In descriptive set theory, a tree on a set is a collection of finite sequences of elements of such that every prefix of a sequence in the collection also belongs to the collection.
Definitions
Trees
The collection of all finite sequences of elements of a set is denoted .
With this notation, a tree is a nonempty subset of , such that if
is a sequence of length in , and if ,
then the shortened sequence also belongs to . In particular, choosing shows that the empty sequence belongs to every tree.
Branches and bodies
A branch through a tree is an infinite sequence of elements of , each of whose finite prefixes belongs to . The set of all branches through is denoted and called the body of the tree .
A tree that has no branches is called wellfounded; a tree with at least one branch is illfounded. By König's lemma, a tree on a finite set with an infinite number of sequences must necessarily be illfounded.
Terminal nodes
A finite sequence that belongs to a tree is called a terminal node if it is not a prefix of a longer sequence in . Equivalently, is terminal if there is no element of such that that . A tree that does not have any terminal nodes is called pruned.
Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by HowardHanson (1960) in connection with tonal music, and then mostly developed in connection with atonal music by theorists such as AllenForte (1973), drawing on the work in twelve-tone theory of Milton Babbitt. The concepts of set theory are very general and can be applied to tonal and atonal styles in any equally tempered tuning system, and to some extent more generally than that.
One branch of musical set theory deals with collections (sets and permutations) of pitches and pitch classes (pitch-class set theory), which may be ordered or unordered, and can be related by musical operations such as transposition, inversion, and complementation. The methods of musical set theory are sometimes applied to the analysis of rhythm as well.
Mathematical set theory versus musical set theory
Although musical set theory is often thought to involve the application of mathematical set theory to music, there are numerous differences between the methods and terminology of the two. For example, musicians use the terms transposition and inversion where mathematicians would use translation and reflection.
Furthermore, where musical set theory refers to ordered sets, mathematics would normally refer to tuples or sequences (though mathematics does speak of ordered sets, and although these can be seen to include the musical kind in some sense, they are far more involved).
A set (pitch set, pitch-class set, set class, set form, set genus, pitch collection) in music theory, as in mathematics and general parlance, is a collection of objects. In musical contexts the term is traditionally applied most often to collections of pitches or pitch-classes, but theorists have extended its use to other types of musical entities, so that one may speak of sets of durations or timbres, for example.
A set by itself does not necessarily possess any additional structure, such as an ordering. Nevertheless, it is often musically important to consider sets that are equipped with an order relation (called segments); in such contexts, bare sets are often referred to as "unordered", for the sake of emphasis.
Two-element sets are called dyads, three-element sets trichords (occasionally "triads", though this is easily confused with the traditional meaning of the word triad). Sets of higher cardinalities are called tetrachords (or tetrads), pentachords (or pentads), hexachords (or hexads), heptachords (heptads or, sometimes, mixing Latin and Greek roots, "septachords"—e.g.,), octachords (octads), nonachords (nonads), decachords (decads), undecachords, and, finally, the dodecachord.
Set construction is the process by which a construction manager undertakes to build full scale scenery suitable for viewing by camera, as specified by a production designer or art director working in collaboration with the director of a production to create a set for a theatrical, film or television production. The set designer produces a scale model, scale drawings, paint elevations (a scale painting supplied to the scenic painter of each element that requires painting), and research about props, textures, and so on. Scale drawings typically include a groundplan, elevation, and section of the complete set, as well as more detailed drawings of individual scenic elements which, in theatrical productions, may be static, flown, or built onto scenery wagons. Models and paint elevations are frequently hand-produced, though in recent years, many Production Designers and most commercial theatres have begun producing scale drawings with the aid of computer drafting programs such as AutoCAD or Vectorworks.
In computing, associative containers refer to a group of class templates in the standard library of the C++ programming language that implement ordered associative arrays. Being templates, they can be used to store arbitrary elements, such as integers or custom classes. The following containers are defined in the current revision of the C++ standard: set, map, multiset, multimap. Each of these containers differ only on constraints placed on their elements.
Key uniqueness: in map and set each key must be unique. multimap and multiset do not have this restriction.
Element composition: in map and multimap each element is composed from a key and a mapped value. In set and multiset each element is key; there are no mapped values.
