Let $\sigma:\mathbb{N}\longrightarrow\mathbb{N}$ be strictly increasing, and consider the power series $$ S_{\sigma}(x)=\sum_{n=0}^{+\infty}(-1)^nx^{\sigma(n)}. $$ Can any real number in $[0,2]$ be obtained as the limit $\lim\limits_{x\rightarrow 1^-}S_{\sigma}(x)$ for some $\sigma$ ? According t...

I have read on different sources that it is not possible to give a simple proof that "two free groups are isomorphic if and only if they have the same rank" using only what "a student who has just read the definition of free group as a set of words over an alphabet" would know. See for example th...

I am following the last module of Differential Calculus on Khan Academy, that deals with Parameteric equations. Here are the parametric equations described in the lecture. $x(t) = 5t + 10$ $y(t) = 50 - 5t^2/2$ However, I really don't understand what parametric equations really mean. How do they d...

I've been reading through the Polytope-Wiki entry on helices. To my understanding, an $n$-gonal helix is a blend of a planar $n$-gon $\{n\}$ with the regular linear apeirogon $\{\infty\}$. The blend of the apeirogon with a line segment produces the only two dimensional "helix", the zig-zag. Final...

Settup So... I kinda handled most of my proof but I need help with some of the stuff I just kinda went with until it worked out. The problem relates to medicine and its decay in the body. We are given that the medicine will release over a period of $b$ hours and another dose is given at time $T$....

Consider the following 2-category $\mathcal K$: objects are finite length abelian categories (i.e., abelian categories where every object has finite length) morphisms are exact functors (preserving finite length) 2-morphisms are all natural transformations. For every finite length abelian categ...

Suppose I have some tensor, for conreteness I will consider a rank 4 tensor with components $R_{abcd}$ where the indices run over 0,1,2,3 (my question arises in a physics context). Under what conditions is the existence of another tensor, call it $G_{abcd}$, which satisfies $G_{ab}^{~~~mn}G_{cdmn...

I have seen two motivations of decomposing an $L^2(\mathbb{T}^n)$ function into a Fourier series with basis functions $e^{2\pi inx}$ with $n$ being integers. The first one is the intuitive frequency representation of a signal; the second one is that the trigonometrical functions are eigenfunction...

If $s_n=\sum_{k=0}^{n}(-4)^k\binom{n+k}{2k}$ how to prove $s_{n+1}+2s_n+s_{n-1}=0$. One of my student had this question in his exam. Honestly to speak I couldn't get any single idea how to even start. I knew some strategies to find binomial sums but they all couldn't help. It would be great if so...

Assume that $u(x)$ is the classical solution solving $$a_{ij}(x)\partial_{ij}u(x)+b_i(x)\partial_iu(x)+c(x)u(x)=f(x)$$ on $\mathbb{R}^n$ for some smooth enough coefficients and uniformly elliptic $a_{ij}$. I am looking for the gradient bound of $u$ explicitly on the behavior of the coefficients. ...

In an art museum, there are $n$ paintings, $n \ge 33$, for which there are used a total of $15$ different colors so that any two paintings have at least one common color and there are no two paintings that have exactly the same colors. Determine all possible values ​​of $n \ge 33 $ so that anyway...

Determine the general term of the sequence $(a_n)_{n\ge1}$, strictly decreasing, of strictly positive numbers, which satisfies the properties: a) $na_n \in \mathbb{N} \setminus \{0\}$ for every $n \in \mathbb{N} \setminus \{0\}$. b) $\sum_{k=1}^{n} (-1)^{k-1} a_k \binom{n}{k} = \sum_{k=1}^{n} a_k...

In order to prove the completeness theorem we obviously need a framework such as ZFC (I'm aware that ZFC isn't the only possibility) so that we can talk about a language $\mathcal{L}$ and also about models of $\mathcal{L}$. Now the completeness theorem makes perfect sense to me in so far as the l...

I'm reading an article and I quote the author here : The condition $\sum_{n=1}^{\infty} n^t L(n) \operatorname{Pr}\left(|X|>n^{1 / r}\right)<\infty$ is equivalent to the moment condition $E\left[|X|^{(t+1) r} L(X)\right]<\infty$. $t \geq 0, 0<r <2$, $X$ is an arbitrary random variable and $L(\cdo...

Consider the following question Consider the Poisson structure $$ \{F, G\}=\int_{\mathbb{R}} \frac{\delta F}{\delta u(x)} \frac{\partial}{\partial x} \frac{\delta G}{\delta u(x)} d x, $$ where $F, G$ are polynomial functionals of $u, u_x, u_{x x}, \ldots$ Assume that $u, u_x, u_{x x}, \ldots$ te...

Given a convex optimization problem $$\min f(x), x \in D$$ $f, D$ convex. The first-order optimality condition says $x$ is the minimizer if and only if $\nabla f(x)^T (x-y) \geq 0, \forall y\in D.$ For unconstrained problems, this is $\nabla f(x) = 0$. This seems to be perfectly suited for findin...

Question Prove that if $C\subseteq \mathbb{R}^d$ is convex, centrally symmetric and bounded, with $vol(C)>k2^d$, then $C$ contains at least $2k$ lattice points (of lattice $\mathbb{Z}^d$). Note Minkowski’s theorem talks about $vol(C)>2^d$ and $C$ contains at least one point apart from the origin....

I have somewhat often found myself in the following situation, especially when self-studying a mathematical subject: I'm reading a book on a certain topic, and at certain parts I don't immediately understand something, for example a proof. But instead of re-reading it I just skip over it and cont...

I'm trying to find a four-digit number that is equal to the product of (the sum of its digits) multiplied by (the square of the sum of the squares of its digits). I've tried running all combinations in Python and found two solutions (2023 and 2400). However, my maths teacher gave it to me and sai...

I have started studying Lebesgue integration and I have a question regarding the Lebesgue integral. In the wikipedia entry of "Lebesgue integration" they define the Lebesgue integral as: Let $f: \mathbb{R} \rightarrow \mathbb{R}^{+}$ be a positive real-valued function. $$\int f d\mu = \int_{0}...

I sort of intuitively see why we care about similar matrices, i.e., when $A=S^{-1}BS$ for some invertible matrix $S$. But I want to make this intuition more precise and abstract. Matrices: First of all, as mentioned here, Because matrices are similar if and only if they represent the same linear...

Let $1 \leqslant p < \infty$ and $0 < \lambda < n$. Consider the function $f_\alpha \colon \mathbb R^n \to \mathbb R$ defined by $$ f_\alpha(x) := \|x\|^{\frac{\lambda - n + \alpha}{p}} \chi_{B(0,1)}(x) + \| x \|^{\frac{\lambda - n}{p}} \chi_{\mathbb R^n \setminus B(0,1)}(x), \quad \forall x \in ...