instruction stringlengths 12 30k |
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Is it possible to go over all lines of a grid with a pencil without lifting it or going over a drawn line? |
Could someone please give me a hint for this problem?
The following sequence is exact:
$0 \rightarrow A \xrightarrow{f} B \xrightarrow{g} C \xrightarrow{h} D \xrightarrow{i} E \xrightarrow{j} F \to 0$.
Prove: If $A, B, D, E$, and $F$ are free R-modules, then $C$ is torsion-free. (R is a PID, particularly: $\ma... |
How would one identify $\textbf{Proj}(\mathcal{O}\oplus \mathcal{O}(1)^{\oplus 2} \oplus \cdots)$ with $\mathbb{P}^1 \times \mathbb{P}^2$?
In general I've been having a hard time going from the global proj of some quasi coherent sheaf to describing an actual space. Any insightful example calculations in this directi... |
Refer to [This book][1] for this discussion.
\
**NOTE**: By Fourier Sine Transform, I am **NOT** referring to complex part of complex Fourier Transform (which is $0$ for even functions), but the single sided sine transform as defined below, which can be non-zero for even functions too.
Definitions from the book... |
Determine if the serie converges or diverges $\sum_{n=2}^{\infty} \left(1 - \frac{\ln(n)}{n}\right)^{n}$
I tried the divergence theorem, the nth root test and the integral test. Unfortunately, I got no answer and don't know what to do. Is there even an analytic answer?
|
So I made some **grave mistake** here: Still I want to show the post and add the real solution at the end:
With Gauss Law: The integral over the divergence on a volume is equal to the integral of that field on the boundary:
(Also see: [Divergence Theorem - Wikipedia](https://en.wikipedia.org/wiki/Divergence_the... |
Is there any way to solve this algebraically and give an exact form of the solution?
WolframAlpha only finds the approximate solution 4.14 |
So I made some **grave mistake** here: Still I want to show the post and add the real solution at the end:
With Gauss Law: The integral over the divergence on a volume is equal to the integral of that field on the boundary:
(Also see: [Divergence Theorem - Wikipedia](https://en.wikipedia.org/wiki/Divergence_the... |
I've been thinking about applications of [regular open sets](https://en.wikipedia.org/wiki/Regular_open_set) in topology, and I came up with the following:
If $X$ is a topological space then the set $\text{RO}(X)$ of regular open sets of $X$ forms a complete Boolean algebra. This Boolean algebra induces a [Stonean s... |
So I made some **grave mistake** here: Still I want to show the post and add the real solution at the end:
With Gauss Law: The integral over the divergence on a volume is equal to the integral of that field on the boundary:
(Also see: [Divergence Theorem - Wikipedia](https://en.wikipedia.org/wiki/Divergence_the... |
I understand the principles of generation that cantor used to create the ordinals but I cannot see what exactly is the property that makes the ordinals an indefinitely open plurality and not the natural numbers for example.
We say that the cardinality of the set of natural numbers is aleph0 but this is not a natural n... |
What exactly makes the ordinals an indefinitely extendible concept? |
Edit: some people have asked me about the **license** of the code below. [The license is MIT as specified in this Github repository][1].
---
I would like to add another solution, after the useful discussion with @JJacquelin. I reworked [my previous solution based on linear algebra and derivatives][2], to work wit... |
As an Idea you can take substitution like $u=\sqrt x$ which makes some simplify
$$\lim_{x \to \infty}\frac {f(x)}{g(x)}=\\\lim_{u \to \infty}\frac {(u^2+u)\log_2{u^2}}{u^2\log_2{(u^2+u)}}=\\\lim_{u \to \infty}\frac {2(u+1)\log_2{u}}{u\log_2{(u^2+u)}}=
\\\lim_{u \to \infty}\frac {2(u+1)}{u}\lim_{u \to \infty}\frac {\l... |
I came up with a solution exactly at 4:20. Your specific case is a lucky one because the term that depends on the independent variable $y$ affects linearly the non-linear exponentials that depend on $x$. So the problem remains linear and you can still apply [the method I proposed in this other answer][1] (with some min... |
Is there any way to solve this equation algebraically and give an exact form of the solution?
$$x^x = (x-1)^{x+1}$$
WolframAlpha only finds the approximate solution 4.14 |
I am reading Nigel Ray's "Universal Constructions in Umbral Calculus" (1998, published in "Mathematical Essays in Honor of Gian-Carlo Rota", page 344). The article reads:
> We define an _umbral calculus_ $(C, r)$ to consist of a coalgebra $C$ in $\mathcal{C}_{R}$ [the category of coalgebras over a commutative ring ... |
I understand the principles of generation that cantor used to create the ordinals but I cannot see what exactly is the property that makes the ordinals an indefinitely open plurality and not the natural numbers for example.
