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I have to give a formal proof that the limit of h(x) = $\frac{\tanh(x)-1}{e^{-2x}}$ is -2 as $x \rightarrow \infty$.
For all ε>0 there must be a N>0 such that |h(x)-(-2)|<ε for all x>N
$$ | \frac{\tanh{x}-1}{e^{-2x}}-(-2) | = | \frac{2}{1+e^{2x}} |$$
Which means that
$$|\frac{2}{1+e^{2x}}|≤ \frac{2}{2x+2... |
The proposed problem was the following (translated into English by me):
"An school has $16$ students interested in participating in a team competition. First, they must split themselves into $8$ pairs. Then, each of those pairs must choose (among themselves) someone to be their leader. Finally, two teams of $8$ peo... |
> This diagram is easy to verify on $S^1$ and $S^2$.
No. You did not define what $\alpha$ is and what attaching maps (vertical arrows) are. It is impossible to deduce anything from that diagram, it is incomplete in the particular case.
There are infinitely many CW structures on a sphere, two of them are "standard... |
I have the following question...
**Consider the sequence defined by the formula $b_k=k^2$ for $k\ge2$. What is the third term in the sequence?**
I have computed the values for $k=2$, $k=3$, and $k=4$ as shown below:
$$b_2=2^2=4$$
$$b_3=3^2=9$$
$$b_4=4^2=9$$
However, I don't know what is considered the third t... |
What is considered the third term in a sequence that starts from 2? |
Let $p>5$ be a prime. For any positive integers $1\leq i,j\leq (p+1)/2$ let $ a_{i,j}=\lfloor ij/p \rfloor $. It is a fact that the determinant $|a_{i,j}|$ where $3\leq i,j\leq (p-1)/2$ is non-zero. The known proof is very indirect and sophisticated. Can you give an elementary proof? |
A long time ago, I developed a technique for estimating irrational square roots. I accepted it for the longest time as accurate as I noticed it produced consistently accurate results, though the calculation was always an underestimate. Only recently have I developed the personal tools to prove this technique works fo... |
My technique for estimating Square roots. Is it already known? |
> This diagram is easy to verify on $S^1$ and $S^2$.
No. You did not define what $\alpha$ is and what attaching maps (vertical arrows) are. It is impossible to deduce anything from that diagram, it is incomplete in the particular case.
There are infinitely many CW structures on a sphere, two of them are "standard... |
If $x$ and $y$ are (non-constant) functions of $z$, is it in general true that $\frac{dy}{dx} = \frac{\frac{dy}{dz}}{\frac{dx}{dz}}$?
If it's not true, can you please provide a counterexample, and if it's true, how would we prove that?
Thanks! |
I have the following question...
**Consider the sequence defined by the formula $b_k=k^2$ for $k\ge2$. What is the third term in the sequence?**
I have computed the values for $k=2$, $k=3$, and $k=4$ as shown below:
$$b_2=2^2=4$$
$$b_3=3^2=9$$
$$b_4=4^2=16$$
However, I don't know what is considered the third ... |
While working on a project, I came across a paper (["On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg][1]) that includes the definition $g ∈ $ on page 15, where $$ is the "tetrahedral group" ("a discrete subgroup of isometries $ ⊂ O(3)$"). The paper then uses $g(z)$ as a function... |
A long time ago, I developed a technique for estimating irrational square roots. I accepted it for the longest time as accurate as I noticed it produced consistently accurate results, though the calculation was always an underestimate. Only recently have I developed the personal tools to prove this technique works fo... |
A long time ago, I developed a technique for estimating irrational square roots. I accepted it for the longest time as accurate as I noticed it produced consistently accurate results, though the calculation was always an underestimate. Only recently have I developed the personal tools to prove this technique works fo... |
If we have some function, for example $$y = ix^2$$, we can express that (or any) function ($x,y$) with some kind of three-dimensional pixel ($t$ represents the pixel base whose length can be changed,and the values of the variables can also be changed, in my example the variable is $i$ ). I posted a link where you ca... |
A long time ago, I developed a technique for estimating irrational square roots. I accepted it for the longest time as accurate as I noticed it produced consistently accurate results, though the calculation was always an underestimate. Only recently have I developed the personal tools to prove this technique works fo... |
I can't find a proper formula for this considering it's almost exclusively made up of implications. Can somebody help me?
just I want to know how to simplify it.
