Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Matlab function input problem, can only see 1 root but supposed to see 2 – Noob question Thank you for reading this question.
I'm studying numerical methods and I'm using Matlab for the practical parts.
The problem:
I'm supposed to find 2 different roots, y = 0, for positive x, x > 0, with the function below. When I ... | Sorry for a late update, been busy.
Thank you very much for the input on my problem. You are correct, the scale and step-size I was using was the culprit.
Using this revised code shows that there is indeed a small positive root very close to x = 0.
x = -0.001:1*exp(-10):0.001;
y = 98.*x-((x.^2+x+ 0.2)./(x+1)).^9 + 5.*x... | {
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"timestamp": "2023-03-29T00:00:00",
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Why is u-equivalence useful? If $X$ is a standard set, we define, for $\alpha\in\ ^*X$,
$U_{\alpha}=\{A\subset X\mid\alpha\in\ ^*A\}$. We can see that $U_\alpha$ is an ultrafilter on $X$.
We thus define an equivalence relation on $^*X$ which is $\alpha\sim\beta$ iff $U_{\alpha}=U_{\beta}$.
Is it possible, using big eno... | The paper by Di Nasso you cite was published in 2002. In a latter paper by Benci, Forti and Di Nasso:
Benci, Vieri; Forti, Marco; Di Nasso, Mauro.
The eightfold path to nonstandard analysis. Nonstandard methods and applications in mathematics, 3–44,
Lect. Notes Log., 25, Assoc. Symbol. Logic, La Jolla, CA, 2006... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Getting the RHS into summation form Some time ago, I wrote down this identity$$\frac 4\pi=1+\left(\frac 12\right)^2\frac 1{1!\times2}+\left(\frac 12\times\frac 32\right)^2\frac 1{2!\times2\times3}+\ldots$$And being the idiot I was, I didn't write down the RHS into a compact sum.
Question: How do you write the RHS with... | If the general term has the form
$$ a_k=\left(\frac{(2k-1)!!}{2^k}\right)^2\frac{1}{k!(k+1)!} = \left(\frac{(2k)!}{4^k k!}\right)^2\frac{1}{k!(k+1)!}$$
then
$$ \sum_{k\geq 0}a_k = \sum_{k=0}\frac{1}{4^k}\binom{2k}{k}\frac{1}{4^k(k+1)}\binom{2k}{k}=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{2 d\theta}{\sqrt{1-e^{i\theta}}\left(... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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$2^n=7x^2+y^2$ solutions My problem is related to the equation from above. It actually is a very particular one. I noticed that for every positive integer $n$ there's ONE SINGLE solution $(x_1,y_1)$ so that $x_1$ and $y_1$ are ODD positive integers ( I didn't prove it, I tested it with a program through roughly 30 test... | 2^n ≡ y^2 mod 7 ⇒ y^2 < 7
It can be seen that y^2 can only have values 1, 2 and 4. Only 1 and 4 are acceptable if y is integer, that is y= (+ or -) 1 or y =(+ or -) 2. With these values of y some powers of n give integers for x, for example y=2, x =6 for n=8 or x=y=2 for n=5.
This is the reason for existing only o... | {
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If $f(0) = 0$ and $|f'(x)|\leq |f(x)|$ for all $x\in\mathbb{R}$ then $f\equiv 0$ Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous and differentiable function in all $\mathbb{R}$. If $f(0)=0$ and $|f'(x)|\leq |f(x)|$ for all $x\in\mathbb{R}$, then $f\equiv 0$.
I've been trying to prove this using the Mean Value ... | Intuitively, the solution to $|f'| \leq |f|$ with $f(0) = c$ can not grow out of the region bounded by the solutions to $f' = +f$ and $f' = -f$ with the same initial condition $f(0) = c$. The boundary solutions are $f_\pm(x) = c e^{\pm x} = 0$ with $c = 0$ we have $f_\pm(x) \equiv 0$.
It's so obvious intuitively, I thi... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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if $X$ is $T_1$ and limit point compact then $X$ is countably compact A space $X$ is said to be countably compact if every countable open covering of $X$ contains a finite subcollection that covers $X$. I want to show that if $X$ is a $T_1$ space and limit point compact then $X$ is countably compact.
| Suppose that the space is limit compact and is not countably compact.
The space is not countably compact open sets $U_1,U_2,\dots$ exist with $X=\bigcup_{i=1}^{\infty}U_i$ and $X\neq\bigcup_{i=1}^{n}U_i$ for every $n\in\mathbb N$.
So the sets $F_i:=U_i^c$ are closed with $\bigcap_{i=1}^{\infty} F_i=\varnothing$ and $\b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Poisson equation using complex analytic method I have the following question in a complex analysis text:
Find a particular solution to the following Poisson equation:
$$\nabla^2u(r, \theta) = r^2 \cos \theta.$$
The solution method outlined in the text uses Wirtinger derivatives to simplify the equation, then integra... | If $u$ is a solution to this one and $v$ is a solution to $\nabla^2 v(r,\theta) = r^2 \sin \theta)$, then $w = u + i v$ satisfies
$$ 4 \dfrac{\partial^2 w}{\partial z \partial \overline{z}} = \nabla^2 w = r^2 \exp(i\theta) = r z= z^{3/2} \overline{z}^{1/2}$$
Integrate with respect to $z$ and $\overline{z}$, and you fin... | {
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Mapping of zero vector under a linear transformation Let $T$ be a linear transformation from a vector space $V$ to $W$ that are defined over a field $\mathbb F$.
Now, $T(\mathbf0)=\mathbf0$.
Which implies the zero vector in $V$ can't move or be transformed to a new vector in $W$. Why is it so?
Does it also have a ge... | Since you're asking for a geometrical interpretation:
First of all, if you have a linear transformation $T$ and use it on all vectors in a vector space $T$, then the mapped vectors will form a vector space again. What's important is that the zero vector is a part of every vector space.
So intuitively, when you map all... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Number of transitive relations We have a set $A$ with cardinality $n$. How to find the number of transitive relations on $A$?
Also how do we get the following results?
Number of reflexive relations on $A=2^{n^2-n}$
Number of symmetric relations on $A=2^{\frac{n(n+1)}{2}}$
| The problem of finding the number of transitive relations on a set of n elements is non-trivial. The number of relations defined on the set itself grows exponentially ($2^{n^2}$)
For finding the other two, lets consider a matrix form of representing relations (assume rows & columns are ordered by the elements - where a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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let $f$ be a differentiable function. Compute $\frac{d}{dx}g(2)$, where $g(x) = \frac{f(2x)}{x}$. let $f$ be a differentiable function and $$\lim_{x\to 4}\dfrac{f(x)+7}{x-4}=\dfrac{-3}{2}.$$
Define $g(x)=\dfrac{f(2x)}{x}$. I want to know the derivative
$$\dfrac{d}{dx}g(2)=?$$
I know that :
$$\dfrac{d}{dx}g(2)=\dfrac{... | HINT
\begin{align*}
\frac{dg}{dx}(2) & = \lim_{x\rightarrow 2} = \frac{g(x) - g(2)}{x - 2} = \lim_{x\rightarrow 2}\frac{\displaystyle\frac{f(2x)}{x} - \frac{f(4)}{2}}{x - 2} = \lim_{x\rightarrow 2}\frac{2f(2x)-xf(4)}{2x(x-2)}\\
& = \lim_{u\rightarrow 4}\frac{2f(u) - u\displaystyle\frac{f(4)}{2}}{u\left(\displaystyle\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2351494",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "4",
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Approximation of Fermi-Dirac integral $\int \text{d}x \frac{f(x,\beta)}{1+e^{\beta x}}$ In physics is quite common to find integrals of the type
\begin{align}
I(\beta) = \int_{-\infty}^{\infty}\text{d}x \frac{f(x)}{1+e^{\beta x}} \tag{1}
\end{align}
where $f(x)$ is some quantity we want to average over the Fermi-Dirac ... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 2
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Prove that number of non-isomorphic ordered tree with 'n' vertices is nth catalan number. according to wikipedia $C$n is the number of non-isomorphic ordered trees with n vertices. But I can't seem to be able to prove this result. How do we do that?
where $n$th catalan number is:
$$
C_n = \frac 1{n+1} \binom{2n}{n}
$$
| We have from basic principles for the species of ordered trees the
species equation
$$\mathcal{T} = \mathcal{Z} +
\mathcal{Z} \mathfrak{S}_{\ge 1}(\mathcal{T}).$$
This yields the functional equation for the generating function $T(z)$
$$T(z) = z + z \frac{T(z)}{1-T(z)}$$
which is
$$T(z) (1-T(z)) = z (1-T(z)) + z T(z... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to calculate $\lim_{x\to\infty}\frac{x}{x-\sin x}$?
