Newest Questions

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_{...

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 ...

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 ...

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 ...

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 ...

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 ...

I came across Kasch's definition of a module that is a generator in a category of modules, namely, that a module $C$ is a generator if for every $M$ in that category of modules $$ 0 = \operatorname{...

For background, say that a centrifuge has $n$ slots arranged in a circle and $k$ tubes are placed within it. This is equivalent to choosing $k$ distinct $n$-th roots of unity. The centrifuge is ...

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, $...

Given a set of non-negative real numbers $c_1, c_2, ..., c_N$, and a positive real number $D$ where $D << 1$, find an upper bound of the function: $Q(x_1, x_2, ..., x_N)$ = $\sum_{i=1}^{N}{x_i\,...

I am trying to calculate the average distance a particle passing through a cylinder experiences. There is both a top and a bottom and the dimensions of the cylinder are known. Particles can exit any ...

I am trying to get a better grasp of how to find the basis of the tangent space. Here is one example I worked on in hopes of practicing it: Consider the chart $(U,\psi)$, the manifold $\mathcal{M} = S^...

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 (...

The associative laws for addition and multiplication means (1): \begin{align*} (a + b) + c &= a + (b + c) \\ (a \cdot b) \cdot c &= a \cdot (b \cdot c) \end{align*} Does the above also ...

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 ...

The height function of the two torus $\mathbb{T}^2$ is a standard example. It has $2$ hyperbolic points and $2$ elliptic points. I was wondering if there exists a reference that computes the classical ...

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 $\...

(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 ...

Let me know if this is more on-topic for physics.se (or more generally, off-topic for mathematics.se). Okay, imagine you have a volume of space with instruments measuring air pressure (and for ...

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 ...

The goal is to find the point $M$ inside a given right triangle $ABC$ such that $\operatorname{Area}(APMN)=\operatorname{Area}(CQMP)=\operatorname{Area}(BQMN)$, where $N$, $P$, and $Q$ are the ...

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 '...

Prove that $0_{1}+1_{2}+12_{3}+123_{4}+\cdots+(1:2:3:\cdots:2025)_{2026}$ is not a square, where ($1:2:3:\cdots:2025)_{2026}$ is the concatenation of all single digit in base $2026$. My first instinct ...

Consider two random variables $y_1$ and $y_2$, where $y_1$ is symmetrically distributed around $0$ and $y_2 = y_1 ^2$. $$E_{y_1,y_2}[y_1 y_2] = \int \int y_1 y_2 p( y_1,y_2) dy_1 dy_2$$ $$= \int \...

Let $M = \{(x_{1}, x_{2}, x_{3}, x_{4}) \in \mathbb{R}^{4}: x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2} = -1\}$. Prove that $M$ is a regular submanifold of $\mathbb{R}^{4}$ with dimension 3. Compute the ...

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^...

Let $n \in \mathbb{Z}_{2^q}$ such that $ n \equiv 2^r m \pmod{2^q}$ for some odd $m$ and $1\leq r<q$. Then the number of solutions to the congruence $x^2 - y^2 \equiv n \pmod{2^q}$ is $(r-1)2^q$. ...

I have a family of functions $f(X, b)$ which are continuous in $b$ for each $X$. Suppose $X \sim P$ I'd like to know what are some general conditions (on $P$ or additional conditions on $f$) $$ g(b) = ...

How do I prove the following inequality for all $k \ge 2$?: $$ \sum_{j=2}^{k+1} \sum_{p=k+1}^\infty \frac{1}{p j^p} + \sum_{j=k+2}^\infty \sum_{p=2}^\infty \frac{1}{p j^p}<\frac{1}{k+1}$$ I've ...