5:40 AM
Purpose
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.
Searching
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: lackadaisical-appeal.glitch.me/hnq/math.stackexchange.com

5:58 AM
26

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...

7

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 $$G=\cfrac{1}{1+\cfrac{1^4}{8+\cfrac{3^4}{16+\cfrac{5^4}{24+\cfrac{7^4}{32+\cfrac{9^4}{40+\ddot... 3 I'm looking for an intuitive motivation for Liouville's theorem from complex analysis. If somebody could illustrate this with a simple example, that would be great. Thank you so much. 1 Let A=\{a,b\} and a\ne b. Let F(A) be the free group constructed on A. Let f_a,f_b be the canonical homomorphisms of \mathbb{Z} into F(A). Let g:F(A)\rightarrow\mathbb{Z}\times\mathbb{Z} be the unique homomorphism for which g(f_a(1))=(1,0) and g(f_b(1))=(0,1). Let h:\mathbb{Z... in Pearl Dive, 17 mins ago, by Martin Sleziak @AlexanderGruber I will need some help from a mod. Would you be willing to rename the feed with HNQs in this room to "HNQ" or some similar name? In this way we would be able to distinguish between bounties (posted by the user "Feed") and HNQs (posted by the user "HNQ"). 1 hour later… 7:15 AM Since I have mentioned various rooms related to bounties, I should have said that for a brief period of time, there was also this one: Bounty room. 7:51 AM 4 This is a 9. grade elementary school quiz problem. There is a bag with 4 marbles. Two of them are yellow and the other two are green. You blindly pick two marbles at once. What is the probability that you get two yellow marbles? I would say the answer is \frac{1}{6}. There is a single combina... 1 hour later… 9:06 AM 2 For example,$$ \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...

9:25 AM
3

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
3

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 ...

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.

5

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 $f(x)=2$ ...

7

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...

0

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 $$\forall u\in W^{1,2}(\Omega)\quad \int_\Omega u^2\le C\int_\Omega |\nab... 3 I am reading through rudin's functional analysis. He makes use of the fact that Y is the smallest closed translation-invariant subspace of L^1(R^n) that contains some K \in L^1(R^n). However, it is not obvious to me that the space Y we are working with are in fact the smallest. More spe... 0 let a=(a_0,a_1,...,a_n)\in \mathbb R^{n+1} and P_a(x)=\displaystyle\sum_{k=0}^{n}a_k \cos (kx)  define b=(a_n,a_{n-1},...,a_0) If Z_a is the number of roots of P_a on [0,2\pi[ then$$Z_a+Z_b \geq 2nin this post Roots of trigonometric polynomial, An answer was given to me.I have... 0 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... 5 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... 6 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...

1

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...

0

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: $f(x)=... 1 Let$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...

0

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 $(S^0)^0 = \operatorn... 2 I am looking for a publicly-available software package (preferably in Python, but I'll take what I can get) capable of performing a decomposition of a real$n\times n$skew-symmetric (sometimes called anti-symmetric) matrix$\textbf{A} = - \textbf{A}^T$. I have seen this decomposition referred to... 1 Let$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 = ...

0

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...

0

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...

1

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 $X... 4 Years ago I had found (and printed) a big list of all equivalences used in modular representation theory (and block theory), with arrows denoting the various implications (for instance, Morita equivalence -> Derived equivalence). At the top there were strong things like isomorphism of source alge... 2 So I need to prove the following: Let$\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... 2 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... 6 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: $$\frac{2\sin{\alpha}}{\lambda} = P\int_{-\alpha_c}^{\alpha_c} d\beta \, ... 1 Suppose I have an equation in some reflexive separable Banach space X:$$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, ... -2 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... -2 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$...

4

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...

3

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...

4

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...

5

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 $Q=\{[z_0:z_1:z_2]\in \Bbb C... 2 Without giving you the lengthy and really nasty computations, let$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... 3 Given any subsetE$of field$\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... 4 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... 2 $$\begin{cases}\min\limits_{{\bf u}(t)}&\displaystyle\int_0^{t_f}ϕ(\mathbf{u}(t))\,{\rm d}t\\&\dot{\bf x}(t)=A{\bf x}(t)+B{\bf u}(t)\\&C{\bf x}(t_f)=α\\&g({\bf x}(t_f))\le β\\&{\bf x}(0)=0\end{cases}$$ I am dealing with an optimal control problem of the above form, where$A$,$B$are matrices and... 0 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 ... 7 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... 8 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,$t=-\frac{1}{b}\ln|a-bv|+C_1$. ... 11 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. 0 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...

18

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...

0

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...

1

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 $$c_{l,j,n}=\frac{1}{2\pi i}\int_{|z+j+1|=\frac{1}{2}} R_n(t) (... 0 Where the map comes from I was reading on The six-vertex model, R-matrices, and quantum groups, where computing the eigenvalues of a transfer matrix is talked about. It is mentioned that Bethe ansatz gives the following formula for some eigenvalues:$$\Lambda = a^NL(z_1)\cdots L(z_n) + b^N M(z...

2

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 $(c_n)_{n \in \mat... 1 Say we have$P:A\to[0,1]$where$A\subseteq \mathbb{R}$$$P(x)=\begin{cases} x^2+5 & x\in\left\{\frac{1}{a}:a\in\mathbb{N}\right\}\cup\left\{\frac{1}{b^{\sqrt{2}}+0.1}+\frac{1}{5}:b\in\mathbb{Z}\right\}\cap[0,1] \\ x & x\in\left\{\frac{1}{c+.1}+0.6:c\in\mathbb{Z}\right\}\cap[0,1] \end{cases}$$ ... 0 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 $$X_t(\omega) = \max \{t-T(\omega), 0\}, \text{ where } T\sim \text{Exp}(1)$$ ... 3 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...

