Independence results | Measure theory
In mathematical analysis, a strong measure zero set is a subset A of the real line with the following property: for every sequence (εn) of positive reals there exists a sequence (In) of intervals such that |In| < εn for all n and A is contained in the union of the In. (Here |In| denotes the length of the interval In.) Every countable set is a strong measure zero set, and so is every union of countably many strong measure zero sets. Every strong measure zero set has Lebesgue measure 0. The Cantor set is an example of an uncountable set of Lebesgue measure 0 which is not of strong measure zero. Borel's conjecture states that every strong measure zero set is countable. It is now known that this statement is independent of ZFC (the Zermelo–Fraenkel axioms of set theory, which is the standard axiom system assumed in mathematics). This means that Borel's conjecture can neither be proven nor disproven in ZFC (assuming ZFC is consistent).Sierpiński proved in 1928 that the continuum hypothesis (which is now also known to be independent of ZFC) implies the existence of uncountable strong measure zero sets. In 1976 Laver used a method of forcing to construct a model of ZFC in which Borel's conjecture holds. These two results together establish the independence of Borel's conjecture. The following characterization of strong measure zero sets was proved in 1973: A set A ⊆ R has strong measure zero if and only if A + M ≠ R for every meagre set M ⊆ R. This result establishes a connection to the notion of strongly meagre set, defined as follows: A set M ⊆ R is strongly meagre if and only if A + M ≠ R for every set A ⊆ R of Lebesgue measure zero. The dual Borel conjecture states that every strongly meagre set is countable. This statement is also independent of ZFC. (Wikipedia).
Maximum and Minimum of a set In this video, I define the maximum and minimum of a set, and show that they don't always exist. Enjoy! Check out my Real Numbers Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCZggpJZvUXnUzaw7fHCtoh
From playlist Real Numbers
Maximum and Minimum Values (Closed interval method)
A review of techniques for finding local and absolute extremes, including an application of the closed interval method
From playlist 241Fall13Ex3
Introduction to Limits at Infinity (Part 1)
This video introduces limits at infinity. https://mathispower4u.com
From playlist Limits at Infinity and Special Limits
Ex: Limits at Infinity of a Function Involving a Square Root
This video provides two examples of how to determine limits at infinity of a function involving a square root. The results are verified graphically. Site: http://mathispower4u.com Blog: http://mathispower4u.wordpress.com
From playlist Limits at Infinity and Special Limits
Listing elements from a set (2)
Powered by https://www.numerise.com/ Listing elements from a set (2)
From playlist Set theory
Determine the Least Element in a Set Given using Set Notation.
This video explains how to determine the least element in a set given using set notation.
From playlist Sets (Discrete Math)
Computing a One Sided Limit with an Absolute Value Function
In this video I do an example of Computing a One Sided Limit with an Absolute Value Function.
