Maybe the question is very trivial in a sense. So, it doesn't work for anyone. A few years ago, when I was a seventh-grade student, I had found a quadratic formula for myself. Unfortunately, I didn't have the chance to show it to my teacher at that time and later I saw that it was "trivial". I sa...

Yesterday I posted this conjecture, but then deleted it thinking it was false. Turns out Python doesn't define $a^b$ as a^b, but rather as a**b. Conjecture: Denote by $G$ Catalan's constant, then $$G=\cfrac{1}{1+\cfrac{1^4}{8+\cfrac{3^4}{16+\cfrac{5^4}{24+\cfrac{7^4}{32+\cfrac{9^4}{40+\ddot...

I'm looking for an intuitive motivation for Liouville's theorem from complex analysis. If somebody could illustrate this with a simple example, that would be great. Thank you so much.

Let $A=\{a,b\}$ and $a\ne b$. Let $F(A)$ be the free group constructed on $A$. Let $f_a,f_b$ be the canonical homomorphisms of $\mathbb{Z}$ into $F(A)$. Let $g:F(A)\rightarrow\mathbb{Z}\times\mathbb{Z}$ be the unique homomorphism for which $g(f_a(1))=(1,0)$ and $g(f_b(1))=(0,1)$. Let $h:\mathbb{Z...

This is a 9. grade elementary school quiz problem. There is a bag with 4 marbles. Two of them are yellow and the other two are green. You blindly pick two marbles at once. What is the probability that you get two yellow marbles? I would say the answer is $\frac{1}{6}$. There is a single combina...

For example, $$ \frac{\int_{0}^{1}f(x)dx}{\int_{0}^{1}dx} = \int_{0}^{1}f(x), \space\space\space (1) $$ The dx disappears since $$ \frac{\int_{0}^{1}f(x)dx}{\int_{0}^{1}dx} = \lim_{N\to\infty}\left(\sum_{k=0}^{k\to N}{\frac{f(\frac{k}{N})\frac{1}{N}}{1\times \frac{1}{N}}}\right) = \lim_{N\to\in...

Q: Given an ordered ring $A$ is the number of automorphisms of $A$ equal to the number of orderings in $A$? An ordering on a ring is totally defined by a subset of $A$ we call $A^+$ that satisfies these conditions: Its closed under addition and multiplication $\forall a \in A$ one and only on...

I came across a question that I couldn't figure out. It was: What is the value of all the letters in this following cryptarithm? $$\begin{array}{ccccccc} &&&&T&H&E\\ +&&&B&E&S&T\\ &S&Y&S&T&E&M\\ \hline &M&E&T&R&I&C\\ \end{array}$$ The problems I can't figure out any of the letters. It ...

Let $f:\mathbb R\to\mathbb R_{\geqslant0}$ be an integrable function with period $1$ such that $\displaystyle\int_0^1 f(x)\,\mathrm dx = 1$ and define$$A:=\left\{y\in[0,1]:\int_y^{y+0.6}f(t)\,\mathrm dt\geqslant0.6\right\}.$$What is the smallest possible Lebesgue measure of $A$? If $f(x)=2$ ...

Let $f:[0,+\infty)$ be a continuous function that satisfies: $\forall_{t\geq0}:\space \lim_{x\to\infty}\space (f(x+t)-f(x))=0$. Does it follow that $f$ is uniformly continuous? I have managed to show that if there exists $\space$ $\lim_{x\to\infty}\space f(x)=G\in\Bbb{R}$ $\space$ then the fun...

Let $\Omega=[0,1]^2$ be the unit square, $\Gamma_1=[0,1]\times\{0;1\}$ its horizontal boundary and $\Gamma_2= \{0;1\}\times[0,1]$ its vertical boundary. I would like to know the optimal Poincaré constants $C$, defined by $$ \forall u\in W^{1,2}(\Omega)\quad \int_\Omega u^2\le C\int_\Omega |\nab...

I am reading through rudin's functional analysis. He makes use of the fact that $Y$ is the smallest closed translation-invariant subspace of $L^1(R^n)$ that contains some $K \in L^1(R^n)$. However, it is not obvious to me that the space $Y$ we are working with are in fact the smallest. More spe...

let $a=(a_0,a_1,...,a_n)\in \mathbb R^{n+1}$ and $P_a(x)=\displaystyle\sum_{k=0}^{n}a_k \cos (kx) $ define $b=(a_n,a_{n-1},...,a_0)$ If $Z_a$ is the number of roots of $P_a$ on $[0,2\pi[$ then $$Z_a+Z_b \geq 2n$$ in this post Roots of trigonometric polynomial, An answer was given to me.I have...

