Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Problem about subgroup of $D_n$ Prove that every subgroup of $D_n$ , either every member of subgroup is a rotation or exactly half of them are rotations.
Intuitively, if every member is a rotation then they will form a subgroup because we can rotate them as much as we like (closure) and other properties will also be sa... | This is what the question is really asking you:
Let $D_n$ be the dihedral group of order $2n$ and $R$ the order $n$ group of rotations.
The if $H$ is any subgroup of $D_n$, either $[H:H \cap R] = 1$ or $[H:H \cap R] = 2$.
If $H \subseteq R$, then $H \cap R = H$, and we have $[H:H \cap R] = [H:H] = 1$.
Otherwise, we can... | {
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"timestamp": "2023-03-29T00:00:00",
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Expand $(a-b)^3$ without formulas When I solve $(a-b)^2$ I take $aa-ab-ab+bb$ I do not use formulas at all because I only forget them. To solve the above example all i do is to multiply one variable or constant at the time but when I ask anyone or anything how to solve $(a-b)^3$ all they tell me is to use this or that ... | Arrange $(a-b)^3$ as $(a-b)(a-b)(a-b)$. Now sum over all ways of picking either $a$ or $-b$ from each factor. So you get
$$aaa+(-b)aa+a(-b)a+aa(-b)+(-b)(-b)a+(-b)a(-b)+a(-b)(-b)+(-b)(-b)(-b)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1794060",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Define a metric for an annulus, which makes it seem like the curved wall of a cylinder.
Can anybody please help me in understanding this question?
| Hint: They want you to define a metric in the annulus so that the shortest path between 2 points with the same radius is not the length of the straight line between them, but the length of the shortest arc joining them, as it is if you're forced to walk along the wall of a cylinder. This can then be extended to a metri... | {
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"url": "https://math.stackexchange.com/questions/1794191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Estimate the sum of alternating harmonic series between $7/12$ and $47/60$ How can I prove that:
$$\frac{7}{12} < \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} < \frac{47}{60}$$
? I don't even know how to start solving this...
| First note that the series converges using Leibniz Test.
Next, denote by $S_N$ the partial sum $\sum_{n=1}^N\frac{(-1)^{n-1}}{n}$. Then, we must have $$S_{2N}<\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}<S_{2N+1}$$
Finally, we see that $\sum_{n=1}^4 \frac{(-1)^{n+1}}{n}=\frac{7}{12}$ and inasmuch as the next term is positiv... | {
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"question_score": "6",
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Finding $z$ for Complex Convergence I am having an issue understanding how to go about solving a problem regarding complex sequences. The problem is as follows:
Find a $z$ for which the following sequence converges: $f_{n} (z) =e^{nz}$
My attempt thus far is:
$$f_{n} (z) = e^{nz} = e^{nx} \cdot (\cos(ny) + i\sin(ny... | Let $z=x+iy$. Then, note that $f_n(z)=e^{nx}e^{iny}$.
The real and imaginary parts of the sequence $f_n(z)$ are given respectively by
$$\text{Re}(f_n(z))=e^{nx}\cos(ny)$$
and
$$\text{Im}(f_n(z))=e^{nx}\sin(ny)$$
Note that if $x< 0$, the exponential term $e^{nx}$ approaches zero as $n\to \infty$. Since both the sin... | {
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Evaluation of Definite Integral
Evaluation of $\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\cos x\sin 2x \sin 3x}{x}dx$
$\bf{My\;Try::}$ Let $\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\cos x\sin 2x \sin 3x}{x}dx = \frac{1}{2}\int_{0}^{\frac{\pi}{2}}\frac{2\sin 3x\cos x\sin 2x}{x}dx$
So we get $$I = \frac{1}{2}\int_... | Hint 1: $$ \int f(x) = F(X) + C \Longrightarrow \int f(ax+b) = \frac{1}{a} \cdot F(ax+b) + C$$
Hint 2: $$ \int \left(f(x) + g(x)\right) = \int f(x) + \int g(x) $$
Hint 3: $$ \int \frac{\cos x}{x} = Ci(x) + C$$
First step:
(use hint 2)
$$
4I=\int_{0}^{\frac{\pi}{2}}\frac{\cos2x-\cos6x+1-\cos 4x}{x}dx =\\
=2\int_{0}^{\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1794469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 1
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Asymptotic behaviours from Fourier transforms I have completely forgotten how one derives the asymptotic behavior in frequency space, given the asymptotic behavior of the function in real space (e.g. time). As an example example, it is often said that when $f(t)\sim t^\alpha$ for $t\to\infty$, then $f(\omega)\sim\omega... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
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\newcommand{\dd}{\mathrm{d}}
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\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
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Uniform convergence of $\sum_{n=-\infty}^{\infty}\frac{1}{n^2 - z^2}$ on any disc contained in $\mathbb{C}\setminus\mathbb{Z}$ I'm currently revising some complex analysis, and need to show that the series $$\sum_{n=-\infty}^{\infty}\frac{1}{n^2 - z^2}$$ defines a holomorphic function on $\mathbb{C}\setminus\mathbb{Z}$... | Let $K \subset (B(0,R) \cap \mathbb{C} \setminus \mathbb{Z})$ be a compact set and set $d(K,\mathbb{Z}) = \delta > 0$.
If $|n| > R$, then for all $x \in K$ we have
$$ |n^2 - x^2| \geq ||n|^2 - |x|^2| = |n|^2 - |x|^2 \geq |n|^2 - R^2. $$
If $|n| \leq R$ then for all $x \in K$ we have
$$ |n^2 - x^2| = |x - n||x - (-n)| \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1794713",
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On changing limits of integration when there are domain problems. As an example say I have $$\int_{\pi/2}^\pi \frac{2}{1- \sin(2x)} dx$$
I would like to perform the substitution $2x = \arcsin(u)$ but I notice this would not be surjective on the interval given by the extremes of integration.
So, to solve the problem, I ... | Your domain of integration should not extend to 1. Instead, work with
$\int_{\pi/2}^{3\pi/4} \frac{2}{1 - \sin(2x)} dx+ \int_{3\pi/4}^{\pi}\frac{2}{1 - \sin(2x)} dx$ such that your substitution of arcsin is well-defined.
Given $x \in [\pi/2, 3\pi/4]$, we know that $\cos(2x)$ is non-positive, so it must be the case tha... | {
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Why is a Lie algebra of a matrix Lie group not closed under complex scalar multiplication? Let the set $\mathcal{g}$ be the Lie algebra of a matrix Lie group $G$. Then my book asserts that $\mathcal{g}$ is a real vector space because it's closed under real scalar multiplication. My question why is it not closed under c... | If $G$ is a real Lie group, so in particular it is a real manifold, then its tangent spaces will be real vector spaces. In particular its Lie algebra will be a real vector space. If, on the other hand, $G$ is a complex Lie group, then its Lie algebra will be a complex vector space.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to prove a complex limit with epsilon delta definition? I have $$\lim_{z \to i} \frac {iz^3-1}{z+i}=0$$
To prove this I am trying to use the epsilon-delta definition. By saying that for any $\delta >0$ and any $\varepsilon >0$ then:
$$0<|z-i|<\delta \quad \mathrm {whenever} \quad \left|\frac {iz^3-1}{z+i}\right|< \... | If we assume $0 < |z-i| < \delta \le 1$, then
$$|z+i| = |-2i - (z-i)| \ge
\Big||2i| - |z-i|\Big| > |2 - \delta| \ge 1$$
(using the reverse triangle inequality), and
$$|z^2 + iz - 1| = |(z-i)^2 + 3i(z-i) - 3| \le \delta^2 + 3 \delta + 3 \le 7$$
(using the regular triangle inequality), so
$$\Big| \frac{(z-i)(z^2 + iz - ... | {
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Factoring a degree 4 polynomial without power of 2 term For my hobby, I'm trying to solve $x$ for $ax^4 + bx^3 + dx + e = 0$. (note there's no $x^2$) I hope there is a simple solution.
I'm trying to write it as $(fx + g)(hx^3+i) = 0$
It follows that
$fh=a; gh=b; if=d; gi=e$
At first sight it looks promising with 4 equa... | I suppose that you are searching a decomposition of the given polynomial in factors with real coefficients.
First note that your condition $db=ae$ means
$$
\frac{a}{d}=\frac{b}{e}=k
$$
so the polynomial is obviously decomposable as:
$$
kdx^4+kex^3+dx+e=kx^3(dx+e)+dx+e=(dx+e)(kx^3+1)
$$
Also note that a degree $4$ poly... | {
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Number of Subgroups of $C_p \times C_p$ and $C_{p^2}$ for $p$ prime As the title says, I am interested in finding all subgroups of $C_p \times C_p$ and $C_{p^2}$ for $p$ prime.
We did not cover the Sylow-theorem so far in the lecture.