I tried to solve
$$
\lim_{x\to\infty}\frac{x}{x-\sin x}.
$$
After dividing by $x$ I got that it equals to:
$$
\lim_{x\to\infty}\frac{1}{1-\frac{\sin x}{x}}.
$$ Now, using L'hopital (0/0) I get that
$$
\lim_{x\to\infty}\frac{\sin x}{x} = \lim_{x\to\infty}\c... | With the sandwich theorem
$\dfrac{x}{x+1}\leq \dfrac{x}{x-\sin x}\leq \dfrac{x}{x-1} $
$\lim \limits_{x \to +\infty}\dfrac{x}{x+1}=1\quad \text{and}\quad \lim \limits_{x \to +\infty}\dfrac{x}{x-1}=1\implies \lim \limits_{x \to +\infty}\dfrac{x}{x-\sin x}=1 $
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2351866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 3
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One-Way Inverse My Algebra $2$ teacher stressed the fact that when you find the inverse $g$ of a function $f$, you must not only check that
$$f \circ g=\operatorname{id}$$
but you must also check that
$$g \circ f=\operatorname{id}$$
For example, if
$$f(x)=x^2$$
then
$$g(x)=\sqrt{x}$$
is not its inverse, because
$$f(g(... | Let $g$ be $\arctan$ and let $f$ be $\tan$ when defined and $17$ for the rest of the inputs (for $\dfrac{\pi}{2}+k\pi$ for integers $k$). Then $f\circ g=\mathrm{id}_{\mathbb R}$ but $g\circ f(x)=x$ only for $x\in\left(-\dfrac{\pi}{2},\dfrac{\pi}2\right)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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"answer_id": 1
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Calculate Derivative of a map
Consider the maps from $R^2 \to R^2$ such that $F(u, v) = (e^{u + v}, e^{u - v})$ and $G(x, y) = (xy, x^2 - y^2)$. Calculate $D(F \circ G)(1, 1)$ by directly composing.
I got
$F \circ G = (e^{x^2 +xy - y^2}, e^{y^2 + xy - x^2})$
But how do I get the derivative matrix?
| Remember that the derivative of a vector field is its Jacobian. If $F:\mathbb{R}^2\to\mathbb{R}^2$ is differentiable function with $F(x,y)=(f_1(x,y),f_2(x,y))$ where both $f_i$ are differentiable, then we have: $$D_{(x,y)}F = \left(
\begin{array}{cc}
\frac{\partial}{\partial x}f_1(x,y) & \frac{\partial}{\partial y}f_1... | {
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"timestamp": "2023-03-29T00:00:00",
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The derivative of absolute value of complex function $f(x,z)$ where $x \in \mathbb{R}$ and $z \in \mathbb{C}$ Let $f: \mathbb{R} \to \mathbb{R}$ be a real function and let $z \in \mathbb{C}$ be a complex number such that
$$
f(x)=|x \cdot z|
$$
Let's calculate the derivative of $f$
if we applicate the derivation rules:
... | I'm going to deal with the general problem: Given a complex valued function
$$g:\quad{\mathbb R}\to {\mathbb C},\qquad x\mapsto g(x)=u(x)+i v(x)\ ,$$
one has $|g(x)|=\sqrt{u^2(x)+v^2(x)}$ and $g'=u'+i v'$. Therefore
$${d\over dx}\bigl|g(x)\bigr|={u(x)u'(x)+v(x)v'(x)\over\sqrt{u^2(x)+v^2(x)}}={{\rm Re}\bigl(g(x) \over... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2352341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The equation $a^x=x$ for $a>1$. The following problem arises when I was sketching the inverse of $f(x)=a^x$ graphically for $a>1$:
1) Is there an $a>1$ such that the equation $a^x=x$ has a unique solution $x\in\mathbb{R}$?
2) If so, then how do we find such $a$ explicitly if possible?
The answer to the first question ... | I guess I will add my two cents
$$a^x=x$$
$$a=x^{{1}/{x}}$$
now
$$e^{x}\ge 1+x$$
$$e^{(x-e)/{e}}\ge x/e$$
$$e^{x/e}\ge x$$
$$e^{1/e} \ge x^{1/x}$$
Therefore maximum of $a=e^{1/e}$ for solution. Thus there exists solution for $a\in\mathbb{(1,e^{1/e})}$
| {
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"timestamp": "2023-03-29T00:00:00",
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Unsure of how to interpret the set $\mathbb{R}^X$ where $X$ is a real vector space? I recently came across the notation $\mathbb{R}^X$ and I'm not exactly sure what it means or how to 'visualize it'. The text it comes from is the following:
Let $X$ be a real vector space and let $\mathcal{F} \subset
\mathbb{R}^X$ be ... | The notation $B^A$ for sets $A$ and $B$ represent the set of all functions from $A$ to $B$.
It is a generalization of the meaning of the notation $X^n:=\overbrace{X\times X\times\cdots\times X}^{n\text{ times}}$ where we can visualize it as the set of functions from $\{1,2,\ldots,n\}$ to $X$.
In your case the set of fu... | {
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what is it called if numbers $a$ and $b$ are such that their sum is equal to unity Question:
Is there a way to definitionally describe rational numbers $a$ and $b$ when $a+b=1$?
Answer:
My guess is that $a$ and $b$ may be defined as `unitary additive complements,' but this is just a guess.
| In the same way that the error function, erf($x$), and the complementary error function, erfc($x$), sum to one (i.e., erf($x$) + erfc($x$) = 1) [1], so too then are $a$ and $b$ complementary under addition if $a + b = 1$.
[1]
https://en.wikipedia.org/wiki/Error_function#Complementary_error_function
| {
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How do you develop a recurrence relation for the function $f(n) = 5n^2 +3$, where $n \in \mathbb{Z}^+$? On an exam of mine, I was asked to find a recurrence relation for the function $f(n) = 5n^2 +3$, where $n \in \mathbb{Z}^+$. I needed to provide a base case and the actual relation itself. I know the base case is fo... | The motivation behind rewriting $f(n)$ this way, is to acquire an equation of the form $$f(n)=g(f(n-1)),$$
i.e. we have to rewrite $f(n)$, so that we get some function of $f(n-1)$.
Here, this function is
$$g(x)=x+10n-5.$$
Now you can directly deduct the recurrence relation, by plugging $g$ into $x_{n+1}=g(x_n)$.
Here, ... | {
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If $T$ is a positive operator then $I+T$ is invertible Let $T$ be a positive operator on a Hilbert space $H$, prove that $I+T:H\to H$ is invertible and $(I+T)^{-1} \in B(H)$.
Now, If I prove $I+T$ is invertible, the bounded inverse theorem implies the second part. Now while proving that $I+T$ is invertible, I have prov... | Note that
$$
I+T-\lambda I=T-(\lambda -1)I.
$$
So $\lambda\in\sigma(I+T)\iff \lambda-1\in\sigma(T)$. In other words,
$$
\sigma(I+T)=\{\lambda+1:\ \lambda\in\sigma(T)\}.
$$
As $T$ is positive, $\sigma(T)\subset[0,\infty)$. Thus $\sigma(I+T)\subset [1,\infty)$.
It follows that $0\not\in\sigma(I+T)$, so $I+T$ is invert... | {
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"timestamp": "2023-03-29T00:00:00",
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Why does the empty set matter? I'm trying to understand why the empty set is a useful tool for mathematicians. Are there any nontrivial theorems that could only be achieved by the existence of the empty set, or is the recognition of the empty set just a conventional standard that mathematicians have adopted?
| The same reason it's a useful tool in everyday life.
There are lots of questions that ask for sets as answers, like...
*
*Which crew members are still aboard the airplane?
*What dishes are in the sink and need to be put away?
*What problems does this piece of software have that must be fixed before we can release ... | {
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"url": "https://math.stackexchange.com/questions/2353071",
"timestamp": "2023-03-29T00:00:00",
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Proof the following! $$a_0(x)\frac{d^2y}{dx^2}+a_1(x)\frac{dy}{dx}+a_2(x)y=0$$
A) Let $f_1$ & $f_2$ are two solutions to the above differential equation. Show that $f_1$ & $f_2$ are linearly independent on $a \leq x \leq b$ and $A_1$,$A_2$,$B_1$&$B_2$ are constants such that $A_1B_2-A_2B_1\neq 0$ then the solution $A_... | This is pure linear algebra: If $f_1$ and $f_2$ are linearly independent vectors in some vector space and $ad-bc\ne0$ then $g_1:=a f_1+b f_2$ and $g_2:=c f_1+d f_2$ are again linearly independent.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Expansion coefficients Suppose that we are given the function $f(x)$ in the following product form:
$$f(x) = \prod_{k = -K}^K (1-a^k x)\,,$$
Where $a$ is some real number.
I would like to find the expansion coefficients $c_n$, such that:
$$f(x) = \sum_{n = 0}^{2K+1} c_n x^n\,.$$
A closed form solution for $c_n$, or a... | Let the function $f_n(x)$ be given by
$$
f_n(x) = \prod_{k=-n}^{n} \left( 1 - a^k x \right)
$$
Since it is clear that it is a polynomial of degree $2 n +1$, it can be expressed as:
$$
f_n(x) = \sum_{k=0}^{2n+1} c_{n,k} x^k
$$
in some yet unknown coefficients $c_{n,k}$. For these coefficients it is easy to see that $c_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2353272",
"timestamp": "2023-03-29T00:00:00",
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Convergence of $\sum_{n=1}^\infty \cos(\pi n^2)(\sqrt{n+1}-\sqrt{n})$ I want to test the convergence of $$\sum_{n=1}^\infty \cos(\pi n^2)(\sqrt{n+1}-\sqrt{n})$$
First of all, $\cos(\pi n^2)=-1$ if $n$ is odd, and $\cos(\pi n^2)=1$ if $n$ is even. That is, $\cos(\pi n^2)=(-1)^n$. So the summation reduces to $$\sum_{n=1}... | Apply alternating series test. You have $a_n:= \sqrt{n+1} - \sqrt n$ decreasing, nonnegative and converging to $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2353362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Limit of quotient of inverse cdfs I am trying to obtain $$\lim_{x\to0}\frac{\Phi^{-1}(1-x)}{\Phi^{-1}(1-x/n)}$$ where $\Phi^{-1}$ is the inverse cdf of the standard normal distribution and $n>0$. As there is an indeterminate form ($\infty/\infty$), I am applying l'Hôpital's rule, but the resulting expression (I mean, o... | Here is another much simpler way to solve.