We say that the cardinality of the set of natural numbers is aleph0 but this is not a natural n... |
What exactly makes the ordinals an indefinitely extensible concept? |
Let $(X, \mathcal{T})$ be the [pseudoarc](https://en.wikipedia.org/wiki/Pseudo-arc), which is a [hereditarily indecomposable continuum](https://en.wikipedia.org/wiki/Indecomposable_continuum). Here hereditary means on every subcontinuum. Subcontinuums of a continuum are exactly its closed and connected subsets.
In a... |
I have to look for a formula which gives the number of i-dimensional singular subspaces of a polar space of order $(s,t)$ and rank $n$. My idea was to prove this by induction on $i$. So I wanted to start with calculating the number of points. Here my idea was to use a double counting on the pairs (point, line through t... |
Is there any way to solve this equation algebraically and give an exact form of the solution: $x^x = (x-1)^{x+1}$? WolframAlpha only finds the approximate solution 4.14. |
**I'm not talking about Stone spaces!**
I've been thinking about applications of [regular open sets](https://en.wikipedia.org/wiki/Regular_open_set) in topology, and I came up with the following:
If $X$ is a topological space then the set $\text{RO}(X)$ of regular open sets of $X$ forms a complete Boolean algebra... |
Let $(X, \mathcal{T})$ be the [pseudoarc](https://en.wikipedia.org/wiki/Pseudo-arc), which is a [hereditarily indecomposable continuum](https://en.wikipedia.org/wiki/Indecomposable_continuum). Here hereditary means on every subcontinuum. Subcontinuums of a continuum are exactly its closed and connected subsets.
In a... |
So I was watching this video https://www.youtube.com/watch?v=cHvYlrud8j8&ab_channel=AndrewMcCrady and along the proof it says that $(L-\epsilon)(1-\frac{g(c)}{g(\alpha)}) + \frac{f(c)}{g(\alpha)} \lt \frac{f(\alpha)}{g(\alpha)} \lt (L + \epsilon)(1 - \frac{g(c)}{g(\alpha)}) + \frac{f(c)}{g(\alpha)}$ implies that
$(L-... |
im trying to find the number of possibilites to distrribute k balls to 2n boxes such that for every i between 1 and n the sum of the balls in the i box and the n+i box isnt equal to 6. i tried writing the equation: $$x_1+x_2+...+x_{2n} = k$$, but i dont know how to progress further and it seems like it should be solves... |
Could someone please give me a hint for this problem?
The following sequence is exact:
$0 \rightarrow A \xrightarrow{f} B \xrightarrow{g} C \xrightarrow{h} D \xrightarrow{i} E \xrightarrow{j} F \to 0$.
Prove: If $A, B, D, E$, and $F$ are free R-modules, then $C$ is torsion-free. (R is a PID, particularly: $\ma... |
Let $(X, \mathcal{T})$ be the [pseudoarc](https://en.wikipedia.org/wiki/Pseudo-arc), which is a [hereditarily indecomposable continuum](https://en.wikipedia.org/wiki/Indecomposable_continuum). Here hereditary means on every subcontinuum. Subcontinuums of a continuum are exactly its closed and connected subsets.
In a... |
Let $(x_n)_{n\in\mathbb N}$ be a real sequence in $[0,1]$. Is there a partition of $\mathbb N$ indexed by the limit points of $(x_n)$? |
Could someone please give me a hint for this problem?
The following sequence is exact:
$0 \rightarrow A \xrightarrow{f} B \xrightarrow{g} C \xrightarrow{h} D \xrightarrow{i} E \xrightarrow{j} F \to 0$.
