Smoothing of bump functions

Let X be a separable Banach space with a Schauder basis, admitting a continuous bump which depends locally on finitely many coordinates. Then X admits also a C^∞-smooth bump which depends locally on finitely many coordinates.     

Smoothing noisy data with spline functions

Summary A procedure for calculating the trace of the influence matrix associated with a polynomial smoothing spline of degree2m–1 fitted ton distinct, not necessarily equally spaced or uniformly weighted, data points is presented. The procedure requires orderm ² n operations and therefore permits efficient orderm ² n calculation of statistics associated with a polynomial smoothing spline, including the generalized cross validation. The method is a significant improvement over an existing method which requires ordern ³ operations.     

Smoothing empirical functions by lp programming

We give three models for smoothing empirical functions. The first model is the well known graduating model given by Whittaker [7], the second and the third model are new.New iterative procedures are presented for solving these models. These iterative procedures are based on the equilibrium conditions of l_p programming, so our solution procedure for solving Whittaker's model is also new.     

Smoothing by spline functions. II

Summary In this paper we generalize the results of [4] and modify the algorithm presented there to obtain a better rate of convergence.     

Smoothing noisy data with spline functions

It is shown how to choose the smoothing parameter when a smoothing periodic spline of degree 2m?1 is used to reconstruct a smooth periodic curve from noisy ordinate data. The noise is assumed “white”, and the true curve is assumed to be in the Sobolev spaceW ₂ ^(2m) of periodic functions with absolutely continuousv-th derivative,v=0, 1, ..., 2m?1 and square integrable 2m-th derivative. The criteria is minimum expected square error, averaged over the data points. The dependency of the optimum smoothing parameter on the sample size, the noise variance, and the smoothness of the true curve is found explicitly.     

Smoothing by L-spline functions of many variables

We investigate the problem of the smoothing of experimental data by cell-like L-spline functions of many variables from the point of view of the theory of such functions proposed by the author. Given values of a function and its derivatives up to some order are smoothed on a rectangular network of nodes. Existence and uniqueness of the solution are proved and equations are derived.     

Smoothing splines using compactly supported, positive definite, radial basis functions

In this paper, we develop a fast algorithm for a smoothing spline estimator in multivariate regression. To accomplish this, we employ general concepts associated with roughness penalty methods in conjunction with the theory of radial basis functions and reproducing kernel Hilbert spaces. It is shown that through the use of compactly supported radial basis functions it becomes possible to recover the band structured matrix feature of univariate spline smoothing and thereby obtain a fast computational algorithm. Given n data points in R ², the new algorithm has complexity O(n ²) compared to O(n ³), the order for the thin plate multivariate smoothing splines.     

Schur complements and statistics

In this paper we discuss various properties of matrices of the type

$\text{S}\text{=}\text{H}\text{?}\text{GE}{}^{\mathrm{?1}}\text{F}$

, which we call the Schur complement of E in

$\text{A}\text{=}\text{E}\text{F}\text{G}\text{H}$

The matrix E is assumed to be nonsingular. When E is singular or rectangular we consider the generalized Schur complements $\text{S}\text{=}\text{H}\text{?}\text{GE}{}^{\mathrm{?}}\text{F}$, where E^? is a generalized inverse of E. A comprehensive account of results pertaining to the determinant, the rank, the inverse and generalized inverses of partitioned matrices, and the inertia of a matrix is given both for Schur complements and for generalized Schur complements. We survey the known results in a historical perspective and obtain several extensions. Numerous applications in numerical analysis and statistics are included. The paper ends with an exhaustive bibliography of books and articles related to Schur complements.     

Stable cohomologies of the complements of the discriminants of deformations of singularities of smooth functions

Complements to discriminants of singularities of smooth functions are far generalizations of the classifying spaces of Artin and Brieskorn braid groups. A group of stable cohomologies (i.e., cohomologies preserved under adjacency of singularities) is described for these spaces. A relationship between these cohomologies and Gauss-Manin connectivity of singularities is indicated. A cellular realization of cohomologies of symmetric groups with coefficients in Z₂ is described.Translated from Itogi Nauki i Tekhniki, Seriya Sovermennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 33, pp. 3–29, 1988.     

Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices

In this paper, we prove that the diagonal-Schur complement of a strictly doubly diagonally dominant matrix is strictly doubly diagonally dominant matrix. The same holds for the diagonal-Schur complement of a strictly generalized doubly diagonally dominant matrix and a nonsingular H-matrix. We point out that under certain assumptions, the diagonal-Schur complement of a strictly doubly (doubly product) γ-diagonally dominant matrix is also strictly doubly (doubly product) γ-diagonally dominant. Further, we provide the distribution of the real parts of eigenvalues of a diagonal-Schur complement of H-matrix. We also show that the Schur complement of a γ-diagonally dominant matrix is not always γ-diagonally dominant by a numerical example, and then obtain a sufficient condition to ensure that the Schur complement of a γ-diagonally dominant matrix is γ-diagonally dominant.     

Sets, complements and boundaries

The relations among a set, its complement, and its boundary are examined constructively. A crucial tool is a theorem that allows the construction of a point where a segment comes close to the boundary of a set in a Banach space. Brouwerian examples show that many of the results are the best possible.     

Pseudo-Schur complements and their properties

The notion of Schur complement of a partitioned matrix with a square nonsingular block is well known and it has many applications in various branches of mathematics. When the block is rectangular or singular, pseudo-Schur complements can be defined and studied. In particular, they satisfy an extension of the quotient property for Schur complements. A new proof of this property is given in this paper and various related topics are discussed.     

Dominating the complements of bounded tolerance graphs and the complements of trapezoid graphs

We investigate the complexity of several domination problems on the complements of bounded tolerance graphs and the complements of trapezoid graphs. We describe an O(n² log⁵ n) time and O(n²) space algorithm to solve the domination problem on the complement of a bounded tolerance graph, given a square embedding of that graph. We also prove that domination, connected domination and total domination are all NP-complete on co-trapezoid graphs.     

Star complements and exceptional graphs

Let G be a finite graph of order n with an eigenvalue μ of multiplicity k. (Thus the μ-eigenspace of a (0,1)-adjacency matrix of G has dimension k.) A star complement for μ in G is an induced subgraph G-X of G such that |X|=k and G-X does not have μ as an eigenvalue. An exceptional graph is a connected graph, other than a generalized line graph, whose eigenvalues lie in [-2,∞). We establish some properties of star complements, and of eigenvectors, of exceptional graphs with least eigenvalue −2.