Note that,
$$\frac{\partial \Phi^{-1}(x)}{\partial x} = \frac{1}{\phi(\Phi^{-1}(x))} \ \ \text{
and } \ \ \frac{\partial \Phi^{-1}(x/n)}{\partial x} = \frac{1}{n\phi(\Phi^{-1}(x/n))}$$
Also,
$$\frac{\partial \phi(\Phi^{-1}(x))}{\partial x} = \frac{-\Phi^{-1}(x)\phi(\Phi^{-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2353441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Galois Theory books (in association with Abstract Algebra books) I know that there have been written similar posts, and I used them as a source for my question.
I ' m looking for a book for Galois Theory (Construction of fields. Algebraic extensions - Classical Greek problems: constructions with ruler and compass. Gal... | I recommend Galois' Theory of Algebraic Equations, by Jean-Pierre Tignol (2nd edition, World Scientific, 2016).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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If $a + \frac{1}{a} = -1$, then the value of $(1-a+a^2)(1+a-a^2)$ is?
If $a + \frac{1}{a} = -1$ then the value of $(1-a+a^2)(1+a-a^2)$ is?
Ans. 4
What I have tried:
\begin{align}
a + \frac{1}{a} &= -1 \\
\implies a^2 + 1 &= -a \tag 1 \\
\end{align}
which means
\begin{align}
(1-a+a^2)(1+a-a^2) &=(-2a)(-2a^2) \\
&=4a^... | Without solving the quadratic:
$$ a^3 = - a^2 - a = a + 1 - a = 1 $$
which was found by using the equation $a^2 + a + 1 = 0$ twice. This means that $a$ is a non-real cube root of unity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2353620",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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How to conclude that the minimal polynomial is the characteristic? I am given the following matrix
$$A=\begin{bmatrix}
0 & 0 & 4 & 1\\
0& 0 & 1 & 4\\
4 & 1 & 0 &0\\
1 & 4 & 0 & 0
\end{bmatrix}$$
And I have to find the minimal polynomial of the matrix. The characteristic polynomial is $$K(\lambda)=-(\lambda -5)(\lambda ... | In this case you can do easily without the characteristic polynomial. It is easy to see that even powers of $A$ will have their nonzero entries in the $2\times2$ blocks at the top left and bottom right, and the odd powers of $A$ have them (like $A$ itself) in the bottom left and top right $2\times2$ blocks. You are loo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2353799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Proving determinant is zero
Let $n\ge 2$ and $A=[\overset{\to}{a_1},\overset{\to}{a_2},\ldots,\overset{\to}{a_n}]$ an $n\times n$ matrix such that there exists $i\neq j$ such that $$\overset{\to}{a_j}=k\overset{\to}{a_i}$$
where $k\neq 0$. Show that $\det(A)=0$.
How would I approach this proof or solve it?
| If $a_i=ka_j$ for some $k\neq 0$ then $a_i$ and $a_j$ are linearly dependent. This imply that the rank of the matrix is less than $n$, thus if $A$ represent a linear operator it cannot be injective, hence it is not invertible and $\det(A)=0$.
P.S.: Im not sure that this answer will be useful for you, I take a little "r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2353921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sum: $\sum\limits_{n=0}^\infty \frac{n!}{(2n)!}$ I'm struggling with the following sum:
$$\sum_{n=0}^\infty \frac{n!}{(2n)!}$$
I know that the final result will use the error function, but will not use any other non-elementary functions. I'm fairly sure that it doesn't telescope, and I'm not even sure how to get $\oper... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
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"question_score": "8",
"answer_count": 4,
"answer_id": 3
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Existence of random variable given first $k$ moments A sequence of real numbers $\{m_k\}$ is the list of moments of some real random variable if and only if the infinite Hankel matrix $$\left(\begin{matrix}
m_0 & m_1 & m_2 & \cdots \\
m_1 & m_2 & m_3 & \cdots \\
m_2 & m_3 & m_4 & \cdots \\
\vdots & \vdots & \vdot... | Yes, this works. There should really be an easy direct argument, but all I can think of right now is the following: The finite moment problem $\int x^n\, d\mu(x)=m_n$, $n=0,1,\ldots , k$, can be solved in the same way as the full problem $n\ge 0$. Namely, run Gram-Schmidt on $1,x,\ldots , x^N$ (with $2N=k$); the orthog... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2354090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Intuitive reason why $\sqrt[n]n\to 1$ as $n\to\infty$? We are aware of the limit
$$
\lim_{n\to\infty}\sqrt[n]n = 1;
$$
is there any geometric or otherwise intuitive reason to see why this limit holds?
Edit: I am adding some context, since this question was previously put on-hold, and I think one of the main reasons wa... | Here the issue is that $n \rightarrow \infty$, but for any fixed $x > 0$ we have $x^{1/n} \rightarrow 1$. So to consider $n^{1/n}$ you have to ask which "wins": the $n$ at the base or the $1/n$ in the exponent. To think about this, it might help to compare to, say, $(2^n)^{1/n}$. This tends to (and is equal to) $2$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 15,
"answer_id": 1
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Why don't previous events affect the probability of (say) a coin showing tails? Why doesn't a previous event affect the probability of (say) a coin showing tails?
Let's say I have a fair and unbiased coin with two sides, heads and tails.
For the first time I toss it up the probabilities of both events are equal to $\fr... | The assumption for a coin is that is has no memory. That means that the chance of heads is the same on every toss. For a fair coin, that chance is $\frac 12$ regardless of the history. If you toss $100$ times and get heads every time (very unlikely, but it could happen) the most probable event after a million tosse... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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To find a limit involving integral How to find the following limit : $\lim _{x \to \infty} \dfrac 1 x \int_0^x \dfrac {dt}{1+x^2 \cos^2 t}$ ? I am not even sure whether the limit exists or not . I tried applying L'Hospital , but then in the numerator we have differentiation under integration, and the derivative comes ... | $$\forall x > 0, \exists n \in \mathbb{N}\ s.t. \ x \in ]2n\pi, 2(n+1)\pi]$$
$$ 0 \leq \frac{1}{x}\int_0^x\frac{dt}{1+x^2\cos^2(t)} \leq \frac{1}{2n\pi}\int_0^{2(n+1)\pi}\frac{dt}{1+(2n\pi)^2\cos^2(t)}$$
$$ \leq \frac{1}{2n\pi}\sum_{k=0}^n\int_0^{2(k+1)\pi}\frac{dt}{1+(2n\pi)^2\cos^2(t)}$$
$$ \leq \frac{1}{2n\pi}\sum_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2354619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Boolean representation of logical statements of more than two extreme conditions I had a question to determine the highest and and lowest and the middle paid of three employees. I tried to solve the problem using logical values T or F, then getting the connection between the truth tables, but I got stuck since the firs... |
Steve would like to determine the relative salaries of three coworkers using two facts. First, he knows that if Fred is not the highest paid of the three, then Janice is.
$$(F{\lt}J\vee F{\lt}M)\to (F{<}M{<}J\vee M{<}F{<}J)\tag 1$$
Clearly that means that: $$F{<}J\to (F{<}M{<}J\vee M{<}F{<}J) \tag{1.1}$$ ... and, le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2354755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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On the evaluation of a limit of a definite integral
Why is it that
$$
\lim_{\epsilon\to 0} \, \frac{2}{\pi} \int_0^\epsilon \frac{f(x)}{\sqrt{\epsilon^2-x^2}} \, \mathrm{d}x = f(0) \, ?
$$
In the particular case when $f(x) = c$ is a constant, the identity follows forthwith. Is it possible to show that this is true ... | For any $\varepsilon>0$
$$ \frac{2}{\pi}\int_{0}^{\varepsilon}\frac{f(x)\,dx}{\sqrt{\varepsilon^2-x^2}}\stackrel{x\mapsto \varepsilon z}{=}\int_{0}^{1}f(\varepsilon z)\frac{2}{\pi\sqrt{1-z^2}}\,dz \tag{1}$$
and we have $\int_{0}^{1}\frac{2\,dz}{\pi\sqrt{1-z^2}}=1$. In particular, if $\lim_{u\to 0^+} f(u)$ exists then
$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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using Laplace transform for advection equation I use Laplace transform to solve an advection-diffusion equation with given boundary and initial conditions. I am stuck on the special case that only advection is considered. The advection equation is,
$$
\frac {\partial{T}}{\partial{t}}+u\frac{\partial{T}}{\partial{x}}=0... | $$
s\overline{T}-T(x,t=0)+u\frac{\partial{\overline{T}}}{\partial{x}}=0 \qquad \text{is OK.}
$$
with condition $\overline{T(0,t)}=\frac{T_0}{s}$
The mistake is in the solving of this equation.
