Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
About a lemma to prove the Cantor-Bernstein-Schroeder theorem. I am reading "Logic in mathematics and set theory" by Kazuyuki Tanaka and Toshio Suzuki.
In this book, there is a lemma to prove the Cantor-Bernstein-Schroeder theorem.
I cannot understand why the equality $$A_0 = (A_0 - B_0) \cup (B_0 - A_1) \cup (A_1 - ... | It doesn't. If the equality is true for some $A_1\subseteq B_0\subseteq A_0$ and $f:A_0\to A_1$, we can simply extend the sets using a new set $C$, and also extend $f$ by making it the identity on $C$, in which case $C$ will be contained within all sets $A_i$ and $B_i$. However you can augment the argument by making $g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3281493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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$f(z)$ is Analytic then $f(\bar z)$ is analytic iff $f$ is constant. let $f_1=u(x,y), f_2=u(x,-y)$ Then is this true that
$(f_2)_x$ at point $(x,y) =(f_1)_x $ at point $(x,-y)$
And
$(f_2)_y $ at point $(x,y) = - (f_1)_y$ at point $(x,-y)$
If yes does this imply if $f(z)$ is Analytic then $f(\bar z)$ is analy... | Without CRD:
Let $g(z):= f(\overline{z})$ and show that
$\lim_{h \to 0 , h \in \mathbb R}\frac{g(z_0+h)-g(z_0)}{h}= f'(\overline{z_0})$
and
$\lim_{h \to 0 , h \in i\mathbb R}\frac{g(z_0+h)-g(z_0)}{h}= -f'(\overline{z_0})$.
Conclusion ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3281652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does $P(A \cap B) = 0$ imply that $A \cap B = \emptyset$? Given that $P(A \cap B) = 0$, where $A$ and $B$ are two events, does this imply that $A \cap B = \emptyset$ ?
Is it not possible to have a probability zero for an event which is not empty?
Regards.
| No. For example, if I pick a random number from $[0,2]$, then $P([0,1]\cap [1,2])=0$, however, $[0,1]\cap[1,2]=\{1\}\neq \emptyset$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3281752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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Determine all singularities and its characters for function:
$f(z) = \frac{\sin (z+1) e^{\frac{1}{z}}}{(z-i)^{2}(z+i)\cos ^{2} (z)}.$
Determine all singularities and its characters for function:
$$f(z) = \frac{\sin (z+1) e^{\frac{1}{z}}}{(z-i)^{2}(z+i)\cos ^{2} (z)}.$$
I have concluded that $z=0$ is essential singu... | Let's write down all points that can be singularities for $f(z)$:
*
*$z = i$
*$z = -i$
*$z = 0$
*$\cos^2 (z) = 0$
*$z = \infty$
As you mentioned, 1 - order 2 pole, 2 - order 1 pole, 3 = is essential singularity. I think, you can easily proof it.
Let's talk about 4. From $\cos^2 (z) = 0$ we get: $z_n = \pi/2 + \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3281843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show that $E^2=E=E^T$.
Let $E$ be an $n\times n$ matrix.
Let $U=\{Ex:x\in \Bbb R^n\}$.
If $\text{proj}_U v=Ev$ $ \forall v\in \Bbb R^n$ show that $E^2=E=E^T$.
I know that if $U$ has a basis $u_1,u_2,\ldots ,u_n$ then
$\text{proj}_U v=\sum_{i=1}^n\langle v,u_i\rangle u_i$
But how to show from here that $E^2=E=E^T$... | Let $\{u_1, \ldots, u_k\}$ be an orthonormal basis for $U$. For each $u_i$, define the projection onto $\operatorname{span}(u_i)$:
$$P_i(x) = \operatorname{proj}_{u_i}(x) = \langle x, u_i \rangle u_i.$$
Note that, for any $i, j$, we have
$$(P_j P_i)(x) = \langle P_i(x), u_j \langle u_j = \langle \langle x, u_i \rangle ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Can this limit be solved with Riemann sum?