Adrian Mathias: Linking descriptive set theory to symbolic dynamics, part I
This talk was held in CUNY's virtual set theory seminar on August 6th, 2021.
published: 12 Nov 2021
Mirna Džamonja - On wide Aronszajn trees
Oxford Set Theory Seminar/ Bristol Logic and Set Theory Seminar
http://jdh.hamkins.org/oxford-set-theory-seminar/
Aronszajn trees are a staple of set theory, but there are applications where the requirement of all levels being countable is of no importance. This is the case in set-theoretic model theory, where trees of height and size ω1 but with no uncountable branches play an important role by being clocks of Ehrenfeucht–Fraïssé games that measure similarity of model of size ℵ1. We call such trees wide Aronszajn. In this context one can also compare trees T and T’ by saying that T weakly embeds into T’ if there is a function f that map T into T’ while preserving the strict order <_T. This order translates into the comparison of winning strategies for the isomorphism player, where any...
published: 06 Nov 2020
Adrian Mathias: Linking descriptive set theory to symbolic dynamics, part II
This talk was held in CUNY's virtual set theory seminar on August 13th, 2021. It is the continuation of the talk given one week earlier.
published: 12 Nov 2021
INTRODUCTION to SET THEORY - DISCRETE MATHEMATICS
We introduce the basics of set theory and do some practice problems.
This video is an updated version of the original video released over two years ago. Hopefully the higher pen quality and refined explanations are beneficial for your learning. If you'd like to see more videos redone in the series, please leave a comment down below.
#DiscreteMath #Mathematics #settheory
Looking for paid tutoring or online courses with practice exercises, text lectures, solutions, and exam practice? http://TrevTutor.com has you covered!
Support me on Patreon: http://bit.ly/2EUdAl3
Visit my website: http://bit.ly/1zBPlvm
Subscribe on YouTube: http://bit.ly/1vWiRxW
*--Playlists--*
Discrete Mathematics 1: https://www.youtube.com/playlist?list=PLDDGPdw7e6Ag1EIznZ-m-qXu4XX3A0cIz
Discrete Mathematics 2: htt...
published: 11 Jul 2017
Introduction to Probability, Basic Overview - Sample Space, & Tree Diagrams
This video provides an introduction to probability. It explains how to calculate the probability of an event occurring in addition to determining the sample space of an event using tree diagrams.
Probability - Free Formula Sheet: https://bit.ly/3zb22rW
______________________________
Introduction to Probability:
https://www.youtube.com/watch?v=SkidyDQuupA
Probability Formulas:
https://www.youtube.com/watch?v=bddckR734aM
Probability Explained:
https://www.youtube.com/watch?v=UORztmWGY6Q
Probability With Geometry:
https://www.youtube.com/watch?v=DeH5aHWxEaI
Probability of Complementary Events:
https://www.youtube.com/...
published: 21 Mar 2019
26/03/2021 - Vincenzo Dimonte, "The role of Prikry forcing in generalized Descriptive Set Theory"
Torino-Udine Logic Seminar recorded on the 26th of March 2021.
published: 31 Mar 2021
Anton Bernshteyn: Distributed Algorithms and Descriptive Combinatorics
ADGA 2021 — Workshop on Advances in Distributed Graph Algorithms
http://adga.hiit.fi/2021/
published: 27 Sep 2021
Grigor Sargsyan - Descriptive inner model theory
Slides: https://drive.google.com/file/d/1hNaqfVSyulytj7RM-DjU8xZrxWPC5xtq/view
Set Theory Today: A conference in honor of Georg Cantor
September 10-14, 2018
Vienna, Austria
Conference website: https://sites.google.com/view/set-theory-today/startseite?authuser=0
Video and editing: Daniel T. Soukup
published: 20 Sep 2018
Slawomir Solecki: Generic measure preserving transformations and descriptive set theory
(06 Mai 2022/May 06, 2022) Colloque des sciences mathématiques du Québec. http://www.crm.umontreal.ca/cal/en/jour20220506.html
Slawomir Solecki (Cornell University): Generic measure preserving transformations and descriptive set theory
Abstract: The behavior of a measure preserving transformation, even a generic one, is highly non-uniform. In contrast to this observation, a different picture of a very uniform behavior of the closed group generated by a generic measure preserving transformation has emerged. This picture included substantial evidence that pointed to these groups being all topologically isomorphic to a single group, namely, $L^0$---the non-locally compact, topological group of all Lebesgue measurable functions from $[0,1]$ to the circle. In fact, Glasner and Weiss asked if ...
