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 ...

I have worked on this problem for a few hours but I completely can't solve it. Prove that for real numbers $a,b,c>0$ that: $$\frac 1 a + \frac 1 b + \frac 1 c + \frac 4 {a+b} + \frac 4 {b+c} + \...

The Henon map $H_{a,b}$ is a smooth one-to-one mapping ${\mathbb R}^2 \ni (x,y) \rightarrow H_{a,b} (x,y):= (1 - a x^2 + y, b x) \in {\mathbb R}^2 $ of the plane to itself. This mapping exhibits a ...

I have following matrix (8 rows, 10 columns) which I have augmented with an 11th constant term to represent 8 equations over 10 variables. ...

I am working on a geometry problem involving a parabola and coordinate transformations. I have solved the preliminary parts, but I am looking for a more elegant or geometric solution for the final ...

Consider a filtered probability space $(\Omega,\mathcal F,\mathbb F=\{\mathcal F_t\}_{t\in[0,T]},\mathbb P)$ where the filtration satisfies the standard assumptions. Let $X=(X_t)_{t\in[0,T]}$ be a ...

Let $\{H_i\}_{i\in I}$ be a directed system of Hilbert spaces over a poset $I$ such that $H_i\subset H_j$ whenever $i\leq j$. We consider the direct limit $H=\cup_{i\in I} H_i$ and equip it with the ...

While absolute convergence allows you to rearrange an infinite amount of terms as you please, generalizing both associativity and commutativity to infinite sums. Here we're only interested in grouping ...

Find all the solutions to $f(f(x) + 2020x + y) = f(2021x) + f(y)$ for all $x,y >0$ when: i) $f: \mathbb{N} \mapsto \mathbb{N}$ ii) $f: \mathbb{R}^{+} \mapsto \mathbb{R}^{+}$ The second part was a ...

A function cl on the power set of $S$ is a Moore closure operator if it satisfies for all $A,B\subseteq S$ $A \subseteq \mathrm{cl}(A)$; $A\subseteq B \Rightarrow \mathrm{cl}(A) \subseteq \mathrm{cl}(...

Consider the integral $$I=\int \sqrt{\frac{1 - \sqrt{x^2 + 1}}{x}} \, dx\qquad(\text{for}\,x<0) $$ My solution is based on Transform 3 and 5 in this draft. $$\begin{aligned}I &= \int \sqrt{\...

Let $E$ be a rank $2$ bundle on a smooth projective surface. Then we know that $E^{*} \otimes \text{det}(E) \cong E$. Does this still hold if the dimension of the variety is strictly bigger than $2$ (...

Picture belos is from the 4.5.1 of Evans' Partial Differential Equations. I want to obtain the equation marked in red from the model. I feel (1) is correct, but have no way to explain it.

I'm reading about duality theory in modal logic, where there's a duality between general frames and modal algebras. I understand one part of the duality, given a general frame $(W,R,A)$, we can get an ...

Let $\mathcal{C}$ be a Grothendieck site, and form a cartesian square with $f:X\to S$, $g: S'\to S$ to get $X'=X\times_S S'$, $f': X'\to S'$ and $g': X'\to X$. Let $\mathcal{F}$ be a sheaf of abelian ...

The classical Aubion Lions Lemma says that if $X_1$ is compactly embedded in $X_2$ and $X_2$ is continuously embedded in $X_3$, then $\{u\in L^p([0,T],X_1), \partial_t u \in L^q([0,T],X_3)\}$ embeds ...

Let $A$ be an $n \times n$ matrix over a field $F$. Let $d(A)$ be the largest non-negative integer such that the matrices $Id, A, A^2,\ldots,A^{d(A)}$ are linearly independent in the vector space of ...

The formula for the nth eigenstate of the QHO can be calculated from the ground state as follows: $$ |n \rangle = \frac{(\hat{a}^{\dagger})^n}{\sqrt{n!}}|0\rangle. $$ Where $\hat{a}^{\dagger}$ is $\...

Suppose we have 2 non-commuting operators $A,B$ that is: $$AB\ne BA$$ If we can express $A,B$ as: $$A=e^{ln(A)}$$ $$B=e^{ln(B)}$$ Then applying "$C+D=D+C$" for any 2 matrices: $$AB=e^{ln(A)+...

This is a theorem from tom Dieck. Proposition (6.10.9). Suppose $\pi_i(X) = 0$ for $i \lt p (\ge 0)$ and $\pi_i(Y) = 0$ for $i \lt q (\ge 0)$. Then $\pi_i(X \star Y) = 0$ for $i \lt p + q + 1$. The ...

I sketch a proof of the following assert: If R is a commutative ring with unit an I is an ideal of R, such that R/I is a flat R-module, then I is a flat R-module. It sounds too nice to be true, and I ...