K e y w o r d s C a l c u l u s of variations, Generalized conjugate points, Nonsingular extremals, SturmLiouville boundary-value problems. 1. I N T R O D U C T I O N Suppose we are g iven an in te rva l T := [t0,Q] in R , two poin ts ~0, ~1 in R n, and a func t ion L m a p p i n g T x R n x R ~ to R . Le t X be the space of p iecewise -C 1 func t ions m a p p i n g T to R ~, let X~ be the set ...

This document describes an infra-structure provided by the R package performanceEstimation that allows to estimate the predictive performance of different approaches (workflows) to predictive tasks. The infra-structure is generic in the sense that it can be used to estimate the values of any performance metrics, for any workflow on different predictive tasks, namely, classification, regression ...

is not trivial to prove. In the one-dimensional case, the triangle inequality is an inequality on absolute values, and can be proven case-by-case. In R, it is best to use the following set-up. The usual inner product (or dot-product) on R is x · y = 〈x, y〉 = 〈(x1, . . . , xn), (y1, . . . , yn)〉 = x1y1 + . . .+ xnyn and |x| = 〈x, x〉. Context distinguishes the norm |x| of x ∈ R from the usual abs...

By using quasi–derivatives we develop a Fourier method for studying the spectral gaps of one dimensional Schrödinger operators with periodic singular potentials v. Our results reveal a close relationship between smoothness of potentials and spectral gap asymptotics under a priori assumption v ∈ H loc (R). They extend and strengthen similar results proved in the classical case v ∈ L loc (R).

Problem (P). Let (Ω ,G ,F,P) be a probability space endowed with the filtration F = (Ft)t∈R+ . Assume that we are given a strictly positive, càdlàg, (P,F)-local martingale N with N0 = 1 and an F-adapted, continuous, increasing process Λ , with Λ0 = 0 and Λ∞ = ∞, and such that Gt := Nte−Λt ≤ 1 for every t ∈ R+. The goal is to construct a random time τ on an extended probability space and a proba...

In the era of huge datasets, the topsearch becomes an effective way to decrease the search time of topobjects. Since we suppose locally accessible data only, the multidimensional indexes containing all attributes together seem to be more effective than a distribution of each attribute to a separate index. Therefore we introduce the topsearch algorithm over grid file – the multidimensional index...

This paper describes the f i r s t stages of an experimental i nves t iga t ion into the e f fec ts of using var ious c a l c u l i to propagate uncer ta in ty in an i n te rac t i ve expert system for in format ion r e t r i e v a l . We in te rp re t uncer ta in ty values as p a r t i a l t ru ths rather than p r o b a b i l i t i e s , and draw upon the mathematics of mul t i -va lued log ic...

In order to produce a low energy effective field theory from a string model, it is necessary to specify a vacuum state. In order that this vacuum be supersymmetric, it is well known that all field expectation values must be along so-called flat directions, leaving the F and D-terms of the scalar potential to be zero. The situation becomes particularly interesting when one attempts to realize su...