I tried this but still no correct answer [enter image description here][1]
[1]: https://i.stack.imgur.com/vUvUn.jpg |
If you mean the [ Weierstrass M-test ][1], you should be careful about the differences between the theorem's statement and your own problem as in your case we are not dealing with sums.
However, please note that for every $n \in \mathbb N$, $0<\frac{1}{n^{\frac{3}{2}}} \leq 1$ and:
$$|S_n(\frac{1}{n^{\frac{3}{2}... |
The proposed problem was the following (translated into English by me):
**"A school has $16$ students interested in participating in a team competition. First, they must split themselves into $8$ pairs. Then, each of those pairs must choose (among themselves) someone to be their leader. Finally, two teams of $8$... |
I am trying to prove a certain claim about a recursive sequence.
The sequence is defined as follows:
$$
S_{n}^{k}(a) =
\begin{cases}
\sqrt{a} & \text{if } n = 1, \quad a \in \mathbb{Z}^{+}, \, a \neq 1 \\
\sum_{i=1}^{k} \sqrt{d_i} & \text{if } n > 1, \quad k \in \mathbb{Z}^{+}
\end{cases}
$$
where $d_i$... |
If you mean the [ Weierstrass M-test ][1], you should be careful about the differences between the theorem's statement and your own problem as in your case we are not dealing with sums.
However, please note that for every $n \in \mathbb N$, $0<\frac{1}{n^{\frac{3}{2}}} \leq 1$ and:
$$|S_n(\frac{1}{n^{\frac{3}{2}... |
Hussemoller's definition of a right G-space confuses me at a few key points:
For a topological group G, a right G-space is a space $X$
together with a map $X \times G \xrightarrow{\quad }X$. The image of $(x, s) \in X \times G$ under this map
is $xs$. We assume the following axioms.
(1) For each $x \in X, s,t ... |
I can't find a proper formula to prove that
$$(p\rightarrow q) \rightarrow ((r ∨ p) \rightarrow (r ∨ q)) $$
Is a tautology; considering it's almost exclusively made up of implications. Can somebody help me?
just I want to know how to simplify it.
I tried [this][1] but still no correct answer.
[1]: http... |
I am using the Pearson Edexcel A-Level maths books for my A-Level studies and stumbled across a question I do not understand (Exercise 3G Question 8d).
The last part of the question confused me since it asked me to calculate the area of the shaded region. I checked the solution bank after attempting the question and... |
A few days ago I was at the math Olympiad and I failed to solve only one rather simple task, it feels like a 9th grade, but I either misunderstand something, or there is something quite interesting hidden there
The problem sounds like this:
> In quadrilateral ABCD the diagonals AC and BD are perpendicular and
>... |
I encountered a problem while working on a mathematical analysis exercise. The problem is as follows:
Given $f\in C^1([0,+\infty])$ such that $|f'(x)|\leq 1/x$ and $\lim_{R\rightarrow +\infty}\frac{1}{R}\int_0^R |f(x)|dx=0$, how can we prove that $\lim_{x\rightarrow +\infty}f(x)=0$?