HINT : To make it more clear, let
$\begin{cases}
f(x)=T(x,t=0) \quad\text{a given function,}\\
\overline{T}=y(x) \quad\text{f... | {
"language": "en",
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Is it possible to determine the value of a matrix element given the dominant eigenvalue and all other elements? I'm working on a population dynamics model and have a matrix of vital rates representing the survival and fecundity of different life stages of the animal which are set out in a 6 x 6 matrix:
\begin{pmatrix}
... | I don't know if negative values of $x$ make sense for your model (perhaps not), but I got $x\simeq-0.22$. For that $x$, there are only two real eigenvalues, one of which is (very nearly) $1$; the other one is about $-0.85$.
| {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
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Linear operator norm equality Let $A:\mathbb R^n \to \mathbb R^m$ be a linear operator.
How can I show this equality in the operator norm?
$$\sup_{x\in \mathbb R^n\setminus\{0\}} \frac{||Ax||}{||x||} = \max_{x\in \mathbb R^n , ||x||=1}||Ax||$$
I've tried to use the fact that for every $x\in \mathbb R^n$ we have $\frac{... | If $x\neq0$, then $\left\|\frac x{\|x\|}\right\|=1$, and therefore$$\frac{\|Ax\|}{\|x\|}=\frac1{\|x\|}\|Ax\|=\left\|A\left(\frac x{\|x\|}\right)\right\|\leqslant\sup_{\|x\|=1}\|Ax\|.$$
On the other hand, if $\|x\|=1$, then$$\|Ax\|=\frac{\|Ax\|}{\|x\|}\leqslant\sup_{x\neq0}\frac{\|Ax\|}{\|x\|}.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Let $T^{m-1}v\neq 0$ but $T^mv=0$. Show that $v,Tv,\ldots,T^{m-1}v$ are linearly independent
Let $T\in\mathcal L(V)$ and $T^{m-1}v\neq 0$ but $T^mv=0$ for some positive integer $m$ and some $v\in V$. Show that $v,Tv,\ldots,T^{m-1}v$ are linearly independent.
I had written a proof but Im not sure if it is correct. And... | More simply put: if you had a linear combination
$c_0 v + c_1 T v + \ldots c_{m-1} T^{m-1} v = 0$
with $c_j$ not all zero, let $c_i$ be the first nonzero coefficient.
Then $$T^i v = - \sum_{j=i+1}^{m-1} (c_j/c_i) T^j v$$ Applying $T^{m-i-1}$ to both sides,
$$T^{m-1} v = - \sum_{j=i+1}^{m-1} (c_j/c_i) T^{m+j-i-1} v = ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is a set of unit vectors uniquely determined by its distances? Let $X = \{x_1, \dots, x_n\}$ be a finite set of points in $\mathbb{R}^d$. We can associate to $X$ its multiset of distances
$$ D_X := \{ \lVert x_i - x_j \rVert : 1 \le i,j \le n \} \qquad \text{(read as a multiset)} $$
where $\lVert \cdot \rVert$ denotes... | Even for $d = 2$ there are such sets of points. Thus, for $ n = 4: \lbrace 0,1,2,5 \rbrace $ and $\lbrace 0,1,5,6 \rbrace \mod 8$.
You can also specify homometric systems with equal sets of distances.
For $n = 6$, you can find at least 5, and for $n = 12$ - at least 18 homometric sets of points on the circle with ... | {
"language": "en",
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Proof problem involving set theory and relations Problem: If $R_1$ is defined on $\mathbb{R}$ by the relation $R_1=\{(a,b):1+ab>0\ , a, b \in \mathbb{R}\}$, then prove that $(a,b) \in R_1$ and $(b,c) \in R_1 \implies (a,c) \in R_1$ is not true for all $a,b,c \in \mathbb{R}$.
My attempt:
$$(a,b) \in R_1 \implies 1+ab>0 ... | That (1) and (2) do not imply (3) is what you were trying to prove - you cannot assume it midway through your proof. Put another way - just because you personally don't see a way to deduce (3) from (1) and (2) doesn't mean there isn't one.
In general, to show that $A$ does not imply $B$, you must give an example of $A$... | {
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Area under a never-continuous function I was thinking about the following function, infamous for being nowhere continuous:
$$f(x\in\mathbb Q)=0$$
$$f(x\notin \mathbb Q)=1$$
How would I calculate the area under this "curve" from $x=0\to1$? Can it be done?
If so, I suspect the result would be very interesting... it would... | The area under this curve is called its Lebesgue integral and is equal to $1$, corresponding to the intuition that most numbers in the interval $[0,1]$ are actually irrationnal.
The Lebesgue integral is an extension of the Riemann integral to a wider class of functions (called "measurable functions"). In your example, ... | {
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"timestamp": "2023-03-29T00:00:00",
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Fourier transformation on $L^2(G)$. We define Fourier transform on $L^1(\mathbb{R})$ and extend the definition to $L^2(\mathbb{R})$.
But for a compact group $G$, we define Fourier transform for $L^2(G)$ and there is no such thing of defining for $L^1(G)$.
My question is why such unusual thing happens for groups? Is th... | If $G$ is a compact group and $\mu$ is the left-invariant Haar measure on $G$, then $\mu(G)$ is finite, so $L^2(G)\subset L^1(G)$ by Holder's inequality and we can directly define the Fourier transform on $L^2(G)$ since all the integrals in question are defined.
However $L^2(\mathbb{R})$ is not a subset of $L^1(\mathbb... | {
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Where did I go wrong? Analyze the logical form of $\{n^2+n+1 | n \in \mathbb{N}\} \subseteq \{2n+1 | n \in \mathbb{N}\}$ This question is taken from Velleman's $\textit{How to Prove it}$. It is in the exercises section of 2.3, Question 1c:
$\{n^2+n+1 | n \in \mathbb{N}\} \subseteq \{2n+1 | n \in \mathbb{N}\}$
My work i... | There is no error. Your solution is correct too. Two things, however:
*
*Although what you wrote is correct, it is always a good idea not to use the same symbol ($n$, in your case) for two different purposes. So, I would have written$$\forall x (\exists n \in \mathbb{N}(x=n^2+n+1) \to \exists m \in \mathbb{N}(x=2m+1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proving uniform convergence on an unbounded interval
Prove that the following series
$$\sum_{n=1}^{\infty}\frac{nx}{1+n^2\log^2(n)x^2}$$
converges uniformly on $[\epsilon,\infty)$ for any $\epsilon>0.$
What I have done:
The function
$$f_n(x)=\frac{nx}{1+n^2\log^2(n)x^2}$$ is a decreasing function on $[1,\infty)... | The function $f_n$ attains its maximum when $x=\frac1{n\log n}$, if $\frac {1}{n\log n}<\varepsilon$; its value there is $\frac1{\log n}$. Of course, if $n$ is large enough, $\frac{1}{n\log n}\leqslant\varepsilon$ and so the maximum of $f_n$ will be then $f_n(\varepsilon)$. But$$f_n(\varepsilon)=\frac{n\varepsilon}{1+n... | {
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Definition of Continuity of Real Valued Functions Definition: Let $F$ be a real valued function defined on a subset $E$ of $\mathbb{R}$. We say that $F$ is continuous at a point $ x \in E$ iff for each $\epsilon > 0$, there is a $\delta > 0$, such that if $x' \in E$ and $|x'-x|<\delta$, then $|f(x') - f(x)| < \epsilon... | A foremost condition for evaluating the limit of a function $f:E\subseteq \Bbb R\to \Bbb R$ at some point say $x_0\in \Bbb R$ is that $x_0$ must be a limit point of $E$.
In the case where $E=\Bbb N$, the set has no limit points. So forget about testing continuity at some point in $\Bbb N$ we cannot even compute the li... | {
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If $ a=\frac{\sqrt{x+2} + \sqrt {x-2}}{\sqrt{x+2} -\sqrt{x-2}}$ then what is the value of $a^2-ax$ is equal to
If $$ a=\frac{\sqrt{x+2} + \sqrt {x-2}}{\sqrt{x+2} -\sqrt{x-2}}$$ then
the value of $a^2-ax$ is equal to:
a)2 b)1 c)0 d)-1
Ans. (d)
My attempt:
Rationalizing $a$ we get,
$ x+ \sqrt {x^2-4}$
$a^2=(x+\sqrt{... | $$\dfrac a1=\dfrac{\sqrt{x+2}+\sqrt{x-2}}{\sqrt{x+2}-\sqrt{x-2}}$$
calling for Componendo and Dividendo
$$\dfrac{a+1}{a-1}=\dfrac{\sqrt{x+2}}{\sqrt{x-2}}$$
Squaring we get $$\dfrac{a^2+1+2a}{a^2+1-2a}=\dfrac{x+2}{x-2}$$
Again apply componendo and dividendo, $$\dfrac{a^2+1}{2a}=\dfrac x2$$
Now simplify
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Term by term integration fourier series
$s= \sigma +it$, for $-1<\sigma<0$, we have
$$ \int_{0}^\infty \sum_{n=1}^\infty \frac{ \sin 2 n \pi x}{n x^{s+1}} \, dx = \sum_{n=1}^\infty \frac{1}{n} \int_0^{\infty}\frac{\sin 2n \pi x }{ x^{s+1}} \, dx $$
For the justification, the author writes,
As $\sum_{n=1}^\inf... | Read again Titchmarsh, he wrote something else. Given what you know, I would sketch a proof of the functional equation :
For $\Re(s) \in (0,1)$
$$\int_0^\infty \sin(nx) x^{-s-1}dx= n^{s}\int_0^\infty \sin(x) x^{-s-1}dx$$
Thus
$$\eta(1-s) \int_0^\infty \sin(x) x^{-s-1}dx = \lim_{N \to \infty} \sum_{n=1}^N (-1)^{n+1} n^... | {
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How to find the vertex form of a cubic? In a calculus textbook, i am asked the following question:
Find a cubic polynomial whose graph has horizontal tangents at (−2, 5) and (2, 3)
A vertex on a function $f(x)$ is defined as a point where $f(x)' = 0$. So the slope needs to be 0, which fits the description given here. ... | $f(x) = ax^3 + bx^2+cx +d\\
f'(x) = 3ax^2 + 2bx + c$
We have some requirements for the stationary points.