Oxford Set Theory Seminar/ Bristol Logic and Set Theory Seminar
http://jdh.hamkins.org/oxford-set-theory-seminar/
Aronszajn trees are a staple of set theory, b...
Oxford Set Theory Seminar/ Bristol Logic and Set Theory Seminar
http://jdh.hamkins.org/oxford-set-theory-seminar/
Aronszajn trees are a staple of set theory, but there are applications where the requirement of all levels being countable is of no importance. This is the case in set-theoretic model theory, where trees of height and size ω1 but with no uncountable branches play an important role by being clocks of Ehrenfeucht–Fraïssé games that measure similarity of model of size ℵ1. We call such trees wide Aronszajn. In this context one can also compare trees T and T’ by saying that T weakly embeds into T’ if there is a function f that map T into T’ while preserving the strict order <_T. This order translates into the comparison of winning strategies for the isomorphism player, where any winning strategy for T’ translates into a winning strategy for T’. Hence it is natural to ask if there is a largest such tree, or as we would say, a universal tree for the class of wood Aronszajn trees with weak embeddings. It was known that there is no such a tree under CH, but in 1994 Mekler and Väänanen conjectured that there would be under MA(ω1).
Oxford Set Theory Seminar/ Bristol Logic and Set Theory Seminar
http://jdh.hamkins.org/oxford-set-theory-seminar/
Aronszajn trees are a staple of set theory, but there are applications where the requirement of all levels being countable is of no importance. This is the case in set-theoretic model theory, where trees of height and size ω1 but with no uncountable branches play an important role by being clocks of Ehrenfeucht–Fraïssé games that measure similarity of model of size ℵ1. We call such trees wide Aronszajn. In this context one can also compare trees T and T’ by saying that T weakly embeds into T’ if there is a function f that map T into T’ while preserving the strict order <_T. This order translates into the comparison of winning strategies for the isomorphism player, where any winning strategy for T’ translates into a winning strategy for T’. Hence it is natural to ask if there is a largest such tree, or as we would say, a universal tree for the class of wood Aronszajn trees with weak embeddings. It was known that there is no such a tree under CH, but in 1994 Mekler and Väänanen conjectured that there would be under MA(ω1).
We introduce the basics of set theory and do some practice problems.
This video is an updated version of the original video released over two years ago. Hopefu...
We introduce the basics of set theory and do some practice problems.
This video is an updated version of the original video released over two years ago. Hopefully the higher pen quality and refined explanations are beneficial for your learning. If you'd like to see more videos redone in the series, please leave a comment down below.
#DiscreteMath #Mathematics #settheory
Looking for paid tutoring or online courses with practice exercises, text lectures, solutions, and exam practice? http://TrevTutor.com has you covered!
Support me on Patreon: http://bit.ly/2EUdAl3
Visit my website: http://bit.ly/1zBPlvm
Subscribe on YouTube: http://bit.ly/1vWiRxW
*--Playlists--*
Discrete Mathematics 1: https://www.youtube.com/playlist?list=PLDDGPdw7e6Ag1EIznZ-m-qXu4XX3A0cIz
Discrete Mathematics 2: https://www.youtube.com/playlist?list=PLDDGPdw7e6Aj0amDsYInT_8p6xTSTGEi2
*--Recommended Textbooks--*
Discrete and Combinatorial Mathematics (Grimaldi): https://amzn.to/2T0iC53
Discrete Mathematics (Johnsonbaugh): https://amzn.to/2Hh7H41
Discrete Mathematics and Its Applications (Rosen): https://amzn.to/3lUgrMI
Book of Proof (Hammack): https://amzn.to/35eEbVg
Hello, welcome to TheTrevTutor. I'm here to help you learn your college courses in an easy, efficient manner. If you like what you see, feel free to subscribe and follow me for updates. If you have any questions, leave them below. I try to answer as many questions as possible. If something isn't quite clear or needs more explanation, I can easily make additional videos to satisfy your need for knowledge and understanding.
We introduce the basics of set theory and do some practice problems.
This video is an updated version of the original video released over two years ago. Hopefully the higher pen quality and refined explanations are beneficial for your learning. If you'd like to see more videos redone in the series, please leave a comment down below.