Can this limit be solved with Riemann sum?
$$ \lim _{n\to \infty }\left[\lim _{x\to 0}\left(\cos x\cdot\cos2x\cdots\cos nx\right)^{\frac{1}{n^3x^2}}\right] $$
What I've tried is to solve it with the Riemann sum but I am getting stuck somewhere , and I am not seeing where .... | $$L=\lim_{n\rightarrow \infty} \lim_{x\rightarrow 0} (\cos x \cos 2x \cos 3x...\cos nx)^{\frac{1}{n^3x^2}}$$
As per @marty cohen, let us use $-y+y^2/2 \le \ln(1-y)\le -y~$ and $(1-t^2/2) \le \cos t \le (1-t^2/2+t^4/24)~$ when $~t~$ and $~y~$ are very small. We get
$$\ln L =\lim_{n\rightarrow \infty} \lim_{x\righta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3282040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solutions to equation (or proof that there is no solution) I am unable to find a solution or prove that there is none. I really need help:
$a, b,$ and $c$ are positive integers such that
$a^2-b^2+c^2=2.$
Is this possible? If possible find the solutions, if not prove that there is no solutions.
| To search for a positive answer to such a question, Wolfram Alpha is your friend. In your case, $a=c=3$ and $b=4$ is a solution (the only positive integer solution Wolfram Alpha was able to find, though that is not proof it is unique).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3282297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Proof of every measurable cardinal carries a normal measure I'm reading the proof of Theorem 10.20 in Set Theory by Jech and I don't understand the last argument.
The theorem says every measurable cardinal carries a normal measure. The proof goes: Let $U$ be a nonprincipal $\kappa$-complete ultrafilter on $\kappa$. For... | By minimality of $f$, we have $$\{\alpha:g(\alpha)\le\gamma\}\in U $$ for some $\gamma,$ so by $\kappa$-completeness, there is a $\beta \le \gamma$ such that $\{\alpha:g(\alpha)=\beta\}\in U.$
(Your other questions can be answered by straightforward definition chasing. For instance that $h$ is constant on $f(Y)$ is a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3282395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Closed cone in the euclidean space $\mathbb{R}^n$ Let $T:\mathbb{R}^n\rightarrow\mathbb{R}^m$ be a linear map and let $\mathcal{C}$ be a closed cone in $\mathbb{R}^n.$ Prove that $T(\mathcal{C})$ is a closed cone in $\mathbb{R}^m$ provided $\ker(T)\cap \mathcal{C}=\{0\}$.
I have no problem to justify that $T(\mathcal{C... | Let $y_n:=Tx_n \to y$ be a sequence in $T(C)$ with $x_n\in C$. If $(x_n)$ contains a bounded subsequence, then $x_{n_k}\to x$ and $y=Tx$. If $(x_n)$ contains no bounded subsequence, then $\|x_n\|\to\infty$.
We can consider the sequence $v_n:= \frac{x_n}{\|x_n\|}\in C$, which is well-defined for all $n$ large enough. I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3282538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Holomorphic extension in 3 complex variables The problem is the following: Suppose $U = \{ z \in \mathbb{D}^3 \ | \ \frac{1}{2} < |z_1| \text{ or } \frac{1}{2} < |z_2| \}$. Prove that every $f \in \mathcal{O}(U)$ extends to $\mathbb{D}^3$.
Here $\mathbb{D}^3$ is the unit polydisc in three complex dimensions.
I've thou... | I figured it out!
Fix $z_3 = \zeta \in \mathbb{D}$ and let $g : \mathbb{D}^2 \to \mathbb{C}$, where $g(z_1,z_2) := f(z_1,z_2,\zeta)$. Then $g$ is holomorphic on the set $U' := \{ z \in \mathbb{D}^2 \ | \ \frac{1}{2} < |z_1| \text{ or } \frac{1}{2} < |z_2| \}$. However, $g$ extends over the complement of $U'$ in $\mathb... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$3^2+2=11$, $33^2+22=1111$, $333^2+222=111111$, and so on. $3^2+2=11$
$33^2+22=1111$
$333^2+222=111111$
$3333^2+2222=11111111$
$\vdots$
The pattern here is obvious, but I could not have a proof.
Prove that $\underset{n\text{ }{3}\text{'s}}{\underbrace{333\dots3}}^2+\underset{n\text{ }{2}\text{'s}}{\underbrace{222\dots... | Hint:
$$\underbrace{aaa\cdots a}_{n\text{ a's}}=\frac{a(10^n-1)}{9}$$
Where $a\in\{0,1,2,3,4,5,6,7,8,9\}$ and $n\in\mathbb{N_0}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3282744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Adjoint of Matrices Defined by Inner Products Let H be a Hilbert space and let $(e_n)_{n \in \mathbb{N}}$ be an orthonormal basis for H. Now define for each $T \in {\bf B}(H)$ the doubly infinite matrix $A = (\alpha_{nm})$ by setting $\alpha_{nm} = (Te_n|e_m)$. I am trying to find the matrix corresponding to $T^*$.
