Functional analysis | Metric geometry
In mathematics, the Kuratowski embedding allows one to view any metric space as a subset of some Banach space. It is named after Kazimierz Kuratowski. The statement obviously holds for the empty space.If (X,d) is a metric space, x0 is a point in X, and Cb(X) denotes the Banach space of all bounded continuous real-valued functions on X with the supremum norm, then the map defined by is an isometry. The above construction can be seen as embedding a pointed metric space into a Banach space. The Kuratowski–Wojdysławski theorem states that every bounded metric space X is isometric to a closed subset of a convex subset of some Banach space. (N.B. the image of this embedding is closed in the convex subset, not necessarily in the Banach space.) Here we use the isometry defined by The convex set mentioned above is the convex hull of Ψ(X). In both of these embedding theorems, we may replace Cb(X) by the Banach space ℓ ∞(X) of all bounded functions X → R, again with the supremum norm, since Cb(X) is a closed linear subspace of ℓ ∞(X). These embedding results are useful because Banach spaces have a number of useful properties not shared by all metric spaces: they are vector spaces which allows one to add points and do elementary geometry involving lines and planes etc.; and they are complete. Given a function with codomain X, it is frequently desirable to extend this function to a larger domain, and this often requires simultaneously enlarging the codomain to a Banach space containing X. (Wikipedia).
Jasna Urbančič (11/03/21):Optimizing Embedding using Persistence
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From playlist AATRN 2021
Like Linux, Kubernetes consists of a core and an assortment of building block components that must be assembled and integrated to create an enterprise production-level platform. Most Kubernetes deployments fail because organizations underestimate the complexity of Kubernetes and overestima
From playlist Containers
Cloud Native Apps with Server-Side WebAssembly
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From playlist WebAssembly
Rasa Algorithm Whiteboard - Understanding Word Embeddings 1: Just Letters
We're making a few videos that highlight word embeddings. Before training word embeddings we figured it might help the intuition if we first trained some letter embeddings. It might suprise you but the idea with an embedding can also be demonstrated with letters as opposed to words. We're
From playlist Algorithm Whiteboard
Attacking and Defending Kubernetes
Many companies have deployed Kubernetes, but few infosec folks have experience attacking it. We aim to address that shortage, culminating in an audience-directed Choose Your Own Adventure, movie-themed demo against an intentionally-vulnerable cluster named Bust-a-Kube. You'll see how to at
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Kubernetes Architecture | Understanding Kubernetes Components | Kubernetes Training | Edureka
🔥 Edureka Kubernetes Certification Training: https://www.edureka.co/kubernetes-certification This Edureka video on "Kubernetes Architecture" will give you an introduction to popular DevOps tool - Kubernetes, and will deep dive into Kubernetes Architecture and its working. The following top
From playlist Kubernetes Tutorial for Beginners | Edureka
The Homework Problem That Started as a Phd Thesis: 14 set theorem
In a handful of introductory topology textbooks, Kuratowski's 14 set theorem is given as an exercise despite it being one of the results proven as a part of his phd thesis in 1922. This homework problem that started out as a phd thesis is not an easy exercise if you don't know how to think
From playlist The New CHALKboard
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From playlist Women in Mathematics - Numberphile
Kubernetes Deployment Tutorial | Kubernetes Tutorial For Beginners | Kubernetes Training | Edureka
( Kubernetes Certification Training: https://www.edureka.co/kubernetes-certification ) This Edureka video on "Kubernetes Deployment Tutorial " will help you understand the various concepts related to Deployment in Kubernetes. The topics included in this session are: 1. What is Kubernetes?
From playlist DevOps Training Videos
MATH1081 Discrete Maths: Chapter 5 Question 27 a
This problem is about planar graphs. The theorem mentioned is Fáry's Theorem (1948); see http://bit.ly/1gmUrXT . Presented by Thomas Britz of the School of Mathematics and Statistics, Faculty of Science, UNSW.
From playlist MATH1081 Discrete Mathematics
A Classification of Planar Graphs - A Proof of Kuratowski's Theorem
A visually explained proof of Kuratowski's theorem, an interesting, important and useful result classifying "planar" graphs. Proof adapted from: http://math.uchicago.edu/~may/REU2017/REUPapers/Xu,Yifan.pdf and: https://www.math.cmu.edu/~mradclif/teaching/228F16/Kuratowski.pdf Also check
From playlist Summer of Math Exposition Youtube Videos
Graph Theory: 61. Characterization of Planar Graphs
We have seen in a previous video that K5 and K3,3 are non-planar. In this video we define an elementary subdivision of a graph, as well as a subdivision of a graph. We then discuss the fact that if a graph G contains a subgraph which is a subdivision of a non-planar graph, then G is non-
From playlist Graph Theory part-10
Dominik Inauen: Isometric Embeddings Flexibility vs Rigidity
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From playlist HIM Lectures: Trimester Program "Evolution of Interfaces"
Rade Zivaljevic (6/27/17) Bedlewo: Topological methods in discrete geometry; new developments
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From playlist Applied Topology in Będlewo 2017
DevOpsDays Boston 2017- Real-World Kubernetes For DevOps by Phil Lombardi
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Ulrich Bauer (3/4/22): Gromov hyperbolicity, geodesic defect, and apparent pairs in Rips filtrations
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MATH1081 Discrete Maths: Chapter 5 Question 33 - Kuratowski's Theorem (part 1)
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From playlist MATH1081 Discrete Mathematics
MATH1081 Discrete Maths: Chapter 5 Question 33 - Kuratowski's Theorem (part 2)
MATH1081 "Discrete Mathematics" Topic 5 Question 33c
From playlist MATH1081 Discrete Mathematics
Graph Theory: 62. Graph Minors and Wagner's Theorem
In this video, we begin with a visualisation of an edge contraction and discuss the fact that an edge contraction may be thought of as resulting in a multigraph or simple graph, depending on the application. We then state the definition a contraction of edge e in a graph G resulting in a
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Excel - Embed a PDF in Excel - Podcast #1466
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From playlist Excel General