Related papers: Intervals and Outer Measure on $\mathbb{R}$

Necessary and sufficient conditions for a measure to be an extreme point of the set of measures (on an abstract measurable space) with prescribed generalized moments are given, as well as an application to extremal problems over such moment…

We construct a Beurling generalized number system satisfying the Riemann hypothesis and whose integer counting function displays extremal oscillation in the following sense. The prime counting function of this number system satisfies…

We propose a measure and a metric on the sets of infinite traces generated by a set of atomic propositions. To compute these quantities, we first map properties to subsets of the real numbers and then take the Lebesgue measure of the…

This paper addresses the problem of choosing a sparse subset of measurements for quick calibration parameter estimation. A standard solution to this is selecting a measurement only if its utility -- the difference between posterior (with…

In this paper, we establish an iterative data-driven approach to derive guaranteed bounds on nonlinearity measures of unknown nonlinear systems. In this context, nonlinearity measures quantify the strength of the nonlinearity of a dynamical…

We present necessary and sufficient conditions for the existence of a countably additive measure on a complete Boolean algebra.

In [1], it is established that a convergent observer with an infinite gain margin can be designed for a given nonlinear system when a Riemannian metric showing that the system is differentially detectable (i.e., the Lie derivative of the…

Verification of temporal logic properties plays a crucial role in proving the desired behaviors of continuous systems. In this paper, we propose an interval method that verifies the properties described by a bounded signal temporal logic.…

A quantitative model of concurrent interaction is introduced. The basic objects are linear combinations of partial order relations, acted upon by a group of permutations that represents potential non-determinism in synchronisation. This…

The paper discusses the question of why intervals, which are the main object of Interval Analysis, have exactly the form that we know well and habitually use, and not some other. In particular, we investigate why traditional intervals are…

Outlier hypothesis testing is studied in a universal setting. Multiple sequences of observations are collected, a small subset of which are outliers. A sequence is considered an outlier if the observations in that sequence are distributed…

We propose a "decomposition method" to prove non-asymptotic bound for the convergence of empirical measures in various dual norms. The main point is to show that if one measures convergence in duality with sufficiently regular observables,…

Two different aspects of parabolic iteration in the complex upper half-plane are considered here. First, from a noncommutative probability perspective, a Berry-Esseen type estimate for the convergence speed of the monotone central limit…

We introduce a universality theorem for functionals of measures on partitions which "behave like" the Ewens measure. Various limit theorems for the Ewens measure, most notably the Poisson-Dirichlet limit for the longest parts, the…

The ordinal sum construction provides a very effective way to generate a new triangular norm on the real unit interval from existing ones. One of the most prominent theorems concerning the ordinal sum of triangular norms on the real unit…

Assuming the Riemann Hypothesis, we derive explicit bounds for the error terms in short interval analogues of the prime number theorem and Mertens' theorems using a smoothing argument. Our results improve upon previous bounds in both…

These draft notes are from a graduate course given by the author in Berkeley during the spring semester of 2005. They cover the basic ideas of a new, geometric approach to geometric measure theory. They begin with a new theory of exterior…

Over extended systems of finite type arithmetic, we utilize a formal representation of the outer measure to define a translation which allows for the systematic formalization of probabilistic statements. As a main result, this translation…

We consider an old question of Slaman and Steel: whether Turing equivalence is an increasing union of Borel equivalence relations none of which contain a uniformly computable infinite sequence. We show this question is deeply connected to…

Interval analysis, when applied to the so called problem of experimental data fitting, appears to be still in its infancy. Sometimes, partly because of the unrivaled reliability of interval methods, we do not obtain any results at all.…