$f'(x) = 3a(x-2)(x+2)\\
f'(x) = 3ax^2 - 12a = 3ax^2 + 2bx + c$
Note, in your work above you assumed that the derivative was monic (leading coefficient equal to 1). This seems to be the cause of your troubles.
$b ... | {
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Make up a reasonable definition for the bipartite complement of a bipartite graph I am shooting from the hip here and seeing what sticks.
I tried this definition below which I am not sure works. If it doesn't, please, suggest more accurate definitions.
The reason I need this is that I want to be able to replace the de... | What you've written really doesn't make sense on a few levels. First of all, your definition of a bipartite complement of a graph is literally just another bipartite graph with the same number of edges. Additionally, in the example in the last paragraph that you give, I don't see have $\{3, 4\}$, $\{2, 5\}$ can be cons... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2356520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the number of triples (a, b, c) of positive integers such that the product $a.b.c=1000$ , and $a \leq b \leq c $?
What is the number of triples (a, b, c) of positive integers such that the product $a.b.c=1000$, and $a \leq b \leq c$?
My try:
The prime factorization of $1000$ is $2^3\cdot 5^3$
$a\cdot b \cdo... | Your computation of $N=10$ is correct and $100$ is the number of ordered triples that have product $1000$. You have failed to account for the condition that $a \le b \le c$. All of the unordered triples that have three distinct elements have shown up six times, so you have overcounted. Those that have two or three e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2356648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Holomorphic function equal to polynomial. I have this question
Let $h$ an holomorphic function in $\mathbb{C}$ and suppose that
$$|h(z+7) -3|^2\leq |z-1|^9 + 42$$
for all $z$ such that $|2z+63|>177$. Prove that $h$ is a polynomial.
How can I attack this problem? I can't see how because I wanted to use the Liouvi... | The meromorphic function $g(z) = h(z+7)/(z-1)^9$ is bounded outside some disk, so its singularity at $\infty$ is removable. Being a meromorphic function on the Riemann sphere, it is a rational function,
and so $h(z)$ is a rational function with no poles in $\mathbb C$, i.e. a polynomial.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2356754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Compactness- Euclidean metric Hi Could you help me to solve this question?
IF $||.||$ be any norm on $\mathbb{R}^m$ and let $B = \{ x \in \mathbb{R}^m : ||x||≤ 1 \}$ . Prove that $B$ is compact. Hint: It suffices to show that $B$ is closed and bounded with respect to the Euclidean metric.
| The euclideian metric induces the norm $d_2(x,0)=||x||_2=\sqrt{x_1^2+x_2^2+...+x_m^2}$.
Now in a finite dimensional space $X$ all the norms are equivalent thus are equivalent with $||.||_2$ i.e: $\exists C_1,C_2>0 $ such that $C_1||x||_2 \leqslant||x|| \leqslant C_2||x||_2, \forall x \in X$.
Now $B$ is bounded with res... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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proof that $n^4+22n^3+71n^2+218n+384$ is divisible by $24$ Let $n$ be a positive integer number.
How do we show that the irreducible polynomial $n^4+22n^3+71n^2+218n+384$ is divisible by $24$ for all $n$?
| Another way to do it is to note:
$n^4+22n^3+71n^2+218n+384 \equiv $
$n^4 - 2n^3 - n^2 + 2n \mod 8$.
If $n \equiv 0, \pm 1, \pm 2, \pm3, 4 \mod 8$ then $n^4 - 2n^3 - n^2 + 2n\equiv$
$0, 1\mp 2 - 1 \pm 1, 16 \mp 16 -4 \pm 4, 81 \mp 54 - 9 \pm 6, 4^4 \mp 2*4^3 - 16 + 8 \equiv$
$0, 0, 72 \mp 48, 0 \equiv 0 \mod 8$.
So $8$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2356982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Riesz representation theorem giving a different result? Evans p298 - Lax Milgram Theorem
Let $H$ be a Hilbert space. Denote the inner product by $(\cdot,\cdot)$ and the natural dual pairing of spaces by $\langle\cdot,\cdot\rangle$.
He gives us bilinear form $B[\cdot,\cdot]:H\times H \to \Bbb R$ and then he says that fo... | Well, $v \to B[u,v]$ is a linear functional on $H$. The RRT tells you that any bounded linear functional $l$ on $H$ can be represented as $l(v)=(u,v)$ for some $v$, and that the map $l \to u$ is in fact an isometric isomorphism. That is why you usually write $l=u^*$.
Hence, since $v \to B[u,v]$ is a bounded linear fun... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Surjections and Injections Please examine following Theorem and the accompanying proof. I understand the idea behind the proof, I am just concerned that I might not have put it in the correct words.
Is the argument correct? If so can the write up be improved?
Given that $f:A\to B$ and that $g:B\to C$. Prove that if $... | The proof is correct. On the other hand you can avoid contradiction, by proving the contrapositive.
Proposition. Given the maps $f\colon A\to B$ and $g\colon B\to C$, if $g$ is one-to-one and $g\circ f$ is onto, then $f$ is onto.
Proof. Let $b\in B$. Since $g\circ f$ is onto, there exists $a\in A$ such that
$$
g(b)=g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2357203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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If $|z|<1$, Is $|z+1|+|z-1|<2√2 $ true? If $|z|<1$, Is $|z+1|+|z-1|<2√2 $ true?
My attempt:-
I got 4 as an upper bound, when I applied triangular inequality on $|z+1|+|z-1|$. I got lower bound as 2, $2≤|z+1|+|z-1|$. I randomly pick complex numbers in the given disk. I got the result correct. How to prove or disprov... | Taking the square of the magnitude of the LHS:
$$\require{cancel}
\begin{align}
\big| |z+1|+|z-1|\big|^2 &= |z+1|^2+|z-1|^2+2 |z+1||z-1| \\
&= (z+1)(\bar z+1)+(z-1)(\bar z -1) + 2 \big|(z+1)(z-1)\big| \\
&= |z|^2+\cancel{z}+\bcancel{\bar z}+1 + |z|^2-\cancel{z}-\bcancel{\bar z} +1 + 2 |z^2-1|\\
&\le 2 |z|^2 + 2 + 2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Doubt in this question involving logarithm If $\log2= 0.301$, then how many number of digits are in $2^{64}$?
What I did:
$$\log(2)^{64}=\log2^{64}=64\log2=64\log2=19.264$$
Number of digits comes out to be $5$. But answer is $20$? I have written $\log 2$ raise to the power $64$
| Because number of digits it's$$[64\log2]+1=20,$$ which gives the answer.
In the general case if $n$ is a number of digits of natural $N$ then $n=[\log{N}]+1$
Indeed, let $N=a_0\cdot10^{n-1}+a_1\cdot10^{n-2}+...+a_{n-1}$, where $a_i\in\{0,1,...,9\}$ and $a_0\neq0$.
Hence, $n$ is a number of digits.
We see that $10^{n-... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Domain of the function $f(x)=\frac{1+\tfrac{1}{x}}{1-\tfrac{1}{x}}$ The domain of the function $f(x)=\frac{1+\tfrac{1}{x}}{1-\tfrac{1}{x}}$ is said to be $\mathbb R-\{0,1\}$, given $f(x)$ is a real valued function. I understand why that is the case, since for both $1$ and $0$ the denimonator becomes $0$ and the value i... | The domain of $f $ is
$$D=\{x\in \mathbb R \;\;: \;x\ne 0 \land \frac {1}{x}\ne 1\}$$
but $$\frac {1}{x}=1\iff x=1$$
hence
$$D=\{x\in\mathbb R \;\;:\;x\ne 0\land x\ne 1\} $$
$$=\mathbb R \backslash \{0,1\} $$
$$=(-\infty,0)\cup (0,1)\cup (1,+\infty) $$
and
$$x\in D \implies f (x)=\frac {x+1}{x-1} $$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 6
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Where is my mistake in proving a language is not regular? Trying to prove that $\big\{0^n1^m | n < m\big\}$ is not a regular language. If I take $w=0^n1^{n+x}=xyz$ with obvious $|w| > n$ then $|xy|\leq n $. Furthemore if I let $z=1^{n+x}$ then $x$ and $y$ contain $0$'s and given that $y\neq\epsilon$ then y must contain... | I hope it can help you
we proof by contradiction. So, we first must assume something, then contradict it. Here we will assume the L
is regular, then get a contradiction.
*
*assume L is regular
*By the pumping lemma there exists a pumping length $p\ge1$
*take string $w=0^p1^{p+1}\in L $ such that $|w|\ge p$
*br... | {
"language": "en",
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"source": "stackexchange",
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Show that $f(x) = |x|$ is continuous on $\mathbb R$ I want to use the sequential criterion to prove that $f(x) = |x|$ is continuous on $\mathbb R$.