What I noticed so far:
As $C_p$ is of order $p$, the elements can only have order $p... | Every element of order $p$ in $G=C_p\times C_p$, and there are $p^2-1$ of them, generates a cyclic subgroup of order $p$, and every such subgroup has $p-1$ generators. This implies that there are $\frac{p^2-1}{p-1}$ subgroups of order $p$, that is, $p+1$. As there are also the trivial group, and the whole group (and no... | {
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Finding if a linear transformation is diagonalisable Hi i am having some trouble tackling this question for my exam revision.
Let $M_{(2,2)}(\mathbb{R})$ denote the vector spce of 2x2 matrices over the real numbers. Also, let A denote the matrix
$$\begin{bmatrix}2&\lambda\\1&0\end{bmatrix}$$
where $\lambda$ is a real n... | $\psi$ is a linear transformation between $M_{2\times 2} \rightarrow M_{2\times 2},$ so you need to find a matrix representation $[\psi]$ of $\psi$ and compute the eigenvalues and eigenvectors of $[\psi]$. Since $M_{2\times 2}$ is 4-dimensional, $[\psi]$ will be a $4 \times 4$ matrix. It will be diagonalizable if and ... | {
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Drawing conclusions from a differential inequality Let $f(x)$ be a smooth real function defined on $x>0$. It is given that:
*
*$f$ is an increasing function ($f'(x)>0$ for all $x>0$).
*$x \cdot f'(x)$ is a decreasing function.
I am trying to prove that:
$$ \lim_{x\to 0}f(x) = -\infty $$
EXAMPLE: $f(x) = -x^{-q}$,... | This is an alternate proof via contradiction, using limits and the greatest lower bound property. If $\lim_{x \to 0}f(x)$ is not $-\infty,$ then since $f$ is increasing for positive $x$ it would follow that $\lim_{x \to 0}f(x)=L$ where $L$ is the greatest lower bound of the set of values $f(x),\ x>0.$
From that it foll... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Lower bound on quadratic form Suppose I have a non-symmetric matrix $A$ and I can prove that $x^T A x = x^T \left(\frac{A+A^T}{2}\right) x>0$ for any $x \ne 0$? Can I then say that $x^T A x \ge \lambda_{\text{min}}(A) \|x\|^2 > 0$?
| Let quadratic form $f$ be defined by $f (x) = x^T A x$, where $A \in \mathbb{R}^{n \times n}$. Since $x^T A x$ is a scalar, then $(x^T A x)^T = x^T A x$, i.e., $x^T A^T x = x^T A x$. There are infinitely many matrix representations of $f$. We take affine combinations of $A$ and $A^T$, and any such combination yields $f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Square root of both sides If you have the equation:
$x^2=2$
You get:
$x=\pm \sqrt{2}$
But what do you do actually do? What do you multiply both sides with to get this answer? You take the square root of both sides, but the square root of what? If you understand what i mean?
| $$x^2=2 \Rightarrow \begin{cases}y=x^2\\y=2 \end{cases}$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Mean-value Theorem $f(x)=\sqrt{x+2}; [4,6]$
Verify that the hypothesis of the mean-value theorem is satisfied for the given function on the indicated interval. Then find a suitable value for $c$ that satisfies the conclusion of the mean-value theorem.
$$f(x)=\sqrt{x+2}; [4,6]$$
So,
$$f'(x) = {1 \over 2} (x+2)^{-{1\ov... | You did everything correctly, let's solve for $c$ together.
You have
$$
\frac{1}{2} (c+2)^{-1/2} = \frac{a}{2}\\
(c+2)^{-1/2} = a \\
\frac{1}{\sqrt{c+2}} = a \\
c+2 = \frac{1}{a^2}
$$
so
$$
\begin{split}
c &= \frac{1}{a^2} - 2 = \frac{1}{\left(2 \sqrt{2} - \sqrt{6}\right)^2} - 2 \\
&= \frac{1}{8 + 6 - 4\sqrt{2}\sqrt{... | {
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"timestamp": "2023-03-29T00:00:00",
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how many distinct values does it have? I solved this problem by manually adding parentheses and counting them, and got correct answer of 32. Is there a simple to find the answer? Thanks.
The value of the expression $1÷2÷3÷5÷7÷11÷13$ can be altered by including parentheses. If we are allowed to place as many parenthese... | In general, inserting parentheses in
$$ a_1 \div a_2 \div a_3 \div \cdots \div a_n $$
can produce every number of the form
$$ a_1^{\strut}a_2^{-1}a_3^{s_3} a_4^{s_4} \cdots a_n^{s_n}$$
(and only those), where each $s_i$ is either $1$ or $-1$. Note that the exponents of $a_1$ and $a_2$ are fixed.
If $a_3$ through $a_n$... | {
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Let $A= \{1,2,3,4,5,6,7,8,9,0,20,30,40,50\}$. 1. How many subsets of size 2 are there? 2.How many subsets are there altogether?
Let $A= \{1,2,3,4,5,6,7,8,9,0,20,30,40,50\}$.
1. How many subsets of size $2$ are there?
2.How many subsets are there altogether?
Answer:
1) I think there are $7$ subsets of size two a... | The total number of subsets of size $2$ is $\binom{14}{2}$. To understand this try to see how many ways are there to pick two distinct elements out of the set.
For the second part using the similar idea from the previous part, there are $\frac{14}{k}$ subsets with size $k$. So the total sum is:
$$\sum_{n=0}^{14} \binom... | {
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Proof that a discrete space (with more than 1 element) is not connected I'm reading this proof that says that a non-trivial discrete space is not connected. I understood that the proof works because it separated the discrete set into a singleton ${x}$ and its complementar. Since they're both open, their intersection is... | It is true that finite sets are closed in every T$_1$ space, and thus they are closed in every discrete space. Also by the definition of the discrete topology, $\textit{every}$ subset of the space is open. So suppose $X$ is discrete and has more than one point. Let $x\in X$. Then $\{x\}$ is open. It is also closed (i... | {
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"timestamp": "2023-03-29T00:00:00",
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Linear connection on a 1-form Let $M$ be a manifold with linear connection $\nabla$ and let $X$ be a vector field on $M$. Given a 1-form $\alpha \in \Omega^{1}(M)$, define $\nabla_{X} \alpha : \scr{X}$ $(M) \to C^{\infty}(M)$ by $$[\nabla_{X} \alpha] (Z) = X(\alpha(Z)) - \alpha(\nabla_{X}Z)$$ for $Z \in \scr{X}$$(M)$. ... | $[\nabla_{X} \alpha] (fY) = X (\alpha(f Y)) - \alpha(\nabla_{X} fY)$
$= X(f \alpha Y) - \alpha ((Xf)Y + f(\nabla_{X}Y)$
$= fX(\alpha Y) + (Xf)\alpha Y - (Xf) \alpha Y + f \alpha(\nabla_{X} Y)$
$= f(X(\alpha Y) - \alpha(\nabla_{X} Y)) = f [\nabla_{X} \alpha](Y)$
| {
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"timestamp": "2023-03-29T00:00:00",
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Examine convergence of $\int_0^{\infty} \frac{1}{x^a \cdot |\sin(x)| ^b}dx$ Examine convergence of $\int_0^{\infty} \frac{1}{x^a \cdot |\sin(x)| ^b}dx$ for $a, b > 0$. There are 2 problems. $|\sin(x)|^b = 0$ for $x = k \pi$ and $x^a = 0$ for $x = 0$. We can write $\int_0^{\infty} \frac{1}{x^a \cdot |\sin(x)| ^b}dx = \i... | To even have a chance at convergence at $\infty$, you need $a > 1$. However, then near $x = 0$, we have $x^a \lvert \sin(x) \rvert^b \approx x^{a+b}$ and since $a+b > 1$, we will have divergence near $x=0$. Thus the integral diverges for all $a,b > 0$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show that the sphere, S, and $\mathbb{R}^2$ is not homeomorphic I am trying to show that the sphere $S^2$ and $\mathbb{R}^2$ are not homeomorphic.I understand that you can't 'compress' a 3D shape into a 2D plane but I don't know how I would express this formally.
$S^2 = \{(x, y, z) ∈ \mathbb{R}^3: x^2 + y^2 + z^2 = 1\}... | Homeomorphism will preserve any "topological" property of spaces - in particular, $S^2$ is compact and $\mathbb R^2$ is not, so they can't be homeomorphic.
In fact, the image of a compact space under a continuous map is compact, so there is not even a surjective continuous map $S^2 \to \mathbb R^2$.
| {
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Show that no line with a y-int of 10 will ever be tangential to the curve with $y=3x^2+7x-2$ Show that no line with a y-int of 10 will ever be tangential to the curve with $y=3x^2+7x-2$.
Having trouble in showing this. So far these are my process.