Prove: If $A, B, D, E$, and $F$ are free R-modules, then $C$ is torsion-free. (R is a PID, particularly: $\ma... |
Let $\large \mathrm {X} \sim \mathcal{U}(0,1)$. $\large \mathrm {(X,Y)}$ have a joint pdf . Define $\large \mathrm{Y=X}$. Then show that $\large \mathrm{X}$ and $\large \mathrm{Y}$ are not jointly continuous but even though marginally each of them is a continuous random variable.
This is how I approach to prove the... |
Proving $\large \mathrm{X}$ and $\large \mathrm{Y}$ are not jointly continuous but marginally each of them are continuous? |
Could someone please give me a hint for this problem?
The following sequence is exact:
$0 \rightarrow A \xrightarrow{f} B \xrightarrow{g} C \xrightarrow{h} D \xrightarrow{i} E \xrightarrow{j} F \to 0$.
Prove: If $A, B, D, E$, and $F$ are free R-modules, then $C$ is torsion-free. (R is a PID, particularly: $\ma... |
I am currently working with the automorphisms of a rooted tree $T$, and I have a question regarding $\text{Aut}(T)$ as a profinite group.
Recall that we can make $\text{Aut}(T)$ into a profinite group by choosing the subgroups $\text{St}_{\text{Aut}(T)}(n)$ as a base of open neighbourhoods of the identity.
I have ... |
Let $\large \mathrm {X} \sim \mathcal{U}(0,1)$. $\large \mathrm {(X,Y)}$ have a joint pdf . Define $\large \mathrm{Y=X}$. Then show that $\large \mathrm{X}$ and $\large \mathrm{Y}$ are not jointly continuous but even though marginally each of them is a continuous random variable.
This is how I approach to prove the... |
Let $(X, \mathcal{T})$ be the [pseudoarc](https://en.wikipedia.org/wiki/Pseudo-arc), which is a [hereditarily indecomposable continuum](https://en.wikipedia.org/wiki/Indecomposable_continuum). Here hereditary means on every subcontinuum. Subcontinuums of a continuum are exactly its closed and connected subsets.
In a... |
The Wikipedia page for memorylessness:
https://en.wikipedia.org/wiki/Memorylessness
goes over arguments that only the geometric and exponential distributions are memoryless, if we stick to just discrete and continuous distributions. What about mixed distributions? Is memorylessness such a nice property that nothi... |
Suppose we have two ODEs:
- $\dot{x}(t) = f(x(t),t)$
- $\dot{y}(t) = f(y(t),t) + g(y(t),t)$
If we have identical starting conditions $x(0) = y(0)$, we see
$$y(t) - x(t) = \int_0^t \left[ f(y(t),t)-f(x(t),t) \right] dt + \int_0^t g(y(t),t)dt $$
Is there analysis giving conditions on $f$ and $g$ that per... |
If we perturb an ODE, with the same starting conditions can we show that they converge together over time or at least do not diverge? |
**I'm not talking about Stone spaces!** A Stonean space is a compact Hausdorff [extremally disconnected](https://en.wikipedia.org/wiki/Extremally_disconnected_space) space i.e. a compact Hausdorff space such that for all open $U$, its closure $\overline{U}$ is also open.
I've been thinking about applications of [reg... |
Are Stonean spaces a reflective subcategory of topological spaces with continuous open maps? |
Let $\large \mathrm {X} \sim \mathcal{U}(0,1)$. $\large \mathrm {(X,Y)}$ have a joint pdf . Define $\large \mathrm{Y=X}$. Then show that $\large \mathrm{X}$ and $\large \mathrm{Y}$ are not jointly continuous but even though marginally each of them is a continuous random variable.
This is how I approach to prove the... |
Let $\large \mathrm {X} \sim \mathcal{U}(0,1)$. $\large \mathrm {(X,Y)}$ have a joint pdf . Define $\large \mathrm{Y=X}$. Then show that $\large \mathrm{X}$ and $\large \mathrm{Y}$ are not jointly continuous but even though marginally each of them is a continuous random variable.
