Abstract algebra | Polynomials

Harmonic polynomial

In mathematics, in abstract algebra, a multivariate polynomial p over a field such that the Laplacian of p is zero is termed a harmonic polynomial. The harmonic polynomials form a vector subspace of the vector space of polynomials over the field. In fact, they form a graded subspace. For the real field, the harmonic polynomials are important in mathematical physics. The Laplacian is the sum of second partials with respect to all the variables, and is an invariant differential operator under the action of the orthogonal group via the group of rotations. The standard states that every multivariate polynomial over a field can be decomposed as a finite sum of products of a and a harmonic polynomial. This is equivalent to the statement that the polynomial ring is a free module over the ring of radial polynomials. (Wikipedia).

Video thumbnail

Classify a polynomial then determining if it is a polynomial or not

👉 Learn how to determine whether a given equation is a polynomial or not. A polynomial function or equation is the sum of one or more terms where each term is either a number, or a number times the independent variable raised to a positive integer exponent. A polynomial equation of functio

From playlist Is it a polynomial or not?

Video thumbnail

Determining if a equation is a polynomial or not

👉 Learn how to determine whether a given equation is a polynomial or not. A polynomial function or equation is the sum of one or more terms where each term is either a number, or a number times the independent variable raised to a positive integer exponent. A polynomial equation of functio

From playlist Is it a polynomial or not?

Video thumbnail

Is it a polynomial with two variables

👉 Learn how to determine whether a given equation is a polynomial or not. A polynomial function or equation is the sum of one or more terms where each term is either a number, or a number times the independent variable raised to a positive integer exponent. A polynomial equation of functio

From playlist Is it a polynomial or not?

Video thumbnail

Learn how to identify if a function is a polynomial and identify the degree and LC

👉 Learn how to determine whether a given equation is a polynomial or not. A polynomial function or equation is the sum of one or more terms where each term is either a number, or a number times the independent variable raised to a positive integer exponent. A polynomial equation of functio

From playlist Is it a polynomial or not?

Video thumbnail

Determining if a function is a polynomial or not then determine degree and LC

👉 Learn how to determine whether a given equation is a polynomial or not. A polynomial function or equation is the sum of one or more terms where each term is either a number, or a number times the independent variable raised to a positive integer exponent. A polynomial equation of functio

From playlist Is it a polynomial or not?

Video thumbnail

Ch4 Pr4: Taylor Polynomial of a polynomial

The Taylor Polynomial to a function about x=a is a polynomial expressed in powers of (x-a). This example is from Chapter 4 Problem 4a,b in the MATH1231/1241 Calculus notes. Presented by Dr Daniel Mansfield from the UNSW School of Mathematics and Statistics.

From playlist Mathematics 1B (Calculus)

Video thumbnail

Classifying a polynomial based on its degree and number of terms

👉 Learn how to classify polynomials. A polynomial is an expression of the sums/differences of two or more terms having different integer exponents of the same variable. A polynomial can be classified in two ways: by the number of terms and by its degree. A monomial is an expression of 1

From playlist Classify Polynomials | Equations

Video thumbnail

Labeling a polynomial based on the degree and number of terms

👉 Learn how to classify polynomials. A polynomial is an expression of the sums/differences of two or more terms having different integer exponents of the same variable. A polynomial can be classified in two ways: by the number of terms and by its degree. A monomial is an expression of 1

From playlist Classify Polynomials | Equations

Video thumbnail

Harmonic measure: Algorithms and applications – Christopher Bishop – ICM2018

Analysis and Operator Algebras Invited Lecture 8.12 Harmonic measure: Algorithms and applications Christopher Bishop Abstract: This is a brief survey of results related to planar harmonic measure, roughly from Makarov’s results of the 1980’s to recent applications involving 4-manifolds,

From playlist Analysis & Operator Algebras

Video thumbnail

8. Quantum Mechanical Harmonic Oscillator

MIT 5.61 Physical Chemistry, Fall 2017 Instructor: Professor Robert Field View the complete course: https://ocw.mit.edu/5-61F17 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP62RsEHXe48Imi9-87FzQaJg This lecture covers the quantum mechanical treatment of the harmonic o

From playlist MIT 5.61 Physical Chemistry, Fall 2017

Video thumbnail

Xavier Ros-Oton: Regularity of free boundaries in obstacle problems, Lecture III

Free boundary problems are those described by PDE that exhibit a priori unknown (free) interfaces or boundaries. Such type of problems appear in Physics, Geometry, Probability, Biology, or Finance, and the study of solutions and free boundaries uses methods from PDE, Calculus of Variations

From playlist Hausdorff School: Trending Tools

Video thumbnail

Sarah Post: Rational extensions of superintegrable systems, exceptional polynomials & Painleve eq.s

Abstract: In this talk, I will discuss recent work with Ian Marquette and Lisa Ritter on superintegable extensions of a Smorodinsky Winternitz potential associated with exception orthogonal polynomials (EOPs). EOPs are families of orthogonal polynomials that generalize the classical ones b

From playlist Integrable Systems 9th Workshop

Video thumbnail

Quadratic differentials and measured foliations on Riemann surfaces by Subhojoy Gupta

Program : Integrable? ?systems? ?in? ?Mathematics,? ?Condensed? ?Matter? ?and? ?Statistical? ?Physics ORGANIZERS : Alexander Abanov, Rukmini Dey, Fabian Essler, Manas Kulkarni, Joel Moore, Vishal Vasan and Paul Wiegmann DATE & TIME : 16 July 2018 to 10 August 2018 VENUE : Ramanujan L

From playlist Integrable​ ​systems​ ​in​ ​Mathematics,​ ​Condensed​ ​Matter​ ​and​ ​Statistical​ ​Physics

Video thumbnail

Classify a polynomial and determine degree and Leading coefficient

👉 Learn how to classify polynomials. A polynomial is an expression of the sums/differences of two or more terms having different integer exponents of the same variable. A polynomial can be classified in two ways: by the number of terms and by its degree. A monomial is an expression of 1

From playlist Classify Polynomials | Equations

Video thumbnail

Chem 131A. Lec 08. Quantum Principles: More on Vibrations and Approximation Techniques

UCI Chem 131A Quantum Principles (Winter 2014) Lec 08. Quantum Principles -- More on Vibrations and Approximation Techniques -- View the complete course: http://ocw.uci.edu/courses/chem_131a_quantum_principles.html Instructor: A.J. Shaka, Ph.D License: Creative Commons BY-NC-SA Terms of U

From playlist Chemistry 131A: Quantum Principles

Video thumbnail

Classify a polynomial and determine degree and leading coefficient

👉 Learn how to classify polynomials. A polynomial is an expression of the sums/differences of two or more terms having different integer exponents of the same variable. A polynomial can be classified in two ways: by the number of terms and by its degree. A monomial is an expression of 1

From playlist Classify Polynomials | Equations

Related pages

Differential operator | Polynomial | Abstract algebra | Orthogonal group | Multilinear polynomial | Mathematics | Vector space | Invariant (mathematics) | Zonal spherical harmonics | Real number | Free module | Spherical harmonics | Group (mathematics) | Harmonic function