I k... | Hint:
$$\langle T^*e_n, e_m\rangle = \langle e_n, Te_m\rangle = \overline{\langle Te_m, e_n\rangle } = \overline{\alpha_{mn}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3282871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Playing the same number at loto game A simple question that was already asked but I didn't understand the modelisation.
If I play always the same number at the loto, it does really look like a binomial distribution, like a die that I throw and I want to get at least a 6. In the case of the die, if I bet on a 6 each ti... | The probability in lotto does increase the same way. As you say, if you roll dice $n$ times trying to get a $6$ your chance of at least one success is $1-(\frac 56)^n$. If you play a lotto with $10^6$ possible outcomes $n$ times the chance of at least one success is $1-(\frac{999,999}{1,000,000})^n$. For small $n$ t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3282972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding zero divisors in a polynomial quotient ring
Is $x^2+x+I$ a zero divisor in $\mathbb{Z}_7[x]/I$, where $I=(x^3+5x^2+2x+5)$?
I know that $\gcd(x^3+5x^2+2x+5,x^2+x)=x+1$, and that it means that $x^2+x$ is indeed a zero divisor.
What I struggle with is finding $g(x)$ such that $$(f(x)+\langle x^3+5x^2+2x+5\rangl... | $ f=x(x\!+\!1)$ is a $0$-divisor mod $g\!\iff\! d:=\gcd(f,g)\neq 1,g$ $\!\iff\! g(0)=0\,$ xor $g(-1)=0\,$
Then we have $\!\bmod g\!:\ (\color{#c00}{g/d})f \equiv (f/d)g\equiv 0,\,$ both $\,g/d,f\not\equiv 0.\,$ Thus you seek $\,\color{#c00}{g/d}$
This is true in the OP: $\,g(0)= 5\neq 0,\,\ g(-1)=7=0\,$ in $\Bbb Z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3283085",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Terence Tao uncountability of $\mathbb{R}$ There is a small detail I would like to understand. In the proof presented by Tao below: I don't understand why do we have the following cancellation:
$\Sigma_{n < n_0 : n \in A} 10^{-n} - \Sigma_{n < n_0 : n \in B} 10^{-n}$ ?
I mean we could have elements $n \in \mathbb{N}$ ... | Our definition of $n_0$ is such that it is the least $k$ such that $k$ is in set $A$ but not $B$ or vice versa. Thus, all $n$ below $n_0$ would appear in both sets, so we can just cancel them.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3283303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Rank of Products of Matrices This is somewhat of a reference request.
In several posts on the rank of products of matrices (e.g. Full-rank condition for product of two matrices), it is stated that
$$ \mathrm{rank}(AB) = \mathrm{rank}(B) - \dim \big(\mathrm{N}(A) \cap \mathrm{R}(B)\big)$$
It appears that this is a clas... | Suppose there exists a $v$ with $B u = v$ and $A v = 0$. What is $AB u$? Can you take it from there?
EDIT: typo.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3283477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Show that no choice of numbers $a$ and $b$ can make $ax + by = (3,0,0)$
Show that no choice of numbers $a$ and $b$ can make $ax + by = (3,0,0)$ when $x = (3,-1,0)$ and $y = (0,1,5)$.
The only materials in the chapter talked about are:
*
*Vector Space Operations
*Standard Basis
*Coordinates of a vector $x$
*Comp... | You may not have learned this yet, but
$n$ vectors in $ \mathbb {R} ^{n}$ are linearly independent if and only if
the determinant of the matrix formed by taking the vectors as its columns is non-zero.