Consider a hollow spherical charge with density $\rho'$ continuously varying only with respect to distance from the center $O$. $V'=$ yellow volume $k \in \mathbb {R}$ $\forall$ point $P$ inside the hollow sphere: \begin{align} \vec{E}_P &=\displaystyle\int_{V'}\rho'\ \vec{f}(r)\ dV'\\ &=\i...

Let $f$ be an infinite times differentiable function. Is it true that: the higher the degree $n$ of the Taylor polynomial $T_{n,f,x_0}$ of $f$ around $x_0$, the better the approximation? Some thoughts. Given $n$, polynomial $T_{n,f,x_0}$ is the best approximation of $f$ near $x_0$ that ful...

Prescribe a map: $$\Psi:\zeta_{\Bbb R^2} \to \Bbb T^2,$$ which gives a transformation of $\zeta-$space in $\Bbb R^2,$ to the flat torus. Let $\zeta_{\Bbb R^2}$ consist of flow lines in which the sources correspond to $(0,0)$ and $(1,0)$ and the sinks correspond to $(0,1)$ and $(1,1)$: $$ \tau...

Let $f:U \subset \mathbb{R^n} \to \mathbb{R}^n$ be totally differentiable and there exists a constant $c > 0$, so that $$\forall x,y \in \mathbb{R^n}: x,y \in U => \left\lVert f(x) - f(y) \right\rVert \geq c \left\lVert x - y \right\rVert$$ Prove that $f:U \to f(U)$ is globally invertible. Ch...

An example of a function that relates to my question: $f(x)=(2x+1)^2$ could be differentiated (at least) 4 different ways using rules Expand and differentiate each term: $f(x)=4x^2+4x+1$ then $f'(x)=8x+4$ Product rule: $f(x)=(2x+1)(2x+1)$ then $f'(x)=(2)(2x+1)+(2x+1)(2)=8x+4$ Chain rule: $f(x)=...

Let $E,F$ vector spaces of finite dimension over $K$ and $f,g$ linear tranforms from $E$ to $F$. Consider the linear tranform $f+g:E \to F$, $(f+g)(u) = f(u)+g(u) \;\;\forall u \in E$. Prove that are equivalent: (i)$\dim Im(f+g) = \dim Im(f) + \dim Im(g)$ (ii)$Im(f)\cap Im(g)=\{0\},f(Nuc(g))=I...

Given that $V$ is a finite dimensional vector space. The annihilator $S^0$ of $S$ is the set $$S^0 = \{f \in V^* :\, (\forall x \in S) \, f(x)=0\}.$$ If $W$ is a subspace of V and $x \notin W$, prove that there exists $f \in W^0$ such that $f(x) \neq 0$. Prove that $(S^0)^0 = \operatorn...

I am looking for a publicly-available software package (preferably in Python, but I'll take what I can get) capable of performing a decomposition of a real $n\times n$ skew-symmetric (sometimes called anti-symmetric) matrix $\textbf{A} = - \textbf{A}^T$. I have seen this decomposition referred to...

Let $V$ be a finite-dimensional vector space and $T:V\rightarrow V$ be linear. (a) Suppose that $V = R(T) + N(T)$. Prove that $V = R(T)\oplus N(T)$. (b) Suppose that $R(T)\cap N(T) = \{0\}$. Prove that $V = R(T)\oplus N(T)$. MY ATTEMPT (a) Let us take a vector $v\in R(T)\cap N(T)$. Thus $v = ...

Problem: Consider a real vector space $V$ of dimension $2m,\ m\in\mathbf{Z}^+$ and a non-singular quadratic form $Q$ on $V$. Suppose $Q$ vanishes on a subspace of dimension $m$, what is the signature of $Q$? Attempts made: by Sylvester's Law of Inertia, choose a basis {$a_1,..a_m,b_1,..b_m$} suc...

Let's say two people are playing a game, where each one flips a coin with some unknown probability of success (e.g. a heads), given by $p_1$ and $p_2$. A player wins the game when they get to 21 heads. We want to know the probability one player gets to 21 heads before the other player (or, more a...