Any assistance would be greatl... |
Notice that $b = a +1$, so $b^2 - a^2 = 2a +1$, and your estimate can be written as
$$ a+ \frac{n-a^2}{2a+1}$$
If we apply Newton's method to $f(x) = x^2 - n$, $f'(x) = 2x$, and make our initial guess $x_0 = a$, then
$$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = a - \frac{a^2-n}{2a}$$
we can see that they are alm... |
If we have some function, for example, $$y = ix^2$$, we can express that (or any) function ($x,y$) with some kind of three-dimensional pixel ($t$ represents the pixel base whose length can be changed, and the values of the variables can also be changed, in my example the variable is $i$ ). I posted a link where you c... |
There is one exercise I proved involving that a smooth map $\varphi$ from $M$ to $\widetilde{M}$ is a linear isometry if and only if (pullback) $(\varphi^* \tilde{g})=g.$
So I am wondering if the proof is correct? Below is both the exercise and my proof:
Exercise:
* Prove that if $(M,g)$ and $(\widetilde{M}, \... |
Proof of linear isometry, Riemannian manifolds. Is my proof correct? |
I am trying to compute an integral which goes as:
\begin{align}\int_{\mathcal{F}}\frac{1}{\left(\sqrt{Im(z)}\eta(z)\bar{\eta(z)\right)^{k}}\frac{dzd\bar{z}}{Im(z)^2}\end{align}
Where $\eta$ is Dedekind's $\eta$, and $\mathcal{F}$ is the fundamental domain of the torus.
I have been checking on procedures to sol... |
I am trying to compute an integral which goes as:
$\int_{\mathcal{F}}\frac{1}{\left(\sqrt{Im(z)}\eta(z)\bar{\eta(z)\right)^{k}}\frac{dzd\bar{z}}{Im(z)^2}$
Where $\eta$ is Dedekind's $\eta$, and $\mathcal{F}$ is the fundamental domain of the torus.
I have been checking on procedures to solve it but wherever I g... |
I am trying to compute an integral which goes as:
$$I=\int_{\mathcal{F}}\frac{1}{\left(\sqrt{Im(z)}\eta(z)\bar{\eta(z)}\right)^{k}}\frac{dzd\bar{z}}{Im(z)^2}$$
Where $\eta$ is Dedekind's $\eta$, and $\mathcal{F}$ is the fundamental domain of the torus.
I have been checking on procedures to solve it but whereve... |
I am trying to compute an integral which goes as:
$$I=\int_{\mathcal{F}}\left(\sqrt{Im(z)}\eta(z)\bar{\eta(z)}\right)^{-k}\frac{dzd\bar{z}}{Im(z)^2}$$
Where $\eta$ is Dedekind's $\eta$, and $\mathcal{F}$ is the fundamental domain of the torus, and $k\in\mathbb{N}^*$.
I have been checking on procedures to solve... |
I am trying to compute an integral which goes as:
$$I=\int_{\mathcal{F}}\left(\sqrt{Im(z)}\eta(z)\overline{\eta(z)}\right)^{-k}\frac{dzd\bar{z}}{Im(z)^2}$$
Where $\eta$ is Dedekind's $\eta$, and $\mathcal{F}$ is the fundamental domain of the torus, and $k\in\mathbb{N}^*$.
I have been checking on procedures to ... |
I am trying to prove a certain claim about a recursive sequence.
The sequence is defined as follows:
$$
S_{n}^{k}(a) =
\begin{cases}
\sqrt{a} & \text{if } n = 1, \quad a \in \mathbb{Z}^{+}, \, a \neq 1 \\
\sum_{i=1}^{k} \sqrt{d_i} & \text{if } n > 1, \quad k \in \mathbb{Z}^{+}
\end{cases}
$$
where $d_i$... |
I am trying to compute an integral which goes as:
$$I=\int_{\mathcal{F}}\left(\sqrt{Im(z)}\eta(z)\overline{\eta(z)}\right)^{-k}\frac{dzd\bar{z}}{Im(z)^2}$$
Where $\eta$ is Dedekind's $\eta$, $\mathcal{F}$ is the fundamental domain of the torus, and $k\in\mathbb{N}^*$.
I have been checking on procedures to solv... |
Suppose $M^n$ is a smooth projective variety. A line bundle $L$ on $M$ is nef (numerically effective) if on any complete curve $C$ in $M$, $L$ has positive degree, i.e.
$$
L\cdot C=\int_{C}R_h\geq 0.
$$
where $R_h$ is the curvature form of any Hermitian metric $h$ on $L$.