For reference, here is the sequential criterion according to Introduction to Real Analysis by Bartle:
$f:A \rightarrow \mathbb R$ is continuous at the point $c\in A$ if a... | This is not appropriate. Notice that the sequential criterion says that for EVERY sequence that converges to c.
You have only looked at a single sequence, the sequence $x+1/n$.
Instead, you would have to start your proof "Let $(x_n)$ be a sequence that converges to $(c)$. Then, from this, use the definition of sequen... | {
"language": "en",
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"source": "stackexchange",
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Integral $\int_{0}^{\infty}x^{-x}dx$ I'm trying to find a closed form for this integral:$$\int_{0}^{\infty}x^{-x}dx$$
Here's the integrand graph:
Clearly it is convergent. My attempt is to obtain a closed form for the area under the curve. Is this possible?
| $\int\limits_0^\infty x^{-x}dx=\int\limits_0^1 x^{-x}dx+\int\limits_0^1 x^{-2+1/x}dx$
The second integral is unpleasant to develope into a series.
But if we can use the solution $\,z_0\,$ for $\,\int\limits_0^1 (x^{xz} - x^{-2+1/x})dx=0$
with $\,z=z_0\approx 1.45354007846425$ , then we get:
$$\displaystyle \int\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2358024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
How to calculate $\lim_{x\to 0^+} \frac{x^x- (\sin x)^x}{x^3}$ As I asked, I don't know how to deal with $x^x- (\sin x)^x$.
Please give me a hint. Thanks!
| $$\lim_{x\rightarrow0^+}\frac{\ln{x}-\ln{\sin{x}}}{x^2}=\lim_{x\rightarrow0^+}\left(\frac{\ln\left(1+\frac{x}{\sin{x}}-1\right)}{\frac{x}{\sin{x}}-1}\cdot\frac{\frac{x}{\sin{x}}-1}{x^2}\right)=$$
$$=\lim_{x\rightarrow0}\left(\frac{x-\sin{x}}{x^3}\cdot\frac{x}{\sin{x}}\right)=\lim_{x\rightarrow0}\frac{1-\cos{x}}{3x^2}=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2358173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Gradient of a matrix? I was following Stephen Boyd's convex optimisation course and came across the following slide:
Can somebody explain to me how the gradient was calculated for the quadratic and least-squares objective. Is there a general method to find the gradient of a matrix?
| $f$ is an normal real valued function. If you want you can write it componentwise as
$$f(x) = {1\over 2}\sum_j\sum_k p_{jk}x_jx_k + \sum_j q_jx_j + r$$
Now the first double sum contains the $x_jx_k$ term twice if $j\ne k$ and if $j=k$ it becomes an $x_j^2$ term, so the derivate with respect to $x_j$ becomes:
$$f'_j(x) ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Show that $X$ is simply connected iff any two paths in $X$ with the same initial and terminal points are path-homotopic
Let $X$ be a path connected space. Show that $X$ is simply connected iff any two paths in $X$ with the same initial and terminal points are path-homotopic
Recall $X$ simply connected means $\pi_1(X,... | Strictly speaking $G\cdot g$ gives you a path homotopy between $g$ and $f\cdot \overline{g}\cdot g$.
It is clear that you can then collapse $\overline{g}\cdot g$ with a further homotopy, but maybe it is worth adding (!?)
In the converse direction you are implicitly using the same idea. $f\cdot \overline{g}$ homotopies ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2358399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Sum of four squares I was looking for numbers who can be expressed as sum of exactly four squares and not less. And I think I have found them. They are all the integers of the form
$$4^{n}\,(7+8k);\;k,\,n\in\mathbb{N}$$
I have no idea how to prove this statement and I wonder if ALL the numbers which need four square... | At least it is not hard to see that $4^n(7+8k)$ cannot be written as sum of three squares: $a^2+b^2+c^2\equiv m\pmod 4$ where $m$ is the number of odd squares on the left. Hence for $n\ge 1$, any representation of $4^n(7+8k)$ must use three even squares. But then this corresponds to the representation $4^{n-1}(7+8k)=(a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2358534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is aleph-omega a Mahlo cardinal? So I've read up a little on Mahlo cardinals and found a definition along the lines of of $\kappa$ is Mahlo iff it is inaccessible and regular cardinals below it form a stationary set.
Using this definition, I can see why $\aleph_0$ and $\aleph_1$ are not Mahlo, in that the first is not ... | A Mahlo cardinal has to be regular, which $\aleph_\omega$ is not. $\aleph_\omega =\bigcup\aleph_n$, so $\operatorname{cf}(\aleph_\omega)=\aleph_0$. Every strong inaccessible $\kappa$ satisfies $\kappa=\aleph_\kappa$, but even that is not enough as the lowest $\kappa$ satisfying that has $\operatorname{cf}(\kappa)=\al... | {
"language": "en",
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Integral of $e^{x^3+x^2-1}(3x^4+2x^3+2x)$ I understand that this looks exactly like a "Do my Homework" kinda question but trust me, I've spent hours(I won't go into detail as it's off topic).
Note: I'm a High School student, our teacher gave this question as a challenge.
I'm struggling with
$$\int e^{x^3+x^2-1}(3x^4+2... | Assume the solution to be of the form $\exp(x^3+x^2-1)Q(x)$ and differentiate this expression to get $\exp(x^3+x^2-1)\left[(3x^2+2x)Q(x)+Q'(x)\right]$. Compare this with $\exp(x^3+x^2-1)\left[3x^4+2x^3+2x\right]$ to obtain the ODE:
$$(3x^2+2x)Q(x)+Q'(x)=3x^4+2x^3+2x.$$
The particular solution is $x^2$, you could also s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2358765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 1
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Domination problem with sets
Let $M$ be a non-empty and finite set, $S_1,...,S_k$ subsets
of $M$, satisfying:
(1) $|S_i|\leq 3,i=1,2,...,k$
(2) Any element of $M$ is an element of at least $4$ sets among
$S_1,....,S_k$.
Show that one can select $[\frac{3k}{7}] $ sets from $S_1,...,S_k$
such that their union is $... | An algorithmic solution:
Clearly we can assume that each element in $M$ appears in exactly $4$ subsets. Let $|M| =n$.
Stage 1. Take maximal subfamily $\mathcal{A} \subseteq
\{S_1,....,S_k\} =:\mathcal{S} $ such that:
$\bullet$ every member of that family $\mathcal{A}$ has 3 elements;
$\bullet$ all sets in $\mathcal{A... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Is a bijective morphism between metric spaces necessarily an isomorphism Does the inverse morphism for a bijective isometry necessarily preserve the metric or should the preservation of the metric for the inverse morphism be stated seperately? To make myself clear my question is that does the inverse morphism in metric... | A metric space isomorphism is an isomorphism on the induced topologies is a bicontinuous bijection. It is possible to have a continuous bijection (at least in general topologies, not sure about metric spaces) that is not bicontinuous.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2358960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Find the condition for $f^{-1}$ to exist.
If $f: \mathbb R \to \mathbb R$ defined by $f(x)= x^3+ px^2 + qx+ k \sin{x}$, where $k,p,q \in \mathbb R$.
Find the condition for $f^{-1}$ to exist.
Can somebody please give me a Hint how to solve this problem.
EDIT(After getting hints):
If we can show that $f$ is strictly i... | Guide:
Check $\lim_{x \rightarrow \infty} f(x)$.
Check $\lim_{x \rightarrow -\infty} f(x)$.
Note that $f$ is continuous.
Find conditions to make it strictly increasing.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2359054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Factorial Inequality Proof I need some help proving the following fact about factorials.
Let $a$ and $b$ be positive integers. Prove that if $a > b$, then $a! > b!$.
| If $a>b$ then $$a!=a \cdot (a-1) \cdot \ \dots \ \cdot (b+1) \cdot b \cdot (b-1) \cdot \ \dots \ \cdot 1$$
so we would have
$$b!=\frac{a!}{a \cdot (a-1) \cdot \ \dots \ \cdot(b+1)} < a! \implies b! < a!.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2359166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How does the trigonometric identity $1 + \cot^2\theta = \csc^2\theta$ derive from the identity $\sin^2\theta + \cos^2\theta = 1$? I would like to understand how would the original identity of $$ \sin^2 \theta + \cos^2 \theta = 1$$ derives into
$$ 1 + \cot^2 \theta = \csc^2 \theta $$
This is my working:
a) $$ \frac{\... | The same way; you start with $\sin^2\theta + \cos^2\theta = 1$ and divide both sides by $\cos^2\theta$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2359280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Polar Coordinates tranformation for Linear Homogeneous Differential Equations (1st order) While studying a book of Differential Equations I found this problem so interesting.