*
*Let line be $y=mx+10$
*$mx+10 = 3x^2+7x-2$
*$3x^2+(7-m)x-12=0$
... | You tagged calculus so with derivatives: the slope of a tangent to the given function is
$$y'=6x+7\implies\;\text{for any point on the graph }\;\;(a, 3a^2+7a-2)$$
the tangent line to the function at that point is
$$y-(3a^2+7a-2)=(6a+7)(x-a)\implies y=(6a+7)x-3a^2-2$$
and thus the $\;y\,-$ intercept is $\;-3a^2-2\;$ , a... | {
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What is $\mathbb{Z_{n}}\left [ x \right ]$
Question: Show that $\mathbb{Z_{n}}\left [ x \right ]$ has characteristic $n$.
What does $\mathbb{Z_{n}}\left [ x \right ]$ stands for? I'm very sure this is not the gaussian ring.
| Let $\mathbb{Z}_n$ be the set of integers $\{0,1,\ldots,n-1\}$ equipped with the operations of addition mod $n$ and multiplication mod $n$. It can be shown this structure is a ring. $\mathbb{Z}_n[x]$ is defined as the set of polynomials of the form $a_n x^n + \cdots + a_1 x + a_0$, where $a_i \in \mathbb{Z}_n$ equipp... | {
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Roots of $x^{101}-100x^{100}+100=0$ I do not know how to prove that $x^{101}-100x^{100}+100=0$ has exactly two positive roots.
Some can give me hint for solving this please. Thanks for your time.
| Descartes' rule of signs indicates that $P(x)=x^{101}-100x^{100}+100$ has either zero or two positive roots.
But $P(0)>0$ and $P(2)<0$ so $P(x)$ has at least one positive root, hence it has exactly two positive roots.
| {
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find the maximum of the function $f(x)=a+b\sqrt{2}\sin{x}+c\sin{2x}$ let $a,b,c\in R$,and such $a^2+b^2+c^2=100$, find the maximum value and minimum value of the function
$$f(x)=a+b\sqrt{2}\sin{x}+c\sin{2x},0<x<\dfrac{\pi}{2}$$
Use Cauchy-Schwarz inequality?
| Use Cauchy-Schwarz inequality:
$$\left(a+b\sqrt{2}\sin{x}+c\sin{2x}\right)^2\le (a^2+b^2+c^2)(1+2\sin^2x+\sin^22x)$$
$$\left(a+b\sqrt{2}\sin{x}+c\sin{2x}\right)^2\le 100\cdot(1+2\sin^2x+\sin^22x)$$
$$|a+b\sqrt{2}\sin{x}+c\sin{2x}|\le 10\cdot\sqrt{1+2\sin^2x+\sin^22x}$$
$$1\le1+2\sin^2x+\sin^22x\le \frac{13}{4}$$
Then $... | {
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A funtion and its fourier transformation cannot both be compactly supported unless f=0 Problem : Suppose that $f$ is continuous on $\mathbb{R}$. Show that $f$ and $\hat f$ cannot both be compactly supported unless $f=0$.
Hint : Assume $f$ is supported in [0,1/2]. Expand $f$ in a Fourier series in the interval [-,1], an... | Suppose the support of $f$ is contained in $[-1,1],$ and $\hat f (y) = 0$ for $|y|>N \in \mathbb N.$ Applying a standard Fourier series argument on $[-\pi,\pi]$ then shows
$$f(x) = \sum_{-N}^{N}\hat f (n) e^{inx}, x \in [-\pi,\pi].$$
Thus $f$ is a trigonometric polynomial that vanishes on $[1,\pi].$ But a trigonometric... | {
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How do I prove something without premises in a Fitch system? If asked “Prove in Fitch: From no premises, derive $A \lor (A \to B)$. Without using Taut Con?"
These are the are the Fitch rules, and this is what I have so far.
Should I aim to use V Elim to isolate both sides and then derive with the method I'm currently ... | Yuck! It looks like somebody is trying to give you a headache.
To solve this there are a couple of general tricks you'll need to implement.
*
*derive $\neg (C\lor D)\vdash \neg C$.
*derive $\neg(C\to D)\vdash C$.
Combining yields a derivation of $\neg(A\lor (A\to B))\vdash A\land\neg A$.
Toward 1, after assumin... | {
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Describe, as a direct sum of cyclic groups, given a map $\phi: \mathbb{Z}^{3} \longrightarrow \mathbb{Z}^{3}$ I'm trying to resolve the next one:
Describe, as a direct sum of cyclic groups, the cokernel of the map $\phi: \mathbb{Z}^{3} \longrightarrow \mathbb{Z}^{3}$ given by left multiplication by the matrix
$$
\left... | Thanks to Derek Holt and SpamIAM for the recomendations and useful links, after a while to read and understand Modules over a PID, I finally got an answer.
Let $\phi$ be a $\mathbb{Z}$-linear map such that can be determined by $\phi(e_{1}) = f_{1}, \dots, \phi(e_{n}) = f_{n}$, where $e_{1}, \dots, e_{n}$ be the basis... | {
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Show that $f(x)=\ln(x)$ is not uniformly continuous on $(0,\infty)$ I'm trying to show that $f(x)=\ln(x)$ is not uniformly continuous on the interval $(0,\infty).$
This is what I have so far:
Let $\epsilon=1.$
Choose $\delta=$
if $x,y\in(0,\infty)$ with $|y-x|<\delta$ then $|f(y)-f(x)|=|\ln\left(\frac{y}{x}\right)|$
I'... | Working with $\epsilon$ and $\delta$ quickly becomes tedious and annoying, it is thus better to learn more convenient and powerful techniques. Remember that $f$ is uniformly continuous on $S$ if and only if for every sequences $(x_n), (y_n) \subseteq S$ with $d(x_n, y_n) \to 0$ we have that $d(f(x_n), f(y_n)) \to 0$ (w... | {
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"source": "stackexchange",
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Large cycles in bridgeless cubic graphs Wikipedia tells us that most cubic graphs have a Hamilton cylce (for instance the proportion of Hamiltonian graphs among the cubic graphs on $2n$ vertices converges to 1 as $n$ goes to infinity) but is also kind enough to provide us with some pictures of non-Hamiltonian cubic gra... | Even though this question seems to be of no interest to anyone I still thought it would be good form to post here a counterexample I found since posting:
Suppose (aiming for a contradiction) that the graph contains a cycle $C$ of the form described in the question. Picking any of the red edges it is easy to conclude t... | {
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Using inverse of transpose matrix to cancel out terms? I am trying to solve the matrix equation $A = B^TC$ for $C$, where $A$, $B$, and $C$ are all non-square matrices. I know that I need to utilize $M^TM$ in order to take the inverse. I'm just not sure how to isolate $C$ in the equation provided.
| We have the matrix equation in $\mathrm C$
$$\mathrm B^\top \mathrm C = \mathrm A$$
Let's left-multiply both sides by $\mathrm B$
$$\mathrm B \mathrm B^\top \mathrm C = \mathrm B \mathrm A$$
If $\mathrm B$ has full row rank, then $\mathrm B \mathrm B^\top$ is invertible. Hence,
$$\mathrm C = (\mathrm B \mathrm B^\top)^... | {
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How many ways are there to select $15$ cookies if at most $2$ can be sugar cookies?
A cookie store sells 6 varieties of cookies. It has a large supply of each kind. How many ways are there to select $15$ cookies if at most $2$ can be sugar cookies?
For my answer, I put $6 \cdot 6 \cdot 5^{13}$. My logic was to assum... | Let $x_k$ be the number of cookies of type $k$, $1 \leq k \leq 6$. Since an order of $15$ cookies is placed,
$$x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 15 \tag{1}$$
where equation 1 is an equation in the non-negative integers. A particular solution to equation 1 corresponds to the placement of five addition signs in a ro... | {
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Estimation of Integral $E(x)$
How can we prove $$\frac{1}{2}e^{-x}\ln\left(1+\frac{2}{x}\right)<\int_{x}^{\infty}\frac{e^{-t}}{t}dx<e^{-x}\ln\left(1+\frac{1}{x}\right)\;, x>0$$
$\bf{My\; Try::}$ Let $\displaystyle f(x)=\int_{x}^{\infty}\frac{e^{-t}}{t}dx\;,$ Then $\displaystyle f'(x) = -\frac{e^{x}}{x}<0\;\forall x>0... | Upper Bound Inequality
Note that we have the elementary inequalities
$$\frac{1}{x+1}\le \log\left(1+\frac1x\right)\le \frac1x \tag 1$$
Using the left-hand side inequality in $(1)$, it is easy to see that
$$\frac{1}{x}\le \log\left(1+\frac1x\right) -\left(\frac{1}{x+1}-\frac{1}{x}\right) \tag 2$$
Multiplying $(2)$ by ... | {
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on finite division subring of a ring
Is there any example of a ring which is not a division ring but any of its subring is a division ring?