A line bundle $L$ on $M$ is big if there... |
My question is in fact the exercise 1.8 page 8 in *Algebraic Geometry* by Robin Hartshone.
> Let $Y$ be an affine variety of dimension $r$ in $\mathbb{A}^n$. Let $H$ be a hypersurface in $\mathbb{A}^n$ and assume that $Y\nsubseteq H$. Then prove that every irrducible component of $Y\cap H$ has dimension $r-1$
He... |
I try to find a balanced exercise on it with Desmos.
It seems we have:
$$\sum_{n=1}^{\infty}\frac{\gcd\left(1+n!,1+n^{2}\right)}{n!}\overset?=e$$
It seems non trivial as you can remark in replacing the square by a cube.
At first glance we can think it's problem with hidden rearrrangement theorem. Keeping ... |
Hussemoller's definition of a right G-space confuses me at a few key points:
For a topological group G, a right G-space is a space $X$
together with a map $X \times G \xrightarrow{\quad }X$. The image of $(x, s) \in X \times G$ under this map
is $xs$. We assume the following axioms.
(1) For each $x \in X, s,t ... |
In full generality, derived functors appear in a situation where our categories of interest admit some notion of *weak equivalences*, without those being isomorphisms per se. In homological algebra, our cochain complex categories have quasi-isomorphisms, which are such a notion of weak equivalences. These quasi-isomorp... |
________________________
1.
Take any separable Hilbert space ($\ell^2$, for example) and a maximal orthonormal set (which must be countable, because our space is separable). This orthonormal set is a Schauder basis. Furthermore we notice that the following series is convergent:
$$S=\sum_{j=1}^\infty \frac{... |
In one of the final problems of the MIT integration bee for this year, $$I=\int_0^\infty\frac{\ln(2e^x-1)}{e^x-1}dx$$was one of the given problems. My try was to let $u=e^x-1$ to get $$I=\int_0^\infty\frac{\ln(2u+1)}{u(u+1)}du=\int_0^\infty\frac{\ln(x+1)}{x(x+2)}dx$$I don't know whether I would be right but I had a fee... |
When I put this function in GeoGebra calculator suite, I am getting the minimum value 4 but when I tried to solve, the minimum value I'm getting is different:
**My attempt:**
27^6 +1/(^2−^2−^2+2) can be written as 27a^6+1/(a^2-(b-c)^2)
**i.e. 27a^6+1/((a+b-c)(a-b+c))**
Observing, we know that, the minimum o... |
A long time ago, I developed a technique for estimating irrational square roots. I accepted it for the longest time as accurate as I noticed it produced consistently accurate results, though the calculation was always an underestimate.
Only recently have I developed the personal tools to prove this technique work... |
Estimating $\sqrt{n}\approx a+\frac{n-a^2}{b^2-a^2}$, where $a^2$ and $b^2$ are the perfect squares nearest non-square $n$. Is it already known? |
> I cannot figure out how I should be allowed to use $11 \choose 2$ at
> denominator since to me there are at most $3$ different ways to get a
> group of two marbles from the $11$ namely (BB, BW, WW) and not $55$ like
> $11 \choose 2$ states.
Well, if you think about it this way, then you should feel uncomforta... |
Hussemoller's definition of a right G-space confuses me at a few key points:
For a topological group G, a right G-space is a space $X$
together with a map $X \times G \xrightarrow{\quad }X$. The image of $(x, s) \in X \times G$ under this map
is $xs$. We assume the following axioms.
(1) For each $x \in X, s,t ... |
The question is: does there exist a linear fractional (Möbius) transformation that maps the set $U = \{z \in \Bbb{C} \mid |z-1|<1, |z-i| < 1\}$ onto the quarter plane $\operatorname{Im}z>0, \operatorname{Re}z>0$.
I thought to take the map $z \mapsto \frac{z}{z-1-i}$. The reason I chose this is because I thought to s... |
Is this conformal map correct? |
________________________
1.
