last day (1506 days later) » 

5:40 AM
HNQ and bountied questions are more likely to be interesting, so it might make sense to have them listed somewhere.
In the case HNQ, perhaps one other thing why people might follow them are moderation purposes.
The HNQs can be also found in the Hot Network Questions room.
You can find them also using SEDE.
Feed in chat has an advantage that the message contains also a brief preview of the question.
Technical aspects
To get a feed for HNQs, I use the feed created by rene: How to add HNQ to chat room feed?
For bounties, it is a bit more complicated - I do not know of any other way than adding a feed for each tag individually. (If there isn't other solution, adding feeds for big tags would be sufficient.)
Other bounty related rooms:
There is a similar room on MO: Listing bounties and HNQs.
As mentioned above, there is a room called Hot Network Questions - where questions from all sites should be available.
For starters, let us try to get here list of HNQs and ask one of the mods to rename the feed:
5:58 AM
Q: A "new" general formula for the quadratic equation?

lone studentMaybe 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...

Q: Conjectured continued fraction formula for Catalan's constant

Mr PieYesterday 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...

Q: What is the intuition behind Liouville's theorem in complex analysis?

T  A OI'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.

Q: Free group over a set of two elements is abelian

pankeLet $A=\{a,b\}$ 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...

in Pearl Dive, 17 mins ago, by Martin Sleziak
@AlexanderGruber I will need some help from a mod. Would you be willing to rename the feed with HNQs in this room to "HNQ" or some similar name?
In this way we would be able to distinguish between bounties (posted by the user "Feed") and HNQs (posted by the user "HNQ").
1 hour later…
7:15 AM
Since I have mentioned various rooms related to bounties, I should have said that for a brief period of time, there was also this one: Bounty room.
7:51 AM
Q: Probability: Drawing two marbles simultaneously from the bag

Matic PotocnikThis 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...

1 hour later…
9:06 AM
Q: Why I can't reduce an integral divided by another integral?

DictatorFor 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...

9:25 AM
Q: Is the number of orderings the same of the number of automorphisms in a ring?

Tomás Pacheco 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...

9:44 AM
Q: Solving the cryptarithm "THE+BEST+SYSTEM=METRIC"

102152111I 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 ...

2 hours later…
11:15 AM
If we add feeds for most of the big tags, we should achieve that most bountied questions will be posted in the room.
Q: Value bound for integrable periodic function

Dexter 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 $f(x)=2$ ...

Q: Is a continuous function defined on $[0,+\infty)$ that satisfies a certain condition uniformly continuous?

DoodLet $f:[0,+\infty)$ 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...

Q: Poincaré optimal constant under multiple conditions

DiegoG7Let $\Omega=[0,1]^2$ 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...

Q: Smallest closed translation-invariant subspace of $L^1(R^n)$ that contains some $K \in L^1(R^n)$.

smaI 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...

Q: Roots of trigonometric polynomial 2

Mohammed M. Zerraklet $a=(a_0,a_1,...,a_n)\in \mathbb R^{n+1}$ 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...

Q: How to prove that $u(r)=k \frac{1}{r}$ is the only solution for the integral equation $\int_{V'}\rho'\ u(r)\ dV' = constant$?

JoeConsider 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...

Q: Taylor polynomial: the higher the degree, the better the approximation?

Nikolaos Skout 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...

Q: Looking to prove that there are infinitely many of these closed orbits

geocalc33Prescribe 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...

Q: $\forall x,y \in \mathbb{R^n}: x,y \in U => \left\lVert f(x) - f(y) \right\rVert \geq c \left\lVert x - y \right\rVert$ globally invertible

JavaTeachMe2018Let $f:U \subset \mathbb{R^n} \to \mathbb{R}^n$ 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...

Q: What are most ways to differentiate the same function?

Robbie_P_mathAn 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)=...

Q: Conditions for the dimension of rank of sum of linear transforms be equal the sum of ranks

WybieLet $E,F$ 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...

Q: How to prove that $(S^{0})^{0} = \operatorname{span}(\psi(S))$, where $\psi: V \to V^{**}$ is the natural isomorphism

Beacon 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...

Q: Implementation of Youla Decomposition for a square skew-symmetric matrix

Surgical CommanderI 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...