According to me if $R$ is a ring and $S$ is a division subring then $1\in S$ and hence $R=S$. Is it true?
| A simple example is the product ring $\mathbb Z_2\times \Bbb Z_2$ (the Klein four group). Every proper nonzero subring is isomorphic to $\Bbb Z_2$, which is a division ring.
| {
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Why is radian so common in maths? I have learned about the correspondence of radians and degrees so 360° degrees equals $2\pi$ radians. Now we mostly use radians (integrals and so on)
My question: Is it just mathematical convention that radians are much more used in higher maths than degrees or do radians have some int... | The answer is simple, it's a distance measure. When you move in a straight line you use inches or metres, in circles it is radians.
If you are at Disneyland and ask how far it was to Anaheim Stadium [go, Angels!] and I tell you that from my house it's about 45º, you are probably not going to be happy.
You want the ... | {
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"source": "stackexchange",
"question_score": "71",
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Compactness of $C([0,1])$ I have to verify if the $C([0,1])$, space of all continuous functions defined on interval $[0,1]$ with supremum metric is compact.
As I know, we have to check if every sequence of functions $f_{n}(x)$ has subsequence that $f_{n_{k}}(x)$ is convergent.
In this metric of course conervgence impli... | The sequence $f_n(x)=x^n$ does not have a convergent subsequence since $f_n$ converges pointwise towards $f(x)=0$ if $x\neq 1$ and $f(1)=1$ which is not continuous. Hence, the space is not compact.
| {
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Proof of a theorem regarding group homomorphisms and kernels I am looking for a proof of the following theorem:
"Let $H<G$ then $H\unlhd G$ $\iff$ there exists a group $K$ and a group homomorphism: $\phi : G \rightarrow K$ such that $ker(\phi) = H$
There is one on a french wikipedia page but I find it incomprehensible.... | The forward direction will be done by considering $K=G/H$ and for the converse part, kernel of a group homomorphism is always a normal subgroup.
| {
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Can someone solve my non-understandable process in proving a theorem? Theorem.
Let $E$ be a subset of $\mathbb{R}^n$.
Then, if $p\gt0$, $\int_E|f-f_k|^p\to0$, and $\displaystyle\int_E|f_k|^p\le{}M$ for all $k$, then $\displaystyle\int_E|f|^p\le{}M$.
For your information, $|\cdot|$ means a Lebesgue measure.
*
*There... | *
*Since $p >0$, we have that
$$ |f-g|^p \leq |f|^p + |g|^p $$
*Using 1, we see
$$ \int |f|^p dx \leq \int | f- f_k|^p dx + \int |f_k|^p dx $$
for all $k \in \mathbb{N}$.
*Apply bound, we see
$$\int |f|^p dx \leq \int | f- f_k|^p dx + M $$
*Apply the limit, we see
$$\int |f|^p dx \leq\underbrace{ \lim_{k \to \infty... | {
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expressing contour integral in different form Hi I have a short question regarding contour integration: Given that $f(z)$ is a continuous function over a rectifiable contour $z = x + iy$. If $f(z) = u(x,y) + iv(x,y)$, why does it follow that the contour integral can be expressed as $$\int_{C}f(z)dz = \int_{C}(udx-vdy) ... | Another way to see it is by expanding the integral
$$
\int_\gamma f(z)dz = \int_a^bf(\gamma(t))\gamma'(t)dt = \int_a^b(u(\gamma_1(t),\gamma_2(t)) + i v(\gamma_1(t),\gamma_2(t)))(\gamma_1(t) + i\gamma_2(t))dt,
$$
where $\gamma(t) = (\gamma_1(t),\gamma_2(t)), t\in [a,b],$
and applying the definition of the line integra... | {
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Does a repeated eigenvalue always mean that there is an eigenplane under the transformation matrix? If you have a 3x3 matrix, if you find that it has repeated eigenvalues, does this mean that there is an invariant plane (or plane of invariant points if eigenvalue=1)?
I always thought that there was an invariant plane i... | If $Av=\lambda v$ and $Aw=\lambda w$, then for any linear combination $\alpha v+\beta w$ we have
$$
A(\alpha v+\beta w)=\alpha Av+\beta Aw=\alpha\lambda v+\beta\lambda w=\lambda(\alpha v+\beta w).
$$
In words, a linear combination of eigenvectors for the same eigenvalue is again an eigenvector for that eigenvalue.
T... | {
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Question in Introductory Linear Algebra I really need help with this question. I am in an introductory linear algebra course. If you guys could help me, I would really appreciate it. Here is the question:
A large apartment building is to be built using modular construction
techniques. The arrangement of apartments o... | Think about this in the context of a system of linear equations..
Plan A = (3,7,8)
Plan B = (4,4,8)
Plan C = (5,3,9)
Total = (66,14,136)
\begin{bmatrix}
3 & 4 & 5 & 66 \\
7 & 4 & 3 & 14 \\
8 & 8 & 9 & 136 \\
\end{bmatrix}
Is this system consistent?
| {
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How do you calculate the smallest cycle possible for a given tile shape? If you connect together a bunch of regular hexagons, they neatly fit together (as everyone knows), each tile having six neighbors. Making a graph of the connectivity of these tiles, you can see that there are cycles all over the place. The small... | I did some experiments for regular polygons, for up to $n=24$ (click to enlarge):
These experiments suggest that
*
*If $6\vert n$ you get a $3$-cycle
*Else if $2\vert n$ you get a $4$-cycle
*Else you get a $6$-cycle
That you can get the $3$-cycle for anything that has edges aligned as a hexagon has them is pret... | {
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2048 Logic Puzzle I thought up this logic problem related to the 2048 game. If all 16 tiles on a 2048 board all had the value 1024, how many ways are there to get to the 2048 tile? Here is what I am talking about in an illustration:
I found a much simpler, but longer way to think about this: There are 3 ways to combin... | I believe you are Correct.
There are 3 lines separating rows horizontally, 4 pairs of numbers across each line, and 2 ways to combine said pairs (top-down or bottom-up), giving $2*3*4=24$ ways of making pairs.
Repeating for the columns and getting the exact same numbers, we now have $24+24 = 48$ total options for mergi... | {
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Find the value of $\frac{a^2}{a^4+a^2+1}$ if $\frac{a}{a^2+a+1}=\frac{1}{6}$ Is there an easy to solve the problem? The way I did it is to find the value of $a$ from the second expression and then use it to find the value of the first expression. I believe there must be an simple and elegant approach to tackle the prob... | From the first equation (inverted),
$$\frac{a^2+a+1}a=6$$ or $$\frac{a^2+1}a=5.$$
Then squaring,
$$\frac{a^4+2a^2+1}{a^2}=25$$
or
$$\frac{a^4+a^2+1}{a^2}=24.$$
| {
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Is a diagonalization of a matrix unique? I was solving problems of diagonalization of matrices and I wanted to know if a diagonalization of a matrix is always unique? but there's nothing about it in the books nor the net.
I was trying to look for counter examples but I found none.
Any hint would be much appreciated
Th... | The diagonal matrix is unique up to a permutation of the entries (assuming we use a similarity transformation to diagonalize). If we diagonalize a matrix $M = U\Lambda U^{-1}$, the $\Lambda$ are the eigenvalues of $M$, but they can appear in any order.
| {
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Geometrical Description of $ \arg\left(\frac{z+1+i}{z-1-i} \right) = \pm \frac{\pi}{2} $
The question is in an Argand Diagram, $P$ is a point represented by the complex number. Give a geometrical description of the locus of $P$ as $z$ satisfies the equation:
$$ \arg\left(\frac{z+1+i}{z-1-i} \right) = \pm \frac{\pi}{2... | Let the points $z_A = -1-i$ and $z_B=1+i$. Then we look for the locus of all points Pwith $z_P=z$ such that $\vert\arg \frac{z-z_A}{A-z_B} \vert = \pi$ (in other words, the angle $\angle APB=\pi$). This is the circle in the complex plane with diameter $AB$, since we know that the angle under which the segment $AB$ is ... | {
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disk of convergence for complex-valued series Find the disk of convergence of $\displaystyle \sum_{k=0}^{\infty} \frac{(z+2)^k}{(k+2)^3 4^{k+1}}$, where $z \in \mathbb{C}$.
I tried applying the ratio test: $\lim_{k \to \infty} \left| \frac{(z+2)^{k+1}}{(k+3)^3 4^{k+2}} \cdot \frac{(k+2)^3 4^{k+1}}{(z+2)^{k}} \right| = ... | I'd rather go directly with the $\;n\,-$ th root test (Cauchy-Hadamard formula) of the general term of the coefficients:
$$\sqrt[k]{|a_k|}=\sqrt[k]{\frac1{(k+2)^34^{k+1}}}=\frac1{4\sqrt[k]{4(k+2)^3}}\xrightarrow[k\to\infty]{}\frac14\;\implies R=4$$
and thus the interval of convergence ( around $\;-2\;$ , of course ) is... | {
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Find the smallest value of the function $F:\alpha\in\mathbb R\rightarrow \int_0^2 f(x)f(a+x)dx$
Let $f -$ fixed continuous on the whole real axis function which is periodic with period $T = 2$, and it is known that the function $f$ decreases monotonically on the segment $[0, 1]$ increases monotonically on the segment ... | We intend to prove that the smallest value of the function $F$ defined by
\begin{equation*}
F({\alpha}) = \int_0^2f(x)f(x+{\alpha})\, dx\tag{1}
\end{equation*}
is $F(1)$.