#DiscreteMath #Mathematics #settheory
Looking for paid tutoring or online courses with practice exercises, text lectures, solutions, and exam practice? http://TrevTutor.com has you covered!
Support me on Patreon: http://bit.ly/2EUdAl3
Visit my website: http://bit.ly/1zBPlvm
Subscribe on YouTube: http://bit.ly/1vWiRxW
*--Playlists--*
Discrete Mathematics 1: https://www.youtube.com/playlist?list=PLDDGPdw7e6Ag1EIznZ-m-qXu4XX3A0cIz
Discrete Mathematics 2: https://www.youtube.com/playlist?list=PLDDGPdw7e6Aj0amDsYInT_8p6xTSTGEi2
*--Recommended Textbooks--*
Discrete and Combinatorial Mathematics (Grimaldi): https://amzn.to/2T0iC53
Discrete Mathematics (Johnsonbaugh): https://amzn.to/2Hh7H41
Discrete Mathematics and Its Applications (Rosen): https://amzn.to/3lUgrMI
Book of Proof (Hammack): https://amzn.to/35eEbVg
Hello, welcome to TheTrevTutor. I'm here to help you learn your college courses in an easy, efficient manner. If you like what you see, feel free to subscribe and follow me for updates. If you have any questions, leave them below. I try to answer as many questions as possible. If something isn't quite clear or needs more explanation, I can easily make additional videos to satisfy your need for knowledge and understanding.
This video provides an introduction to probability. It explains how to calculate the probability of an event occurring in addition to determining the sample sp...
This video provides an introduction to probability. It explains how to calculate the probability of an event occurring in addition to determining the sample space of an event using tree diagrams.
Probability - Free Formula Sheet: https://bit.ly/3zb22rW
______________________________
Introduction to Probability:
https://www.youtube.com/watch?v=SkidyDQuupA
Probability Formulas:
https://www.youtube.com/watch?v=bddckR734aM
Probability Explained:
https://www.youtube.com/watch?v=UORztmWGY6Q
Probability With Geometry:
https://www.youtube.com/watch?v=DeH5aHWxEaI
Probability of Complementary Events:
https://www.youtube.com/watch?v=0T-CaQCiSf4
Conditional Probability:
https://www.youtube.com/watch?v=sqDVrXq_eh0
__________________________________
Independent and Dependent Events:
https://www.youtube.com/watch?v=lWAdPyvm400
Probability of Mutual Exclusive Events:
https://www.youtube.com/watch?v=X6usGgwXFyU
Multiplication and Addition Rule:
https://www.youtube.com/watch?v=94AmzeR9n2w
Compound Probability:
https://www.youtube.com/watch?v=EHU6pVSczb4
Expected Value:
https://www.youtube.com/watch?v=b6VK2VPMXNI
Probability Tree Diagrams:
https://www.youtube.com/watch?v=w4wKXVwtGac
___________________________________
Bayes Theorem:
https://www.youtube.com/watch?v=OByl4RJxnKA
Probability - Binomial Distribution:
https://www.youtube.com/watch?v=3PWKQiLK41M
Probability - Geometric Distribution:
https://www.youtube.com/watch?v=d5iAWPnrH6w
Probability - Poisson Distribution:
https://www.youtube.com/watch?v=m0o-585xwW0
Continuous Probability Distributions:
https://www.youtube.com/watch?v=QxqxdQ_g2uw
Probability Density Functions:
https://www.youtube.com/watch?v=3xAIWiTJCvE
__________________________________
Probability - Uniform Distributions:
https://www.youtube.com/watch?v=KfunVw-0AH0
Probability - Exponential Distributions:
https://www.youtube.com/watch?v=J3KSjZFVbis
Probability - Normal Distributions (Calculus):
https://www.youtube.com/watch?v=gHBL5Zau3NE
Probability - Standard Normal Distributions:
https://www.youtube.com/watch?v=CjF_yQ2N638
Probability - The Law of Large Numbers:
https://www.youtube.com/watch?v=ihTpK6dXSas
___________________________________
Final Exams and Video Playlists:
https://www.video-tutor.net/
Full-Length Videos and Worksheets:
https://www.patreon.com/MathScienceTutor/collections
This video provides an introduction to probability. It explains how to calculate the probability of an event occurring in addition to determining the sample space of an event using tree diagrams.
