Measure theory | Calculus of variations | Real analysis
In mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is a hypersurface in this case), but can be every intersection of the graph itself with a hyperplane (in the case of functions of two variables, a plane) parallel to a fixed x-axis and to the y-axis. Functions of bounded variation are precisely those with respect to which one may find Riemann–Stieltjes integrals of all continuous functions. Another characterization states that the functions of bounded variation on a compact interval are exactly those f which can be written as a difference g − h, where both g and h are bounded monotone. In particular, a BV function may have discontinuities, but at most countably many. In the case of several variables, a function f defined on an open subset Ω of is said to have bounded variation if its distributional derivative is a vector-valued finite Radon measure. One of the most important aspects of functions of bounded variation is that they form an algebra of discontinuous functions whose first derivative exists almost everywhere: due to this fact, they can and frequently are used to define generalized solutions of nonlinear problems involving functionals, ordinary and partial differential equations in mathematics, physics and engineering. We have the following chains of inclusions for continuous functions over a closed, bounded interval of the real line: Continuously differentiable ⊆ Lipschitz continuous ⊆ absolutely continuous ⊆ continuous and bounded variation ⊆ differentiable almost everywhere (Wikipedia).
What are Bounded Sequences? | Real Analysis
What are bounded sequences? We go over the definition of bounded sequence in today's real analysis video lesson. We'll see examples of sequences that are bounded, and some that are bounded above or bounded below, but not both. We say a sequence is bounded if the set of values it takes on
From playlist Real Analysis
Convergent sequences are bounded
Convergent Sequences are Bounded In this video, I show that if a sequence is convergent, then it must be bounded, that is some part of it doesn't go to infinity. This is an important result that is used over and over again in analysis. Enjoy! Other examples of limits can be seen in the
From playlist Sequences
Definite Integral Using Limit Definition
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Definite Integral Using Limit Definition. In this video we compute a definite integral using the limit definition.
From playlist Calculus
Monotonic Sequences and Bounded Sequences - Calculus 2
This calculus 2 video tutorial provides a basic introduction into monotonic sequences and bounded sequences. A monotonic sequence is a sequence that is always increasing or decreasing. You can prove that a sequence is always increasing by showing that the next term is greater than the p
From playlist New Calculus Video Playlist
Absolute Value Definition of a Bounded Sequence | Real Analysis
The definition of a bounded sequence is a very important one, and it relies on a sequence having a lower an upper bound. However, we can also state the definition of a bounded sequence with only a single bound - namely an upper bound on the absolute value of the terms of the sequence. If t
From playlist Real Analysis
Bounded and Monotonic Sequences
We define bounded sequences and monotone sequences, then look at the Monotone Sequence Theorem. This video is part of a Calculus II course taught at the University of Cincinnati.
From playlist Older Calculus II (New Playlist For Spring 2019)
Proof: Convergent Sequence is Bounded | Real Analysis
Any convergent sequence must be bounded. We'll prove this basic result about convergent sequences in today's lesson. We use the definition of the limit of a sequence, a useful equivalence involving absolute value inequalities, and then considering a maximum and minimum will help us find an
From playlist Real Analysis
Free ebook http://tinyurl.com/EngMathYT An introduction to the convergence property of monotonic and bounded sequences. The main idea is known as the "Monotonic convergence thoerem" and has important applications to approximating solutions to equations. Several examples are presented to
From playlist A second course in university calculus.
Math 131 092816 Continuity; Continuity and Compactness
Review definition of limit. Definition of continuity at a point; remark about isolated points; connection with limits. Composition of continuous functions. Alternate characterization of continuous functions (topological definition). Continuity and compactness: continuous image of a com
From playlist Course 7: (Rudin's) Principles of Mathematical Analysis
Descartes Rule of Signs - Upper and Lower Bounds
TabletClass Math http://www.tabletclass.com . This explains Descartes Rule of Signs. The lesson is designed to focus on on how Descartes Rule of Signs helps finds the zeros of a polynomial.
From playlist Pre-Calculus / Trigonometry
Y. Tonegawa - Analysis on the mean curvature flow and the reaction-diffusion approximation (Part 1)
The course covers two separate but closely related topics. The first topic is the mean curvature flow in the framework of GMT due to Brakke. It is a flow of varifold moving by the generalized mean curvature. Starting from a quick review on the necessary tools and facts from GMT and the def
From playlist Ecole d'été 2015 - Théorie géométrique de la mesure et calcul des variations : théorie et applications
DeepMind x UCL | Deep Learning Lectures | 11/12 | Modern Latent Variable Models
This lecture, by DeepMind Research Scientist Andriy Mnih, explores latent variable models, a powerful and flexible framework for generative modelling. After introducing this framework along with the concept of inference, which is central to it, Andriy focuses on two types of modern latent
From playlist Learning resources
On minimizers and critical points for anisotropic isoperimetric problems - Robin Neumayer
Variational Methods in Geometry Seminar Topic: On minimizers and critical points for anisotropic isoperimetric problems Speaker: Robin Neumayer Affiliation: Member, School of Mathematics Date: February 19, 2019 For more video please visit http://video.ias.edu
From playlist Variational Methods in Geometry
Regularized Functional Inequalities and Applications to Markov Chains by Pierre Youssef
PROGRAM: TOPICS IN HIGH DIMENSIONAL PROBABILITY ORGANIZERS: Anirban Basak (ICTS-TIFR, India) and Riddhipratim Basu (ICTS-TIFR, India) DATE & TIME: 02 January 2023 to 13 January 2023 VENUE: Ramanujan Lecture Hall This program will focus on several interconnected themes in modern probab
From playlist TOPICS IN HIGH DIMENSIONAL PROBABILITY
[Variational Autoencoder] Auto-Encoding Variational Bayes | AISC Foundational
A.I. Socratic Circles For details including slides, visit https://aisc.a-i.science/events/2019-03-28 Lead: Elham Dolatabadi Facilitators: Chris Dryden , Florian Goebels Auto-Encoding Variational Bayes How can we perform efficient inference and learning in directed probabilistic models,
From playlist Math and Foundations
New Approximations of the Total Variation, and Filters in Image Processing - Haim Brezis
Haim Brezis Institute for Advanced Study February 19, 2013 I will present new results concerning the approximation of the BV-norm by nonlocal, nonconvex, functionals. The original motivation comes from Image Processing. Numerous problems remain open. The talk is based on a joint work with
From playlist Mathematics
Beyond Homogenization by Graeme Milton
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)
Adaptive Estimation via Optimal Decision Trees by Subhajit Goswami
Program Advances in Applied Probability II (ONLINE) ORGANIZERS: Vivek S Borkar (IIT Bombay, India), Sandeep Juneja (TIFR Mumbai, India), Kavita Ramanan (Brown University, Rhode Island), Devavrat Shah (MIT, US) and Piyush Srivastava (TIFR Mumbai, India) DATE: 04 January 2021 to 08 Januar
From playlist Advances in Applied Probability II (Online)
Continuous implies Bounded In this video, I show that any continuous function from a closed and bounded interval to the real numbers must be bounded. The proof is very neat and involves a straightforward application of the Bolzano-Weierstraß Theorem, enjoy! Bolzano-Weierstraß: https://yo
From playlist Limits and Continuity