At first we give $f$ the additional property to be differentiable with a continuous derivative. At the end we will fill that gap.
The function $F$ wi... | {
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Simplifying sum of powers of conjugate pairs The result of summing a conjugate pair of numbers each raised to the power $n$:
$$
(a + bi)^n + (a - bi)^n
$$
Produces a real number where $a + bi$ is a complex number.
Given the result is real, is there a simplified way to express the above expression in terms of $a$ and $b... | Let $z=|z|(cos (\alpha) + isin(\alpha))$ therefore $z^n=|z|^n(cos (n \alpha) + isin (n\alpha))$ so the expression becomes $2|z|^ncos (n\alpha)$. We know $|z|=\sqrt {a^2+b^2}$, also $\alpha$ can be expressed in terms of $a,b$ (using cotangent).
| {
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Lower bound for norm of matrix I have the following problem: $A$ is a positive definite, symmetric matrix.
Firstly I was required to find a matrix $B$ such that $B^n = A$. I believe this to be $C(D^{\frac1n}) C'$ where C is the orthogonal matrix of eigenvectors of $A$, and $A = CDC'$.
After this I am asked to find a l... | For the spectral norm, you can write a direct relation between the norm of $B$ and the norm of $A$. Since $A$ is symmetric and positive-definite, the spectral norm of $A$ is just the maximal eigenvalue of $A$. Your $B$ is also symmetric and positive-definite and so its norm also equals to the maximal eigenvalue which w... | {
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Example to show that unconditional convergence does not imply absolute convergence in infinite-dimensional normed spaces. I tried to show that unconditional convergence does not imply absolute convergence in infinite-dimensional normed spaces using a direct proof, but unfortunately I did not succeed. The definition of ... | Denote by $y = (y^n)_{n \in \mathbb{N}}$ the sequence $y = (1, \frac{1}{2}, \frac{1}{3}, \dots)$ (we will use upper indices for the terms of an element in $\ell^{\infty}(\mathbb{N})$ in order to not confuse ourselves when considered sequences of elements in $\ell^{\infty}(\mathbb{N})$). Then we have
$$ \left( y - \sum_... | {
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Functor is of the form Set(-,A) Let $F: Set^{op} \rightarrow Set$ be a functor such that for corresponding functor $\overline{F}: Set \rightarrow Set^{op}$ we have $\overline{F} \dashv F$. With corresponding functor I mean that $F$ and $\overline{F}$ are just basically the same functor (just written differently).
An ex... | Yoneda's lemma doesn't help to characterize when a given functor is representable or not.
Here just write the fact that you have an adjunction :
$$Hom_{Set^{op}}(\overline{F}(A),B) = Hom_{Set}(A,F(B))$$
which means in $Set$:
$$Hom(B,F(A)) = Hom(A,F(B)).$$
Now take $B = \ast$ (a point) :
$$F(A)\simeq Hom(\ast,F(A)) = H... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1799732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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ordered set notation in functions Do please forgive me, if this question is a duplicate.
How does one correctly notate a function $f$, which takes a ordered subset $S$ from the field $\mathbb{K}$ and returns an other (ordered) subset from the same field in question?
Obviously, the notation $f:S \mapsto T : \text{ ...} ... | $$f:\mathcal{P}(\mathbb{K})\to \mathcal{P}(\mathbb{K})$$
Where $\mathcal{P}(\mathbb{K})$ denotes the power set of $\mathbb{K}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1799843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How do I show that a contraction mapping in a metric space is continuous? I start out by letting $V$ be an arbitrary open set in $X$. Then
$$
f^{-1}(V) = \{x\in X\mid f(x) \in B_\epsilon(f(a))\}.
$$
This can be re-written as:
$$
f^{-1}(V) = \{x\in X\mid d(f(a), f(x)) < \epsilon \}.
$$ I realize that contraction mapp... | Let $(X, d)$ be a metric space, $S \subset X$, and $f:S \longrightarrow S$ a function be such that $d(f(x), f(y)) \leq c d(x, y)$, for all $x, y \in S$, where $0 \leq c < 1$ is given.
Fix $\epsilon > 0$ and choose $a \in S$.
The case where $c = 0$ is trivial. Assume $c > 0$ and let $\delta = \epsilon / c$.
For all $x ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1799921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Integrate $\int_0^\infty \frac{e^{-x/\sqrt3}-e^{-x/\sqrt2}}{x}\,\mathrm{d}x$ I can't solve the integral
$$\int_0^\infty \frac{e^{-x/\sqrt3}-e^{-x/\sqrt2}}{x}\,\mathrm{d}x$$
I tried it by using Beta and Gamma function and integration by parts. Please help me to solve it.
| An alternative approach to Marco Cantarini's perfectly sound answer.
If we set, for any $\alpha>1$,
$$ I(\alpha) = \int_{0}^{+\infty}\frac{e^{-x}-e^{-\alpha x}}{x}\,dx $$
differentation under the integral sign/Feynman's trick gives
$$ I'(\alpha) = \int_{0}^{+\infty} e^{-\alpha x}\,dx = \frac{1}{\alpha}, $$
and since $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1800042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
Ito formula when g(t,x) is an integral Suppose we have a stochastic process which is written as an Ito process.
$$dX_t=\mu_t\ dt +\sigma_t\ dB_t$$.
If $Y_t$ is defined as a stochastic process as a function of $X_t$, then we can find $dY_t$ using the Ito formula. The key is to have the function $g(t,x)$ which relates $X... | Express $Y$ as an Ito process:
$$
dY_t=X_t\,dt = X_t\,dt +\ 0\,dB_t.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1800180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Derivation of Dirac delta function Is there anyone could give me a hint how to find the distributional derivative of the delta function $\delta$? I don't know how to deal with the infinite point.
| Upon request in the comments:
There is a large class of distributions which are given by integration against locally integrable functions. Specifically, given a locally integrable $f$ and a smooth compactly supported $g$, one can define $T_f(g)=\int_{-\infty}^\infty f(x) g(x) dx$. This leads to a common abuse of notati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1800257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Prove the fractions aren't integers
Prove that if $p$ and $q$ are distinct primes then $\dfrac{pq-1}{(p-1)(q-1)}$ is never an integer. Is it similarly true that if $p,q,r$ are distinct primes then $\dfrac{pqr-1}{(p-1)(q-1)(r-1)}$ is also never an integer?
I think using a modular arithmetic argument here would help. I... | Suppose, for the sake of contradiction, such distinct $p$ and $q$ exist.
First of all observe that the statement implies that $p-1|pq-1$. So,
$$p-1|pq-1-q(p-1) \implies p-1|q-1$$
Similarly we get,
$$q-1|p-1$$
These observations imply that $p-1 = q-1$. This implies that $p = q$. Contradiction. They aren't distinct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1800384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 5,
"answer_id": 3
} |
Is the Lie derivative $L_{X}(\omega \wedge \mu)$ an exact form?
Let $\omega$ be an $n$-form and $\mu$ be an $m$-form where both are acting on a manifold $M$. Is the Lie derivative $L_{X}(\omega \wedge \mu)$ where $X$ is a smooth vector field acting on $M$ an exact form?
I think it is but I've been unable to prove it... | In general this is not true. Recall that
$$
L_X(\omega) = i_x d\omega + d i_x\omega
$$
where you see that the right part is exact and the left part mustn't be. As an example for your case take $N$ a manifold with a non exact form $\mu$
and let $\omega$ be a 0-form (function) on $\mathbb{R}$ and define $M=N\times\math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1800519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Probability of having a Girl A and B are married. They have two kids. One of them is a girl. What is the probability that the other kid is also a girl?
Someone says $\frac{1}{2}$, someone says $\frac{1}{3}$. Which is correct?
Now A and B have 4 children and all of them are boys. B is pregnant. So what is the probabilit... | Conditional probability:
Let $A$ and $B$ be two events.
$$P(A|B) = \frac{P(A\cap B)}{P(B)}$$
which means, the probability of $A$ occuring given that $B$ occured, is the probability of both $A$ and $B$ occuring, divided by the probability that $B$ occurs.
In this case, $A$ is the event that the other kid is a girl, and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1800658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
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Eigen-values of a matrix $P^{-1}AP$
QUESTION: If A and P be $2$ non-singular $n\times n$ matrices and $\lambda$ is the eigen-value of $A$, then show that $\lambda$ is also
the eigen-values of a matrix $P^{-1}AP$.