Take any separable Hilbert space ($\ell^2$, for example) and a maximal orthonormal set (which must be countable, because our space is separable). This orthonormal set is a Schauder basis. Furthermore we notice that the following series is convergent:
$$S=\sum_{j=1}^\infty \frac{... |
I have a very long cylindrical mass. Now say I have some cylindrical Gaussian surface $\Omega$ centered around this cylinder, visualized in purple below (ignore everything else in the diagram),
[![enter image description here][1]][1]
for which I apply Gauss's law of gravity
$$\oint_\Omega\mathbf g\,\cdot\,\tex... |
How to properly (rigorously?) treat a dot product integral? |
$
f(x)=
\begin{cases}
x^2&\text{if $x$ is irrational}\,\\
2x+1&\text{if $x$ is rational}\\
\end{cases}
$
I want to calculate the limit of $f(x)$ as $x$ tends to $0$. Is it enough to just say that when $x$ tends to $0$ and $x$ is irrational, the limit of $x^2$ is clearly $0$, but when $x$ is rational we get ... |
Hints towards a solution. If you're stuck, explain what you've tried.
The setup is quite complicated. How can we simplify it?
1. Point $E$ is quite troublesome, introduced only at the end, and not as relevant to the other points. Is there a way to remove it? Can we find a condition not involving $E$, that would... |
I am reading a paper and they mention the following: "Two isometries of the hyperbolic plane are said to be co-parallel if they have disjoint axes and the motions along these axes induce the opposite orientations on the region they bound"
I am very confused by what is meant by that (visually w.r.t to the fixed point... |
What does "Motions along translation axes induce the opposite orientations on the region they bound" mean? |
I'm aware of the many answers to evaluating the integral of $\frac{\sin x}{x}$ over $[0,\infty)$ given [here][1]. No answer seems to mention the approach below, and neither have I found anything elsewhere, and hence I'm doubting if it's possible.
I stumbled upon this in connection with a problem I was working on.
... |
Can $\int_0^\infty \frac{\sin x}{x}dx$ be evaluated using Fourier series? |
I want to prove that a sequence is not Cauchy by proving something like
$$ \exists \epsilon >0 \space s.t \space \space \forall n_0 \in N \space \space \exists n, n+p > n_0 \space=> |x_{n+p} - x_n| > \epsilon $$
For that specific question I think I can prove that by taking sequences $$x_n, x_{n+3}$$ as an example... |
When I put this function in GeoGebra calculator suite, I am getting the minimum value 4 but when I tried to solve, the minimum value I'm getting is different:
**My attempt:**
${27^6}$ +$\frac{1}{(^2−^2−^2+2)}$ can be written as ${27a^6}$+$\frac{1}{a^2-(b-c)^2}$
**i.e. ${27a^6}$+$\frac{1}{(a+b-c)(a-b+c)}$**
... |
**HINT**
Alternatively, you can proceed as follows:
\begin{align*}
\frac{1 + x^{3}}{(1 + x)^{3}} & = \frac{(1 + x)(1 - x + x^{2})}{(1 + x)^{3}} = \frac{1 - x + x^{2}}{(1 + x)^{2}} = -\left(1 - x + x^{2}\right)\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{1}{1 + x}\right)
\end{align*}
where the derivative's argument ... |
Hints towards a solution. If you're stuck, explain what you've tried.
The setup is quite complicated. How can we simplify it?
1. Point $E$ is quite troublesome, introduced only at the end, and not as relevant to the other points. Is there a way to remove it? Can we find a condition not involving $E$, that would... |
I want to prove that a sequence is not Cauchy by proving something like
$$ \exists \epsilon >0 \space s.t \space \space \forall n_0 \in N \space \space \exists n, n+p > n_0 \space=> |x_{n+p} - x_n| > \epsilon $$
For that specific question I think I can prove that by taking sequences $$x_n, x_{n+3}$$ as an example... |
I am looking to a reference to a solution of Exercise 13 (page 91) of L. Fuchs, Infinite Abelian Groups, Vol. 1, Academic Press, 1970. There is no solution to this exercise in a collection Exercises in Abelian Group Theory by Calugraenu at al.
