Tom Kojar website

Microscopic regularity structures

by thomaskojar February 1, 2023

Brownian motion constructions

Interpolation (Levy-Ciesielski construction) Cameron-Martin space eigenfunctions White noise Limit of Random walk Hyperreals construction see “A Non-standard Construction of Multi-Dimensional Brownian Motions and Option Pricing””. Heat equation The $u(x,t)=E_{x}\left[g(B_{t})\chi\{t\leq \tau_{\Omega}\}\right]$ solves the heat equation with initial data $g$ in domain $\Omega$ . So conversely by starting with the heat semigroup, we construct the…

by thomaskojar August 24, 2022August 24, 2022

Elliptic Explorer model

In the Arxiv article A Left passage explorer model, we explore a possible extension of the Harmonic explorer.

by thomaskojar August 15, 2022February 1, 2023

Quantum Heisenberg spin-1/2 chain

Classical hydrodynamics is a remarkably versatile description of the coarse-grained behaviour of many-particle systems once local equilibrium has been established1. The form of the hydrodynamical equations is determined primarily by the conserved quantities present in a system. Some quantum spin chains are known to possess, even in the simplest cases, a greatly expanded set of…

by thomaskojar December 4, 2021December 4, 2021

History of math symbols

Forwarding the article from Earliest Uses of Symbols of Operation (tripod.com) ADDITION AND SUBTRACTION SYMBOLS Plus (+) and minus (-). Nicole d’ Oresme (1323-1382) may have used a figure which looks like a plus symbol as an abbreviation for the Latin et (meaning “and”) in Algorismus proportionum, believed to have been written between 1356 and 1361. The symbol appears in a manuscript…

by thomaskojar May 13, 2021February 1, 2023

Modeling SLE

Based on Tom Kennedy’s C++ code for Fast SLE. The code was only available in C++ for speed purposes, so now it can run SLE algorithms in Matlab. https://www.mathworks.com/matlabcentral/fileexchange/63415-fast-sle-algorithm

by thomaskojar April 4, 2020February 1, 2023

Hausdorff dimension of the zero level set of alpha-Stochastic heat equation

Consider the equation $dz+\nu A^{\alpha}z=C^{1/2}dW, z(0)=0$ where for $x:=\sum_{k\in Z^{2}\setminus \{(0,0)\}} x_{k}e_{k}$ , with $\sum |x_{k}|^{2}<\infty$ , we set $A^{\alpha}(x):=\sum_{k\in Z^{2}_{*}}|k|^{\alpha} x_{k}e_{k}$ for $\alpha>1$ and $C(x):=\sum_{k\in Z^{2}_{*}}\sigma_{k}^{2}x_{k}e_{k}$ such that for $\delta\in (1-\alpha,2-\alpha)$ and $\frac{c_{2}}{|k|^{\delta}}\leq |\sigma_{k}|\leq \frac{c_{3}}{|k|^{\delta}}$ . RO: The dimension of the zero level set is $dim(\{x\in T_{2}: z_{t}(x)=0\})=3-\alpha-\delta$ .…

by thomaskojar March 23, 2020April 4, 2020

Parabolic as infinite dimensional elliptic

This intuition was used by Perelman. Quote from Tao: “We now come to an interesting (but still mostly heuristic) correspondence principle between elliptic theory and parabolic theory, with the latter being viewed as an infinite-dimensional limit of the former, in a manner somewhat analogous to that of the central limit theorem in probability” References: “Parabolic theory as a…

by thomaskojar February 5, 2020April 4, 2020

Course exams written by someone else other than the instructors/coordinators. Good or Bad?

As you might know already evaluating learning outcomes is very difficult. For example, student feedback is not always reliable; there are studies showing inverse relations between student learning and student ratings (eg. a course that managed to challenge students). One potential idea is to have the final exam (or even midterms) in the math course…

by thomaskojar November 24, 2019February 1, 2023

Contour integration

1. Global nature An important feature to note about Cauchy’s theorem is the global nature of its hypothesis on the analytic functions f. Cauchy’s theorem is the powerful technique of contour shifting, which allows one to compute a contour integral by replacing the contour with a homotopic contour on which the integral is easier to…

by thomaskojar November 3, 2019April 4, 2020

Wald’s equation and bullet projectile quality

References: Milton Friedman suggesting problem to Wald and the math in sequential analysis. In order to understand the story, it is necessary to have an idea of a simple statistical problem, and of the standard procedure for dealing with it. The actual problem out of which sequential analysis grew will serve. The Navy has two…

by thomaskojar October 30, 2019October 30, 2019

Dimers and Amoebae

by thomaskojar April 16, 2019April 16, 2019

Zero level set of 2D NSE

Conformal invariance (CI) In “Conformal invariance in two-dimensional turbulence” by Bernard etal, it was guessed that numerically the zero level set (ZLS) for the vorticity field has the same properties as an SLE curve. Since NSE and Euler equations are volume preserving, the equations themselves are not conformally invariant. So one has to study the…

by thomaskojar March 28, 2019April 4, 2020

Periodic TASEP

KPZ Fixed point Finding a transition probability formula that is amenable to taking limit as the number of TASEP particles goes to infinity. Prize: 50$ Is there an analogous backwards heat equation problem as in the KPZ fixed point for Z-TASEP? Identifying the KPZ fixed point in the subcritical region Prize: 50$ . …

by thomaskojar March 28, 2019March 28, 2019

Elliptic divergence operator isomorphism for Lp spaces p greater than 2

For the problem $Lu:=div(A\nabla u)=div(g)$ , we will show $L: W_{0}^{1,p}\to W^{-1,p}$ (cf. Meyers “An Lp-estimate for the gradient of solutions of second order elliptic divergence equations “). Assume that we have coercivity: $\inf_{\phi\in L^{1/q’}}\sup_{\psi\in L^{1/q}}|B_{A}(\psi,\phi) |\geq \frac{1}{K}> 0$ . Suppose that $g\in L^{q’}$ and consider sequence $g_{k}\in L^{2}$ such that…

by thomaskojar March 2, 2019April 4, 2020

Proofs of CLT

Proofs of Central limit theorem Method of characteristics http://www.cs.toronto.edu/~yuvalf/CLT.pdf Method of Moments http://www.cs.toronto.edu/~yuvalf/CLT.pdf Minimizing Entropy “An information-theoretic proof of the central limit theorem with the Lindeberg condition” Heat equation Kolmogorov Petrovsky Stein’s method https://normaldeviate.wordpress.com/2013/11/16/steins-method/

by thomaskojar March 31, 2017February 8, 2019