In your particular case, that matrix has a form (upper triangular) that makes it very easy to compute
the determin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3283575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is the following subset of space of matrices connected, open? If $S=\left\{ A=\begin{bmatrix}A_1&0 \\ 0&A_2 \end{bmatrix} \in \mathbb{M}_4(\mathbb{C}): det A_1=detA_2 \right\}$. Then is this set open in $\mathbb{M}_{4}(\mathbb{C})$ with usual topology. If not then what are the interior points. I would like some hints ... | An idea: denote by $\;\mathcal B\;$ the set of all blocks-matrices in $\;M_4(\Bbb C)\;$, and define
$$f:\mathcal B\to\Bbb C\;,\;\;f\begin{pmatrix}A_1&0 \\ 0&A_2 \end{pmatrix}:=\det A_1-\det A_2$$
The above map is continuous (wrt the usual topologies in domain and codomain: the Euclidean one in $\;\Bbb C\;$ and the on... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let X be a $T_1$-space satisfying the conclusion of Tietze's Extension Theorem then prove that X is normal I am unsure what exactly the "conclusion" of the theorem means, as in, is it the entire theorem without the assumption of normality of X? So I am assuming that we have to go the other way around as compared to the... | Let $A$ and $B$ be disjoint closed sets. Define $f:A \cup B \to \mathbb R$ by $f(x)=0$ if $x \in A$ and $f(x)=1$ if $x\in B$. Then we can verify that $f$ is continuous by showing that the inverse image of any closed set is closed. We are told that Tietze's Theorem can applied. This means any continuous function defined... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3283799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Composition of matrices - eigenvectors Say $\lambda$ is an eigenvalue of $\textsf{ST}$. There exists $ \ne \textbf{0}$ such that
$\textsf{ST}x= \lambda x$.
Multiply both sides by $\textsf T$:
$$\textsf{TST}x=\textsf{T} (\lambda x)$$
$$\textsf{TS}(\textsf{T}x)=\lambda(\textsf{T}x)$$
Thus $\textsf{T}x$ is an eigenvector... | $\require{AMScd}$
Maybe more convoluted than necessary, but let me try.
We have the following:
$$
S: W \to V\\
T: V \to W
$$
where $V, W$ are vector spaces. Thus we have $ST: V \to W \to V$ and $TS: W \to V \to W$. A way to combine all these is to consider the following diagram:
\begin{CD}
V @>T>> W @>S>> V\\
@... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3283922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Approximating the product of two real numbers Let $a,b$ two positive real numbers. For example I want to calculate approximate value of $45.11\times 67.89$ only to 2 decimal places. Note that any calculators or other such devices aren't allowed. Also suppose I want to calculate $\frac{11789}{234558}$ only approximately... | It's $$(45+0.11)(68-0.11)=3060+23\cdot0.11-0.0121=$$
$$=3062.53-0.0121=3062.5179\approx3062.52$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3283992",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
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A signed measure is bounded
If $\nu$ is a signed measure defined over $(X,\mathcal{M})$ such that $\nu(E)\in\Bbb{R}$, for all $E\in\mathcal{M}$, then $\nu$ is bounded.
This looks weird to me. Of course, $\nu(E)\in\Bbb{R}$ implies that $\nu(E)<+\infty$, once we defines a signed measure to take values at $\Bbb{R}\cup\{... | Assume that $E_1,E_2,...$ satisfy that $\nu(E_n)\to+\infty$. By excluding finitely many terms from the beginning, we can assume that $\nu(E_n)>0$ and by taking differences of sets of a subsequence for which $\nu(E_{n_k})>2^k$ that the $E_n$ are dijoint. Then $\mathbb{R}\ni\nu\left(\bigcup_nE_n\right)=\sum_{n=1}^{\infty... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $M=\sum_{g\in G}\rho(g)$ then $\operatorname{tr}(M)=0$ implies $M=0$ Let G be a finite group and $\rho:G\to GL_n(\mathbb C)$ a representation.
a) If $M=\sum_{g\in G} \rho (g) \neq 0$ then prove that there is a non-zero vector $v$ such that $\rho(g)v=v $ for every $g\in G$.
b) If $\sum_{g\in G} \chi(g)=0$ then prov... | Notice that
$$ \sum_{g\in G} \chi(g)$$
is the inner product of the representation with the trivial representation, hence if it equals $0$ it means that the decomposition in irreducible representation does not contain the trivial representation. But if $M\neq 0$ then $\rho$ has a one dimensional invariant subspace, i.e.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Why does $\binom{n}{k} = 0$, if $k > n$? I have came across this in a textbook that I am currently studying, but I don't understand how I should proof this.
A short explanation or proof would be nice.
| For example expand the binomial
$$
(1+x)^n
$$
and the coefficient of $x^k$ is $\binom{n}{k}$ for all $k$. [This is why it is called a binomial coefficient.] Of course (when $n$ is a positive integer), this coefficient is $0$ for $k > n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3284361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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change of base for $M_{22}$ I have been stuck on this question for a while now. I can easily do the change of basis matrix if the entries in the basis are vectors as opposed to a matrix.