I could simply show that $\lambda$ being the eigen-value of $A$, we have that
$$det (A-\lambda I_n)=0... | Actually $A$ and $P^{-1}AP$ share the same characteristic polynomial, hence they have the same eigenvalues. Note that
$$\begin{align*}
\det(P^{-1}AP-\lambda I_n) & = \det(P^{-1}(A-\lambda I_n)P))\\
& = \det(P^{-1})\det(A-\lambda I_n)\det(P)\\
& = \det(A-\lambda I_n).
\end{align*}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1800757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding coefficient of $x^n$ in this series I was doing an assignment question, I came across this:
I understand everything (I know the theorem used in the answer), but I don't get how solution switched from first "i.e." to second "i.e.". I mean how did it happen? And what is happening after that Geometric Progression... | They are using the fact that $(1-x^6)^4=1-4x^6+{4\choose2}x^{12}-\cdots$ and noting that since you're only looking for the coefficient of $x^8$, you can drop all those higher-order terms, including the ${4\choose2}x^{12}$.
As for the rest, it's a matter of observing that
$${1\over(1-x)^4}={1\over3!}\left(1\over1-x\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1800867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How quickly can we detect if $\sqrt{(-1)}$ exists modulo a prime $p$? How quickly can we detect if $\sqrt{(-1)}$ exists modulo a prime $p$? In other words, how quickly can we determine if a natural, $n$ exists where $n^2 \equiv -1 \bmod p$?
NOTE
This $n$ is essentially the imaginary unit modulo $p$.
| Let's do some experimentation.
$p = 3$: no.
$p = 5$: yes, $2^2 \equiv -1$.
$p = 7$: no.
$p = 11$: no.
$p = 13$: yes, $5^2 \equiv -1$.
$p = 17$: yes, $4^2 \equiv -1$.
$p = 19$: no.
$p = 23$: no.
It appears that only those prime numbers which are congruent to $1$ modulo $4$ have this property.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1801264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Induced homomorphism of a covering space How can I determine what's the induced homology homomorphism of a covering $S^{n} \rightarrow RP^{n}$? I suppose that a Hurewicz homomorpism would be pretty effective, but since I know nothing about higher homotopy groups of spheres and their generators I'd rather avoid it.
| All of the homology groups of $\mathbb{S}^n$ are trivial, except of top and bottom one. The induced map $H_0(\mathbb{S}^n) \to H_0(\mathbb{R}\mathbb{P}^n)$ will always be isomorphism (this is very easy to calulate). The top homology group $H_n(\mathbb{R}\mathbb{P}^n)$ is either trivial or $\mathbb{Z}$, depending on whe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1801361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Surface of the intersection of $n$ balls Suppose there are $n$ balls (possibly, of different sizes) in $\mathbb R^3$ such that their intersection $\mathfrak C$ is non-empty and has a positive volume (i.e. is not a single point). Apparently, $\mathfrak C$ is a convex body with a piecewise smooth surface — a "quilt" of s... | Surface and volumes can be calculated analytically for any n value. For intersections of n=5 spheres and more, they can be calculated from the 4 by 4 intersections: see theorems 4.5 and 4.6 in my book chapter:
Spheres Unions and Intersections and some of their Applications in Molecular Modeling, In: Distance Geometry: ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Prove that $\det(A)=p_1p_2-ba={bf(a)-af(b)\over b-a}$ Let $f(x)=(p_1-x)\cdots (p_n-x)$ $p_1,\ldots, p_n\in \mathbb R$ and let $a,b\in \mathbb R$ such that $a\neq b$
Prove that $\det A={bf(a)-af(b)\over b-a}$ where $A$ is the matrix:
$$\begin{pmatrix}p_1 & a & a & \cdots & a \\ b & p_2 & a & \cdots & a \\ \vdots & \vdot... | Here is a possible proof without induction. The idea is to consider $\det A$ as a function of $p_n$.
We define the function $F: \Bbb R \to \Bbb R$ as
$$
F(p) = \begin{vmatrix}
p_1 &a &\ldots &a &a \\
b &p_2 &\ldots &a &a \\
\vdots &\vdots &\ddots &\vdots &\vdots \\
b &b &\ldots &p_{n-1} &a\\
b &b &\ldots &b &p
\end{vm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1801627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 1
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Statistical significance and sample size I have a device that is said to succeed at doing some task at least 99% of attempts, and fails no more than 1% of attempts.
If I want to be 98% sure that it achieves that success rate, how many results would I need to check at minimum?
And what would be the maximum number of fai... | This is how I would approach the question (Even if I made a mistake, you'll get the idea):
At first, we should choose the model. Lets do hypothesis testing for binomial distribution. Our device has binomial distribution with some constant probability of success $p$ and probability of failure $1-p$. We are checking hyp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1801724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$a_n = \frac{1}{n}b_n$, $\lim b_n = L>0, L\in\mathbb{R}$, prove $\sum a_n$ diverges I have to prove that if
$$a_n = \frac{1}{n}b_n$$for $n\ge 1$ and $$\lim_{n\to\infty} b_n = L>0, L\in\mathbb{R}$$ then $$\sum_{n=1}^{\infty} a_n$$ diverges.
My idea was to show that it's not true that $a_n\to 0$ but I guess it's true bec... | hint : what is $$\lim_{ n\rightarrow\infty}\sum_{x=n}^{x=kn}a_{x}$$ ? where k is some integer
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1801828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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Determinant of determinant is determinant? Looking at this question, I am thinking to consider the map $R\to M_n(R)$ where $R$ is a ring, sending $r\in R$ to $rI_n\in M_n(R).$ Then this induces a map. $$f:M_n(R)\rightarrow M_n(M_n(R))$$
Then we consider another map $g:M_n(M_n(R))\rightarrow M_{n^2}(R)$ sending, e.g. $... | (Edit: the OP has modified their question; this answer no longer applies.)
Your question is not well posed because determinant is defined on commutative rings only, but $M_n(R)$ in general is not a commutative ring. But there is indeed something similar to what you ask. See
*
*M. H. Ingraham, A note on determinants,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find two fractions such that their sum added to their product equals $1$ This is a very interesting word problem that I came across in an old textbook of mine. So I managed to make a formula redefining the question, but other than that, the textbook gave no hints really and I'm really not sure about how to approach it.... | Simple solution may be $(a,b,c,d) = (1,1,0,1)$.
$$
\frac ab + \frac cd + \frac ab \cdot\frac cd = 1 \Leftrightarrow
\frac{ad+cb+ac}{bd}=1 \Leftrightarrow\\
a(d+c) = b(d-c)\Leftrightarrow
\frac{a}{b} = \frac{d - c}{d+c} \
$$
Answer to:
Find two fractions so that their sum added to their product equals 1.
Take some $d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1801986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Enough projectives and $F$ preserves limits implies $G$ preserves epi's. Exercise: Let $\mathcal{C}, \mathcal{D}$ be categories, $G : \mathcal{C} \to \mathcal{D}$ and $F : \mathcal{D} \to \mathcal{C}$ an adjunction $F \dashv G$. Suppose $\mathcal{D}$ has enough projectives and $F$ preserves projectives. Prove that $G$ ... | Suppose $f : X \to Y$ is an epimorphism in $\mathcal{C}$. We wish to show that $G f : G X \to G Y$ is an epimorphism.
Let $q : B \to G Y$ be an epimorphism in $\mathcal{D}$ where $B$ is projective. Then $F B$ is projective, so we have a morphism $x : F B \to X$ in $\mathcal{C}$ such that $f \circ x = \epsilon_Y \circ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1802108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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How to differentiate product of vectors (that gives scalar) by vector? I'm trying to understand derivation of the least squares method in matrices terms:
$$S(\beta) = y^Ty - 2 \beta X^Ty + \beta ^ T X^TX \beta$$
Where $\beta$ is $m \times 1$ vertical vector, $X$ is $n \times m$ matrix and $y$ is $n \times 1$ vector.
Th... | Recall that the multiple regression linear model is the equation given by
$$Y_i = \beta_0 + \sum_{j=1}^{p}X_{ij}\beta_{j} + \epsilon_i\text{, } i = 1, 2, \dots, N\tag{1}$$
where $\epsilon_i$ is a random variable for each $i$. This can be written in matrix form like so.
\begin{equation*}
\begin{array}{c@{}c@{}c@{}c@{}c@... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1802243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Prove similarity of matrix $A^{-1}$ to matrix $A^{*}$ which is Hermitian adjoint Let $A \in \mathcal M_{n}(\Bbb C)$ and $A$ is similar to unitary matrix.
Prove that $A^{-1}$ is similiar to $A^{*}$, where $A^{*}$ is Hermitian adjoint.
$A = C^{-1}UC$, where $U$ is unitary matrix
So $A^{-1} = (C^{-1}UC)^{-1} = C^{-1}U^{-... | What you have is correct. You can now say that
$$
A^*= (C^*C)A^{-1}(C^*C)^{-1}
$$
By definition, this means $A^*$ is similar to $A^{-1}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1802417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Check if $f(x)=2 [x]+\cos x$ is many-one and into or not? If $f(x)=2 [x]+\cos x$
Then $f:R \to R$ is:
$(A)$ One-One and onto
$(B)$ One-One and into
$(C)$ Many-One and into
$(D)$ Many-One and onto
$[ .]$ represent floor function (also known as greatest integer function
)
Clearly $f(x)$ is into as $2[x]$ is an even int... | You are right with respect to surjectiveness (it is not onto).