This is how I approach to prove the... |
If one wants to reconstruct a function $f$ which, we assume is an element of a Hilbertspace $(\mathcal{H}(\Omega,K),(\cdot,\cdot)_K)$ of functions $\Omega\to\mathbb{R}$ with a reproducing Kernel (a.k.a. symmetric positive definite Function) $K:\Omega\times\Omega\to\mathbb{R},\,\Omega\subset\mathbb{R}^d$, on a finite se... |
Let $\large \mathrm {X} \sim \mathcal{U}(0,1)$. $\large \mathrm {(X,Y)}$ have a joint pdf . Define $\large \mathrm{Y=X}$. Then show that $\large \mathrm{X}$ and $\large \mathrm{Y}$ are not jointly continuous but even though marginally each of them is a continuous random variable.
This is how I approach to prove the... |
Let $e_1, \ldots, e_D$ be a given orthonormal basis for $\mathbb{C}^D$. We define the phase profile of a vector $u \in \mathbb{C}^D$ to be $( \text{arg} (e_1^* u), \ldots \text{arg} (e_D^* u) )$. We take $\text{arg} (0) := 0$.
Let $A: \mathbb{C}^D \rightarrow \mathbb{C}^D$ be an arbitrary positive semi-definite op... |
I have a smooth (not necessarily connected) submanifold $M \subset R^n$ of dimension $m$ with the property that the tangent space to $M$ at each point $p$ is "the same". Now I understand that in general it makes no sense to talk about equality of tangent spaces (just about isomorphisms) but in this Euclidean case by "s... |
Is a sub manifold of Euclidean space with the same tangent space at each point a hyperplane? |
Hint 1:
>! If $A$ can win, their penultimate move must setup $11$ different squares so $B$ can't block them all. What would this look like?
Hint 2:
>! The first part of the question is a hint for the second part.
Solution:
>! First $A$ creates the configuration from a). Imagine a box around all the poin... |
Hint 1:
>! If $A$ can win, their penultimate move must setup $11$ different squares so $B$ can't block them all. What would this look like?
Hint 2:
>! The first part of the question is a hint for the second part.
Solution:
>! First $A$ creates the configuration from a). Imagine a box around all the poin... |
Bra-ket notation?
I believe some use bra-ket notation as implying certain things like being strictly for wavefunctions or implying normalization, but in reality it is just a vector notation which has two main advantages:
- it defines an explicit notation for dual vectors, making things like the Dirac's delta be... |
Could someone please tell me what is known about the real non-trivial zeros of Riemann zeta function inside critical strip? Do we know there is none?
I want to know what is known about the above questions without the assumption of Riemann hypothesis.
Any hint or help would be appreciated. Thanks in advance. |
>Consider a function $f:[0,1]\to\mathbb{R}$ satisfying
$$\int_{\ 0}^1 f(x)(x-f(x))\,dx=\frac{1}{12}.$$
Then find the value of $f'(1)$
After banging my head on the table for half an hour, I assumed the function to be of the form $\frac{x}n$ where $n\in\mathbb{R}$. Solving it gives the result $n=2$. So the value o... |
The motivation here comes from [this MSE question][1] involving a set of divisibility constraints. In particular, it requires finding $n,p,q$ such that $p \mid 2n^2-1$ and $q \mid 2n^2-1$. Since $2n^2-1$ is always odd, $2$ is invertible, and these can be rearranged to the following:
$$n^2 \equiv 2^{-1} \pmod m \equi... |
When is $(m+1)/2$ a quadratic residue mod $m$? |
I am currently working on the following problem :
> Let $X$ be a topological space such that its suspension $\Sigma X$ is a n-manifold. Show that $H_k(X) \cong H_k(\mathcal{S}^{n-1})$ for all $k$.
I already proved that $H_k(X) \cong H_{k+1}(\Sigma X)$ for all $k$, using Mayer-Vietoris. I thought it could help me... |
I am interested in the asymptotics for large order and large argument for the modified Bessel function of first kind
$$I_{\frac{N-1}{2}} (Nz) $$
where z is a fixed positive real numer $z>0$ and $N$ is natural. I want to pass $N$ to $+\infty$.
I checked Nist and Abramowitz and Stegun, but I did not find anything use... |
Let the function space $A$ denote all functions $f : [0, 1) \to [0, 1)$ such that, for some set $Z$ of Lebesgue measure zero, the derivative $f'$ exists on $[0, 1) \setminus Z$ and $|f'| = 1$ there.