From playlist One-sided Limits
Ex: Limits of the Floor Function (Greatest Integer Function)
This video explains how to determine limits of a floor function graphically and numerically using a graphing calculator. Site: http://mathispower4u.com
From playlist Limits
Homogenization and Correctors for Linear Stochastic Equations in.... by Mogtaba A. Y. Mohammed
DISCUSSION MEETING Multi-Scale Analysis: Thematic Lectures and Meeting (MATHLEC-2021, ONLINE) ORGANIZERS: Patrizia Donato (University of Rouen Normandie, France), Antonio Gaudiello (Università degli Studi di Napoli Federico II, Italy), Editha Jose (University of the Philippines Los Baño
From playlist Multi-scale Analysis: Thematic Lectures And Meeting (MATHLEC-2021) (ONLINE)
Equidistribution of Measures with High Entropy for General Surface Diffeomorphisms by Omri Sarig
PROGRAM : ERGODIC THEORY AND DYNAMICAL SYSTEMS (HYBRID) ORGANIZERS : C. S. Aravinda (TIFR-CAM, Bengaluru), Anish Ghosh (TIFR, Mumbai) and Riddhi Shah (JNU, New Delhi) DATE : 05 December 2022 to 16 December 2022 VENUE : Ramanujan Lecture Hall and Online The programme will have an emphasis
From playlist Ergodic Theory and Dynamical Systems 2022
On the structure of measures constrained by linear PDEs – Guido De Philippis – ICM2018
Partial Differential Equations | Analysis and Operator Algebras Invited Lecture 10.3 | 8.3 On the structure of measures constrained by linear PDEs Guido De Philippis Abstract: The aim of this talk is to present some recent results on the structure of the singular part of measures satisfy
From playlist Partial Differential Equations
37 Sundar - Invariant measures and ergodicity for stochastic Navier-Stokes equations
PROGRAM NAME :WINTER SCHOOL ON STOCHASTIC ANALYSIS AND CONTROL OF FLUID FLOW DATES Monday 03 Dec, 2012 - Thursday 20 Dec, 2012 VENUE School of Mathematics, Indian Institute of Science Education and Research, Thiruvananthapuram Stochastic analysis and control of fluid flow problems have
From playlist Winter School on Stochastic Analysis and Control of Fluid Flow
D. Tewodrose - Limits of Riemannian manifolds satisfying a uniform Kato condition
I will present a joint work with G. Carron and I. Mondello where we study Kato limit spaces. These are metric measure spaces obtained as Gromov-Hausdorff limits of smooth n-dimensional Riemannian manifolds with Ricci curvature satisfying a uniform Kato-type condition. In this context, stri
From playlist Ecole d'été 2021 - Curvature Constraints and Spaces of Metrics
D. Tewodrose - Limits of Riemannian manifolds satisfying a uniform Kato condition (vt)
I will present a joint work with G. Carron and I. Mondello where we study Kato limit spaces. These are metric measure spaces obtained as Gromov-Hausdorff limits of smooth n-dimensional Riemannian manifolds with Ricci curvature satisfying a uniform Kato-type condition. In this context, stri
From playlist Ecole d'été 2021 - Curvature Constraints and Spaces of Metrics
Ahlfors-Bers 2014 "On isomorphism and disjointness of interval exchanges and flows on flat surfaces"
Jon Chaika (University of Utah): A basic question in dynamical systems is when are two systems isomorphic. Starting from rotations of the circle and flows on tori we will talk about the fact that typical interval exchanges and flows on flat surfaces are not isomorphic. In fact, they satisf
From playlist The Ahlfors-Bers Colloquium 2014 at Yale
The Abel lectures: Hillel Furstenberg and Gregory Margulis
0:30 Welcome by Hans Petter Graver, President of the Norwegian Academy of Science Letters 01:37 Introduction by Hans Munthe-Kaas, Chair of the Abel Prize Committee 04:16 Hillel Furstenberg: Random walks in non-euclidean space and the Poisson boundary of a group 58:40 Questions and answers
From playlist Gregory Margulis
Strong Stationarity and Multiplicative Functions - Nikos Frantzikinakis
Special Year Learning Seminar Topic: Strong Stationarity and Multiplicative Functions Speaker: Nikos Frantzikinakis Affiliation: Member, School of Mathematics Date: February 15, 2023 The notion of strong stationarity was introduced by Furstenberg and Katznelson in the early 90's in orde
From playlist Mathematics
Pere Ara: Crossed products and the Atiyah problem
Talk by Pere Are in Global Noncommutative Geometry Seminar (Americas) https://globalncgseminar.org/talks/crossed-products-and-the-atiyah-problem/ on March 19, 2021.
From playlist Global Noncommutative Geometry Seminar (Americas)
Extreme Value Theorem Using Critical Points
Calculus: The Extreme Value Theorem for a continuous function f(x) on a closed interval [a, b] is given. Relative maximum and minimum values are defined, and a procedure is given for finding maximums and minimums. Examples given are f(x) = x^2 - 4x on the interval [-1, 3], and f(x) =
From playlist Calculus Pt 1: Limits and Derivatives