This is a challenge problem that my AP Stat teacher can't solve, so I am hoping that I can find an answer here. I am aware that you could use a computer to run simulations to get an approximate, but I am looking for a more definitive answer. The question is: Assume 2 points are placed in rectangl...

Was exploring successive differences of primes and noticed an interesting pattern of the histogram of counts for the sixth and ninth difference. The ninth is more pronounced, code and image below. Mathematica code: n = 5000000; (* First n primes *) p = Prime[Range[2, n]]; (* Excluding first prime...

Motivation: $2$ branches of Lambert $\text W_k(z)$ is a limit of the inverse of $x^n(x+c)$ which is expressible in terms of FoxH in Mathematica. $\text W_0(x)=\text W(x):$ $$-\lim_{a\to0}e^{\frac{(-x)^a((a+1)x+1)-1}a-1}=xe^x$$ Using the inverse function of $-e^{\frac{(-x)^a((a+1)x+1)-1}a-1}$: Bl...

In this question it is shown that $\mathbb{E}_B[\|\bar{\mathbf{y}}_B-\bar{\mathbf{y}}\|_2^2] = \frac{N-|B|}{|B|}\frac{1}{N}e$ where $\mathbf{y}_1, \mathbf{y}_2, \dots, \mathbf{y}_N$ is a sequence of $N$ vectors in $\mathbb{R}^d$ and $\bar{\mathbf{y}}$ is the overall average, i.e., $\bar{\mathbf{y...

Let $k$ be a field and consider the following two $k$-algebras: $R_1 = k[a,b,c,d] / (ab - cd). $ $R_2$ is the unital $k$-subalgebra of $k(t)[x,y]$ generated (as an algebra) by $x,y,tx,t^{-1}y$. Are $R_1$ and $R_2$ isomorphic? Clearly we have a surjective map $\varphi \colon k[a,b,c,d] \rightarr...

Suppose $G$ is an abelian finite group, and the number of order-2 elements in $G$ is denoted by $N$. I have found that $N= 2^n-1$ for some $n$ that satisfy $2^n| \ |G|$. I write my proof. Would you tell me if this proof is correct? Moreover, Would you tell me can we say more about the number $N$?...

I'm preparing for an exam currently, and I came across this question: I have noticed that A can be constructed from the matrix on the left by a series of row operations, so I had the idea maybe to express A as a product of elementary matrices as well as the matrix on the left and, maybe there wa...

I (non-mathematician) asked a similar kind of question 5 days ago. Now I revisit the case in a different manner. The powerful numbers may be written in the form $A^2B^3$, where $A$ and $B$ are positive integers. Erdös conjectured in 1975 that there do not exist three consecutive powerful integers...

I am reading some materials about noncommuative algebra which made me very confused. Let $f: A\to B$ be a injective ring homomorphism between two rings, $A$, $B$ are not necessarily commutative. Then the author writes that: Let $r:B\to E=\mathrm{End}_{A}(B)$ denote the canonical inclusion $b\to ...

Hi it's a follow up of Prove $\sum_{cyc}{\sqrt{\frac{x+1}{x^2+16x+1}}}\geqslant 1$ and $ \sum_{cyc}{\sqrt{\frac{x+1}{4x^2+10x+4}}}\leqslant 1$ for $x,y,z>0,xyz=1$ : Problem : Let $x_i>0$ and $n$ an even natural number such that $\prod_{i=1}^{n}x_i=1$ with $n\geq 3$ and $x_i\leq 1$ , $x_{i+1}\geq ...

Maschke's theorem tells us that any representation of a finite group $G$ can be decomposed into a direct sum of irreducible representations. The proof does make intuitive sense to me (Intuition behind Maschke's theorem), but my question is really about why we should expect it to be true in the fi...

Back in late April I arrived at the following formula for when $x+b < -1$: $$\boxed{ \sum_{a=1}^{\infty}\sum_{k=0}^{\infty}\frac{(k+a)^{x+b}}{a} = \sum_{k=0}^{\infty}\sum_{g=0}^{k}\frac{b^{g}}{(g!)^{2}}\binom{k}{g}\sum_{a=1}^{\infty}\frac{(k+a-g)^{x}}{a}\ln^{g}(k+a-g). }$$ I presented my work t...

Relevant Theorems: $(a)$ There is no continuous function $f$ on $\mathbb{R}$ which takes on every value exactly twice. $(b)$ There is no continuous function $f$ on $\mathbb{R}$ which takes on each value either $0$ times or $2$ times. $(c)$ Find a continuous function $f$ on $\mathbb{R}$ which tak...

Let $u=u(r)$ be radially symmetric, nonnegative and decreasing function. Let $s\in (0, 1)$ and $p, q\in\mathbb{R}$ such that $1<q<p$ and $ps<N$ with $N\in\mathbb{N}, N\ge 2$. Assume that $u\in D^{s, p}(\mathbb{R}^N)$, where $$D^{s, p}(\mathbb{R}^N) = \left\lbrace u\in L^{Np/(N-sp)}(\mathbb{R}^N):...

If an inner product is linear by definition, i.e., $\langle\mathbf{v+w},\mathbf{u}\rangle=\langle\mathbf{v},\mathbf{u}\rangle+\langle\mathbf{w},\mathbf{u}\rangle $ and $\langle a\mathbf{v},\mathbf{w}\rangle=a\langle\mathbf{v},\mathbf{w}\rangle$, and the dot product is an inner product, then why i...

Prove or disprove. Suppose that $\left(M_{n}\right)_{n}$ is a martingale with $M_{n} \geqslant-10 \quad \forall n$, a.s. Is it true that $$ \sum_{i=1}^{\infty}\left(M_{i}-M_{i-1}\right)^{4}<\infty \quad \text { a.s. ? } $$ This was my attempt which I'm unsure of: Proved by contradiction: Suppose ...

I am studying Jun Shao's Mathematical Statistics and got a bit stuck in the proof of Theorem 5.4, which states: Let $u$ be a Borel function on $\mathbb{R}^d$ satisfying $\int u(x)dF = 0$ and $\hat{F}$ be the MELE of F. Suppose that $U = Var(u(X_1))$ is positive definite. Then for any $m$ fixed d...

If $\mathsf{X}$ is a set, a binary operation on it is a map $*:\mathsf{X}\times\mathsf{X}\to\mathsf{X}$. Examples of these operations are addition and multiplication in a field. The scalar multiplication in a vector space $\mathsf{V}$ over a field $\mathsf{F}$ is defined as a map $\cdot:\mathsf{F...

I was wondering if it was indeed possible to perform a transposed vector multiplication with another transposed vector. And if so how I'm supposed to do so. Background: From https://en.wikipedia.org/wiki/Complex_normal_distribution I saw As you can see in the exponential there is a multiplicatio...

This is the question from my textbook: Represent the vector with respect to the basis $$x^2+x^3,\; \mathbf{B} = \langle 1, 1+x, 1+x+x^2, 1+x+x^2+x^3 \rangle$$ The explanation from the solution is that it is "easily solved by eye to give $c_{4}=1, c_{3}=0, c_{2}=-1 \;and \;c_{1}=0$" I don't unders...