In diatonic set theory structure implies multiplicity is a quality of a collection or scale. This is that for the interval series formed by the shortest distance around a diatonic circle of fifths between members of a series indicates the number of unique interval patterns (adjacently, rather than around the circle of fifths) formed by diatonic transpositions of that series. Structure being the intervals in relation to the circle of fifths, multiplicity being the number of times each different (adjacent) interval pattern occurs. The property was first described by John Clough and in "Variety and Multiplicity in Diatonic Systems" (1985). Structure implies multiplicity is true of the diatonic collection and the pentatonic scale, and any subset. For example, cardinality equals variety dictates that a three member diatonic subset of the C major scale, C-D-E transposed to all scale degrees gives three interval patterns: M2-M2, M2-m2, m2-M2. On the circle of fifths: C G D A E B F (C) 1 2 1 2 1 2 3 E and C are three notes apart, C and D are two notes apart, D and E two notes apart. Just as the distance around the circle of fifths between forms the interval pattern 3-2-2, M2-M2 occurs three times, M2-m2 occurs twice, and m2-M2 occurs twice. Cardinality equals variety and structure implies multiplicity are true of all collections with Myhill's property or maximal evenness. (Wikipedia).
Fun with lists, multisets and sets III | Data Structures in Mathematics Math Foundations 154
We continue our discussion of data structures in mathematics, now treating the case of multisets or msets. This is an unordered structure in which repetitions are allowed. It turns out that multisets support interesting algebraic structures: notable we are able to add two multisets to get
From playlist Math Foundations
The algebra of natural number multisets | Data structures in Mathematics Math Foundation 157
We introduce some deceptively simple but important notation to deal with multisets/msets from n, for some natural number n. In particular we augment addition of msets with multiples of an mset, and use that to give a list-theoretic description of the multiplicity of various elements that a
From playlist Math Foundations
Multivariable Calculus | Differentiability
We give the definition of differentiability for a multivariable function and provide a few examples. http://www.michael-penn.net https://www.researchgate.net/profile/Michael_Penn5 http://www.randolphcollege.edu/mathematics/
From playlist Multivariable Calculus
11_3_6 Continuity and Differentiablility
Prerequisites for continuity. What criteria need to be fulfilled to call a multivariable function continuous.
From playlist Advanced Calculus / Multivariable Calculus
Not-So-Close Packed Crystal Structures
A description of two crystal structures that are created from not-so-close packed structures.
From playlist Atomic Structures and Bonding
Local linearity for a multivariable function
A visual representation of local linearity for a function with a 2d input and a 2d output, in preparation for learning about the Jacobian matrix.
From playlist Multivariable calculus
Sets and other data structures | Data Structures in Mathematics Math Foundations 151
In mathematics we often want to organize objects. Sets are not the only way of doing this: there are other data types that are also useful and that can be considered together with set theory. In particular when we group objects together, there are two fundamental questions that naturally a
From playlist Math Foundations
Degree Lowering Along Arithmetic Progressions - Borys Kuca
Special Year Informal Seminar Topic: Degree Lowering Along Arithmetic Progressions Speaker: Borys Kuca Affiliation: University of Crete Date: March 06, 2023 Ever since Furstenberg proved his multiple recurrence theorem, the limiting behaviour of multiple ergodic averages along various se
From playlist Mathematics
Pre-recorded lecture 8: Differentially non-degenerate singular points and global theorems
MATRIX-SMRI Symposium: Nijenhuis Geometry and integrable systems Pre-recorded lecture: These lectures were recorded as part of a cooperation between the Chinese-Russian Mathematical Center (Beijing) and the Moscow Center of Fundamental and Applied Mathematics (Moscow). Nijenhuis Geomet
From playlist MATRIX-SMRI Symposium: Nijenhuis Geometry companion lectures (Sino-Russian Mathematical Centre)
Multivariable Calculus | The scalar multiple of a vector.