3

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{... 2 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...

1

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...

0

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...

5

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 ...

6

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 ...

5

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...

6

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...

0

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 $P_{n-1}\mapsto V_{P_n}(T_i)$, $[z_o,...,z... 5 Reminder : Given a set$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$appears exactly once in each row and each column. For example,... 0 I am trying to work with graphs whose adjacency matrix$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...

1

Consider the scalar conservation law $u_t+f(u)_x=0,$ whose conservative and consistent first order numerical scheme is given by $$\label{1}u_i^{n+1}=u_i^n-\lambda\left(F(u_i^n,u_{i+1}^n)-F(u_{i-1}^n,u_{i}^n) \right)$$ Suppose we modify the above scheme by $$u_i^{n+1}=u... 3 Edit: I know the question is very vague; this is partly due to the fact that I stumbled upon this map rather accidentaly and that I just wondered what can be said about this. So if you have some thoughts about this, please share them :) Let R be a unital ring, possibly non-commutative. Let M... 3 Let R\subseteq S be commutative rings in the sense that R is a subring of S with the same unity. So canonically, S has an R-module structure. We say that the inclusion map i: R\to S splits as a map of R-modules iff there exists an R-module map f: S\to R such that f\circ i=id_R... 1 I am an undergraduate student of mathematics and we have in out linear algebra course,a brief introduction of Adjoint operators and unitary operators.Now I understand that adjoint of a linear operator in finite inner product space corresponds to the conjugate transpose of a matrix.And a self-adjo... 1 Let V, W, and Z be finite-dimensional vector spaces. Let T: V \rightarrow W and S: W \rightarrow Z be linear mpas. If S\circ T is an isomorphism, then T and S are isomorphisms. I am told that this is in general not true, but I am not sure where to start to prove untrue. Can ... 12:07 PM 12:22 PM 1 Let \Lambda be a lattice in \mathbb{C} and X = \mathbb{C}/\Lambda be a complex torus. Exercise 6 of chapter 3 of Tamás Szamuely's book "Galois Groups and Fundamental Groups" (actually, the updated version in the erratum here) asks to show that X have a Galois branched cover Y \to X, ram... 1 For a commutative ring R, \alpha \in \tilde{H}^p(X;R), \beta \in \tilde{H}^q(X;R) and  u \in \tilde{H}_{p+q}(X;R), Verify \langle\alpha \cup \beta, u\rangle = \langle\beta, \alpha \cap u\rangle. We were asked to solve this after reading section 22.5 of "Modern Classical Homotopy Theory " b... 0 Here is the question: Show that is natural in both variables. That is suppose f: X \rightarrow Y, u \in \tilde{H^{*}}(Y), \alpha \in \tilde{H_{*}}(X). Then we can form$$<u, f_{*}(\alpha)> \in \tilde{H}_{n-k}(Y)$$And$$<f^{*}(u), \alpha)> \in \tilde{H}_{n-k}(X).$$Show that$$f_{*}(<f^{*}(u...

4

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...

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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 $\partial_t f(t,s) \in L^1(a,b)_... 1:12 PM 3 We have a square matrix$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... 1:24 PM 2 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... 1 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 $$J = J_0 \cdot \left[F_{\frac{d-1}{t}}\left(\eta\right)-F_{\frac{d-1}{t}}\left(\eta-v_d\right)\right]\\ n = n_0 \cdot\left[F_{\frac{d-t}{t}}\left(\eta\r... 1 Say \mathcal{C} is some conic and P\in \mathcal{C} is fixed point on it. For each X on \mathcal{C} let Y be such on \mathcal{C} that \angle XPY = 90^{\circ}. Prove that the line XY goes through a fixed point. I can prove this with projective geometry: Transformation \Pi:... 1 Let (M,d) be a Riemannian manifold and let T_1M=\{v\in TM: \|v\|=1\}. Define a map,$$s:T_1M\to \hat{\mathbb{R}},~~ s(v)= \sup\{t:d(\pi(v),\operatorname{exp}(tv))=t\}$$where \pi is the projection map from the the tangent bundle to M (\pi(p,v)=p,~(p,v)\in TM). I need to prove that the ... 0 Context. Let X be a Banach space with a Schauder basis (x_n)_{n=1}^\infty, let q_X:X\to X^{**} be the canonical isometric embedding, and let \mathcal{U} be a free ultrafilter on \mathbb{N}. Due to Banach-Alaoglu together with the basic facts about limits along a free ultrafilter, for ... 1 I know that if f is a conformal mapping of \mathbb{D} onto some domain D such that \partial D is a Jordan curve, then f has a continuous extension up to \partial \mathbb{D} such that f(\partial \mathbb{D}) = \partial D. This is, as far as I know, called Caratheodory's extension theo... 0 I ham given the following problem to solve: 1.9. The program should take three numbers: a; b; c and find the roots of the quadratic equation in the form: If the value of the determinant of the quadratic equation is negative (i.e. ∆ <0), the program should write an appropriate me... 2:31 PM 0 Consider the following linear programming problem$$ \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$J\times 1$... 3:01 PM 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? ... 0 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 ... 1 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) $. How could I extend this concept to a Cartesian product of two sets? For example, woul... 1 hour later… 4:28 PM 7 I know that a sequence of measures on$\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...

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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...