Let $P = \mathbb{R}P^2$ be the projective plane, I am attempting to compute the singular homology groups of $P$. I can easily show $H_n(P) = 0$ for $n>1$, $H_0(P) = \mathbb{Z}$. However, I am ...

Im following Serge Lang 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 elementos of $F$ is a vector space ...

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-...

I'm trying to learn the concept of nilpotent groups. On the one hand, there's this formal definition:$Z_0(G)=1, \; Z_{i+1}(G)/Z_i(G) = Z(G/Z_i(G))$. Least $i$ for which $Z_i(G) = G$ (if exists) is ...

Let $f$ be a decreasing function on $[0,1]$ and $a\in(0,1)$. Prove that$$\int_0^af(x)\mathrm dx\ge a\int_0^1f(x)\mathrm dx.$$This would be quite obvious if $f$ were continuous. But for non-...

I was recently reading I.N.Herstein's "Topics in algebra" and stumbled across interesting proposition and it's proof:For any three sets, $A, B, C$ we have:$$A \cap (B \cup C) = (A \cap B) ...

Let $S$ be a smooth complex projective surface. We consider the double cover $\pi:X\to S$ branched along $B\equiv 2L$(suppose $B$ is smooth). Now let $C\subset X$ be an irreducible curve.If $C$ is ...

In chapter 3 of Analysis I by Terence Tao, the following definition of empty set is given:(Empty set). There exists a set $\phi$, known as the empty set, which contains no elements, i.e., for every ...

I’ve been exploring a measurement approach for NP and NP-complete problems based on average time per logical step.I define:...

The action\begin{align}\label{action}S=\int d^4x\sqrt{-g}\bigg[\frac{1}{2\kappa}\bigg(R-\varepsilon\, B^{\mu\lambda}B^\nu\, _\lambda R_{\mu\nu}\bigg)-\frac{1}{12}H_{\lambda\mu\nu}H^{\lambda\mu\nu}-V(...