Let the function space $B$ denote all functions $f : [0, 1) \to [0, 1)$ such that for all $a < b$ in $[0, 1)$, the por... |
I read a book about orthogonal polynomials written by Gabor Szego. There is a conclusion:
Let the real-valued functions $f_0(x),f_1(x),\dots,f_l(x)$, where $l$ is finite or infinite, be of the class $L_\alpha^2[a,b]$ and linearly independent. By orthogonalization, there exists one orthogonal set $\phi_0(x),\phi_1(x)... |
I have a smooth (not necessarily connected) submanifold $M \subset R^n$ of dimension $m$ with the property that the tangent space to $M$ at each point $p$ is "the same". Now I understand that in general it makes no sense to talk about equality of tangent spaces (just about isomorphisms) but in this Euclidean case by "s... |
Is a sub manifold of Euclidean space with the same tangent space at each point an affine subspace? |
I know that the tree $T$ has $n-1$ edges, one component (it is connected), and does not contain a cycle (that is, from each vertex to each there is a single path, or between every two vertices of the tree there is a single path, otherwise there would be a cycle) and we want to show the equivalence between the statement... |
Let $P = \mathbb{R}^2 \setminus \{(0,0)\}$. We define a relation on $P$ as follows: $(x,y) \sim (z,w)$ if and only if there exist $\lambda \in \mathbb{R}\setminus \{0\}$ such that $x = \lambda z$ and $y = \lambda w$.
Define the function $\phi : \mathbb{R} \to {^{P}\big/_{\sim}}$ by $\phi(t) = [(t,1)]$ for $t \in \ma... |
I have a smooth (not necessarily connected) submanifold $M \subset R^n$ of dimension $m$ with the property that the tangent space to $M$ at each point $p$ is "the same". Now I understand that in general it makes no sense to talk about equality of tangent spaces (just about isomorphisms) but in this Euclidean case by "s... |
Six schools participate in a youth sports conference and each school is represented by three players a cricketer, a soccer player, and a hockey player. It is required to select a committee of six members from among these players. Find the number of different ways in which this committee can be formed,
i) if each fr... |
Six schools participate in a youth sports conference and each school is represented by three players a cricketer, a soccer player, and a hockey player.
It is required to select a committee of six members from among these players. Find the number of different ways in which this committee can be formed,
i) if each... |
I am reading the Gortz's Algebraic Geometry, proof of Lemma 6.26 and trying to prove some statement in it.
First I propose a question. Let $R:=k[T_1, \dots,T_n]$ be a polynomial ring with closed point $y\in \operatorname{Spec}R$ so that $\mathfrak{p}_y$ is the maximal ideal in $R$. Let $\mathfrak{m}_y$ be the ma... |
I'll assume you already know or have established that $\sim$ is in fact an equivalence relation.
Assume $\phi(t)=\phi(s)$. Then by the definition of $\phi$, we have that $[(t, 1)]=[(s, 1)]$. By the definition of an equivalence class, this tells us that $(t, 1) \sim (s, 1)$. (This is the step that allows you to go... |
I am currently working through Krylov's *Lectures on Elliptic and Parabolic Equations in Holder Spaces*. One of the key points of chapter 8 is that the two seminorms
$$[u]_{1+\delta/2,2+\delta;U} := \sup_{X \neq Y \in U}\frac{|u(X)-u(Y)|}{\rho(X,Y)^{\delta}}$$
and
$$[u]'_{1+\delta/2,2+\delta;U} := \sup_{X\in Q}\... |
How do we show that on a parabolic holder space, the polynomial and holder seminorms are equivalent? |
# The question #
This question was taken from an International Junior Mathematical Olympiad exam for 9th graders. The question is as follows:
> Let $x$ satisfies the equation $\dfrac{1}{x}=\dfrac{1}{2017^2}+\dfrac{1}{2018^2}+\ldots+\dfrac{1}{4030^2}$. Which of the following numbers is the nearest to $x$?