Suppose
\begin{equation}
M(x,y)dx+N(x,y)dy=0 \tag 1
\end{equation}
is a homogeneous ODE. Show that the transformation $x=r \cos (\t... | This is what I've got:
Given this fancy thing (At least I think so..)
$$ \frac{dy}{dx}=\frac{dy}{d\theta} \times \frac{d\theta}{d x} = \frac{ \frac{dy}{d\theta} } { \frac{dx}{d\theta}} $$
So $x=r\cos(\theta)$ and $y=r \sin(\theta)$ ,yield:
$$\frac{dx}{d\theta}= \cos (\theta) \frac{dr}{d\theta}-r \sin(\theta)$$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2359363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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How to find $ \lim_{n\to\infty}\frac{\sin(1)+\sin(\frac{1}{2})+\dots+\sin(\frac{1}{n})}{\ln(n)}$
$$ \lim_{n\to\infty}\frac{\sin(1)+\sin(\frac{1}{2})+\dots+\sin(\frac{1}{n})}{\ln(n)}
$$
I tried applying Cesaro Stolz and found its $(\sin 1/(n+1))/\ln(n+1)/n$ where $\ln$ is $\log_e$ and it would be $1$ and so the limit ... | If your aim is to compute
$$
\lim_{n\to\infty}\frac{\sin1+\sin\frac{1}{2}+\dots+\sin\frac{1}{n}}{\ln n}
$$
you can indeed try and apply Stolz-Cesàro with
$$
a_n=\sin1+\sin\frac{1}{2}+\dots+\sin\frac{1}{n}
\qquad
b_n=\ln n
$$
This leads to computing
$$
\lim_{n\to\infty}\frac{a_{n+1}-a_n}{b_{n+1}-b_n}
=
\lim_{n\to\infty}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2359492",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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What does $\lim\limits_{(x,y)\rightarrow0}$ mean and how to show $ \lim\limits_{(x,y)\rightarrow0}\frac{x^3}{x^2+y^2}=0$? Consider $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ where
$$f(x,y):=\begin{cases}
\frac{x^3}{x^2+y^2} & \textit{ if } (x,y)\neq (0,0) \\
0 & \textit{ if } (x,y)= (0,0)
\end{cases} $$
... | $\lim_\limits{(x,y)\to 0}$ likely means $\lim_\limits{(x,y)\to(0,0)}$, which means that $x$ and $y$ are both tending to $0$. One could use polar coordinates where $x=r\cos(\theta)$ and $y=r\sin(\theta)$ to obtain:
$$\lim_{(x,y)\to(0,0)}\frac{x^3}{x^2+y^2}=\lim_{r\to 0}\frac{r^3\cos^3(\theta)}{r^2}=\lim_{r\to 0} r\cos^3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2359621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 1
} |
Proving the multiplicative property of limits.
If $\lim_{x\rightarrow a}f(x) =L$ and $\lim_{x\rightarrow a}g(x) =M$ then prove $\lim_{x\rightarrow a}f(x)g(x) =LM$
Attempt
(Should there be any statement that I should write at start of this proof?)
$$|f(x)g(x) -LM|=|f(x)g(x)+Lg(x)-Lg(x) -LM|=|g(x)[f(x)-L]+L[g(x)-M]<|g... | Your approach is fine but a few details are not correct.
You should replace
$$|f(x)-L|<\frac{\epsilon}{2g(x)}$$
with something like
$$|f(x)-L|<\frac{\epsilon}{2M'}.$$
where $M'$ is a positive constant to be decided.
Recall that $\lim_{x\rightarrow a}g(x) =M$ implies that $g$ is bounded in a neighbourhood of $a$.
Moreo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2359711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Limit Definition for Half-Derivative The derivative of a function $f$ is defined as
$$\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$
Let
$$d_1(f,h)=\frac{f(x+h)-f(x)}{h}$$
and, in fact, let all $d_n$ be defined by
$$\lim_{h\to 0} d_n(f,h)=f^{(n)}(x)$$
In order to obtain $d_2$, we can plug $d_1$ into itself to get
$$\frac{\frac{... | One can extend this in a similar manner that the binomial expansion theorem is extended. Note that $\binom nk=0$ if $k>n$ and $n$ is a natural number. Thus, we have the Grunwald-Letnikov derivative,
$$f^{(\alpha)}(x)=\lim_{h\to0}\frac1{h^\alpha}\sum_{k=0}^\infty(-1)^k\binom\alpha kf(x+(\alpha-k)h)\tag{$\alpha\ge0$}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2359871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to integrate: $\int_0^{\infty} \frac{1}{x^3+x^2+x+1}dx$ I want to evaluate $\int_0^{\infty} \frac{1}{x^3+x^2+x+1}$. The lecture only provided me with a formula for $\int_{-\infty}^{\infty}dx \frac{A}{B}$ where $A,B$ are polynomials and $B$ does not have real zeros. Unfortunately, in the given case $B$ has a zero at... | $$\int_{0}^{+\infty}\frac{x-1}{x^4-1}\,dx = \int_{0}^{1}\frac{1-x}{1-x^4}\,dx +\int_{0}^{1}\frac{x^2-x}{x^4-1}\,dx = \int_{0}^{1}\frac{dx}{1+x^2}=\frac{\pi}{4}$$
by just breaking the integration range as $(0,1)\cup (1,+\infty)$ and applying the substitution $x\mapsto\frac{1}{x}$ on the second "half".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2359994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Find $\lim\limits_{x \rightarrow \infty} \sqrt{x} (e^{-\frac{1}{x}}-1)$
Find $$\lim\limits_{x \rightarrow \infty} \sqrt{x} (e^{-\frac{1}{x}}-1)$$
$$\lim\limits_{x \rightarrow \infty} \sqrt{x} (e^{-\frac{1}{x}}-1)
= \lim\limits_{x \rightarrow \infty} \frac{e^{-\frac{1}{x}}-1}{x^{-0.5}}
= \lim\limits_{x \rightarrow \i... | Note the fundamental limit formula $$\lim_{t\to 0}\frac{a^{t}-1}{t}=\log a$$ and put $x=1/t,a=1/e$ to get $$\lim_{x\to \infty} x(e^{-1/x}-1)=-1$$ and then dividing by $\sqrt{x} $ we can see that the desired limit is $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2360103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Why are vectors and <-b,a> perpendicular? If i have the vectors:
X = {a, b}
Y = {-b, a}
How can I explain that these vectors will always be perpendicular? I know I can prove this very easily via the dot product, but I need to explain it in a layman's way.
| One way: Show that the triangle built on them satisfies the Pythagorean theorem, which implies that it's a right triangle
Another way: Show that the four points $(a,b)$, $(-b,a)$, $(-a,-b)$, and $(b,-a)$ form a rhombus, each of whose vertices are equidistant from the center. This implies that they form a square.
A thir... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2360146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 8,
"answer_id": 6
} |
Find the number of points of differentiability for the following function. If
$$f(x)=\begin{cases}
\cos x^3&;x\lt0\\
\sin x^3 - |x^3-1|&;x\ge0
\end{cases}$$
then find the number of points where $g(x)=f(|x|) \text { is non differentiable.}$
| The question is asking about $f(|x|)$, by symmetry, since $g(x)$ is not differentiable at $x=1$, it is not differentiable at $x=-1$ as well.
To show that it is not differentiable at $x=1$:
If it is differentiable at $x=1$, then the following limit exists.
\begin{align}\lim_{x \rightarrow 1} \frac{f(x)-f(1)}{x-1}&= \li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2360233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Fractional linear functions of complex analysis - don't understand this section In my notes under "Fractional Linear Transformations" of Complex Numbers, it says the following:
Let $M$ be a $2\times 2$ complex matrix
$$M=\begin{pmatrix} a & b \\ c & d\end{pmatrix}.$$
We write $T_M$ for the associated fractional linear ... | It's simple! $\;\lambda\begin{pmatrix}a&b\\c&d \end{pmatrix}=\begin{pmatrix}\lambda a&\lambda b\\\lambda c&\lambda d \end{pmatrix}$, so
$$T_{\lambda M}(z)=\frac{\lambda az+\lambda b}{\lambda cz+\lambda d}=\frac{az+b}{cz+d}.$$
There indeed results from the considerations in the post that
$$\det M'=\det\bigl((\det M)^{-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2360335",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Find biholomorphism between these domains I am pretty stuck on constructing a biholomorphism between:
$G_1 = \{z \in \mathbb{C}: Re(z) >0, Im(z)>0 \}$ and
$G_2 =\{z \in \mathbb{C}:|z|<1, Re(z) >0, Im(z)>0 \}$
I am able to visualize this as some kind of stretching, so first I thought about something like
$$f(z)=\frac{z... | So the map you have given $g(z)$ is the Cayley Transform which maps the upper half plane to the unit disc. Now observe that if you restrict this map, to the first quadrant which is essentially $G_{1}$, then the image is the lower semi-circle.
Now you take a map from the lower semi circle to the upper semcircle(just ro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2360480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Value of the double sum of product of cosines For $m \neq n$, $a \in (0,1/2)$
\begin{equation}
\sum_{n\geq 1}^{\infty} \sum_{m\geq 1}^{\infty}\int_{a \pi}^{\pi/2} \frac{\cos(2mx)\cos(2nx)}{mn} \, dx
\end{equation}
I know that the value of the integral is
\begin{equation}
\sum_{n\geq 1}^{\infty} \sum_{m\geq 1}^{\infty}... | Since $\sum_{m\geq 1}\frac{\cos(2mx)}{m}$ is pointwise convergent to $-\log(2\sin x)$ on $\left(0,\frac{\pi}{2}\right)$ we have:
$$\int_{a\pi}^{\pi/2}\sum_{m,n\geq 1}\frac{\cos(2mx)\cos(2nx)}{mn}\,dx= \int_{a\pi}^{\pi/2}\log^2(2\sin x)\,dx \tag{1}$$
and
$$ \int_{a\pi}^{\pi/2}\sum_{m=1}^{+\infty}\frac{\cos^2(2mx)}{m^2}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2360600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Formula for Natural logarithm of $\pi$ Does any formula or expansion exist that gives $\ln \pi$ ?