Hint:
For injectiveness (one to one), look in a neighbourhood around $x = 3\pi$ for example.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1802497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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$((n-K)s^2)/\sigma^2$ what is this in terms of matrix linear regression? $$
\frac{(n-K)s^2}{\sigma^2}
$$
what is this in terms of matrix linear regression? Has Chi Squared Distribution with (n-K) df
| I don't know what you mean by "matrix" linear regression, and your question isn't all that clear. However, suppose you're doing multiple linear regression with $K$ predictors (including the constant predictor) and $n$ cases.
Suppose the errors (not to be confused with the (observable) residuals) all are independent an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1802569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Awesome riddle including independence and exponential distribution The life cycles of 3 devices $A, B$ and $C$ are independent and exponentially distributed with parameters $\alpha,\beta,\gamma$. These three devices form a system that fails if not only device A fails but also device B or C fails too. Maybe $a \land (b... | We first go after the complementary event, the event the system is still alive at time $t$. This event can happen in two disjoint ways: (i) $A$ is alive or (ii) $A$ is dead but $B$ and $C$ are alive.
The probability of (i) is $e^{-\alpha t}$.
The probability of (ii) is $(1-e^{-\alpha t})e^{-\beta t}e^{-\gamma t}$.
Thu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1802665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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When a set of functions becomes complete? I know that a set of functions are said to form a complete basis on an inteval if any function on that interval can be expressed as a linear combination of the functions in the set. I also know that every function in the set are orthogonal.
Now what is what is the condition(s)... | A metric space is said to be "complete" if every Cauchy sequence converges.
For example: Let $(X, \mu)$ be a measure space. Then $L^P(X)$ is complete under the $L^P$ norm, for $p \in [1,\infty]$. [It is a Banach space.]
Every finite dimensional normed vector space is also complete. (This this can be explained by the Li... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How is $[Q(\sqrt2, \sqrt3 ) : Q(\sqrt2)]=2$? $\mathbb{Q}$ is the rationals. I know that $\sqrt3 \notin \mathbb{Q}(\sqrt2)$ but so what? The answer to this question seems to be based upon that. Really don't understand what that means in finding the minimal polynomial.
| If you know that $\sqrt{3}$ is not in $\mathbb{Q}(\sqrt{2})$, then you know the degree is greater than $1$. But $\sqrt{3}$ is a root of the equation $x^2-3=0$, which has coefficients in $\mathbb{Q}(\sqrt{2})$, so the degree is exactly $2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1802841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$\sum_{n=0}^{\infty}\frac{a^2}{(1+a^2)^n}$ converges for all $a\in \mathbb{R}$ $$\sum_{n=0}^{\infty}\frac{a^2}{(1+a^2)^n}$$
Can I just see this series as a geometric series? Since $c = \frac{1}{1+a^2}<1$, we can see this as the geometric series:
$$\sum_{n=0}^{\infty}bc^n = \sum_{n=0}^{\infty}a^2\left(\frac{1}{1+a^2}\ri... | Let $a=0$. Then the series obviously converges to $0$.
Now suppose that $a\ne 0$. Then our series is the geometric series
$$a^2+a^2r+a^2r^2+a^2r^3+\cdots,$$
where $r=\frac{1}{1+a^2}\lt 1$. Since $|r|\lt 1$, the series converges. It is probably by now a familiar fact that when $|r|\lt 1$ the series $1+r+r^2+r^3+\cdots$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1802887",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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is the rotation matrix is unique for one rotation I have a test for rotation , and found two rotation behave the same at one point
rot1 = [ 0.8736 0.2915 -0.3897;
-0.4011 0.8848 -0.2373;
0.2756 0.3636 0.8898]
rot2 = [ 0.9874 -0.1420 -0.0700;
0.0700 0.7880 -0.6117;
0.1420 ... | Any two rotation matrices about a point should be distinct when acting on some arbitrary vector. That being said, two distinct rotations could certainly map some specific vector into another specific vector. That's what JeanMarie's answer addresses.
I think your issue, though, may have to do with the limitations of com... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1802967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Why is the index $i(\mathcal{L})$ of an ample line bundle on an abelian variety equal to $0$? I've seen that here https://www.math.uchicago.edu/~ngo/Shimura.pdf there's a theorem called Mumford's Vanishing Theorem (Theorem 2.2.2) which says:
Let $\mathcal{L}$ be a line bundle on $X$ (abelian variety) such that $K(\mat... | If $\mathcal{L}$ is ample, then it is nondegenerate (cf. page 84 of Mumford's book) and $h^0(\mathcal{L}^n)>0$ for some $n>0,$ where $h^q(\mathcal{L})=dim_k H^q(A, \mathcal{L}),$ so $i(\mathcal{L}^n)=0$ and hence also $i(\mathcal{L})=0$ by the Corollary of Mumford in page 159.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1803105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Probability of $m$ failed trades in series of $n$ trades This is a trading problem:
Let's say I have an automated trading system with a probability of
success of $70\%$ on any individual trade. I run $100$ trades a year.
What is the probability of getting $5$ or more consecutive failed
trades?
More generally, fo... | Let your trades be $t_1\cdots t_n$, with $t_i\in \{0,1\}$. What is the probability that this sequence contains $k$ consequtives zeroes, if the probability of zero is $p$? Well, the sequence could start at $t_1$, or $t_2,\ldots, t_{n-k+1}$. The probability that a sequence of $k$ zeroes starts at $t_i$ is $p^k$, and ther... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1803184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
A question about a route of a point that travels in a particular way through the plane I don't know exactly how to classify this question. It is not from any homeworks, just something I've been wondering about.
Let's say that in the beginning of an experiment ( the beginning is $t=0$ secs) we have two points on the pla... | Let the first point's position be $(t,0)$, and the second $(x(t),y(t))$.
We have that the direction of the second point is proportional to the difference between the points:
$$(x'(t), y'(t))\propto(t - x(t), -y(t))$$
But, since the speed is constant, we must have that:
$$(x'(t), y'(t))=\left(\frac{t-x(t)}{\sqrt{(t - x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1803293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
the connection between $\gamma_m(a)$ and $\gamma_m(b)$ when $a\cdot b\equiv 1\pmod m$
show the connection between the order of $a$ $\gamma_m(a)$ and the order of $b$ $\gamma_m(b)$ when
$$a\cdot b\equiv 1\pmod m$$
I took $a=5$ and $b=4$
$$5\cdot 4\equiv 1\pmod{19}$$
$$\gamma_m(a)=9\text{ and } \gamma_m(b)=9$$
So I thi... | One has $a^{\gamma_m(a)}\equiv 1$ and $b^{\gamma_m(b)}\equiv 1\pmod{m}$. Multiplying the two and assuming without loss of generality that $\gamma_m{a}\gt \gamma_m(b)$ we can write
$$a^{\gamma_m(a)-\gamma_m(b)}\left(a\cdot b\right)^{\gamma_m(b)}\equiv 1\pmod{m}$$
This means $a^\alpha\equiv 1\pmod{m}$ with $\gamma_m(a)-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1803381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Chevalier de mere paradox with game with three dice Chevalier de Mere asked Blaise Pascal why in a game with three dice the sum $11$ is more favorable than $12$, when both sums have exactly the same possible combinations:
For $11$ we have $(5,5,1), (5,4,2), (5,3,3), (4, 4, 3), (6,4,1), (6,3,2)$ and for $12$ we have $(6... | You're making the very common mistake of confusing issues of distinguishability with issues of equiprobability. If you roll three indistinguishable dice, the probabilities of the various sums are exactly the same as if you roll three distinguishable dice. The dynamics of the dice are not influenced by your ability to d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1803493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that $x-a \sin(x)=b$ has one real solution, where $0\lt a \lt 1 $ $a,b \in \mathbb{R}$. Prove that $x-a\sin(x)=b$ has one real solution, where $0\lt a \lt 1 $.
I need some sort of starting hint as to how to prove this.