I am studying my probability course and came up with the following question: Let $X, Y, Z$ be three independent geometrically distributed random variables with parameter $p, q, l$ which represents the number of failures before the first success. In addition, those variables are conditioned by $X...

Years ago I had found (and printed) a big list of all equivalences used in modular representation theory (and block theory), with arrows denoting the various implications (for instance, Morita equivalence -> Derived equivalence). At the top there were strong things like isomorphism of source alge...

So I need to prove the following: Let $\chi:G\rightarrow \mathbb{C}$ be a character of group $G$, with the property that $N \leq \ker \chi$, show that: $$\tilde\chi: G/N \rightarrow \mathbb{C} \qquad \tilde\chi(gN)=\chi(g) $$ is a character of the $G/N$ group. My attempt: Let's choo...

Consider $P:A\to[a,b]$ where $A\subseteq[a,b]$. I want an average which satisfied the following properties When $P$ is arbitrary The average must always be between infimum and supremum of $P$'s range. Countably-additive measures with integrals won't work for countable $A$ since the...

Case 1: We will look at an easier problem first. Let $|\alpha|, |\beta| \leq \alpha_c, \alpha_c \leq \pi$. I want to solve for $\rho(\beta)$ in the following equation, where $P$ denotes the principal value of the integral: $$\frac{2\sin{\alpha}}{\lambda} = P\int_{-\alpha_c}^{\alpha_c} d\beta \, ...

Suppose I have an equation in some reflexive separable Banach space $X$: $$Au=f$$ for given data $f$ and $A\colon X \to X^*$ a pseudo-monotone operator. Existence can be proved via Galerkin approximations where one takes a finite dimensional subspace $X_n \subset X$ and considers $$\langle Au_n, ...

I am working on #3 of this problem set: https://www.cpp.edu/~hspc/problems/HSPC_2019_Middle_School_Questions.pdf and I wanted to try solving it numerically. My approach was this: number of mats = total different combinations of RGB - (number of those combinations that fail the column conditio...

How to find the number of ways in which the above $9$-by-$9$ grid can be filled using the digits $(1-9)$ (repetition is allowed) such that all of the following conditions are satisfied: Any $3$-by-$3$ grid is not totally empty. Any $3$-by-$3$ grid is not totally filled. Any $...

We know that a function $f: R^2\to R$ that is section-continuous (that is each $x\mapsto f(x,y)$ and $ y\mapsto f(x,y)$ are continuous) need not be continuous. $f(x,y)=\frac{xy}{x^2+y^2}\chi_{\{0\}^c}$ is a counterexample for such a claim. However apparently if add the condition that $f$ maps com...

Let $X_{i\le n}$ be compact but not necessarily $T_2$, each with equivalence relation $\sim_k\ $. Is it possible to find equivalence relation $\sim$ such that $\ \ ^{\prod X_i}/\ _{\sim}\ \simeq\prod\big(\ ^{X_k}/_{\sim_k}\big)\ ?$ Disproof: We show first that possible equivalence relation...

I have a very vague intuitive idea which I have been struggling to give a meaningful definition to, and was wondering if this idea I am working on has been tried before. I apologize if I am unable to formulate these abstract intuitive idea's better than this; I am trying to work on my exposition...

Consider the complex projective plane $\Bbb CP^2$. We can embed $\Bbb RP^2$ in $\Bbb CP^2$ in a natural way, namely, $[x_0:x_1:x_2]\in \Bbb RP^2 \mapsto [x_0:x_1:x_2] \in \Bbb CP^2$. We can thus consider $\Bbb RP^2$ as a subspace of $\Bbb CP^2$. On the other hand, let $Q=\{[z_0:z_1:z_2]\in \Bbb C...

Without giving you the lengthy and really nasty computations, let $n\in\mathbb{N}$, $\beta(n):=\frac{n(n+1)(2n+1)}{6}$ and $\alpha(n):=\sqrt{1/\beta(n)}$. The $(n\times n)$-matrix $A(n)$ is then given by the following entries: The first row of $A(n)$ is given by \begin{align*} a_{1,i}=\f...

Given any subset $E$ of field $\mathbb{F}$ (real or complex), does there exist a normed linear space $X$ over $\mathbb{F}$ and a bounded linear operator $$A:X\rightarrow X$$ such that spectrum of $A$ is precisely the set $E$. NOTE : It is known that this is true for compact sets as we can use th...

