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I was unsatisfied with the many proofs of the Pythagorean theorem in which it's not clearly apparent which axioms are specifically needed, or because said axioms seem too geometrically motivated in ...

I've been studying the properties of rectangles for a little while on my own so I don't know what are the actual terms or what are the formulas, but I've noticed that if you take $A$ as a "degree ...

Let $(X, \mathcal{B}, T)$ be a measureable dynamical system, meaning that $T : X \rightarrow X$ is a Borel automorphism over a standard Borel space $(X, \mathcal{B})$. A measureable set $W$ is called ...

The Problem A duck has two legs. When a duck folds one leg, only one leg is visible. When a duck is sitting, neither of its legs is visible. When Roman went to the lake, there were 33 ducks. He ...

Taking the indefinite integral of the infinite series definition of the Riemann-Zeta function gives this generalized antiderivative: $$ \int\sum\limits_{n=1}^\infty \frac{1}{n^x} dx = x - \sum\limits_{...

For a recent project, I have had to read a little bit about linear time invariant systems. In the process of educating myself, I, of course, was introduced to the Dirac delta functional/distribution, $...

I've been playing around with an idea about composite numbers and the digits of their factors. I've noticed a certain pattern, and for lack of a better term, I've started calling numbers that exhibit ...

Considering the space, $ X = S^{1} \times \partial D^{2} \,\cup\, \{x, y\} \times D^{2}. $ the subspace of the solid torus $ S^{1} \times D^{2} $ given by the union of the boundary of the boundary ...

The space of smooth embeddings of a manifold $M$ into $\mathbb{R}^\infty$ is contractible by the Whitney embedding argument. Question: Is the space of topological embeddings of a manifold $M$ into $\...

Suppose we have the following structure: there is $1$ cell in the first row, $2$ cells in the second row, ..., $k$ cells in the $k$-th row, ... (first picture): A mouse stays in the cell in the first ...

(Hahn decomposition theorem) Let $\mu$ be a signed measure. Then one can find a partition $X = X_+ \cup X_-$ such that $\mu\downharpoonright_{X_+} \geq 0$ and $\mu\downharpoonright_{X_-} \leq 0$. A ...

I have the following eleven equations, where each variable is a positive integer: \begin{align} b-c &= 2u^2 \\ b+c &= 4v^2 \\ b-2c &= w^2 \\ d-a &= 2r^2 \\ d+a &= 4s^...

On p. 353 of Algebraic Geometry, Hartshorne poses the question of whether a curve of degree $7$ with $g=5$ exists in $\mathbb{P}^3$. He then says ``We need a very ample divisor $D$ of degree $7$, with ...

The problem statement is: $A \land B \land A \land (B \lor C) \lor B \land (B \lor C)$ and my solution is $ = A \land B \land A \land (B \lor C) \lor (B \land (B \lor C))$ => Now since $(B \land (...

Let $p, q\in n^2 = \{0, 1, \dots, n-1\}^2$ be points on the plane. Say "$p$ covers $q$" if the line segment from $p$ to $q$ intersect $n^2$ in no points other than $p$ or $q$ (they are in '...