The exercise is as folows: Let $A=A_1\oplus\dots\oplus A_k$, where $A_i$ ... |
Given a (autonomous) dynamical system, one can prove instability of a point via the Lyapunov method, by simply finding a Lyapunov function that increases in a neighbourhood of the point. This ensures a stronger property than instability. It ensures the existence of a small enough neighbourhood, such that all orbits sta... |
I am studying *Multiplicative number theory I: Classical theory by Hugh L. Montgomery, Robert C. Vaughan* ([E-book][1]). In Appendix A, Theorem A.1. states that $I = \int f dg$ exists if $f$ is continuous and $g$ is of bounded variation. Let $a=x_0 \le x_1 \le \dots \le x_N=b$ is a partition of $[a,b]$, and $S(x_n, \xi... |
Inverses arise by $\rm\color{#0a0}{RTD}$ = Rationalizing The Denominator. $ $ Put $\, F = \mathbb Q,\ D \!=\! \mathbb Q[\sqrt{c}],\ \alpha = \sqrt{c}\ $ in
**Lemma** $\ $ TFAE $\,$ where $\,F\,$ is a subfield of domain $\,D,\,$ and $\, \alpha \in D,\,$ $\ {\rm and}\,\ R = \alpha F + F\:\! :=\:\! \{\:\! a\,\alpha... |
We first give an elementary proof requiring little field theory background, followed by a shorter proof using a little vector space knowledge.
---
Inverses arise by $\rm\color{#0a0}{RTD}$ = Rationalizing The Denominator. $ $ Put $\, F = \mathbb Q,\ D \!=\! \mathbb Q[\sqrt{c}],\ \alpha = \sqrt{c}\ $ in
**Lem... |
I'm aware of the many answers to evaluating the integral of $\frac{\sin x}{x}$ over $[0,\infty)$ given [here][1]. No answer seems to mention the approach below, and neither have I found anything elsewhere, and hence I'm doubting if it's possible.
I stumbled upon this in connection with a problem I was working on.
... |
There is one exercise I proved involving that a smooth map $\varphi$ from $M$ to $\widetilde{M}$ is a local isometry if and only if (pullback) $(\varphi^* \tilde{g})=g.$
So I am wondering if the proof is correct? Below is both the exercise and my proof:
Exercise:
* Prove that if $(M,g)$ and $(\widetilde{M}, \t... |
Proof of local isometry, Riemannian manifolds. Is my proof correct? |
I need to find the volume bounded by:$\\$
$0\leq z\leq 10-x^2-y^2 $, $x+1-y^2 \geq 0$ and $x+y^2-1 \leq 0$. I dont really understand how to go about this problem. What is the projection on the xy-plane?. My first idea was:
$\int \int \int_K dzdydz = \int \int_D ( \int_0^{10-x^2-y^2}dz)dxdy = \int \int_D(10-x^2-y^2)dx... |
I played around with the following series
$$S(x) = \frac{1}{x^2}+\frac{1}{x^3}+\frac{1}{x^5}+\frac{1}{x^7}+\frac{1}{x^{11}}+...=\sum_{p\in primes}\frac{1}{x^p}$$
And it seems, that for $x\ge2$ the value of this sum is pretty close numerically to the following expression:
$$S(x) \approx \big(\sqrt{2}-1\big)\exp\big[-... |
As the title says, why is $$\wedge ^2 E[p] \cong \mu_p,$$ where $E[p]$ refers to the $p$-torsion points of an Elliptic curve over a number field $K$, $\wedge$ refers to the exterior product and $\mu_p$ refers to the group of $p^{th}$ roots of unity.