We define scalar multiplication in the context of 2 and 3 dimensional vectors. We also present a few properties of scalar multiplication and vector addition. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Vectors for Multivariable Calculus
Math 032 Multivariable Calculus 01 090514: Vectors
Vectors; equality, magnitude, direction; parallel vectors. Basic algebraic operations; laws (commutative, etc.); interaction of magnitude with basic operations. Triangle Inequality.
From playlist Course 4: Multivariable Calculus (Fall 2014)
Lec 13 | MIT RES.6-008 Digital Signal Processing, 1975
Lecture 13: Network structures for finite impulse response (FIR) systems and parameter quantization effects in digital filter structures Instructor: Alan V. Oppenheim View the complete course: http://ocw.mit.edu/RES6-008S11 License: Creative Commons BY-NC-SA More information at h
From playlist MIT RES.6-008 Digital Signal Processing, 1975
Mod-01 Lec-06 Introduction to Nanomaterials
Nanostructures and Nanomaterials: Characterization and Properties by Characterization and Properties by Dr. Kantesh Balani & Dr. Anandh Subramaniam,Department of Nanotechnology,IIT Kanpur.For more details on NPTEL visit http://nptel.ac.in.
From playlist IIT Kanpur: Nanostructures and Nanomaterials | CosmoLearning.org
Volume in Seiberg-Witten theory and the existence of two Reeb orbits - Daniel Cristofaro-Gardiner
Daniel Cristofaro-Gardiner University of California, Berkeley; Member, School of Mathematics November 1, 2013 I will discuss recent joint work with Vinicius Gripp and Michael Hutchings relating the volume of any contact three-manifold to the length of certain finite sets of Reeb orbits. I
From playlist Mathematics
Fundamentals of Mathematics - Lecture 14: Signatures, Formulas, Structures, Theories, and Models
course page: http://www.uvm.edu/~tdupuy/logic/Math52-Fall2017.html videography - Eric Melton handouts - DZB, Emory
From playlist Fundamentals of Mathematics
Daniel Huybrechts: Algebraic and arithmetic aspects of twistor spaces
I will recall the well-known notion of twistor spaces for K3 surfaces (and Hyperkähler manifolds) and discuss some natural questions relating to the algebraic and arithmetic geometry of their fibres. Recording during the meeting "The Geometry of Algebraic Varieties" the October 02, 2019 a
From playlist Algebraic and Complex Geometry
Perfectoid spaces (Lecture 3) by Kiran Kedlaya
PERFECTOID SPACES ORGANIZERS: Debargha Banerjee, Denis Benois, Chitrabhanu Chaudhuri, and Narasimha Kumar Cheraku DATE & TIME: 09 September 2019 to 20 September 2019 VENUE: Madhava Lecture Hall, ICTS, Bangalore Scientific committee: Jacques Tilouine (University of Paris, France) Eknath
From playlist Perfectoid Spaces 2019
Mod-03 Lec-13 Second Order Linear Equations Continued I
Ordinary Differential Equations and Applications by A. K. Nandakumaran,P. S. Datti & Raju K. George,Department of Mathematics,IISc Bangalore.For more details on NPTEL visit http://nptel.ac.in.
From playlist IISc Bangalore: Ordinary Differential Equations and Applications | CosmoLearning.org Mathematics
Stack Data Structure - Algorithm
This is an explanation of the dynamic data structure known as a stack. It includes an explanation of how a stack works, along with pseudocode for implementing the push and pop operations with a static array variable.
From playlist Data Structures
Joseph Ayoub - 1/5 Sur la conjecture de conservativité
La conjecture de conservativité affirme qu'un morphisme entre motifs constructibles est un isomorphisme s'il en est ainsi de l'une des ses réalisations classiques (de Rham, ℓ-adique, etc.). Il s'agit d'une conjecture centrale dans la théorie des motifs ayant des conséquences concrètes sur
From playlist Joseph Ayoub - Sur la conjecture de conservativité