In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra. There are numerous ways to multiply two Euclidean vectors. The dot product takes in two vectors and returns a scalar, while the cross product returns a pseudovector. Both of these have various significant geometric interpretations and are widely used in mathematics, physics, and engineering. The dyadic product takes in two vectors and returns a second order tensor called a dyadic in this context. A dyadic can be used to contain physical or geometric information, although in general there is no direct way of geometrically interpreting it. The dyadic product is distributive over vector addition, and associative with scalar multiplication. Therefore, the dyadic product is linear in both of its operands. In general, two dyadics can be added to get another dyadic, and multiplied by numbers to scale the dyadic. However, the product is not commutative; changing the order of the vectors results in a different dyadic. The formalism of dyadic algebra is an extension of vector algebra to include the dyadic product of vectors. The dyadic product is also associative with the dot and cross products with other vectors, which allows the dot, cross, and dyadic products to be combined to obtain other scalars, vectors, or dyadics. It also has some aspects of matrix algebra, as the numerical components of vectors can be arranged into row and column vectors, and those of second order tensors in square matrices. Also, the dot, cross, and dyadic products can all be expressed in matrix form. Dyadic expressions may closely resemble the matrix equivalents. The dot product of a dyadic with a vector gives another vector, and taking the dot product of this result gives a scalar derived from the dyadic. The effect that a given dyadic has on other vectors can provide indirect physical or geometric interpretations. Dyadic notation was first established by Josiah Willard Gibbs in 1884. The notation and terminology are relatively obsolete today. Its uses in physics include continuum mechanics and electromagnetism. In this article, upper-case bold variables denote dyadics (including dyads) whereas lower-case bold variables denote vectors. An alternative notation uses respectively double and single over- or underbars. (Wikipedia).
The Two-Dimensional Discrete Fourier Transform
The two-dimensional discrete Fourier transform (DFT) is the natural extension of the one-dimensional DFT and describes two-dimensional signals like images as a weighted sum of two dimensional sinusoids. Two-dimensional sinusoids have a horizontal frequency component and a vertical frequen
From playlist Fourier
Calculus 3: Tensors (3 of 28) What is a Dyad? A Graphical Representation
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain the physical graphical representation of a tensor of rank 2, or a dyad. A tensor of rank 2 has 9 components, which means there will be 3 vectors each representing a force or stress or somethin
From playlist CALCULUS 3 CH 10 TENSORS
Physics 2 - Motion In One-Dimension (7 of 22) Definition of dx/dt
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain the definition of dx/dt.
From playlist PHYSICS - MECHANICS
Calculus 3: Tensors (4 of 28) The Dyad: 3 Vectors Define "Stress" at the 3 Planes
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain a dyad, a tensor of rank 2, by drawing 3 vectors, each on the surface of a cube representing a piece of a beam. Next video in the series can be seen at: https://youtu.be/EgMpMKXZ4lo
From playlist CALCULUS 3 CH 10 TENSORS
The Discrete Fourier Transform
This video provides a basic introduction to the very widely used and important discrete Fourier transform (DFT). The DFT describes discrete-time signals as a weighted sum of complex sinusoid building blocks and is used in applications such as GPS, MP3, JPEG, and WiFi.
From playlist Fourier
This video discusses how to compute the Discrete Fourier Transform (DFT) matrix in Matlab and Python. In practice, the DFT should usually be computed using the fast Fourier transform (FFT), which will be described in the next video. Book Website: http://databookuw.com Book PDF: http:
From playlist Data-Driven Science and Engineering
Calculus 9.1 Modeling with Differential Equations
My notes are available at http://asherbroberts.com/ (so you can write along with me). Calculus: Early Transcendentals 8th Edition by James Stewart
From playlist Calculus
This video introduces the concept of phased arrays. An array refers to multiple sensors, arranged in some configuration, that act together to produce a desired sensor pattern. With a phased array, we can electronically steer that pattern without having to physically move the array simply b
From playlist Understanding Phased Array Systems and Beamforming
Gerard Cornuejols: Dyadic linear programming
A finite vector is dyadic if each of its entries is a dyadic rational number, i.e. if it has an exact floating point representation. We study the problem of finding a dyadic optimal solution to a linear program, if one exists. This is joint work with Ahmad Abdi, Bertrand Guenin and Levent
From playlist Workshop: Continuous approaches to discrete optimization
On the dyadic Hilbert transform – Stefanie Petermichl – ICM2018
Analysis and Operator Algebras Invited Lecture 8.10 On the dyadic Hilbert transform Stefanie Petermichl Abstract: The Hilbert transform is an average of dyadic shift operators. These can be seen as a coefficient shift and multiplier in a Haar wavelet expansion or as a time shifted operat
From playlist Analysis & Operator Algebras
Types Of Differential Equations
Introduction to the Order of a differential equation and the idea of Linear and Non-linear differential equations. Discussion of how the number of constants of a general solution is related to the order of a differential equation.
From playlist Mathematical Physics I Uploads
Tuomas Hytönen: Dyadic representation meets operator-valued kernels
Abstract: The dyadic representation theory of singular integrals arose from the work on the A_2 conjecture on sharp weighted inequalities. In this original application, it has now been largely surpassed by the sparse domination technology. I will discuss some other types of applications in
From playlist Follow-up Workshop to TP "Harmonic Analysis and Partial Differential Equations"
Part 8 of the APL study group. Details here: https://forums.fast.ai/t/apl-array-programming/97188 Discuss this session here: https://youtu.be/bRr7V38Oa7o
From playlist fast.ai APL Study Group
Ben Jaye: Reflectionless measures for singular integral operators
The lecture was held within the framework of the (Junior) Hausdorff Trimester Program Harmonic Analysis and Partial Differential Equations. 15.7.2014
From playlist HIM Lectures: Trimester Program "Harmonic Analysis and Partial Differential Equations"
Karma Dajani - An introduction to Ergodic Theory of Numbers (Part 2)
In this course we give an introduction to the ergodic theory behind common number expansions, like expansions to integer and non-integer bases, Luroth series and continued fraction expansion. Starting with basic ideas in ergodic theory such as ergodicity, the ergodic theorem and natural ex
From playlist École d’été 2013 - Théorie des nombres et dynamique
Alexander the Great Family Tree
Get the poster: https://usefulcharts.com/products/ancient-history-family-trees CREDITS: Chart: Matt Baker Script/Narration: Matt Baker Editing: Jack Rackam Intro animation: Syawish Rehman Intro music: "Lord of the Land" by Kevin MacLeod and licensed under Creative Commons Attribution l
From playlist Royal Family Trees
Part 12 of the APL study group. Details here: https://forums.fast.ai/t/apl-array-programming/97188 Discuss this session here: https://forums.fast.ai/t/fast-ai-apl-study-session-11/97949
From playlist fast.ai APL Study Group
Tuomas Hytonen: Two-weight inequalities meet R-boundedness
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Analysis and its Applications
Lecture: Numerical Differentiation Methods
From simple Taylor series expansions, the theory of numerical differentiation is developed.
From playlist Beginning Scientific Computing
Part 9 of the APL study group. Details here: https://forums.fast.ai/t/apl-array-programming/97188 Discuss this session here: https://forums.fast.ai/t/fast-ai-apl-study-session-9/97510
From playlist fast.ai APL Study Group