I know that for the formal multiplicative group $\mathbb{G}_m$ ove... |
Why is $\wedge ^2 E[p] \cong \mu_p$? |
The vector-parametric description of the tangent plane to the graph of a function $f: \mathbb{R}^2 \to \mathbb{R}$ at a point $(a, b, f(a, b))$ is the following:
$$\begin{bmatrix}
x\\
y\\
z\\
\end{bmatrix} =
\begin{bmatrix}
a\\
b\\
f(a, b)\\
\end{bmatrix} + \lambda
\begin{bmatrix}
1\\
0\\
f_x(a, b)\\... |
While reading Donald Knuth's "Sorting and Searching" I have come across a table of inversions which lists the numbers of inversions $k$ for a number of permutations of lengths $n$ (varying from 1 to 6).
There is a zigzag line which separates $I_{n}(n-1)$ from $I_{n}(n)$ as $n$ increases from 1 to 6 down the table.
... |
This is an exercise in a book.
[![pic1][1]][1]
So we need to find the most general function $g$ such that this curve from $\mathbb{R}^3$
$f(t) = (a\cos(t), a\sin(t), g(t))$
lies in a plane.
It's not mentioned but there, but I guess we can assume that
1) $g$ is e.g. 3 times differenti... |
This is an exercise in a book.
[![pic1][1]][1]
So we need to find the most general function $g$ such that this curve from $\mathbb{R}^3$
$f(t) = (a\cos(t), a\sin(t), g(t))$
lies in a plane.
It's not mentioned but there, but I guess we can assume that
1) $g$ is e.g. 3 times differenti... |
Given a topological space $(\mathsf{X}, \mathcal{T})$ where $\mathsf{X}$ is a set and $\mathcal{T}$ is a topology on it, one can generate a sigma-algebra using this topology: the Borel sigma algebra
$$
\sigma(\mathcal{T}) = \mathcal{B}(\mathsf{X}).
$$
The classic example is $\mathsf{X} = \mathbb{R}$ in which case ... |
What topology $\mathcal{T}$ on $\mathbb{R}$ generates the Borel sigma-algebra on $\mathbb{R}$? |
Given a topological space $(\mathsf{X}, \mathcal{T})$ where $\mathsf{X}$ is a set and $\mathcal{T}$ is a topology on it, one can generate a sigma-algebra using this topology: the Borel sigma algebra
$$
\sigma(\mathcal{T}) = \mathcal{B}(\mathsf{X}).
$$
The classic example is $\mathsf{X} = \mathbb{R}$ in which case ... |
This is not an answer... Just to give a simple example why it is not always solvable. Imagine a function $\mathbb R^2 \to \mathbb R$.
Let us prescribe any non-closed (importante!) contour of gradient orthogonal to the direction of the contour with $\nabla g = \|\nabla g\|\hat n$. Now let every other point in the pl... |
I encountered this theorem some time ago.
[![pic1][1]][1]
I think I was able to prove it pretty easily.
But it made me thinking about the following.
For some curve and for its parametrization, could we have that at some fixed point $s = s_0$ the curvature $\kappa(s_0) = 0$ but there is no neighbo... |
________________________
1.
Take any separable Hilbert space ($\ell^2$, for example) and a maximal orthonormal set (which must be countable, because our space is separable). This orthonormal set is a Schauder basis. Furthermore we notice that the following series is convergent:
$$S=\sum_{j=1}^\infty \frac{... |
Say $E_1 : \mathbb{P}_3 \rightarrow \mathbb{R}$ by $f(x) \mapsto f(1)$.
Is it sufficient to say there exists right inverse $S_R : \mathbb{R} \rightarrow \mathbb{P}_3 $ from the following?
We see $\mathbb{P}_3$ has basis $B = \{1,x,x^2,x^3\}$.
So we can say $f(x) = a+bx+cx^2+dx^3$ where $f(1)=a+b\cdot1+c\cdo... |
This is a question I ended up with while trying to make a program that would find an (at least locally-) minimal-energy configuration for a piecewise linear 1-dimensional object in $\mathbb{R}^3$ in relation to a practical knot theory type of question. If it helps at all for the purposes of context or intuition, the se... |
I want to show that if $z\in\mathbb{C}$ such that $|z|=1$ and $1+z^{k_1}+z^{k_2}=0$ for integers $k_1<k_2$, than $z$ is a root of unity.