A. $2... |
I'm trying to find the number of possibilities to distribute $k$ balls to $2n$ boxes such that for every $i$ between $1$ and $n$ the sum of the balls in the $i$ box and the $n+i$ box isn't equal to $6$. I tried writing the equation: $$x_1+x_2+...+x_{2n} = k$$, but I don't know how to progress further and it seems like ... |
If I have two 3D vectors as shown by the blue and red vector in the image below, where the red vector is at an angle of more than 20 degrees from the blue vector, how could I calculate a new vector (green in the example) that is exactly 20 degrees away from the blue, in the same direction as the red one? It doesnt need... |
How to get a 3D vector in the same direction as another vector limited by an angle? |
As presented in the answer of this [post](https://math.stackexchange.com/questions/25723/what-is-the-right-norm-for-the-banach-space-tensor-product-in-this-situation), the projective tensor norm on the algebraic tensor product of two Banach spaces $X$ and $Y$ is given by
\\[
\Vert \omega\Vert_{\pi} = \inf\left\\{\sum... |
The projective tensor norm on tensor product of Banach spaces implies the inner product on tensor product of Hilbert spaces? |
I am interested in the asymptotics for large order and large argument for the modified Bessel function of first kind
$$I_{\frac{N-1}{2}} (Nz) $$
where z is a fixed positive real numer $z>0$ and $N$ is natural. I want to pass $N$ to $+\infty$.
I checked NIST and Abramowitz and Stegun, but I did not find anything use... |
Rectangle $ABCD$ is tilted such that its base $AB$ makes an angle of $\theta$ with the horizontal floor. It has its vertex $A$ in contact with the horizontal floor, and the rectangle is released from this position, to fall freely and the vertex at $A$ is free to slide along the floor.
What will be the rectangle's ... |
Consider a function $f(x)$ with the property $f(a)f(b) = f(a+b)$, where $a,b$ are numbers (or matrices, or operators). Is there a name for this property?
For example, $f(x)=c^x$ for a fixed $c \in \mathbb{R}$ will do this.
This comes up often enough in my physics research that I would really like to refer to it ... |
Let the function space $A$ denote all functions $f : [0, 1) \to [0, 1)$ such that, for some set $Z$ of Lebesgue measure zero, the derivative $f'$ exists on $[0, 1) \setminus Z$ and $|f'| = 1$ there.
Let the function space $B$ denote all functions $f : [0, 1) \to [0, 1)$ such that for all $a < b$ in $[0, 1)$, the por... |
I am currently working through Krylov's *Lectures on Elliptic and Parabolic Equations in Holder Spaces*. One of the key points of chapter 8 is that the two seminorms
$$[u]_{1+\delta/2,2+\delta;U} := \sup_{X \neq Y \in U}\frac{|u(X)-u(Y)|}{\rho(X,Y)^{\delta}}$$
and
$$[u]'_{1+\delta/2,2+\delta;U} := \sup_{X\in Q}\... |
I am currently working through Krylov's *Lectures on Elliptic and Parabolic Equations in Holder Spaces*. One of the key points of chapter 8 is that the two seminorms
$$[u]_{1+\delta/2,2+\delta;U} := \sup_{X \neq Y \in U}\frac{|u(X)-u(Y)|}{\rho(X,Y)^{\delta}}$$
and
$$[u]'_{1+\delta/2,2+\delta;U} := \sup_{X\in Q}\... |
How do we show that on a parabolic Hölder space, the polynomial and Hölder seminorms are equivalent? |
As title states, I'm a bit confused as to how to be certain a statement that is intuitively true for $\mathbb{N}$ is provable in $\mathsf{PA}$.
Specifically, I'm looking to prove the equivalence between two formulae $\phi_0$ and $\phi_1$. The first formula, $\phi_0$, is a functional formula in $\mathscr{L}_\mathsf{P... |
>Show
$$\left(A_{1} * A_{2}\right) \vartriangle\left(B_{1} * B_{2}\right) \subset\left(A_{1} \vartriangle B_{1}\right) \cup\left(A_{2} \vartriangle B_{2}\right)$$
where $*$ can be $\cup, \cap, \setminus.$
I proved this inclusion for $\cup, \cap$ but I have no clue how to prove it for $\setminus$.
