A note on a one-sided Jacobi algorithm

Summary We propose a one-sided or implicit variant of the Jacobi diagonalization algorithm for positive definite matrices. The variant is based on a previous Cholesky decomposition and currently uses essentially one square array which, on output, contains the matrix of eigenvectors thus reaching the storage economy of the symmetric QL algorithm. The current array is accessed only columnwise which makes the algorithm attractive for various parallelized and/or vectorized implementations. Even on a serial computer our algorithm shows improved efficiency, in particular if the Cholesky step is made with diagonal pivoting. On matrices of ordern=25–200 our algorithm is about 2–3.5 times slower than QL thus being almost on the halfway between the standard Jacobi and QL algorithms. The previous Cholesky decomposition can be performed with higher precision without extra time or storage costs thus offering considerable gains in accuracy with highly conditioned input matrices.This work was partly done during the first author's visit to the Department of Mathematics, The University of Tennessee-Knoxville, while participating in the Special Year on Numerical Linear Algebra sponsored by the UTK Departments of Computer Science and Mathematics, and the ORNL Mathematical Sciences Section, Engineering Physics and Mathematics Division as well as during a second author visit to the Fernuniversität Hagen. Both authors gratefully acknowledge the support of the respective institutions     

Kolmogorov-Smirnov isometries of the space of generalized distribution functions

In this paper we describe the structure of surjective isometries of the space of all generalized probability distribution functions on ? with respect to the Kolmogorov-Smirnov metric.     

Kolmogorov-Smirnov isometries and affine automorphisms of spaces of distribution functions

In this paper we describe the structure of surjective isometries of the spaces of all absolutely continuous, singular, or discrete probability distribution functions on R equipped with the Kolmogorov-Smirnov metric. We also study the structure of affine automorphisms of the space of all distribution functions.     

A note on the distribution of some additive functions

Let f denote an additive arithmetical function with continuous limiting distribution F on the integers. Then f also has a limiting distribution G on shifted primes. Under some growth conditions on the values of f at primes, we provide optimal lower bounds for the modulus of continuity of F and G, at all points from a specified infinite set.     

On a Kolmogorov-Smirnov type aligned test

For testing the hypothesis that two (symmetric) distributions may differ only in locations, a Kolmogorov-Smirnov type test based on the aligned observations is considered and its properties are studied.     

Goodness of fit test for small diffusions by discrete time observations

We consider a nonparametric goodness of fit test problem for the drift coefficient of one-dimensional small diffusions. Our test is based on discrete time observation of the processes, and the diffusion coefficient is a nuisance function which is “estimated” in some sense in our testing procedure. We prove that the limit distribution of our test is the supremum of the standard Brownian motion, and thus our test is asymptotically distribution free. We also show that our test is consistent under any fixed alternative.     

A note on value distribution of difference polynomials of meromorphic functions

In this paper, we investigate the value distribution of the difference counterpart Δf(z)-afn(z)of f^′(z) -afn(z)$\Delta f(z)-af{(z)}^n\text{o}\text{f}\hspace{0.17em}{f}^{\prime }(z)\hspace{0.17em}-af{(z)}^n$ and obtain an almost direct difference analogue of result of Hayman.     

A note on one-sided interval edge colorings of bipartite graphs

For a bipartite graph G with parts X and Y, an X-interval coloring is a proper edge coloring of G by integers such that the colors on the edges incident to any vertex in X form an interval. Denote by ${\chi }_{int}^{\prime }(G,X)$ the minimum k such that G has an X-interval coloring with k colors. Casselgren and Toft (2016) [12] asked whether there is a polynomial $P(\mathrm{\Delta })$ such that if G has maximum degree at most Δ, then ${\chi }_{int}^{\prime }(G,X)\le P(\mathrm{\Delta })$. In this short note, we answer this question in the affirmative; in fact, we prove that a cubic polynomial suffices. We also deduce some improved upper bounds on ${\chi }_{int}^{\prime }(G,X)$ for bipartite graphs with small maximum degree.     

