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A list of all the posts and pages found on the site. For you robots out there is an XML version available for digesting as well.
Pages
Junekey Jeon's personal website
About me
Posts
Circle is the only shape with the smallest maximum arc-chord ratio
Published:
Given a nontrivial rectifiable closed plane curve $\gamma\colon\mathbb{R}/\ell\mathbb{Z}\to\mathbb{R}^{2}$ of length $\ell$ parameterized by the arclength, the arc-chord constant of $\gamma$ is defined as
A generalization of the Lax-Milgram Theorem
Published:
This is a short note on a generalization of the Lax-Milgram theorem, which is a common tool for proving existence of weak solutions to PDE.
The Fourier transform of the Heaviside step function
Published:
Back in 2010, as an electrical engineering major student I was taking an introductory course on signals and systems taught by my formal advisor. The course was mainly about four different kinds of Fourier transforms (Discrete-Time Fourier Series a.k.a. Discrete Fourier Transform, Discrete-Time Fourier Transform, Continuous-Time Fourier Series, and Continuous-Time Fourier Transform) and some additional topics like the famous sampling theorem.
How to quickly factor out a constant factor from integers
Published:
This post revisits the topic of integer division, building upon the discussion in the previous post. Specifically, I’ll delve into removing trailing zeros in the decimal representation of an input integer, or more broadly, factoring out the highest power of a given constant that divides the input. This exploration stems from the problem of converting floating-point numbers into strings, where certain contemporary algorithms, such as Schubfach and Dragonbox, may yield outputs containing trailing zeros.
On the optimal bounds for integer division by constants
Published:
It is well-known that the integer division is quite a heavy operation on modern CPU’s - so slow, in fact, that it has even become a common wisdom to avoid doing it at ALL cost in performance-critical sections of a program. I do not know why division is particularly hard to optimize from the hardware perspective. I am just guessing, maybe (1) every general algorithm is essentially just a minor variation of the good-old long division, (2) which is almost impossible to parallelize. But whatever, that is not the topic of this post.
Fixed-precision formatting of floating-point numbers
Published:
TL;DR
Faster integer formatting - James Anhalt (jeaiii)’s algorithm
Published:
This post is about an ingenious algorithm for printing integers into decimal strings. It sounds like an extremely simple problem, but it is in fact quite more complicated than one might imagine. Let us more precisely define what we want to do: we take an integer of specific bit-width and a byte buffer, and convert the input integer into a string consisting of its decimal digits, and then write it into the given buffer. For simplicity, we will assume that the integer is unsigned and is of $32$-bits. So, we want to implement the following function written in C++:
char* itoa(std::uint32_t n, char* buffer) {
// Convert n into decimal digit string and write it into buffer.
// Returns the position right next to the last character written.
}
There are numerous algorithms for doing this, and I will dig into a clever algorithm invented by James Anhalt (jeaiii), which seems to be the fastest known algorithm at the point of writing this post.
Continued fractions and their application into fast computation of \(\lfloor nx\rfloor\)
Published:
When I was working on Dragonbox and Grisu-Exact (which are float-to-string conversion algorithms with some nice properties) I had to come up with a fast method for computing things like $\lfloor n\log_{10}2 \rfloor$ or $\lfloor n\log_{2}10 \rfloor$, or more generally $\lfloor nx\rfloor$ for some integer $n$ and a fixed positive real number $x$. Actually, the sign of $x$ isn’t extremely important, but let us just assume $x>0$ for simplicity.
publications
A Generalized Typicality for Abstract Alphabets
Junekey Jeon. 2014. "A Generalized Typicality for Abstract Alphabets." 2014 IEEE International Symposium on Information Theory (ISIT), 2649-2653
A Bayesian sensor fusion scheme for attitude tracking
Junekey Jeon, Hwa-Suk Kim, Woo-Sug Jung and Sun-Joong Kim. 2017. "A Bayesian sensor fusion scheme for attitude tracking." 2017 19th International Conference on Advanced Communication Technology (ICACT)
Spectral clustering with brainstorming process for multi-view data
Jeong-Woo Son, Junekey Jeon, Alex Lee and Sun-Joong Kim. 2017. "Spectral clustering with brainstorming process for multi-view data." Proceedings of the AAAI Conference on Artificial Intelligence, 2548–2554
On evolution of corner-like gSQG patches
Junekey Jeon and In-Jee Jeong. "On Evolution of Corner-Like gSQG Patches." J. Math. Fluid Mech. 25, 35 (2023)
An Improved Regularity Criterion and Absence of Splash-like Singularities for g-SQG Patches
Junekey Jeon and Andrej Zlatoš. "An Improved Regularity Criterion and Absence of Splash-like Singularities for g-SQG Patches." Anal. PDE 17, 3 (2024)
talks
An Improved Regularity Criterion and Absence of Splash-like Singularities for g-SQG Patches
We prove that splash-like singularities cannot occur for sufficiently regular patch solutions to the generalized surface quasi-geostrophic equation on the plane or half-plane with parameter $\alpha \le 1/4$. This includes potential touches of more than two patch boundary segments in the same location, an eventuality that has not been excluded previously and presents nontrivial complications (in fact, if we do a priori exclude it, then our results extend to all $\alpha \in (0,1)$). As a corollary, we obtain an improved global regularity criterion for H3 patch solutions when $\alpha \le 1/4$, namely that finite time singularities cannot occur while the H3 norms of patch boundaries remain bounded.
Abstract Level-Set Dynamics of gSQG Equation
We develop an abstract measure-theoretic notion of solutions to gSQG equation based on the dynamics of level-sets of solutions, and prove an $H^{2}$ local-wellposedness result in this setting using a sequence of regularized contour equations.
teaching
MATH 110. Introduction to Partial Differential Equations
Undergraduate course, University of California San Diego, 2019
MATH 140B. Foundations of Real Analysis II
Undergraduate course, University of California San Diego, 2020
MATH 140C. Foundations of Real Analysis III
Undergraduate course, University of California San Diego, 2020
MATH 144. Introduction to Fourier Analysis
Undergraduate course, University of California San Diego, 2020
MATH 20D. Introduction to Differential Equations
Undergraduate course, University of California San Diego, 2020
MATH 130. Differential Equations and Dynamical Systems
Undergraduate course, University of California San Diego, 2021
MATH 20D. Introduction to Differential Equations
Undergraduate course, University of California San Diego, 2021
MATH 142A. Introduction to Analysis I
Undergraduate course, University of California San Diego, 2021
MATH 20D. Introduction to Differential Equations
Undergraduate course, University of California San Diego, 2022
MATH 10B. Calculus II
Undergraduate course, University of California San Diego, 2022
MATH 144. Introduction to Fourier Analysis
Undergraduate course, University of California San Diego, 2022
MATH 140B. Foundations of Real Analysis II
Undergraduate course, University of California San Diego, 2023
MATH 140C. Foundations of Real Analysis III
Undergraduate course, University of California San Diego, 2023
MATH 240C. Real Analysis III
Graduate course, University of California San Diego, 2023
MATH 20D. Introduction to Differential Equations
Undergraduate course, University of California San Diego, 2023
MATH 20E. Vector Calculus
Undergraduate course, University of California San Diego, 2024
MATH 148. Analysis of Partial Differential Equations
Undergraduate course, University of California San Diego, 2024