In Banach spaces, the following result is well-known: (1) Let $X$ be a Banach space. Let $\{x_n\}\subset X$ and $\{x^*_n\}\subset X^*$ be such that $x_n \rightarrow x$ (convergence with respect to strong topology on $X$) and $x^*_n\overset{\ast}{\rightharpoonup} x^*$ (convergence with respect to...

$$\begin{cases}\min\limits_{{\bf u}(t)}&\displaystyle\int_0^{t_f}ϕ(\mathbf{u}(t))\,{\rm d}t\\&\dot{\bf x}(t)=A{\bf x}(t)+B{\bf u}(t)\\&C{\bf x}(t_f)=α\\&g({\bf x}(t_f))\le β\\&{\bf x}(0)=0\end{cases}$$ I am dealing with an optimal control problem of the above form, where $A$, $B$ are matrices and...

Let $A$ be an unbounded self-adjoint operator acting on a Hilbert space $H$ (typically $L^2(\mathbb{R}^d)$). Then, using Stone's theorem, the operator $A^{\otimes 2}:=A\otimes 1+1\otimes A$ defines a self-adjoint operator on $H\otimes H$, the domain of which might be difficult to determine. Is ...

Consider two non-empty, bounded and convex domains $A \subset B \subset \mathbb{R}^d$ with $C^2$-boundaries. Define a function $\phi:\partial A \to \partial B$ by $x\mapsto y$ such that $y-x=\lambda_x\vec{n}_A(x)$ for some $\lambda_x >0$ where $\vec{n}_A(x)$ is the outer normal of $\partial A$ at...

In not a few textbooks, we have some version of this example of a falling body: Problem. Solve $\frac{dv}{dt} = a - bv$ with the initial condition $(t,v)=(0,0)$. Solution. Rearrange to get $\frac{dt}{dv} \overset 1= \frac{1}{a - bv}.$ Integrating, $t=-\frac{1}{b}\ln|a-bv|+C_1$. ...

Thinking about Goldbach conjecture, I have the following question: Is every even number greater than 4 the sum of a prime number with another number that belongs to the set of twin primes? For example, as 31 and 17 belong to the set of twin primes, 38=31+7 and 40=17+23.

This question is an offshoot of the following answer to a closely related MSE question. Let $N$ be a deficient-perfect number, i.e. $N$ is a positive integer such that $D(N) \mid N$ where $D(N)=2N-\sigma(N)$ is the deficiency and $\sigma(N)$ is the sum of divisors of $N$, respectively. Since $N...

Let $\sigma(x)$ denote the sum of the divisors of $x$, and denote the abundancy index of $x$ as $$I(x) = \dfrac{\sigma(x)}{x},$$ and the deficiency of $x$ as $$D(x) = 2x - \sigma(x).$$ If the equation $I(a)=b/c$ has no solution $a \in \mathbb{N}$, then $b/c$ is said to be an abundancy outlaw. St...

I'm following Neukirch's algebraic number theory. The situation is as follows: Let $K$ be a number field of degree $n$. Then $n = r + 2s$, where $r$ is the number of real embeddings $\rho : K \to \Bbb C$ (i.e. those embeddings $\rho$ such that $\rho(K) \subseteq \Bbb R$) and $s$ is the number of...

I am given a coefficient to estimate as a assignment question. But I am unable to think how it must be true. Notations Let $l$ belongs to $\{1, 2,\ldots,a\}$ and $j$ belongs to $\{0,1,\ldots,n\}$. Assume that this equation holds $$ c_{l,j,n}=\frac{1}{2\pi i}\int_{|z+j+1|=\frac{1}{2}} R_n(t) (...

Where the map comes from I was reading on The six-vertex model, $R$-matrices, and quantum groups, where computing the eigenvalues of a transfer matrix is talked about. It is mentioned that Bethe ansatz gives the following formula for some eigenvalues: $$\Lambda = a^NL(z_1)\cdots L(z_n) + b^N M(z...

This is part of the proof of theorem 28.27 from Rene Schilling's Measures, Integrals and Martingales. Let $(X, \mathscr{A},P)$ be a probability space and $(e_n)_{n \in \mathbb{N}_0} \subset L^2(P)$ be independent random variables such that $E(e_n)=0$ and $E(e_n^2) = 1$ and let $(c_n)_{n \in \mat...

