponding sequence of dual functionals converges in an analogous sense to the dual to functional .     

On the Ricci and Einstein equations on the pseudo-euclidean and hyperbolic spaces

We consider tensors T=fg on the pseudo-euclidean space Rn and on the hyperbolic space Hn, where n?3, g is the standard metric and f is a differentiable function. For such tensors, we consider, in both spaces, the problems of existence of a Riemannian metric , conformal to g, such that , and the existence of such a metric which satisfies , where is the scalar curvature of . We find the restrictions on the Ricci candidate for solvability and we construct the solutions when they exist. We show that these metrics are unique up to homothety, we characterize those globally defined and we determine the singularities for those which are not globally defined. None of the non-homothetic metrics , defined on Rn or Hn, are complete. As a consequence of these results, we get positive solutions for the equation , where g is the pseudo-euclidean metric.     

Several remarks on dimensions modulo ANR-compacta

We investigate a dimension function L-dim (L is a class of ANR-compacta). Main results are as follows.Let L be an ANR-compactum.(1) If L*L is not contractible, then for every n?0 there is a cube Im with .(2) If L is simply connected and f:X→Y is an acyclic mapping from a finite-dimensional compact Hausdorff space X onto a finite-dimensional space Y, then .(3) If L is simply connected and L*L is not contractible, then for every n?2 there exists a compact Hausdorff space such that , and for an arbitrary closed set either or .     

Kripke semantics for provability logic GLP

A well-known polymodal provability logic due to Japaridze is complete w.r.t. the arithmetical semantics where modalities correspond to reflection principles of restricted logical complexity in arithmetic. This system plays an important role in some recent applications of provability algebras in proof theory. However, an obstacle in the study of is that it is incomplete w.r.t. any class of Kripke frames. In this paper we provide a complete Kripke semantics for . First, we isolate a certain subsystem of that is sound and complete w.r.t. a nice class of finite frames. Second, appropriate models for are defined as the limits of chains of finite expansions of models for . The techniques involves unions of n-elementary chains and inverse limits of Kripke models. All the results are obtained by purely modal-logical methods formalizable in elementary arithmetic.     

Open selections on smooth fans

Given a dendroid X, an open selection is an open map such that s(A)∈A for every A∈C(X). We show that a smooth fan X admits an open selection if and only if X is locally connected.     

Solution of a nonsymmetric algebraic Riccati equation from a two-dimensional transport model

For the steady-state solution of an integral-differential equation from a two-dimensional model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B^--XF^--F⁺X+XB⁺X=0, where , and with a nonnegative matrix P, positive diagonal matrices D^±, and nonnegative parameters f, and . We prove the existence of the minimal nonnegative solution X^∗ under the physically reasonable assumption , and study its numerical computation by fixed-point iteration, Newton’s method and doubling. We shall also study several special cases; e.g. when and P is low-ranked, then is low-ranked and can be computed using more efficient iterative processes in U and V. Numerical examples will be given to illustrate our theoretical results.     

On spaces which are D, linearly D and transitively D

In this note, we comment on D-spaces, linearly D-spaces and transitively D-spaces. We show that every meta-Lindelöf space is transitively D. If X is a weak -refinable TD-scattered space, then X is transitively D, where TD is the class of all transitively D-spaces. If X is a weak -refinable -scattered space, then X is a D-space, where is the class of all D-spaces, and hence every weak -refinable (or submetacompact) scattered space is a D-space. This gives a positive answer to a question mentioned by Martínez and Soukup. In the last part of this note, we show that if X is a weak -refinable space then X is linearly D.     

Large energy entire solutions for the Yamabe equation

We consider the Yamabe equation in Rn, n?3. Let k?1 and . For all large k we find a solution of the form , where , for n?4, for n=3 and o(1)→0 uniformly as k→+∞.     

Lifting fixed points of completely positive mappings

Let 1?n?∞, and let be a row contraction on some Hilbert space H. Let F(T) be the space of all X∈B(H) such that . We show that, if non-zero, this space is completely isometric to the commutant of the Cuntz part of the minimal isometric dilation of .     

Invariant algebraic curves for the cubic Liénard system with linear damping

We show that the system , with f,g polynomials of degree 1 and 3 respectively cannot have simultaneously an algebraic invariant curve and a limit cycle.