I can define $g(x)= x-a\sin(x)-b$ but more than that I'm having difficulties proving. What theo... | Consider the function:
$$
f(x)=x-a\sin(x)-b
$$
This has derivative:
$$
f'(x)=1-a\cos(x)>0
$$
since $0<a<1$, so $f(x)$ is increasing on $\mathbb{R}$. Hence as $f(x)$ is continuous increasing and is negative for sufficiently negative $x$ and positive for sufficiently positive $x$ it has a zero on $\mathbb{R}$, and it is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1803599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Solution of $4 \cos x(\cos 2x+\cos 3x)+1=0$ Find the solution of the equation:
$$4 \cos x(\cos 2x+\cos 3x)+1=0$$
Applying trigonometric identity leads to
$$\cos (x) \cos \bigg(\frac{x}{2} \bigg) \cos \bigg(\frac{5x}{2} \bigg)=-\frac{1}{8}$$
But I can't understand what to do from here. Could some suggest how to proceed... | Thinking about the answer, we might notice that if $\theta=\frac{2\pi k}9$, then $\cos9\theta=1$. We can write this as
$$\begin{align}\cos9\theta-1&=4(4\cos^3\theta-3\cos\theta)^3-3(4\cos^3\theta-3\cos\theta)-1\\
&=(16\cos^4\theta+8\cos^3\theta-12\cos^2\theta-4\cos\theta+1)^2(\cos\theta-1)=0\end{align}$$
From this we c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1803682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Show that the sequence of norms of inverses of a convergent sequence of matrices diverges to infinity. This is a question I found while working on the book "Analysis in Euclidean Spaces" by Ken Hoffman.
Suppose $(A_n)$ is a sequence of invertible matrices from $\mathbb{R}^{k \times k}$ that converges to the matrix $A... | Since all norms are equivalent here (you are in finite dimension), you are free to pick the one that is the most convenient. In particular, let $\lVert\cdot \rVert\colon \mathbb{R}^{k\times k}\to [0,\infty)$ be a the norm defined by $\lVert A\rVert = \max_{1\leq i,j\leq k} \lvert A_{i,j}\rvert$.
Since for all $n\geq 1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1803802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Let $A = \{ \frac{1}{n} : n \in \Bbb{N} \}$, show that $\overline{A} = A \cup \{0\}$
Let $A = \{ \frac{1}{n} : n \in \Bbb{N} \}$, show that $\overline{A} = A \cup \{0\}.$
We have $A \subseteq \overline{A}$ by the definition of closures. To show that $\{0\} \subset \overline{A}$ we need to show that for every open set... | Here is another way to do it which I find easier:
Since $\overline{A}$ is defined as the set $A$ and all its limit points, by definition $\overline{A}=A\cup L$, where $L$ is the set of limit points of $A$.
Now you only need to show that $L=\{0\}$, i.e. that $A$ only has only $0$ as its limit point. The result then fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1803917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
$\sum (-1)^{n+1}\log\left(1+\frac{1}{n}\right)$ convergent but not absolutely convergent I need to prove that:
$$\sum_{n=1}^{\infty} (-1)^{n+1}\log\left(1+\frac{1}{n}\right)$$
is convergent, but not absolutely convergent.
I tried the ratio test:
$$\frac{a_{n+1}}{a_n} = -\frac{\log\left(1+\frac{1}{n+1}\right)}{\log\left... | First i would show the absolute converge by the test for alternating series. First show that it is a alternating series (pretty obvious since your log only gives positive values). Then since $$ log(1+ \frac{1}{n} ) \rightarrow 0 \ , n \rightarrow \infty $$ it is convergent. To show the absolute i would remove $ (-1)^{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1803975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
If (X,T) is perfect and A is a dense subset of X, then A has no isolated points. If $(X,T)$ is perfect and $A \subseteq X$ is a dense subset of X, then A has no isolated points.
Since $A$ is dense $\Rightarrow (\forall U \in T)(A \cap U \neq \emptyset)$ and
since $(X,T)$ is perfect $\Rightarrow$ $(\forall x \in X)(\{x\... | This is true if $X$ is $T_1$. Otherwise it need not be true: $X = \{0,1,2,3\}$, with topology $\left\{\emptyset, X, \{0,1\},\{2,3\}\right\}$ is perfect but $A = \{0,2\}$ is dense and consists of two isolated points. For a $T_0$ example, consider $X = [0,\infty)$ in the topology generated by all sets $[0,a) ,a > 0$. Her... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1804053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Weight Modification for Computationally-Efficient Nonlinear Least Squares Optimization There was a time where I could figure this out for myself, but my math skills are rustier than I thought, so I have to humbly beg for help. Thank you in advance.
I am solving a weighted nonlinear least-squares problem of the usual fo... | It's not going to work, I think:
$$
\mathbf{\theta}^* = \arg \min_\mathbf{\theta} \sum_i \left[ \frac{y_i^2-\hat{y_i}^2\left(\mathbf{\theta}\right)}{w_i^\prime} \right]^2
= \arg \min_\mathbf{\theta} \sum_i \left[ \frac{y_i+\hat{y_i}\left(\mathbf{\theta}\right)}{w_i^\prime}\right]^2\left[
\frac{y_i-\hat{y_i}\left(\mathb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1804126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Probabilities ant cube I have attached a picture of the cube in the question.
An ant moves along the edges of the cube always starting at $A$ and never repeating an edge. This defines a trail of edges. For example, $ABFE$ and $ABCDAE$ are trails, but $ABCB$ is not a trail. The number of edges in a trail is known as it... | Your answers for a) to c) are correct (except for your loose use of the equals sign).
For d), note that any path of length $3$ to G will contain exactly two horizontal steps and one vertical step, the vertical step can come at any of the three steps, the probabilities are fully determined by when the vertical step come... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1804231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Variable Change In A Differential Equation If I have the following differential equation:
$\dfrac{dy}{dx} = \dfrac{y}{x} - (\dfrac{y}{x})^2$
And if I make the variable change: $\dfrac{y}{x} \rightarrow z$
I know have $\dfrac{dy}{dx} = z-z^2$
What is $\dfrac{dx}{dy}$ after the variablechange?
| Suppose you have a differential equation that looks like this: $$y'=F\left ( \frac{y}{x}\right )$$
then you can make a substitution $v(x)=\frac{y}{x} \iff y=vx \implies y'=v+xv'$ to transform your ODE into an ODE in $v$ $$\implies v+xv'=F(v) \iff \frac{dv}{F(v)-v}=\frac{dx}{x}$$
This equation is separated and you can s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1804336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Unitization of Suspension Let $A$ a C*-algebra (unital or not). Its suspension is defined to be: $$ S(A) \equiv A\otimes C_0((0,1);\,\mathbb{C}) $$where $C_0$ denotes all continuous functions which vanish at infinity.
We know that if $X$ is locally compact and if $X^{+}$ is its one-point compactification, then $$ \wid... | As noted in the comments, this is false. Intuitively, on the left-hand side in 1) you add one point to both spaces and then take the cartesian product whereas on the right-hand side you first take the product and then add one point.
For example let $A=B=C_0((0,1))$. Then $\widetilde{A\otimes B}=\widetilde{C_0((0,1)\tim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1804399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Can integration relate real and complex numbers? eg, considering $\int\frac{dx}{1+x^2}$ vs $\int\frac{dx}{(1+ix)(1-ix)}$ We all know that
$$\int\frac{dx}{1+x^2}=\tan^{-1}x+C$$
Let's evaluate this a bit differently,
$$\int\frac{dx}{1+x^2}=\int\frac{dx}{(1+ix)(1-ix)}$$
$$=\frac{1}{2}\int\frac{(1+ix+1-ix)dx}{(1+ix)(1-ix)}... | Let $\,\,\mathsf{y=tan^{-1}(x)}$
$\mathsf{\implies\,x=tan(y)}$
$\mathsf{\implies\,x=\dfrac{sin(y)}{cos(y)}}$
$\mathsf{\implies\,x=\dfrac{\dfrac{e^{iy}-e^{-iy}}{2i}}{\dfrac{e^{iy}+e^{-iy}}{2}}}$
$\mathsf{\implies\,x=\dfrac{e^{iy}-e^{-iy}}{i\left(e^{iy}+e^{-iy}\right)}}$
$\mathsf{\implies\,ix=\dfrac{e^{iy}-e^{-iy}}{e^{iy... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1804517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to evaluate $\int_0^1\frac{\ln(1-2t+2t^2)}{t}dt$? The question starts with:
$$\int_0^1\frac{-2t^2+t}{-t^2+t}\ln(1-2t+2t^2)dt\text{ = ?}$$
My attempt is as follows:
$$\int_0^1\frac{-2t^2+t}{-t^2+t}\ln(1-2t+2t^2)dt$$
$$=2\int_0^1\ln(1-2t+2t^2)dt+\int_0^1\frac{-t}{-t^2+t}\ln(1-2t+2t^2)dt$$
$$=-4+\pi-\frac{1}{2}\int... | As OP has found
\begin{equation*}
I = \int_0^1\dfrac{\ln(1-2t+2t^2)}{t}\, dt = \dfrac{1}{2}\int_0^1\dfrac{\ln(1-2t+2t^2)}{t(1-t)}\, dt.
\end{equation*}
Via the transformation $s= \dfrac{t}{1-t}$ and a partial integration we get
\begin{equation*}
I = \dfrac{1}{2}\int_0^{\infty}\dfrac{\ln(1+s^2)-2\ln(1+s)}{s}\, ds = \int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1804604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 2
} |
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