Proof: Supremum and Infimum are Unique | Real Analysis
If a subset of the real numbers has a supremum or infimum, then they are unique! Uniqueness is a tremendously important property, so although it is almost complete trivial as far as difficulty goes in this case, we would be ill-advised to not prove these properties! In this lesson we'll be
From playlist Real Analysis
Definition of Supremum and Infimum of a Set | Real Analysis
What are suprema and infima of a set? This is an important concept in real analysis, we'll be defining both terms today with supremum examples and infimum examples to help make it clear! In short, a supremum of a set is a least upper bound. An infimum is a greatest lower bound. It is easil
From playlist Real Analysis
Epsilon Definition of Supremum and Infimum | Real Analysis
We prove an equivalent epsilon definition for the supremum and infimum of a set. Recall the supremum of a set, if it exists, is the least upper bound. So, if we subtract any amount from the supremum, we can no longer have an upper bound. The infimum of a set, if it exists, if the greatest
From playlist Real Analysis
Supremum of a set In this video, which is the most important video of the chapter, I define the supremum of a set of real numbers. It is like a maximum, except that it always exists, and will be super useful in the rest of our analysis adventure. Check out my Real Numbers Playlist: https
From playlist Real Numbers
Infimum of a set In this video, I define sup's little cousin, the infimum of a set. It is like a minimum, except that it always exists. Supremum of a set: https://youtu.be/lZEcsOn6qUA Check out my Real Numbers Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCZggpJZvUXnUzaw7f
From playlist Real Numbers
Proof: Supremum of {1/n} = 1 | Real Analysis
The supremum of the set containing all reciprocals of natural numbers is 1. That is, 1 is the least upper bound of {1/n | n is natural}. We prove this supremum in today's real analysis lesson using the epsilon definition of supremum! Definition of Supremum and Infimum of a Set: https://ww
From playlist Real Analysis
inf(S) = -sup(-S) In this video, I present a neat identity relating inf and sup. This means that, from now on, everything that we say for sup will hold for inf as well. Moreover, using this, we can prove the greatest lower bound property. Enjoy! Check out my Real Numbers Playlist: https:
From playlist Real Numbers
Bound for Supremum of the Intersection of Sets | Real Analysis Exercises
If A and B are two bounded and nonempty subsets of the real numbers, then what is the supremum of their intersection? What is sup(A intersect B)? We cannot say for sure in general, but we can place an upper bound on the supremum. If A and B are bounded nonempty subsets of the reals then we
From playlist Real Analysis Exercises
Richard Kraaij: A Lagrangian formalism for large deviations of independent copies...
Richard Kraaij: A Lagrangian formalism for large deviations of independent copies of Feller processes Dawson and Gaertner (1987) showed that the path of the empirical process of n independent identically distributed diffusion processes satisfy a large deviation principle in n with a rate
From playlist HIM Lectures 2015
Lecture 4: The Characterization of the Real Numbers
MIT 18.100A Real Analysis, Fall 2020 Instructor: Dr. Casey Rodriguez View the complete course: http://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/ YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP61O7HkcF7UImpM0cR_L2gSw An introduction to properties of fields and
From playlist MIT 18.100A Real Analysis, Fall 2020
Gamma Convergence (Lecture 3) by Nandakumar
PROGRAM: MULTI-SCALE ANALYSIS AND THEORY OF HOMOGENIZATION ORGANIZERS: Patrizia Donato, Editha Jose, Akambadath Nandakumaran and Daniel Onofrei DATE: 26 August 2019 to 06 September 2019 VENUE: Madhava Lecture Hall, ICTS, Bangalore Homogenization is a mathematical procedure to understa
From playlist Multi-scale Analysis And Theory Of Homogenization 2019
Guillaume Aubrun: Asymptotic tensor powers of Banach spaces
HYBRID EVENT Recorded during the meeting "Randoms Tensors and Related Topics" the March 14, 2022 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's Audio
From playlist Analysis and its Applications
Definition of infinity In this video, I define the concept of infinity (as used in analysis), and explain what it means for sup(S) to be infinity. In particular, the least upper bound property becomes very elegant to write down. Check out my real numbers playlist: https://www.youtube.co
From playlist Real Numbers
Proof: Minimum of a Set is the Infimum | Real Analysis
The minimum of a set is also the infimum of the set, we will prove this in today's lesson! This also applies to functions, since the range of a function is just a set of values. So if a function takes on a minimum value m, then the minimum m is also the infimum of the function. Recall th
From playlist Real Analysis
Dynamic equations on time scales
An introductory presentation on dynamic equations on time scales and uniqueness of solutions including new research resutls. The basic ideas of time scale calculus are presented and then a new theorem is discussed under which general initial value problems have, at most, one solution. T
From playlist Mathematical analysis and applications
Quenched large deviations for random motions in degenerate random media by Chiranjib Mukherjeer
Large deviation theory in statistical physics: Recent advances and future challenges DATE: 14 August 2017 to 13 October 2017 VENUE: Madhava Lecture Hall, ICTS, Bengaluru Large deviation theory made its way into statistical physics as a mathematical framework for studying equilibrium syst
From playlist Large deviation theory in statistical physics: Recent advances and future challenges
Quadratic forms and Hermite constant, reduction theory by Radhika Ganapathy
Discussion Meeting Sphere Packing ORGANIZERS: Mahesh Kakde and E.K. Narayanan DATE: 31 October 2019 to 06 November 2019 VENUE: Madhava Lecture Hall, ICTS Bangalore Sphere packing is a centuries-old problem in geometry, with many connections to other branches of mathematics (number the
From playlist Sphere Packing - 2019
Real Analysis | Riemann Integrability
We give the definition of the lower integral, upper integral, and what it means for a function to be integrable. A few results are also proven including the fact that any continuous function is Riemann integrable. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=
From playlist Real Analysis
Prove Infimums Exist with the Completeness Axiom | Real Analysis
The completeness axiom asserts that if A is a nonempty subset of the reals that is bounded above, then A has a least upper bound - called the supremum. This does not say anything about if greatest lower bounds - infimums exist for sets that are bounded below, but we can use the completenes
From playlist Real Analysis Exercises