Q: Let $V$ be a finite-dimensional vector space and $T:V\rightarrow V$ be linear. Suppose that $V = R(T) + N(T)$. Prove that $V = R(T)\oplus N(T)$.

BrickByBrickLet $V$ 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 = ...

Q: Signature of quadratic form on vector space of even dimension

Harry AlliProblem: 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...

Q: Probability of winning a coin flipping game

Ryan SimmonsLet'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...

Q: Conditional expectation for three variables

volondI 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...

Q: Diagram of equivalences in modular representation theory

AnalysisStudent0414Years 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...

Q: Showing $\tilde\chi: G/N \rightarrow \mathbb{C} $ is a character of group $G/N$.

LukaSo 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...

Q: How do we create a better definition for my average which is easier to compute and gives exact values?

ArbujaConsider $P:A\to[a,b]$ 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...

Q: How do I solve this integral equation?

Bruce LeeCase 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 \, ...

Q: Non-uniqueness of (Galerkin) approximations and convergent subsequences without the axiom of choice?

MMMLSuppose 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, ...

Q: Combinatorics with constraints and repeating elements

robertI am working on #3 of this problem set: 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...

Q: The number of ways in which a $9$-by-$9$ grid can be filled given some conditions.

Hussain-Alqatari 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 $...

Q: Section continuity implies continuity

user593295We 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...

Q: Product of quotient is quotient of product for compact spaces?

Local Kleinian ManifoldLet $X_{i\le n}$ 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...

Q: Has anyone tried relating the projective plane to the Riemann Zeta function?

Matt CalhounI 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...

Q: Homotopy type of $\Bbb CP^2 - \Bbb RP^2$

TroaroConsider 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...

Q: Is it possible to determine the eigenvalues of this rather complicated matrix? How many eigenvalues are stable/ unstable?

math12Without 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...

Q: Spectrum can be an arbitrary subset.

Akash YadavGiven 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...

Q: weak star and strong convergence of net in Banach spaces

BlindIn 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...

Q: Optimal control with final state inequality constraint

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

Q: Domain of essential self-adjointness for $A\otimes 1+1\otimes A$

ChrLet $A$ 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 ...

Q: Show that function is a diffeomorphism

MimimiConsider 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...

Q: Separation of variables -- what to do about the case where we might be dividing by zero?

user20311In 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, $t=-\frac{1}{b}\ln|a-bv|+C_1$. ...

Q: Is every even number greater than 4 the sum of a prime number with another number that belongs to the set of twin primes?

mathspikaThinking 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.

Q: If $N$ is deficient-perfect, under what conditions does this inequality hold?

Jose Arnaldo Bebita-DrisThis 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...

Q: When $p$ is an odd prime, is $(p+2)/p$ an outlaw or an index?

Jose Arnaldo Bebita-DrisLet $\sigma(x)$ 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...

Q: Minkowski theory: an isomorphism of $\Bbb R$-vector spaces induces a scalar product on $\Bbb R^{r + 2s}$

Edward EvansI'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...

Q: Unable to deduce an inequality related to an integral

Yannic MullerI 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) (...

Q: Is a parametrization $\mathbb{C}^3 \ni (\rho, u, \eta) \mapsto (\rho \sin(u+\eta), \rho \sin u, \rho \sin \eta) \in \mathbb{C}^3 $ surjective?

seraWhere 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...

Q: Question about a square integrable uniformly bounded orthonormal sequence $(e_n)$ such that $\sum c_n e_n$ converges a.e.

nomadicmathematicianThis 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...

Q: How do we define an "intuitive" average on this function?

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

Q: Markov but not strong Markov: $\max\{t-T,0\}$ for $T\sim \text{Exp}(1)$

D.R.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 $$X_t(\omega) = \max \{t-T(\omega), 0\}, \text{ where } T\sim \text{Exp}(1)$$ ...

Q: On the least number of factors $\sigma(q^{e_q})$ to get the least multiple of $\operatorname{rad}(n)$, on assumption that $n$ is an odd perfect number

user759001I 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...

Q: A function with big eigenvalues acting on the unit ball.

Mad MaxLet $B(0,1)=\{(x,y,z):x^2+y^2+z^2< 1\}$ 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{...