Say we have $P:A\to[0,1]$ where $A\subseteq \mathbb{R}$ $$P(x)=\begin{cases} x^2+5 & x\in\left\{\frac{1}{a}:a\in\mathbb{N}\right\}\cup\left\{\frac{1}{b^{\sqrt{2}}+0.1}+\frac{1}{5}:b\in\mathbb{Z}\right\}\cap[0,1] \\ x & x\in\left\{\frac{1}{c+.1}+0.6:c\in\mathbb{Z}\right\}\cap[0,1] \end{cases}$$ ...

Inspired by Process with Markov property but not strong Markov property. I'm having trouble rigorously verifying that this is a Markov process but not a strong Markov process. From the definition, I want to show that $$X_t(\omega) = \max \{t-T(\omega), 0\}, \text{ where } T\sim \text{Exp}(1)$$ ...

I don't know if my question can be answered easily from your reasonings and knowledeges of the theory of odd perfect numbers. I wondered about it yesterday. Definitions and notation. We denote the sum of divisors function $\sum_{1\leq d\mid m}d$ as $\sigma(m)$ and the radical of an integer $m>1...

Let $B(0,1)=\{(x,y,z):x^2+y^2+z^2< 1\}$ denote the unit ball in $\mathbb{R}^3$ and $C$ denote a cylinder around the $z$-axis with radius $2$. Suppose that on each point $w$ in $B(0,1)$ we attach a pair of linearly independent vectors $ u_w,v_w$. Suppose now that we can find a function $f:\mathbb{...

This was a task that gave $9$ points in an exam I failed. Since our professor doesn't provide solutions I thought I'd ask here. Let $f:\mathbb{R^2} \to \mathbb{R}$ be twice continuous partially differentiable and $x_0 \in \mathbb{R^2}$ random. Deduce the following formula for $x \in \mathbb{R...

Consider the Kernel regression estimator: $$\hat{y}(x)=\frac{\sum_{i=1}^n{K(\|x-x_i\|)y_i}}{\sum_{i=1}^n{K(\|x-x_i\|)}},$$ where $x,x_1,\dots,x_n\in\mathbb{R}^d$, $y_1,\dots,y_n\in\mathbb{R}$, where $K:[0,\infty)\rightarrow(0,\infty)$ is a strictly positive valued, differentiable kernel functio...

I was reading the paper Distance function and cut loci on a complete Riemannian manifold. I found two problems: How is he getting $$d(g_1(l-\varepsilon),g_2(l+\tau))=\sqrt{\varepsilon^2+\tau^2+2\varepsilon\tau \cos \omega }~(1+O(\tau^2))?$$ (Any reference will be hepful. How is he claiming that...

Let $X$ be a smooth manifold and let $Y$ be a smooth submanifold. Denote by $$ TY^0=\{(q,p)\in T^*X\colon q\in Y,p|_{T_qY}=0\}\subseteq T^*X $$ The annihilator of $Y$. Is is it true that the annihilator is a smooth Lagrangian submanifold of $T^*X?$, and how to see this? The given hint is too use ...

Let me start with the following tl;dr version of my question What is a higher-order derivative, in general? How does it relate to the exterior derivative and to differential forms? Suppose we have a bundle and a section $E \overset{\sigma}{\underset{\pi}{\leftrightarrows}} M$. Assume we ...

Let $n$ be an even postive integer, and $a,b\in (-1,1)$, $a+b\ge0$. Show that $$\left(\frac12+\sum_{k=1}^na^{k^2}\right)\left(\frac12+\sum_{k=1}^nb^{k^2}\right)\ge\left(\frac12+\sum_{k=1}^n(ab)^{k^2}\right)^2\tag{1}$$ It seems promising to use Cauchy-Schwarz inequality to prove it or other...

Hi it's a problem of mine : Prove that (without calculating the integral) : $$\int_{0}^{e}\operatorname{W(x)^{\pi}}>1$$ Where $\operatorname{W(x)}$ denotes the Lambert's function . I have tested during 2 hours some methods but it fails always because the inequality is pretty...

so I just started studying projective varieties (over algebraically closed fields) and I simply want to understand why $$V_{P_n}(T_i) \simeq P_{n-1}$$ whereas $V_{P_N}(T_i):= \{[z_0,..., z_n]\in P_n| z_i=0\}$ $i=0,...,n$. So clearly I can get a morphism $P_{n-1}\mapsto V_{P_n}(T_i)$, $[z_o,...,z...