Probability - Free Formula Sheet: https://bit.ly/3zb22rW
______________________________
Introduction to Probability:
https://www.youtube.com/watch?v=SkidyDQuupA
Probability Formulas:
https://www.youtube.com/watch?v=bddckR734aM
Probability Explained:
https://www.youtube.com/watch?v=UORztmWGY6Q
Probability With Geometry:
https://www.youtube.com/watch?v=DeH5aHWxEaI
Probability of Complementary Events:
https://www.youtube.com/watch?v=0T-CaQCiSf4
Conditional Probability:
https://www.youtube.com/watch?v=sqDVrXq_eh0
__________________________________
Independent and Dependent Events:
https://www.youtube.com/watch?v=lWAdPyvm400
Probability of Mutual Exclusive Events:
https://www.youtube.com/watch?v=X6usGgwXFyU
Multiplication and Addition Rule:
https://www.youtube.com/watch?v=94AmzeR9n2w
Compound Probability:
https://www.youtube.com/watch?v=EHU6pVSczb4
Expected Value:
https://www.youtube.com/watch?v=b6VK2VPMXNI
Probability Tree Diagrams:
https://www.youtube.com/watch?v=w4wKXVwtGac
___________________________________
Bayes Theorem:
https://www.youtube.com/watch?v=OByl4RJxnKA
Probability - Binomial Distribution:
https://www.youtube.com/watch?v=3PWKQiLK41M
Probability - Geometric Distribution:
https://www.youtube.com/watch?v=d5iAWPnrH6w
Probability - Poisson Distribution:
https://www.youtube.com/watch?v=m0o-585xwW0
Continuous Probability Distributions:
https://www.youtube.com/watch?v=QxqxdQ_g2uw
Probability Density Functions:
https://www.youtube.com/watch?v=3xAIWiTJCvE
__________________________________
Probability - Uniform Distributions:
https://www.youtube.com/watch?v=KfunVw-0AH0
Probability - Exponential Distributions:
https://www.youtube.com/watch?v=J3KSjZFVbis
Probability - Normal Distributions (Calculus):
https://www.youtube.com/watch?v=gHBL5Zau3NE
Probability - Standard Normal Distributions:
https://www.youtube.com/watch?v=CjF_yQ2N638
Probability - The Law of Large Numbers:
https://www.youtube.com/watch?v=ihTpK6dXSas
___________________________________
Final Exams and Video Playlists:
https://www.video-tutor.net/
Full-Length Videos and Worksheets:
https://www.patreon.com/MathScienceTutor/collections
Slides: https://drive.google.com/file/d/1hNaqfVSyulytj7RM-DjU8xZrxWPC5xtq/view
Set Theory Today: A conference in honor of Georg Cantor
September 10-14, 2018
Vi...
Slides: https://drive.google.com/file/d/1hNaqfVSyulytj7RM-DjU8xZrxWPC5xtq/view
Set Theory Today: A conference in honor of Georg Cantor
September 10-14, 2018
Vienna, Austria
Conference website: https://sites.google.com/view/set-theory-today/startseite?authuser=0
Video and editing: Daniel T. Soukup
Slides: https://drive.google.com/file/d/1hNaqfVSyulytj7RM-DjU8xZrxWPC5xtq/view
Set Theory Today: A conference in honor of Georg Cantor
September 10-14, 2018
Vienna, Austria
Conference website: https://sites.google.com/view/set-theory-today/startseite?authuser=0
Video and editing: Daniel T. Soukup
(06 Mai 2022/May 06, 2022) Colloque des sciences mathématiques du Québec. http://www.crm.umontreal.ca/cal/en/jour20220506.html
Slawomir Solecki (Cornell Univer...
(06 Mai 2022/May 06, 2022) Colloque des sciences mathématiques du Québec. http://www.crm.umontreal.ca/cal/en/jour20220506.html
Slawomir Solecki (Cornell University): Generic measure preserving transformations and descriptive set theory
Abstract: The behavior of a measure preserving transformation, even a generic one, is highly non-uniform. In contrast to this observation, a different picture of a very uniform behavior of the closed group generated by a generic measure preserving transformation has emerged. This picture included substantial evidence that pointed to these groups being all topologically isomorphic to a single group, namely, $L^0$---the non-locally compact, topological group of all Lebesgue measurable functions from $[0,1]$ to the circle. In fact, Glasner and Weiss asked if this was the case.