Q: Deducing $f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac12(x-x_0)^TH_f(x_0)(x-x_0)+o(\lVert x-x_0\rVert^2)$

JavaTeachMe2018This 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...

Q: Does kernel regression preserve monotonicity?

cfpConsider 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...

Q: $d^2(p,\cdot)$ is not directional differentiable

Sachchidanand PrasadI 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...

Q: Annihilator is a smooth submanifold of the cotangent bundle

JamesLet $X$ 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 ...

Q: Higher covariant derivatives and the exterior derivative

Eric AuldLet 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 ...

Q: Proving $\def\n#1{\left(\frac12+\sum\limits_{k=1}^n{#1}^{k^2}\right)}\n{a}\n{b}\ge{\n{(ab)}}^2$

communnites 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...

Q: Integral inequality $\int_{0}^{e}\operatorname{W(x)^{\pi}}>1$

The.old.boyHi 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...

Q: Morphism from projective varieties

Vasco1008so 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...

Q: Graph built from orthogonal Latin squares

Thomas LesgourguesReminder : 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,...

Q: Can we say something on the stationary distribution of almost-undirected graphs?

Enric FloritI 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...

Q: Second order conservative scheme with piecewise linear approximations

RosyConsider 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...

Q: What can be said about $ \hat{\otimes}_R:f\otimes g\mapsto f\otimes_R g$?

Redundant AuntEdit: 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 $M$...

Q: On subring $R\subseteq S$ such that the inclusion map $i: R\to S$ splits as an $R$-module map

user521337Let $R\subseteq S$ 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...

Q: Reference request for studying and visualizing dual vector spaces.

Kishalay SarkarI 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...

Q: Let $T: V \rightarrow W$ and $S: W \rightarrow Z$ be linear maps. If $S\circ T$ is an isomorphism, are $S$ and $T$ isomorphisms?

Beacon 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 ...

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Q: Branch points of a dihedral Galois branched cover of a complex torus

Robin CarlierLet $\Lambda$ 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...

Q: Verify that $\langle\alpha \cup \beta, u\rangle = \langle\beta, \alpha \cap u\rangle.$

SecretlyFor 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...

Q: Problem 22.35 in "Modern Classical Homotopy Theory " by Jeffery Strom on pg.510.

EmptymindHere 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...

Q: If $M_1, M_2$ are two manifolds such that $M_1 \subset M_2$, show that $ \dim(M_1) \leq \dim(M_2)$

OmerWe 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...

Q: Prove $\partial_t \int f(t,s) ds = \int \partial_t f(t,s)ds$ if $t \mapsto \partial_t f(t,s)$ exists almost everywhere for each $s$

JZSLet $f(t,s)$ 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)_...

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Q: Prove a property of a determinant with integer entries with one variable

oldWe 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...

1:24 PM
Q: Missing finitely many points by rational curve with parametrization by rational functions and rational curve self intersection

user45765I 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...

Q: Analytically solving (finding a maxima) a system of equations involving PolyLog functions (Fermi-Dirac Integrals)

IndeterminateI 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...

Q: Prove that the line $XY$ goes through a fixed point where $X,Y$ are on fixed conic so that $\angle XPY = 90$ where $P$ is fixed on the conic.

Aqua 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:...

Q: The map $f:T_1M\to \hat{\mathbb{R}}$ is continuous.

XYZABCLet $(M,d)$ 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 ...

Q: gaps in my Banach space ultrapower proof

Ben WContext. 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 ...

Q: Analytic continuation of a conformal map across the unit circle

SPSI 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...

Q: How can I solve this system of equation?

user366312I 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...

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Q: Uniqueness of Lagrangian Multipliers

STFConsider 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 $J\times 1$ ...

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Q: What should be the mental process for quickly evaluating representative Sylow subgroups of $A_5$?

S.D.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? ...

Q: 3-term arithmetic progression in a set of integers

Kiên Phùng HữuChoose 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 ...

Q: Submodularity for Cartesian product

ryan80We 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...

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Q: Lévy's metric on $\mathbb{R}^d$

mathexI 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...

4 hours later…
8:52 PM
Q: The set of weights of a module in the $\mathcal O$ category has a maximal element

math.hFirst, 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...


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