Two-sample Kolmogorov-Smirnov test using a Bayesian nonparametric approach

In this paper, a Bayesian nonparametric approach to the two-sample problem is proposed. Given two samples \(\text{X} = {X_1}, \ldots ,{X_{m1}}\;\mathop {\text~}\limits^{i.i.d.} F\) and \(Y = {Y_1}, \ldots ,{Y_{{m_2}}}\mathop {\text~}\limits^{i.i.d.} G\), with F and G being unknown continuous cumulative distribution functions, we wish to test the null hypothesis H ₀: F = G. The method is based on computing the Kolmogorov distance between two posterior Dirichlet processes and comparing the results with a reference distance. The parameters of the Dirichlet processes are selected so that any discrepancy between the posterior distance and the reference distance is related to the difference between the two samples. Relevant theoretical properties of the procedure are also developed. Through simulated examples, the approach is compared to the frequentist Kolmogorov–Smirnov test and a Bayesian nonparametric test in which it demonstrates excellent performance.     

A note on the weighted norm inequality for the one-sided maximal operator

Let be the one-sided maximal function. In this note we obtain some necessary and sufficient conditions in order that the weighted weak type inequality holds for . Meanwhile, some necessary or sufficient conditions for the weighted inequality for are given.

     

Isometries of the space of distribution functions with respect to the Kolmogorov-Smirnov metric

In this paper we describe the general forms of surjective isometries of the space of all probability distribution functions on R with respect to the Kolmogorov-Smirnov metric.     

Bounded probability properties of Kolmogorov-Smirnov and similar statistics for discrete data

Summary A somewhat general class of situations, that include Kolmogorov-Smirnov type results as special cases, is considered. These situations, which are described in the following sections, are required to have uniquely determined probability properties when the sample values used are from continuous populations of any nature. If the populations sampled are discrete, however, these probability values are not uniquely determined. This paper shows that the values for the continuous case represent bounds for the values that occur in any discrete case. The method used to show that these bound relations hold consists in noting that any discrete data situation can be interpreted as a situation involving the grouping of continuous data. Then bound relationships are established between the values of probabilities for the grouped data situations and the corresponding ungrouped data situations, which are the situations considered for the case of the continuous data. These bounds on probabilities for discrete data cases should be useful for practical applications. In practice, all data are discrete (due to limitations in measurement accuracy).     

A note on discrete heat conduction

If the longitudinal line method is applied to the Cauchy problem u_t = u_xx, u(0, x) = u₀(x) with a bounded function u₀, one is led to a linear initial value problem v￠(t)=A v(t), v(0)=wv'(t)=A v(t),\, v(0)=w in l^￥ (\Bbb Z)l^\infty (\Bbb Z). Using Banach limit techniques we study the asymptotic behaviour of the solutions of these problems as t tends to infinity.     

A note on discrete Borg-type theorems

We consider the discrete versions of the well-known Borg’s theorem and use simple linear algebraic techniques to obtain new versions of the discrete Borg-type theorems. To be precise, we prove that the periodic potential of a discrete Schrödinger operator is almost a constant if and only if the possible spectral gaps of the operator are of small width. This result is further extended to more general settings and the connection to the well-known Ten Martini problem is also discussed.     

A note on some joint distribution functions involving the time of ruin

In a recent paper, Willmot (2015) derived an expression for the joint distribution function of the time of ruin and the deficit at ruin in the classical risk model. We show how his approach can be applied to obtain a simpler expression, and by interpreting this expression by probabilistic reasoning we obtain solutions for more general risk models. We also discuss how some of Willmot’s results relate to existing literature on the probability and severity of ruin.     

A note on maximum-type functions

The paper studies the differential properties of functions of the form

$g(x) = \mathop {\max }\limits_{y \in Y} f(x,y),$

where x ∈ X (X is an open convex set from ? ^m ) and y ∈ Y (Y is a compact from ? ⁿ ). Apart from the conventional smoothness conditions imposed on f(x, y), the condition of the concavity of g(x) on X is also imposed.

The differentiability of function g(x) on X is proved.The results of the study facilitate the derivation of the conditions ensuring the sufficiency of Pontryagin’s maximum principle.     

A note on polyharmonic functions

In this note we prove the following theorem:Suppose 0<p<∞ and α>−1. Then there is a constant C=C(p,m,n,α) such that