Reminder : Given a set $S$ of $n$ elements (we will use $[n]$ in the following for simplicity), a Latin square $L$ is a function $L : [n]\times [n] \to S$, i.e., an $n\times n$ array with elements in $S$, such that each element of $S$ appears exactly once in each row and each column. For example,...

I am trying to work with graphs whose adjacency matrix $A$ have the property $$A_{ij} > 0 \iff A_{ji} > 0,$$ but $A_{ij} \neq A_{ji}$ in general. In particular, I am interested in saying something about the stationary distribution of the corresponding transition matrix, i.e., $D^{-1}A$ with $D=\o...

Consider the scalar conservation law $u_t+f(u)_x=0,$ whose conservative and consistent first order numerical scheme is given by \begin{equation}\label{1}u_i^{n+1}=u_i^n-\lambda\left(F(u_i^n,u_{i+1}^n)-F(u_{i-1}^n,u_{i}^n) \right) \end{equation} Suppose we modify the above scheme by $$u_i^{n+1}=u...

Edit: I know the question is very vague; this is partly due to the fact that I stumbled upon this map rather accidentaly and that I just wondered what can be said about this. So if you have some thoughts about this, please share them :) Let $R$ be a unital ring, possibly non-commutative. Let $M$...

Let $R\subseteq S$ be commutative rings in the sense that $R$ is a subring of $S$ with the same unity. So canonically, $S$ has an $R$-module structure. We say that the inclusion map $i: R\to S$ splits as a map of $R$-modules iff there exists an $R$-module map $f: S\to R$ such that $f\circ i=id_R...

I am an undergraduate student of mathematics and we have in out linear algebra course,a brief introduction of Adjoint operators and unitary operators.Now I understand that adjoint of a linear operator in finite inner product space corresponds to the conjugate transpose of a matrix.And a self-adjo...

Let $V$, $W$, and $Z$ be finite-dimensional vector spaces. Let $T: V \rightarrow W$ and $S: W \rightarrow Z$ be linear mpas. If $S\circ T$ is an isomorphism, then $T$ and $S$ are isomorphisms. I am told that this is in general not true, but I am not sure where to start to prove untrue. Can ...

Let $\Lambda$ be a lattice in $\mathbb{C}$ and $X = \mathbb{C}/\Lambda$ be a complex torus. Exercise 6 of chapter 3 of Tamás Szamuely's book "Galois Groups and Fundamental Groups" (actually, the updated version in the erratum here) asks to show that $X$ have a Galois branched cover $Y \to X$, ram...

For a commutative ring $R, \alpha \in \tilde{H}^p(X;R), \beta \in \tilde{H}^q(X;R)$ and $ u \in \tilde{H}_{p+q}(X;R),$ Verify $\langle\alpha \cup \beta, u\rangle = \langle\beta, \alpha \cap u\rangle.$ We were asked to solve this after reading section 22.5 of "Modern Classical Homotopy Theory " b...

Here is the question: Show that is natural in both variables. That is suppose $f: X \rightarrow Y, u \in \tilde{H^{*}}(Y), \alpha \in \tilde{H_{*}}(X).$ Then we can form $$<u, f_{*}(\alpha)> \in \tilde{H}_{n-k}(Y)$$ And $$<f^{*}(u), \alpha)> \in \tilde{H}_{n-k}(X).$$ Show that $$f_{*}(<f^{*}(u...

We are given two manifolds $M_1 \subset M_2 \subset \mathbb{R}^n$. I need to prove that $\dim(M_1) \leq \dim(M_2)$. This is part of calculus 4 course, and I haven't taken a course in topology yet, so the proof should be elementary. I was able to show that if $k_1 = \dim(M_1), k_2 = \dim(M_2)$ t...

Let $f(t,s)$ be a (jointly) continuous bounded function on $(a,b) \times (a,b) \subseteq \mathbb{R}^2$. Suppose further than for each fixed $s \in (a,b)$, the continuous function $t \mapsto f(t,s)$ is differentiable (Lebesgue) almost everywhere on $(a,b)$ and that $\partial_t f(t,s) \in L^1(a,b)_...

We have a square matrix $A$ of size $2n\times 2n$ (where $n>2$) with entries from the set $\{-1,0,1,t,-t\}$. Such that: 1) matrix $A$ has $n$ pairs of rows (in the picture a pair is bordered blue), where one (upper) row from a pair has exactly two nonzero entries $-t, t$. The second row from the...