I know that th... |
Let $X$ be a smooth projective irreducible variety of pure dimension $d$ over an algebraically closed field of positive characteristic $p$. Let $\ell \not = p$ be a prime number. There is an isomorphism $\mathrm{Tr}: H^{2d}(X,\mathbb Q_{\ell}) \xrightarrow{\sim} \mathbb Q_{\ell}(-d)$, and cup-product define a pairing
... |
Why is the bilinear form on $H^d(X,\mathbb Q_{\ell})$ afforded by Poincaré duality alternating when $d = \dim(X)$ is odd? |
Does there exist a map $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ such that $f$ is bijective and continuous but not open?
Intuition tells me that this would contradict the [invariance of domain](https://en.wikipedia.org/wiki/Invariance_of_domain). However, I stumble in arguing so. |
Suppose $\{r_i\}_{i\in \mathbb{N}} \subseteq (0,1)$ is a $p$-summable sequence for some $p\geq 1$ (namely $\sum_{i\in \mathbb{N}}r_i^p < \infty$). In general, depending on the choice of $p$ or not, can we find $\lambda\in (0,1)$ such that $\lambda^n r_1 < r_n$ whenever $n$ is large enough?
What inspired this quest... |
A Vitali set is a subset $V$ of $[0,1]$ such that for every $r\in \mathbb R$ there exists one and only one $v\in V$ for which $v-r \in \mathbb Q$. Equivalently, $V$ contains a single representative of every element of $\mathbb R / \mathbb Q$.
For any $\epsilon>0$, is it possible to construct a Vitali set $H$ so th... |
Is it possible to construct a Vitali set $H$ so that the outer measure $\lambda^*(H)<\epsilon$? |
I want to solve the following exercise:
John tells the truth two out of every three times he is asked, while Peter tells the truth four out of every five times. Both agree in assuring that from a box of marbles containing 6 marbles of different colors a red one was taken. Determine the probability that this statemen... |
I am confused by the following question:
Let $f(x, y) = (xy, x^2 +y
^2
)$ and show that $f^{-1}$
exists and is differentiable
in some open set around $(2, 5)$. Compute $Df^{−1}(2, 5)$. At which other values can you say that is $f^{−1}$
differentiable?
Given that $f$ is differentiable, I know I can use the fa... |
Collapsing either one of the circles in the bouquet of two circles to the basepoint, how can I describe this by a map (in the free product) and how is this related to that $\mathbb{R}^2\backslash\{p,q\}$ is **not** homeomorphic to a bouquet of two circles! However there is a **deformation retraction** of that space to ... |
How will the map be described? |
Find all values of A, B, C and C such that:
$$
\frac{x-1}{(x-1)(x-2)(x-2)} = \frac{A}{x+1} + \frac{B}{x+2} + \frac{C}{(x-2)^2}
$$
I keep getting into a loop in which:
$$
x - 1 = Ax^2 - 4Ax + 4A + Bx^2 - Bx - 2B + Cx + C
$$
Which then creates a system which will create equal versions of existing equations in... |
if $F_{n+1}$ is the the $n+1$th Fibonacci number, is it possible to derive the identity
$F_{n+1} = \sum_{k=0}^{n}\binom{n-k}{k}$
by using the generating function $F(x) = \frac{x}{(1-x-x^2)}$?
The only way that I could think of doing this was maybe by showing that
$\frac{F^{(n+1)}(0)}{(n+1)!} = \sum_{k=0}^{... |
I want to prove that if the length of a line from the origin to the line $Ax + By + C$ is less than the radius of a circle with center $(0, 0)$, then the line $Ax + By + C = 0$ intersects the circle $x^2 + y^2 = r^2$ at two points. I want to prove it because just drawing a graph isnt enough.
To tackle this problem, ... |
I want to prove that if the length of a line from the origin to the line $Ax + By + C$ is less than the radius of a circle with center $(0, 0)$, then the line $Ax + By + C = 0$ intersects the circle $x^2 + y^2 = r^2$ at two points. I want to prove it because just drawing a graph isnt enough.
To tackle this problem, ... |
This comes straight from problem 1.5 (p. 12) of *Handbook of Product Graphs, Second Edition* by Hammack, Imrich, and Klavžar.
It even comes with a hint in the appendix:
> **Exercise 1.5** Hint: Show that the homomorphic image of an odd cycle is a closed walk that contains an odd cycle.
But I think maybe there'... |
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