We will describe the background touched on above, including the connections with Descriptive Set Theory. Further, we will indicate a proof of the following theorem that answers the Glasner--Weiss question in the negative: for a generic measure preserving transformation $T$, the closed group generated by $T$ is { f not} topologically isomorphic to $L^0$.
(06 Mai 2022/May 06, 2022) Colloque des sciences mathématiques du Québec. http://www.crm.umontreal.ca/cal/en/jour20220506.html
Slawomir Solecki (Cornell University): Generic measure preserving transformations and descriptive set theory
Abstract: The behavior of a measure preserving transformation, even a generic one, is highly non-uniform. In contrast to this observation, a different picture of a very uniform behavior of the closed group generated by a generic measure preserving transformation has emerged. This picture included substantial evidence that pointed to these groups being all topologically isomorphic to a single group, namely, $L^0$---the non-locally compact, topological group of all Lebesgue measurable functions from $[0,1]$ to the circle. In fact, Glasner and Weiss asked if this was the case.
We will describe the background touched on above, including the connections with Descriptive Set Theory. Further, we will indicate a proof of the following theorem that answers the Glasner--Weiss question in the negative: for a generic measure preserving transformation $T$, the closed group generated by $T$ is { f not} topologically isomorphic to $L^0$.
Oxford Set Theory Seminar/ Bristol Logic and Set Theory Seminar
http://jdh.hamkins.org/oxford-set-theory-seminar/
Aronszajn trees are a staple of set theory, but there are applications where the requirement of all levels being countable is of no importance. This is the case in set-theoretic model theory, where trees of height and size ω1 but with no uncountable branches play an important role by being clocks of Ehrenfeucht–Fraïssé games that measure similarity of model of size ℵ1. We call such trees wide Aronszajn. In this context one can also compare trees T and T’ by saying that T weakly embeds into T’ if there is a function f that map T into T’ while preserving the strict order <_T. This order translates into the comparison of winning strategies for the isomorphism player, where any winning strategy for T’ translates into a winning strategy for T’. Hence it is natural to ask if there is a largest such tree, or as we would say, a universal tree for the class of wood Aronszajn trees with weak embeddings. It was known that there is no such a tree under CH, but in 1994 Mekler and Väänanen conjectured that there would be under MA(ω1).
We introduce the basics of set theory and do some practice problems.
This video is an updated version of the original video released over two years ago. Hopefully the higher pen quality and refined explanations are beneficial for your learning. If you'd like to see more videos redone in the series, please leave a comment down below.
#DiscreteMath #Mathematics #settheory
Looking for paid tutoring or online courses with practice exercises, text lectures, solutions, and exam practice? http://TrevTutor.com has you covered!
Support me on Patreon: http://bit.ly/2EUdAl3
Visit my website: http://bit.ly/1zBPlvm
Subscribe on YouTube: http://bit.ly/1vWiRxW
*--Playlists--*
Discrete Mathematics 1: https://www.youtube.com/playlist?list=PLDDGPdw7e6Ag1EIznZ-m-qXu4XX3A0cIz
Discrete Mathematics 2: https://www.youtube.com/playlist?list=PLDDGPdw7e6Aj0amDsYInT_8p6xTSTGEi2
*--Recommended Textbooks--*
Discrete and Combinatorial Mathematics (Grimaldi): https://amzn.to/2T0iC53
Discrete Mathematics (Johnsonbaugh): https://amzn.to/2Hh7H41
Discrete Mathematics and Its Applications (Rosen): https://amzn.to/3lUgrMI
Book of Proof (Hammack): https://amzn.to/35eEbVg
Hello, welcome to TheTrevTutor. I'm here to help you learn your college courses in an easy, efficient manner. If you like what you see, feel free to subscribe and follow me for updates. If you have any questions, leave them below. I try to answer as many questions as possible. If something isn't quite clear or needs more explanation, I can easily make additional videos to satisfy your need for knowledge and understanding.
This video provides an introduction to probability. It explains how to calculate the probability of an event occurring in addition to determining the sample space of an event using tree diagrams.