The expansion should not just be any formula of $\pi$ with a $\ln$ before it. For example $\ln \pi$ = k + $\sum f(x)$ or something of this type.
| One could use this formula to get
$$\ln(\pi)=\gamma+\sum_{n=1}^\infty\left[2\ln(n)-2\ln(n-0.5)-\frac1n\right]$$
Using the Euler product of the Riemann zeta function,
$$2\ln(\pi)=\ln(6)-\sum_{p\in\mathbb P}\ln(1-p^{-2})$$
where $p$ is prime.
Another interesting series:
$$\ln(\pi)=\ln(2)+2\lim_{x\to-1^+}\sum_{n=2}^\infty... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2360710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Order of element in field extension Given a field extension $\mathbb{F}_{p^n}$ and some element $\alpha \in \mathbb{F}_{p^n}$ which is not contained within any proper subfield of $\mathbb{F}_{p^n}$, is there a lower bound on the order of $\alpha$?
I understand that the nonzero elements of a finite field form a cyclic g... | Well, for any $m\mid n$, the subfield $\mathbb{F}_{p^m}$ consists of all elements of order dividing $p^m-1$. So the possible orders of elements of $\mathbb{F}_{p^n}$ which are not in any proper subfield are all the factors of $p^n-1$ that are not factors of $p^m-1$ for any proper divisor $m$ of $n$.
This gives a simpl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2360794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Set containing open ball and contained in closed ball has the same boundary I'm having some difficult time trying to figure out how to prove that if $U$ is an open unit ball and $\overline{U}$ is a closed unit ball in $\mathbb{R}^n$ and $U\subseteq A\subseteq \overline{U}$ then the boundary of $A$ is the same as the bo... | Say that $U=B(0,1)$. Since $\overline U$ is closed, we have that $\overline A\subset\overline U$. Since $U$ is an open ball, every point of $U$ is an interior point of $A$. Hence, $\partial A\subset \overline U\setminus U=\{x:\,|x|=1\}=\partial U$. Taking $x\in\partial U$, for every $r>0$, $B(x,r)$ intersects both $U\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2360923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proving convergence in $L^p$ Suppose $f \in L^p(\mathbb{R}) , 1 \leq p < \infty$. Prove that $$\lim_{h\rightarrow 0} \int_{\Bbb R} |f(x+h)-f(x)|^p = 0$$
I was thinking about showing that the sequence $\{f(x+\frac{1}{n}) - f(x)\}_n$ converges to zero in $L^p(\mathbb{R})$ so that I could pass the limit under the integra... | Let $f=1_[a,b]$ where $A \subseteq \mathbb{R}$.
$\forall h \in (0,1)$ we have that $(\int_{\mathbb{R}}|1_{[a,b]}(x+h)-1_{[a,b]}(x)|^p)^{1/p}=(\int_{\mathbb{R}}|1_{[a,b]-h}(x)-1_{[a,b]}(x)|^p)^{1/p}=(\int_{([a,b]-h) \triangle [a,b]})^{1/p}=m(([a,b]-h) \triangle [a.b])^{1/p} \leqslant m([a-h,a) \cup(b-h,b])^{1/p} \leqsla... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2361135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What practical applications had the number pi in the ancient worlds and what was the motivation for calculating it? I know that modern sciences have many many applications for the number PI, many of them outside of geometry, but I do not understand what practical applications had this constant in the ancient world.
Wha... | For all practical applications in ancient time daily life, the various rational approximations in vogue where more than enough, even considering that the instruments to measure the length where prone to higher errors.
I think the interest in finding ever more precise value of $\pi$ was actually due to its irrationality... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2361236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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How to use density argument to obtain inequality? Let $(X, \|\cdot \|)$ and $(Y,\|\cdot\|)$ be a Banach spaces. Assume that $T$ is a linear operator from $X$ to $Y$ ($T:X\to Y$). Assume that $D\subset X$ is dense in $X.$ The operator $T$ satisfies the inequality $\|Tf\|_{Y} \leq \|f\|_{X}$ for all $f\in D.$
Questio... | The answer to your question is no. There are linear operators $T$ that are bounded on a dense subspace, but not bounded.
See this example which has $Y=\mathbb R$, but is not constructive.
For your edit:
Here, the operator is not given on $X$, but only a dense subset $D$. Therefore we can choose which values $T$ will ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2361381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Direct products - Is this an abuse of notation?
Let $G=\langle x\rangle\times\langle y\rangle$ where $|x|=8$ and $|y|=4$. Find all pairs $a,b$ in $G$ s.t. $G=\langle a\rangle\times\langle b\rangle$, where $a$ and $b$ are expressed in terms of $x$ and $y$. (Abstract Algebra: Dummit & Foote, Direct products, Ex. 15)
Is... | You must just find all pairs of elements $a, b\in G$ such that $|a| \cdot |b| = |G|$ and $\langle a \rangle \cap \langle b \rangle = \{0\}$.
Since $|G|=32$ and $G$ obvioulsy does not contain elements of order greater than $8$, the only possibilities are $|a|=4, \, |b|=8$ and $|a|=8, \, |b|=4$.
Every element of $G$ is o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2361563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Why does $\frac{\cos\Delta x - 1 }{\Delta x} \to 0$ I'm watching Lecture 3 in MIT single variable calculus.
https://www.youtube.com/watch?v=kCPVBl953eY&list=PL590CCC2BC5AF3BC1&index=3
And at one point the instructor does the following:
I was under the impression that when evaluating limits we need to avoid having $0/... | hint
$$1-\cos (d)=2\sin^2(\frac {d}{2}) $$
$$|\cos (d)-1|\le \frac {d^2}{2} $$
if we know that
$$|\sin (A)|\le | A |$$
hence
$$\Bigl |\frac {\cos (\Delta x)-1}{\Delta x}\Bigr |\le \frac {|\Delta x|}{2} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2361672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 2
} |
Why dense subsets are convenient to prove theorems Could you please explain the following concept (preferably by examples) about dense subsets:
If you want to prove that every point in $A$ has a certain property that is preserved under limits, then it suffices to prove that every point in a dense subset $B$ of $A$ has... | Let $B$ be a dense subset of $A$. Suppose property $P$ is preserved under limits and we know that every point in $B$ satisfies property $P$.
Now pick an arbitrary point of $A$, say $a$. Then $a=\lim_n b_n$ of elements in $B$ each of which have the property and since our property even holds for the limit, it holds for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2361816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 2
} |
Laurent series of $\frac{z}{z^2+1}$ I am interested in the Laurent series of $f(z)=\frac{z}{z^2+1}$ in $D_{0,2}(i)$.
What I did:
We have $\displaystyle f(z)=\frac{1}{2}\frac{1}{z+i}+\frac{1}{2}\frac{1}{z-i}$, so
*
*$\displaystyle \frac{1}{2}\frac{1}{z+i}=\frac{1}{2i}\frac{1}{1-\left(-\frac{z}{i}\right)}=\frac{1}{2i}... | It is the right idea to use the geometric series. Try it this way:
$$
\frac1{z+i}=\frac1{2i+z-i}=\frac1{2i}\cdot\frac1{1+\frac{z-i}{2i}}=\frac1{2i}\cdot\frac1{1-\left(-\frac{z-i}{2i}\right)}=\frac1{2i}\sum_{n=0}^\infty\left(\frac{i}2\right)^n(z-i)^n.
$$
The term $\frac1{z-i}=(z-i)^{-1}$ is already a part of the Laurent... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2361969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How many points $(x, y)$ with integer coordinates satisfy the inequality $x^2+y^2 \leq 25$? I had this questions from a previous exam that I couldn't answer, I am apologizing for any English mistakes or for any stupid questions, I tried to solve them and I searched the internet and I couldn't find answers or at least o... | This is not an answer but a comment or a hint (so please don't down-vote it).
Consider this figure:
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2362063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 4
} |
Pre-calculus $\frac 27 = \frac 1a + \frac 1b$ I had this questions from a previous exam that I couldn't answer, I am apologizing for any English mistakes or for any stupid questions, I tried to solve them and I searched the internet and I couldn't find answers.
4-if $\frac 27$ could be written in a unique way in the f... | The given problem boils down to finding lattice points on a hyperbola. The equation
$$ 2ab=7(a+b) \tag{1}$$
is equivalent to
$$ (2a-7)(2b-7) = 49\tag{2}$$
and $49=7^2$ cannot be written as a product of integers in too many ways. From the assumption $(2a-7)=1$ and $(2b-7)=49$ we get the non-trivial solution $\color{red}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2362140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Evaluating the determinants of matrices A and B I am stuck on a homework question that asks to evaluate the det(A) and det(B). We are only given the following information: A and B are 3x3 matrices, such that det((2A)⁻¹Bᵀ)=1 and det(4B⁻¹A³)=1/2, how could I solve this?
| I'm going to work this out for $n \times n$ matrices, since there is really no more effort involved.
The solution depends almost entirely on the three properties
$\det(XY) = (\det X)(\det Y), \tag{1}$
and
$\det X^T = \det X, \tag{2}$
which hold for any two $n \times n$ matrices $X$, $Y$,and the fact that $\det D$, wher... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2362217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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