Suppose we have an interval say $D\to D+\Delta$.Let $N(D)$ be the number of set of 3 natural numbers $(a,b,c)$ such that $ D\le a^2+b^2+c^2\le D+\Delta$.In physics we very easily assume that$$\lim_{D\...

Lately, I was doing problems on the book "The Art of Mathematics : Coffee time in Memphis".One certain problem caught my attention, the problem 21, Neighbors in a matrix. Basically it ...

Here's a theorem in Le Gall's Measure theory, probability and stochastic processes. He omits the proof, but refers to a more general result in Kallenberg's Foundations of modern probability (Theorem 6....

I want to calculate the area of a region using a change of variables. For example, if I setu=x+y,v = x/yI’m not sure how to correctly compute the Jacobian. I often make mistakes in finding it, which ...

Let $\mu$ be a Radon measure on a compact Hausdorff space $X$ such that $\mu(\{x\}) = 0$ for all $x \in X$. Let $A$ be a Borel subset of $X$ with $\mu(A) > 0$. I want to show that for every $0 < ...

I have some doubts from the book 'Homological algebra' by Cartan and Eilenberg. On page 253, the authors describe a resolution as follows:Fix a presentation of the group $$Q_{4t}=\langle x,y:x^t=y^2,...

Several of the most basic results of Lebesgue integration are stated for nonnegative (Lebesgue measurable) maps $X \to [0,+\infty]$ and may involve the order structure $\leq$ of $[0,+\infty]$:Fatou'...

Reference image ^^^Ok so I think I might have found a new theorem or maybe rediscovered an old one. Also please note that I am just a 13 year old so I don't really understand super complex notation ...

I'm following Serge Lang's Linear Algebra third edition for the definition of a vector space. A set $V$ of objects over a field $F$ which can be added and multiplied by elements of $F$ is a vector ...

Let $n = 2025$. We are given a sequence of positive integers $a_1, a_2, \dots, a_n$.Let the cyclic ratios be defined as:$$r_i = \frac{a_i}{a_{i+1}} \quad \text{for } 1 \le i \le n-1, \quad \text{and}...

Problem Statement: Let $f(x) = x^4 + ax^3 + bx^2 + cx + d$ be an irreducible polynomial over the field of rational numbers $\mathbb{Q}$, where $a, b, c, d \in \mathbb{Q}$. Let $r_1, r_2, r_3, r_4$ be ...

Let $R_1, \dots, R_n$ be rings and consider their direct product $R = R_1 \times \cdots \times R_n$. What do the irreducible elements of $R$ look like?To get some intuition, I started with the ...

(I'm actually learning calculus.)Before I started working on this problem,I went to read this proof:Partial Fractions ProofI think I understand what the proof tried to do(And I can complete some ...

Given integers $n$ and $k$, Alice is given $k$ numbers $1 \le a_1 < a_2 < \cdots < a_k \le n$. She then writes down a message $x\ (1 \le x \le m)$. Bob is given the message $x$ and one ...

first time poster so I'm sorry if any of the formatting is slightly off.I am trying to use this equation to find the inverse of a 3x3 matrix.$$\mathbf A^{-1} = \frac{1}{det(\mathbf A)} \sum_{s=0}^{n-...