In mathematics, more precisely in the theory of functions of several complex variables, a pseudoconvex set is a special type of open set in the n-dimensional complex space Cn. Pseudoconvex sets are important, as they allow for classification of domains of holomorphy. Let be a domain, that is, an open connected subset. One says that is pseudoconvex (or Hartogs pseudoconvex) if there exists a continuous plurisubharmonic function on such that the set is a relatively compact subset of for all real numbers In other words, a domain is pseudoconvex if has a continuous plurisubharmonic . Every (geometrically) convex set is pseudoconvex. However, there are pseudoconvex domains which are not geometrically convex. When has a (twice continuously differentiable) boundary, this notion is the same as Levi pseudoconvexity, which is easier to work with. More specifically, with a boundary, it can be shown that has a defining function; i.e., that there exists which is so that , and . Now, is pseudoconvex iff for every and in the complex tangent space at p, that is, , we have If does not have a boundary, the following approximation result can be useful. Proposition 1 If is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains with (smooth) boundary which are relatively compact in , such that This is because once we have a as in the definition we can actually find a C∞ exhaustion function. (Wikipedia).
Metacognition and speaking | Introduction | Part 1
In this video, I provide an overview of metacognition and discuss its role in speaking.
From playlist Metacognition
Convolution Theorem: Fourier Transforms
Free ebook https://bookboon.com/en/partial-differential-equations-ebook Statement and proof of the convolution theorem for Fourier transforms. Such ideas are very important in the solution of partial differential equations.
From playlist Partial differential equations
Is the function continuous or not
👉 Learn how to determine whether a function is continuos or not. A function is said to be continous if two conditions are met. They are: the limit of the function exist and that the value of the function at the point of continuity is defined and is equal to the limit of the function. Other
From playlist Is the Functions Continuous or Not?
Miroslav Englis: Analytic continuation of Toeplitz operators
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
What is multiplicity and what does it mean for the zeros of a graph
👉 Learn about zeros and multiplicity. The zeroes of a polynomial expression are the values of x for which the graph of the function crosses the x-axis. They are the values of the variable for which the polynomial equals 0. The multiplicity of a zero of a polynomial expression is the number
From playlist Zeros and Multiplicity of Polynomials | Learn About
Berit Stensønes: Sup-norm estimates for $\overline{\partial }$ in $\mathbb{C}^{3}$
I will talk about recent joint work with Dusty Grundmeier and Lars Simon. We have proved sup-norm estimates for dbar on a wide class of pseudoconvex domains in $\mathbb{C}^{3}$, including all known examples of bounded, pseudoconvex domains with real-analytic boundary of finite D’Angelo typ
From playlist Numerical Analysis and Scientific Computing
Dror Varolin - Minicourse - Lecture 5
No Audio Dror Varolin Variations of Holomorphic Hilbert spaces Traditional complex analysis focuses on a single space, like a domain in Euclidean space, or more generally a complex manifold, and studies holomorphic maps on that space, into some target space. The typical target space for
From playlist Maryland Analysis and Geometry Atelier
Proof of the Convolution Theorem
Proof of the Convolution Theorem, The Laplace Transform of a convolution is the product of the Laplace Transforms, changing order of the double integral, proving the convolution theorem, www.blackpenredpen.com
From playlist Convolution & Laplace Transform (Nagle Sect7.7)
Bruno Iochum: Spectral triples and Toeplitz operators
I will give examples of spectral triples constructed using the algebra of Toeplitz operators on smoothly bounded strictly pseudoconvex domains in Cn, or the star product for the Berezin-Toeplitz quantization. The main tool is the theory of generalized Toeplitz operators on the boundary of
From playlist HIM Lectures: Trimester Program "Non-commutative Geometry and its Applications"
In this video, I provide some intuition behind the concept of convolution, and show how the convolution of two functions is really the continuous analog of polynomial multiplication. Enjoy!
From playlist Real Analysis
Franc Forstnerič - Non singular holomorphic foliations on Stein manifolds (Part 3)
Non singular holomorphic foliations on Stein manifolds (Part 3)
From playlist École d’été 2012 - Feuilletages, Courbes pseudoholomorphes, Applications
What is the multiplicity of a zero?
👉 Learn about zeros and multiplicity. The zeroes of a polynomial expression are the values of x for which the graph of the function crosses the x-axis. They are the values of the variable for which the polynomial equals 0. The multiplicity of a zero of a polynomial expression is the number
From playlist Zeros and Multiplicity of Polynomials | Learn About
Math 139 Fourier Analysis Lecture 05: Convolutions and Approximation of the Identity
Convolutions and Good Kernels. Definition of convolution. Convolution with the n-th Dirichlet kernel yields the n-th partial sum of the Fourier series. Basic properties of convolution; continuity of the convolution of integrable functions.
From playlist Course 8: Fourier Analysis
What are zeros of a polynomial
👉 Learn about zeros and multiplicity. The zeroes of a polynomial expression are the values of x for which the graph of the function crosses the x-axis. They are the values of the variable for which the polynomial equals 0. The multiplicity of a zero of a polynomial expression is the number
From playlist Zeros and Multiplicity of Polynomials | Learn About
The Monge - Ampère equations, the Bergman kernel... (Lecture 3)by Kengo Hirachi
PROGRAM CAUCHY-RIEMANN EQUATIONS IN HIGHER DIMENSIONS ORGANIZERS: Sivaguru, Diganta Borah and Debraj Chakrabarti DATE: 15 July 2019 to 02 August 2019 VENUE: Ramanujan Lecture Hall, ICTS Bangalore Complex analysis is one of the central areas of modern mathematics, and deals with holomo
From playlist Cauchy-Riemann Equations in Higher Dimensions 2019
Dror Varolin - Minicourse - Lecture 3
Dror Varolin Variations of Holomorphic Hilbert spaces Traditional complex analysis focuses on a single space, like a domain in Euclidean space, or more generally a complex manifold, and studies holomorphic maps on that space, into some target space. The typical target space for a domain i
From playlist Maryland Analysis and Geometry Atelier
Dror Varolin - Minicourse - Lecture 4
Dror Varolin Variations of Holomorphic Hilbert spaces Traditional complex analysis focuses on a single space, like a domain in Euclidean space, or more generally a complex manifold, and studies holomorphic maps on that space, into some target space. The typical target space for a domain i
From playlist Maryland Analysis and Geometry Atelier
Dror Varolin - Minicourse - Lecture 2
Dror Varolin Variations of Holomorphic Hilbert spaces Traditional complex analysis focuses on a single space, like a domain in Euclidean space, or more generally a complex manifold, and studies holomorphic maps on that space, into some target space. The typical target space for a domain i
From playlist Maryland Analysis and Geometry Atelier
Overview of zeros of a polynomial - Online Tutor - Free Math Videos
👉 Learn about zeros and multiplicity. The zeroes of a polynomial expression are the values of x for which the graph of the function crosses the x-axis. They are the values of the variable for which the polynomial equals 0. The multiplicity of a zero of a polynomial expression is the number
From playlist Zeros and Multiplicity of Polynomials | Learn About
Dror Varolin - Minicourse - Lecture 1
Dror Varolin Variations of Holomorphic Hilbert spaces Traditional complex analysis focuses on a single space, like a domain in Euclidean space, or more generally a complex manifold, and studies holomorphic maps on that space, into some target space. The typical target space for a domain i
From playlist Maryland Analysis and Geometry Atelier