analysis

analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … … constructive mathematics, realizability, computability propositions as types, proofs as programs, computational trinitarianism basic constructions: strong axioms further In real ana…

I am trying to understand the meaning of the Cauchy principal value in the context of real analysis. I should mention that I have not formally studied this concept before, and I have not taken a ...

Just a brief announcement that I have been working with Quanta Books to publish a short book in popular mathematics entitled “Six Math Essentials“, which will cover six of the fundamental concepts in mathematics — numbers, algebra, geometry, probability, analysis, and dynamics — and how they connect with our real-world intuition, the history of math and science, and to modern practice of mathemat…

Many problems in analysis (as well as adjacent fields such as combinatorics, theoretical computer science, and PDE) are interested in the order of growth (or decay) of some quantity that depends on one or more asymptotic parameters (such as ) – for instance, whether the quantity grows or decays linearly, quadratically, polynomially, exponentially, etc. in […]

Suppose a function f(z) equals 0 at z = 0, 1, 2, 3, …. Under what circumstances might you be able to conclude that f is zero everywhere? Clearly you need some hypothesis on f. For example, the function sin(πz) is zero at every integer but certainly not constantly zero. Carlson’s theorem says that if […] The post When zeros at natural numbers implies zero everywhere first appeared on John D. Cook .

I was reading a theorem giving conditions for a divergent series to have a convergent subseries and had a sort of flashback. I studied nonlinear PDEs in grad school, which amounted to applied functional analysis. We were constantly proving or using theorems about sequences having convergent subsequences, often subsequences that converged in a very weak […] The post Convergent subsequence first ap…

The Elias M. Stein Prize for New Perspectives in Analysis is awarded for the development of groundbreaking methods in analysis which demonstrate promise to revitalize established areas or create new opportunities for mathematical discovery. The current prize amount is US$5,000 and the prize is awarded every three years for work published in the preceding six […]

Let f be an analytic function on the unit disk with f(0) = 0 and derivative f ′(0) = 1. If f is one-to-one (injective) then this puts a strict limit on the size of the series coefficients. Let an be the nth coefficient in the power series for f centered at 0. If f is one-to-one […] The post Bounds on power series coefficients first appeared on John D. Cook .

A bump function is a smooth (i.e. infinitely differentiable) function that is positive on some open interval (a, b) and zero outside that interval. I mentioned bump functions a few weeks ago and discussed how they could be used to prevent clicks in radio transmissions. Today I ran into a twitter thread that gave a […] The post Bump functions first appeared on John D. Cook .

A sine wave is the canonical periodic function, so an obvious way to create a periodic function of two variables would be to multiply two sine waves: f(x, y) = sin(x) sin(y) This function is doubly periodic: periodic in the horizontal and vertical directions. Now suppose you want to construct a doubly periodic function of […] The post Doubly periodic but not analytic first appeared on John D. Coo…

A few months ago I posted a question about analytic functions that I received from a bright high school student, which turned out to be studied and resolved by de Bruijn. Based on this positive resolution, I thought I might try my luck again and list three further questions that this student asked which do […]

I was asked the following interesting question from a bright high school student I am working with, to which I did not immediately know the answer: Question 1 Does there exist a smooth function which is not real analytic, but such that all the differences are real analytic for every ? The hypothesis implies that […]

Just a short note that the memorial article “Analysis and applications: The mathematical work of Elias Stein” has just been published in the Bulletin of the American Mathematical Society.  This article was a collective effort led by Charlie Fefferman, Alex Ionescu, Steve Wainger and myself to describe the various mathematical contributions of Elias Stein, who […]

analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … … The intermediate value theorem (IVT) is a fundamental principle of analysis which allows one to find a desired value by interpolation. It says that a continuous function from an int…

Also known as Rudin’s Infamous Tiny Torture Box. Trigger Warning: Baby Rudin. I. For my undergrad, I majored in Applied Mathematics at a university that was fairly well known for its mathematics department. One of the constants in “Applied Mathematics” is the use of calculus for a variety of applications. “The real world”. In statistics and … Continue reading Book Review: Principles of Mathematic…

analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … … constructive mathematics, realizability, computability propositions as types, proofs as programs, computational trinitarianism basic constructions: strong axioms further In real ana…

Summary of Research The analysis research group focuses on complex analysis, geometric analysis, geometric measure theory, harmonic analysis, operator theory, and partial differential equations. Faculty Members Simon Bortz Prof. Bortz’ research is at the interface of harmonic analysis, partial differential equations and geometric measure theory. In particular, he is interested in free boundary pr…

The gauge integral (Henstock–Kurzweil) The current (pure) mathematics curriculum at the university is well-established. Most of the choices made are sensible, but some important topics are still usually not taught. Some of these topics are obscure and not well known even to many professional mathematicians; others are known to specialists but for some reason are...

Prerequisites If you wish to follow this guide, you should be familiar with analysis on ##\mathbb{R}## and ##\mathbb{R}^n##. See my previous insight for the list of prerequisite topics and book suggestions: https://www.physicsforums.com/insights/self-study-analysis-part-intro-analysis/ You should also be comfortable with linear algebra; see my insight on that: https://www.physicsforums.com/insigh…

This is Part 2 of a series of articles in which the goal is to exhibit a continuous function that is nowhere differentiable and to explore some interesting properties of a special type of Fourier series. In Part 1, we defined the Weierstrass function, $$w(x) = \sum_{k=0}^{\infty}\alpha^k \cos(\beta^k)$$ We showed that the series is absolutely...