Here is my approach:
Suppose $z=\cos(\theta)+i\sin(\theta)$. Then we have $$1+\cos(k_1\theta)+\cos(k_2\theta)+i(\sin(k_1\theta)+\sin(k_2\theta))=0.$$
This would require $$\sin(k... |
In the Regular season of the MIT Integration bee, the following integral was given $$I=\int_1^3\frac{1+\frac{1+...}{x+...}}{x+\frac{1+...}{x+...}}dx$$If the integrand is called $f(x)$ then I am guessing that $f(x)=\frac{1+f(x)}{x+f(x)}$ implying $f(x)=\frac{1-x+\sqrt{(x-1)^2+4}}2$. So we now have $$I=\int_2^4\sqrt{x^2+... |
I recently have thought of a proof but I can’t tell if it is correct or not.
The proof is of $n \pi$ being irrational if n is an integer and non zero. The proof is below:
We assume that $n \pi$ is not irrational, thus meaning that $n \pi \in \mathbb Q$. Since $\sin(x)$ is an irrational number if x $\in \mathbb Q$ w... |
Is my proof of this fact about pi correct? |
Why not try this?
$$S_R:k\rightarrow kx^3, \mathbb{R}\rightarrow\mathbb{P}_3$$
Then:
$$E(S_R(k))=E(kx^3)=k*1^3=k$$
Therefore, $E(S_R(k))=k$ for all $k\in \mathbb{R}$, so that's a right inverse.
|
Let's say I have these 2 matrices:
$$A = \begin{bmatrix} c \\ d \end{bmatrix} $$
and
$$B = \begin{bmatrix} e \\ f \end{bmatrix}$$
$A'B = ce + df $ and $B'A = ec + fd$
As shown above, $A'B = B'A$. But is there a matrix product property that could've told me this without having to manually check? I haven... |
Let $d\mu$ be a Gaussian measure on the space of tempered distributions $\mathcal{S}'(\mathbb{R}^n)$. The characteristic functional of this measure is computed as
$$S[f] = \int e^{i\phi(f)}d\mu(\phi), \quad f \in \mathcal{S}(\mathbb{R}^n)$$
where $\mathcal{S}(\mathbb{R}^n)$ denotes the Schwartz space of functions.
... |
I have studied Hypothesis testing from [thinkstats2][1] which does simulation to estimate p value.
For e.g. In coin toss example, we run the simulation, to generate the test statistic i.e. Difference between the number of Heads and Tails. p value is the proportion of test statistics greater than observed. So intuiti... |
regarding the gamma 1/3 cube:
$253^{1/6}(\Gamma\left(\frac{1}{3}\right))^3/(2 * 6^{1/2}\pi^2)=_2F_1\left(\frac{1}{12},\frac{7}{12};1;\frac{3^2}{253^2}\right)$
[testing it ][1]
source: https://iamned.com/math
[1]: https://www.wolframalpha.com/input?i=253%5E%281%2F6%29%28gamma%281%2F3%29%29%5E3%2F%282*6... |
Is there a known description of the set of multiplicative orders of a fixed unit $a$ modulo all divisors of some modulus $n$, i.e. of $\text{ord}_d(a)$ with $d\mid n$? It is easy to see that it is a subset of the set of all divisors of $\text{ord}_n(a)$. Indeed, if $a^m\equiv1\pmod{n}$ then $a^m\equiv1\pmod{d}$ for any... |
This is problem `6.3.8` from *Trigonometry: A Complete Introduction*, by Hugh Neill (ISBN 1473678498). It's in the surveying chapter, which is a practical continuation of the sine and cosine rules of the chapter before that. So I'm assuming you don't need much more than those two rules. It's a simple problem, but I'm ... |
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