Related papers: Discrepancies and their means

A class of distortions termed functional Bregman divergences is defined, which includes squared error and relative entropy. A functional Bregman divergence acts on functions or distributions, and generalizes the standard Bregman divergence…

We consider the problem of finding, for a given quadratic measure of non-uniformity of a set of $N$ points (such as $L_2$ star-discrepancy or diaphony), the asymptotic distribution of this discrepancy for truly random points in the limit…

Experimental designs intended to match arbitrary target distributions are typically constructed via a variable transformation of a uniform experimental design. The inverse distribution function is one such transformation. The discrepancy is…

We investigate the observability of a general class of linear dispersive equations on the torus $\mathbb{T}$. We take one line segment or two line segments in space-time region as the observable set. We give the characteristic on the slopes…

Kolmogorov's invariant torus theorem is proved using a simple fixed point theorem.

Fixed a continuous kernel K on the $d$-dimensional torus, we consider a generalization of the univariate $sk$-spline to the torus, associated with the kernel K. It is proved an estimate which provides the rate of convergence of a given…

In the geometric theory of defects, media with a spin structure, for example, ferromagnet, is considered as a manifold with given Riemann--Cartan geometry. We consider the case with the Euclidean metric corresponding to the absence of…

The law of the iterated logarithm for discrepancies of geometric progressions with small ratios is proved.

An important tool to quantify the likeness of two probability measures are f-divergences, which have seen widespread application in statistics and information theory. An example is the total variation, which plays an exceptional role among…

In the present paper we prove several results concerning the existence of low-discrepancy point sets with respect to an arbitrary non-uniform measure $\mu$ on the $d$-dimensional unit cube. We improve a theorem of Beck, by showing that for…

The formal term-by-term differentiation with respect to parameters is demonstrated to be legitimate for the Mittag-Leffler type functions. The justification of differentiation formulas is made by using the concept of the uniform…

In this work we prove the Stepanov differentiation theorem for multiple-valued functions. This theorem is proved in the wide generality of metric-space-multiple-valued functions without relying on a Lipschitz extension result. General…

We look at the values of two Dirichlet $L$-functions at the Riemann zeros (or a horizontal shift of them). Off the critical line we show that for a positive proportion of these points the pairs of values of the two $L$-functions are…

We show that the Fourier transform on the Jacobian of a curve interchanges "$\delta$ functions" at the curve and the theta divisor. The Torelli theorem is an immediate consequence.

We investigate the joint distribution of $L$-functions on the line $ \sigma= \frac12 + \frac1{G(T)}$ and $ t \in [ T, 2T]$, where $ \log \log T \leq G(T) \leq \frac{ \log T}{ ( \log \log T)^2 } $. We obtain an upper bound on the discrepancy…

We establish a link between the phenomenon of Taylor dispersion and the theory of empirical distributions. Using this connection, we derive, upon applying the theory of large deviations, an alternative and much more precise description of…

In this note we provide a quick proof of the Sklar's Theorem on the existence of copulas by using the generalized inverse functions as in the one dimensional case, but a little more sophisticated.

In this paper we prove a criterion of convergence in distribution in Skorokhod space. We apply this criterion to some special Levy processes and obtain almost-sure versions of limit theorems for these processes.

We prove that functions defined on a lattice in a finite dimensional torus with bounded finite differences can be smoothly extended to the whole torus, and relate the bounds on the extension's derivatives with bounds on the original…

We prove several results concerning the discrepancy, tested on balls in the $d$-dimensional torus $\mathbb{T}^{d}$, between absolutely continuous measures and finite atomic measures.