Related papers: Set Theory and the Analyst
There are various approaches to the problem of how one is supposed to conduct a statistical analysis. Different analyses can lead to contradictory conclusions in some problems so this is not a satisfactory state of affairs. It seems that…
In order theory, partially ordered sets are only equipped with one relation which decides the entire structure/Hasse diagram of the set. In this paper, we have presented how partially ordered sets can be studied under simultaneous partially…
The methodology used here might provide a neat method of examining paradoxes and ways to circumvent them. Most of the known set theoretic paradoxes (Russell's, Cantor's, Burali-Forti's,..) can be paralleled here and examined. This account…
This text grew out of notes I have used in teaching a one quarter course on integration at the advanced undergraduate level. My intent is to introduce the Lebesgue integral in a quick, and hopefully painless, way and then go on to…
Information theory is widely accepted as a powerful tool for analyzing complex systems and it has been applied in many disciplines. Recently, some central components of information theory - multivariate information measures - have found…
This paper introduces a new theory which encompasses concepts and ideas from set theory, type theory, and Le\'{s}niewski's mereology and describes its possibility as an alternative foundation for mathematics. In the introduction section I…
When it is acknowledged that all candidate parameterised statistical models are misspecified relative to the data generating process, the decision maker (DM) must currently concern themselves with inference for the parameter value…
We provide a formal framework accounting for a widespread idea in the theory of economic design: analytically established incompatibilities between given axioms should be qualified by the likelihood of their violation. We define the degree…
We study relations between subsets of integers that are large, where large can be interpreted in terms of size (such as a set of positive upper density or a set with bounded gaps) or in terms of additive structure (such as a Bohr set). Bohr…
Reasoning about observed effects and their causes is important in multi-agent contexts. While there has been much work on causality from an objective standpoint, causality from the point of view of some particular agent has received much…
The purpose of this paper is two-fold. First, we would like to get rid of common assumption that causal set is bounded and attempt to model its scalar field action under the assumption that it isn't. Secondly, we would like to propose…
Given a non-archimedean real closed field with archimedean value group which contains the reals, we establish for the category of semialgebraic sets and functions a full Lebesgue measure and integration theory such that the main results…
Latent semantic similarity (LSS) is a measure of the similarity of information exchanges in a conversation. Challenging the assumption that higher LSS bears more positive psychological meaning, we propose that this association might depend…
A question is proposed whether or not set theory is consistent.
The object of contextuality analysis is a set of random variables each of which is uniquely labeled by a content and a context. In the measurement terminology, the content is that which the random variable measures, whereas the context…
In this paper, we prove a result on non-Baire sets in category bases which when applied together with a result of Grzegorek yeilds a comparatively stronger version of a decomposition theorem due to Ulam.
Similarity functions measure how comparable pairs of elements are, and play a key role in a wide variety of applications, e.g., notions of Individual Fairness abiding by the seminal paradigm of Dwork et al., as well as Clustering problems.…
In many applications in biology, engineering and economics, identifying similarities and differences between distributions of data from complex processes requires comparing finite categorical samples of discrete counts. Statistical…
These expository notes are a somewhat embellished version of two rather informal evening review sessions given by the second author at the 2008 Bedlewo summer school on "Dynamical Systems - Geometric Structures and Rigidity"; they provide a…
Let $K\subset\mathbb{R}$ be a self-similar set defined on $\mathbb{R}$. It is easy to prove that if the Lebesgue measure of $K$ is zero, then for Lebesgue almost every $t$, $$K+t=\{x+t:x\in K\}$$ only consists of irrational or…