Probability - Free Formula Sheet: https://bit.ly/3zb22rW
______________________________
Introduction to Probability:
https://www.youtube.com/watch?v=SkidyDQuupA
Probability Formulas:
https://www.youtube.com/watch?v=bddckR734aM
Probability Explained:
https://www.youtube.com/watch?v=UORztmWGY6Q
Probability With Geometry:
https://www.youtube.com/watch?v=DeH5aHWxEaI
Probability of Complementary Events:
https://www.youtube.com/watch?v=0T-CaQCiSf4
Conditional Probability:
https://www.youtube.com/watch?v=sqDVrXq_eh0
__________________________________
Independent and Dependent Events:
https://www.youtube.com/watch?v=lWAdPyvm400
Probability of Mutual Exclusive Events:
https://www.youtube.com/watch?v=X6usGgwXFyU
Multiplication and Addition Rule:
https://www.youtube.com/watch?v=94AmzeR9n2w
Compound Probability:
https://www.youtube.com/watch?v=EHU6pVSczb4
Expected Value:
https://www.youtube.com/watch?v=b6VK2VPMXNI
Probability Tree Diagrams:
https://www.youtube.com/watch?v=w4wKXVwtGac
___________________________________
Bayes Theorem:
https://www.youtube.com/watch?v=OByl4RJxnKA
Probability - Binomial Distribution:
https://www.youtube.com/watch?v=3PWKQiLK41M
Probability - Geometric Distribution:
https://www.youtube.com/watch?v=d5iAWPnrH6w
Probability - Poisson Distribution:
https://www.youtube.com/watch?v=m0o-585xwW0
Continuous Probability Distributions:
https://www.youtube.com/watch?v=QxqxdQ_g2uw
Probability Density Functions:
https://www.youtube.com/watch?v=3xAIWiTJCvE
__________________________________
Probability - Uniform Distributions:
https://www.youtube.com/watch?v=KfunVw-0AH0
Probability - Exponential Distributions:
https://www.youtube.com/watch?v=J3KSjZFVbis
Probability - Normal Distributions (Calculus):
https://www.youtube.com/watch?v=gHBL5Zau3NE
Probability - Standard Normal Distributions:
https://www.youtube.com/watch?v=CjF_yQ2N638
Probability - The Law of Large Numbers:
https://www.youtube.com/watch?v=ihTpK6dXSas
___________________________________
Final Exams and Video Playlists:
https://www.video-tutor.net/
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Set Theory Today: A conference in honor of Georg Cantor
September 10-14, 2018
Vienna, Austria
Conference website: https://sites.google.com/view/set-theory-today/startseite?authuser=0
Video and editing: Daniel T. Soukup
(06 Mai 2022/May 06, 2022) Colloque des sciences mathématiques du Québec. http://www.crm.umontreal.ca/cal/en/jour20220506.html
Slawomir Solecki (Cornell University): Generic measure preserving transformations and descriptive set theory
Abstract: The behavior of a measure preserving transformation, even a generic one, is highly non-uniform. In contrast to this observation, a different picture of a very uniform behavior of the closed group generated by a generic measure preserving transformation has emerged. This picture included substantial evidence that pointed to these groups being all topologically isomorphic to a single group, namely, $L^0$---the non-locally compact, topological group of all Lebesgue measurable functions from $[0,1]$ to the circle. In fact, Glasner and Weiss asked if this was the case.
We will describe the background touched on above, including the connections with Descriptive Set Theory. Further, we will indicate a proof of the following theorem that answers the Glasner--Weiss question in the negative: for a generic measure preserving transformation $T$, the closed group generated by $T$ is { f not} topologically isomorphic to $L^0$.
In descriptive set theory, a tree on a set is a collection of finite sequences of elements of such that every prefix of a sequence in the collection also belongs to the collection.
Definitions
Trees
The collection of all finite sequences of elements of a set is denoted .
With this notation, a tree is a nonempty subset of , such that if
is a sequence of length in , and if ,
then the shortened sequence also belongs to . In particular, choosing shows that the empty sequence belongs to every tree.
Branches and bodies
A branch through a tree is an infinite sequence of elements of , each of whose finite prefixes belongs to . The set of all branches through is denoted and called the body of the tree .
A tree that has no branches is called wellfounded; a tree with at least one branch is illfounded. By König's lemma, a tree on a finite set with an infinite number of sequences must necessarily be illfounded.
Terminal nodes
A finite sequence that belongs to a tree is called a terminal node if it is not a prefix of a longer sequence in . Equivalently, is terminal if there is no element of such that that . A tree that does not have any terminal nodes is called pruned.