I have two questions: Can a irreducible rational curve have infinitely self intersections? If the solution for $0=f(x,y)\in k[x,y]$ exists and $x$ and $y$ are parametrized by rational functions of some parameter $t$, then we miss only finitely many points. It is not clear to me why only fini...

I have the following system of equations involving PolyLog functions (Fermi Dirac Integrals) where $d,t\in \mathbb{Z}$ and $d,t >0$ such that $$ J = J_0 \cdot \left[F_{\frac{d-1}{t}}\left(\eta\right)-F_{\frac{d-1}{t}}\left(\eta-v_d\right)\right]\\ n = n_0 \cdot\left[F_{\frac{d-t}{t}}\left(\eta\r...

Say $\mathcal{C}$ is some conic and $P\in \mathcal{C}$ is fixed point on it. For each $X$ on $\mathcal{C}$ let $Y$ be such on $\mathcal{C}$ that $\angle XPY = 90^{\circ}$. Prove that the line $XY$ goes through a fixed point. I can prove this with projective geometry: Transformation $\Pi:...

Let $(M,d)$ be a Riemannian manifold and let $T_1M=\{v\in TM: \|v\|=1\}$. Define a map, $$s:T_1M\to \hat{\mathbb{R}},~~ s(v)= \sup\{t:d(\pi(v),\operatorname{exp}(tv))=t\}$$ where $\pi$ is the projection map from the the tangent bundle to $M$ $(\pi(p,v)=p,~(p,v)\in TM)$. I need to prove that the ...

Context. Let $X$ be a Banach space with a Schauder basis $(x_n)_{n=1}^\infty$, let $q_X:X\to X^{**}$ be the canonical isometric embedding, and let $\mathcal{U}$ be a free ultrafilter on $\mathbb{N}$. Due to Banach-Alaoglu together with the basic facts about limits along a free ultrafilter, for ...

I know that if $f$ is a conformal mapping of $\mathbb{D}$ onto some domain $D$ such that $\partial D$ is a Jordan curve, then $f$ has a continuous extension up to $\partial \mathbb{D}$ such that $f(\partial \mathbb{D}) = \partial D$. This is, as far as I know, called Caratheodory's extension theo...

I ham given the following problem to solve: 1.9. The program should take three numbers: a; b; c and find the roots of the quadratic equation in the form: If the value of the determinant of the quadratic equation is negative (i.e. ∆ <0), the program should write an appropriate me...

Consider the following linear programming problem $$ \min_{x}\sum_{j=1}^J x_ja_j\\ s.t.\\ x_j\geq 0 \text{ }\forall j=1,...,J\\ \sum_{j=1}^J x_jb_{j,r}-c_r=0 \text{ }\forall r=1,...,R $$ where $a_j, b_{j,r}, c_r$ are known scalars $\forall j=1,...,J$ and $\forall r=1,...,R$. $x$ is a $J\times 1$ ...

This Groupprops Wiki page has a nice chart classifying the subgroups of $A_5$ upto automorphism. It shows the various representative subgroups. However, say if I were told to manually find the representative subgroups corresponding to 2-Sylow, 3-Sylow and 5-Sylow, how should I go about it? ...

Choose a set S consisting of $\frac{n+1}{2}$ numbers from the first $n$ natural numbers($1,2,3,...,n$) ($n\geq 2017$, $n$ is odd). Prove that there must be three numbers in S which are a 3-term arithmetic progression. I am thinking of using recursion but I find the condition $n \geq 2017$ quite ...

We define a set function $f:2^E \rightarrow \mathbb{R}$ to be submodular if for every $ S,T\subseteq E $ with $ S\subseteq T $ and for every $ x\in E\setminus T : f(S\cup \{x\})-f(S)\geq f(T\cup\{x\}) - f(T) $. How could I extend this concept to a Cartesian product of two sets? For example, woul...

I know that a sequence of measures on $\mathbb{R}$ converges in distribution if and only if the corresponding Lévy's metric converges (Relationship to weak toplogy (Lévy metric)). According to this article : "The concept of the Lévy metric can be extended to the case of distributions in $\mathb...

First, some definitions: Given a Kac-Moody algebra $\mathfrak g$, the category $\mathcal O$ of $\mathfrak g$ is the category whose objects are $\mathfrak g$-modules $V$ which are weight modules ($V = \bigoplus_{\lambda \in \mathfrak h^*} V_\lambda),$ every weight-space $V_\lambda$ is finite...