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SMASHMATH

i am a math student and tutor

Taylor Grant

discord: taylornix

e

california CA

this is not actually a picture of me (see photo credit below)

exp⁡(x)=∑n=0∞xnn!

math phd student, dork extraordinaire.

i like differential equations and linear algebra.

here are some math posts ive written if you want to look at them i guess

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personal favorites:

 * The Alpha Method (Generalized Exponential Response Formula)
 * Shortcuts for Finding Eigenvalues and Eigenvectors
 * Solving systems of first-order ODEs like a baller

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my research stuff:

things ive discovered independently derived. i think they’re all cool, but only
a few of them are actually useful, in my opinion.

 * Function Interpolation: a method to get a function (which is a linear
   combination of some given set of basis functions) that satisfies certain
   conditions using determinants, given that one exists and is unique. for
   example, a determinant which gives the unique lowest degree polynomial that
   passes through a certain set of points.

 * A formula for some particular solutions to certain ODEs: a determinant
   formula which gives a particular solution to any linear constant-coefficient
   ordinary differential equation which has a forcing function of exponential
   nature (ex. g(t)=tneαtcos⁡(βt)). Uses results from Function Interpolation.

 * Constructing integer systems of differential equations with integer
   solutions: methods to construct nice systems with nice solutions. useful for
   professors/textbook authors to make good lecture examples or exam problems.
   somewhat of a work in progress. i also have a post for second order systems.

 * matrix exponential stuff: i really love matrix exponentials.
   1. Matrix Exponential Formulas for 2x2 Matrices
   2. Matrix Exponentials Using Differential Equations
   3. Exponentials of Symmetric Matrices Using the Spectral Theorem
   4. Matrix Exponential Formulas for 2x2 Matrices Using Laplace Transforms
   5. Another approach to matrix exponential formulas: coming soon…
 * New Ways to Calculate Normalized Solutions to Linear Constant-Coefficient
   Differential Equations: solve just one set of n first-order initial value
   problems to get the n normalized solutions to an n-th order differential
   equation. this should be the fastest way to find them using a computer.
   alternatively, find one normalized solution and get the others recursively.

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the picture is not me. it is a stock photo that i bought and have the license to
use.

credit: iStock.com/ajr_images


NEWS

Jun 12, 2022 back with a few new posts Dec 9, 2021 2x2 matrix exponentials post
rewritten Dec 6, 2021 updating posts to the distill theme Dec 6, 2021 i have a
twitter now which i might update maybe Dec 6, 2021 new news now


LATEST POSTS

Nov 11, 2023 Constant Coefficient ODEs Made Simple with Linear Operators Jun 27,
2023 In Defense of Cramer's Rule Oct 6, 2022 Change of Basis

best way to contact me is through discord (taylornix). please send a message
when adding.
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