In functional analysis, a branch of mathematics, the distortion problem is to determine by how much one can distort the unit sphere in a given Banach space using an equivalent norm. Specifically, a Banach space X is called λ-distortable if there exists an equivalent norm |x| on X such that, for all infinite-dimensional subspaces Y in X, (see distortion (mathematics)). Note that every Banach space is trivially 1-distortable. A Banach space is called distortable if it is λ-distortable for some λ > 1 and it is called arbitrarily distortable if it is λ-distortable for any λ. Distortability first emerged as an important property of Banach spaces in the 1960s, where it was studied by and . James proved that c0 and ℓ1 are not distortable. Milman showed that if X is a Banach space that does not contain an isomorphic copy of c0 or ℓp for some 1 ≤ p < ∞ (see sequence space), then some infinite-dimensional subspace of X is distortable. So the distortion problem is now primarily of interest on the spaces ℓp, all of which are separable and uniform convex, for 1 < p < ∞. In separable and uniform convex spaces, distortability is easily seen to be equivalent to the ostensibly more general question of whether or not every real-valued Lipschitz function ƒ defined on the sphere in X stabilizes on the sphere of an infinite dimensional subspace, i.e., whether there is a real number a ∈ R so that for every δ > 0 there is an infinite dimensional subspace Y of X, so that |a − ƒ(y)| < δ, for all y ∈ Y, with ||y|| = 1. But it follows from the result of that on ℓ1 there are Lipschitz functions which do not stabilize, although this space is not distortable by . In a separable Hilbert space, the distortion problem is equivalent to the question of whether there exist subsets of the unit sphere separated by a positive distance and yet intersect every infinite-dimensional closed subspace. Unlike many properties of Banach spaces, the distortion problem seems to be as difficult on Hilbert spaces as on other Banach spaces. On a separable Hilbert space, and for the other ℓp-spaces, 1 < p < ∞, the distortion problem was solved affirmatively by , who showed that ℓ2 is arbitrarily distortable, using the first known arbitrarily distortable space constructed by . (Wikipedia).
B04 Example problem of simple harmonic oscillation
Solving an example problem of simple harmonic oscillation, which requires calculating the solution to a second order ordinary differential equation.
From playlist Physics ONE
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In this video, I find the particular solution to the equation of motion for a driven damped harmonic oscillator.
From playlist Intermediate Classical Mechanics
Dynamics : An overview of the cause of mechanics
Dynamics is a subset of mechanics, which is the study of motion. Whereas kinetics studies that motion itself, dynamics is concerned about the CAUSES of motion. In particular, it involves the concepts of force, momentum and energy. This video gives an overview of what dynamics is, and is u
From playlist Dynamics
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Keywords 👉 Learn how to solve particle motion problems. Particle motion problems are usually modeled using functions. Now, when the function modeling the position of the particle is given with respect to the time, we find the speed function of the particle by differentiating the function
From playlist Particle Motion Problems
B15 Example problem with a linear equation using the error function
Solving an example problem for a linear equation with the error function.
From playlist Differential Equations
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From playlist Differential Equations: Complete Set of Course Videos
C60 Example problem involving free damped motion
Example problem using a linear differential equation to solve for damped harmonic motion.
From playlist Differential Equations
C58 Example problem using the alternate form
Solving a problem of simple harmonic motion and writing the solution in alternate form.
From playlist Differential Equations
Variational Surface Cutting - SIGGRAPH 2018
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From playlist Research
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From playlist Distinguished Visitors Lecture Series
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From playlist Nexus Trimester - 2016 - Fundamental Inequalities and Lower Bounds Theme
Gitta Kutyniok: "An Information Theoretic Approach to Validate Deep Learning-Based Algorithms"
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From playlist Machine Learning for Physics and the Physics of Learning 2019
Lecture 21: Relative Orientation, Binocular Stereo, Structure, Quadrics, Calibration, Reprojection
MIT 6.801 Machine Vision, Fall 2020 Instructor: Berthold Horn View the complete course: https://ocw.mit.edu/6-801F20 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP63pfpS1gV5P9tDxxL_e4W8O In this lecture, Prof. Horn discusses relative orientation, interior orientation,
From playlist MIT 6.801 Machine Vision, Fall 2020
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Seminar about distortion functions and Lorenz curves in finance.
From playlist Talks and Interviews
C67 The physics of simple harmonic motion
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From playlist Differential Equations
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From playlist Astronomy