Research Opportunity

by dinowc, Apr 21, 2025, 10:17 PM

Hi everyone, my name is William Chang and I'm a second year phd student at UCLA studying applied math. Over the past year, I've mentored many undergraduates at UCLA to finished papers (currently under review) in reinforcement learning (see here. :juggle:

)

I'm looking to expand my group (and the topics I'm studying) so if you're interested, please let me know. I would especially encourage you to reach out to me chang314@g.ucla.edu if you like math. :wow:

This post has been edited 1 time. Last edited by dinowc, Today at 12:07 AM
Reason: typo

advanced fields

x^{2s}+x^{2s-1}+...+x+1 irreducible over $F_2$?

by khanh20, Apr 21, 2025, 6:18 PM

With $s\in \mathbb{Z}^+; s\ge 2$ , whether or not the polynomial $P(x)=x^{2s}+x^{2s-1}+...+x+1$ irreducible over $F_2$ ?

algebra

polynomial

Advice on Statistical Proof

by ElectrickyRaikou, Apr 21, 2025, 6:12 PM

Suppose we are given i.i.d.\ observations $X_i$ from a distribution with probability density function (PDF) $f(x_i \mid \theta)$ for $i = 1, \ldots, n$ , where the parameter $\theta$ has a prior distribution with PDF $\pi(\theta)$ . Consider the following two approaches to Bayesian updating:

(1) Let $X = (X_1, \ldots, X_n)$ be the complete data vector. Denote the posterior PDF as $\pi(\theta \mid x)$ , where $x = (x_1, \ldots, x_n)$ , obtained by applying Bayes' rule to the full dataset at once.

(2) Start with prior $\pi_0(\theta) = \pi(\theta)$ . For each $i = 1, \ldots, n$ , let $\pi_{i-1}(\theta)$ be the current prior and update it using observation $x_i$ to obtain the new posterior:

$\pi_i(\theta) = \frac{f(x_i \mid \theta) \pi_{i-1}(\theta)}{\int f(x_i \mid \theta) \pi_{i-1}(\theta) \, d\theta}.$
Are the final posteriors $\pi(\theta \mid x)$ from part (a) and $\pi_n(\theta)$ from part (b) the same? Provide a proof or a counterexample.

Here is the proof I've written:

Proof

Do you guys think this is rigorous enough? What would you change?

Computational Calculus - SMT 2025

by Munmun5, Apr 21, 2025, 9:35 AM

1. Consider the set of all continuous and infinitely differentiable functions $f$ with domain $[0,2025]$ satisfying $f(0)=0,f'(0)=0,f'(2025)=1$ and $f''$ is strictly increasing on $[0,2025]$ Compute smallest real M such that all functions in this set , $f(2025)<M$ .
2. Polynomials $A(x)=ax^3+abx^2-4x-c$ $B(x)=bx^3+bcx^2-6x-a$ $C(x)=cx^3+cax^2-9x-b$ have local extrema at $b,c,a$ respectively. find $abc$ . Here $a,b,c$ are constants .
3. Let $R$ be the region in the complex plane enclosed by curve $f(x)=e^{ix}+e^{2ix}+\frac{e^{3ix}}{3}$ for $0\leq x\leq 2\pi$ . Compute perimeter of $R$ .

This post has been edited 4 times. Last edited by Munmun5, Yesterday at 2:56 PM

calculus computations

calculus

We know that $\frac{d}{dx}\bigg(\frac{dy}{dx}\bigg)=\frac{d^2 y}{dx^2}.$ Why we

by Vulch, Apr 21, 2025, 9:15 AM

We know that $\frac{d}{dx}\bigg(\frac{dy}{dx}\bigg)=\frac{d^2 y}{dx^2}.$ Why we can't write $\frac{d^2 y}{dx^2}$ as $\frac{d^2 y}{d^2 x^2}?$

This post has been edited 2 times. Last edited by Vulch, Yesterday at 9:16 AM

calculus

Why is this series not the Fourier series of some Riemann integrable function

by tohill, Apr 21, 2025, 8:08 AM

$\sum_{n=1}^{\infty}{\frac{\sin nx}{\sqrt{n}}}$ (0<x<2π)

This post has been edited 2 times. Last edited by tohill, Yesterday at 9:19 AM

real analysis

fourier series

Finite solution for x

by Rohit-2006, Apr 21, 2025, 4:19 AM

$P(t)$ be a non constant polynomial with real coefficients. Prove that the system of simultaneous equations —
$\int_{0}^{x} P(t)sin t dt =0$ $\int_{0}^{x}P(t) cos t dt=0$ has finitely many solutions $x$ .

This post has been edited 1 time. Last edited by Rohit-2006, Yesterday at 4:20 AM

calculus

high school math

by aothatday, Apr 10, 2025, 2:20 PM

Let $x_n$ be a positive root of the equation $x^n=x^2+x+1$ . Prove that the following sequence converges: $n^2(x_n-x_{ n+1})$

This post has been edited 4 times. Last edited by aothatday, Apr 16, 2025, 2:39 PM

from my teacher

How to solve this problem

by xiangovo, Mar 19, 2025, 11:54 AM

How many nonzero points are there on x^3y + y^3z + z^3x = 0 over the finite field \mathbb{F}_{5^{18}} up to scaling?

algebraic geometry

superior algebra

complex analysis

by functiono, Jan 15, 2024, 11:49 AM

find the real number $a$ such that

$\oint_{|z-i|=1} \frac{dz}{z^2-z+a} =\pi$

complex analysis

Submit

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math_exploration

by dinowc, Apr 21, 2025, 10:17 PM

by khanh20, Apr 21, 2025, 6:18 PM

by ElectrickyRaikou, Apr 21, 2025, 6:12 PM

by Munmun5, Apr 21, 2025, 9:35 AM

by Vulch, Apr 21, 2025, 9:15 AM

by tohill, Apr 21, 2025, 8:08 AM

by Rohit-2006, Apr 21, 2025, 4:19 AM

by aothatday, Apr 10, 2025, 2:20 PM

by xiangovo, Mar 19, 2025, 11:54 AM

by functiono, Jan 15, 2024, 11:49 AM