Let
$$B_1 = \{\begin{bmatrix}
1 & 1 \\
1 & -1
\end{bmatrix}
,\begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}
,\begin{bmatrix}
0 & ... | Take the first vector of $B_1$ (in this case, by vector I mean the matrix) and write it as a linear combination of the elements in $B_2$, like this:
$$\begin{pmatrix} 1&1 \\ 1&-1\end{pmatrix}=0\begin{pmatrix} 1&1 \\ 0&-1\end{pmatrix}+1\begin{pmatrix} 1&0 \\ 1&-1\end{pmatrix}+1\begin{pmatrix} 0&1 \\ 0&0\end{pmatrix}$$
t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Can $\mathbb{Q×Q}$ be embedded in $\mathbb{R}$ as group? I think ans is NO : if possible let that is true hence there is a monomorphism from $H= \mathbb{Q×Q}$ to $\mathbb{R}$. as $\mathbb{R} $ has only subgroups which is cyclic or dense and $H$ is not cyclic hence dense but it's proper subgroup
$\mathbb{Z×Z}$ is nei... | The map $(a,b)\mapsto a+b\sqrt2$ is an injection $\mathbb{Q×Q} \to \mathbb{R}$ because $\sqrt2$ is irrational.
$\sqrt2$ is not special here; any irrational number works, for instance $\pi$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3284665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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Can every closed differential form be expressed via constant coefficients? Let $M$ be a smooth $n$ dimensional manifold, and let $1 \le k < n$. Let $\omega \in \Omega^k(M)$ be a closed $k$-form on $M$.
Let $p \in M$. Do there exist coordinates around $p$, such that $\omega=a_{i_1i_2\dots i_k}dx^{i_1} \wedge dx^{i_2} \... | I believe you don't have such coordinates around $0$ for $\omega=xdx$ in $\mathbb{R}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3284768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
How many 5-letter words can we make if the letters are in order? Using the $26$ English letters, the number of $5$-letter words that can be made if the letters are distinct is determined as follows:
$26P5=26\times25\times24\times23\times22=7893600$ different words.
What if the letters in each word are in alphabetical ... | Hint. How many ways can you choose the five different letters? Once you have them, in how many ways can you organize them in alphabetical order?
(This assumes the letters are distinct.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3284875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Solve for $(5x-1)(x-3)<0$ The inequality $(5x-1)(x-3)<0$ is true when
$(5x-1)<0$ and $(x-3)>0$
or
$(5x-1)>0$ and $(x-3)<0$.
If I solve for $x $ in the first scenario, $x < \frac{1}{5}$ and $x > 3$ which is wrong. But if I solve for $x$ in the second scenario, $x > \frac{1}{5}$
and $x < 3$ which is correct.
Why is ... | As $x$ increases, the two linear factors are negative, then positive, and each changes sign once, at a root. So one changes sign before the other, and the combinations $--,+-,++$ are possible, but not $-+$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3284981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 3
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Calculating $\int_0^\infty \frac{\cos(t)}{(1+t^2)^3}\text{d}t$ I am very new to the Residuetheorem and now I am asked to calculate the following integral:
$$\int_0^\infty \frac{\cos(t)}{(1+t^2)^3}\text{d}t$$
I know it has poles of order $3$ at $x=\pm i$ and that I have to find a closed curve in order to calculate it.
... | Note that your integral is equal to$$\frac12\operatorname{Re}\left(\int_{-\infty}^\infty\frac{e^{it}}{(1+t^2)^3}\,\mathrm dt\right).$$And$$\int_{-\infty}^\infty\frac{e^{it}}{(1+t^2)^3}\,\mathrm dt=2\pi i\operatorname{res}_{z=i}\frac{e^{iz}}{(1+z^2)^3}.$$Finally, this last residue is equal to $-\frac{7i}{16e}$. Therefor... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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proof of second smallest eigenvalue using Lagrange equations Without the use of Spectral Theorem, assume that $A$ is a symmetric $n\times n$ matrix, and define $f: \mathbb{R^n}\to\mathbb{R}$, $g_0: \mathbb{R^n}\to\mathbb{R}$, and $g_1: \mathbb{R^n}\to\mathbb{R}\;$ by $$f(\mathbf{x}) = \mathbf{x}\cdot A\mathbf{x},\ \ \... | Notice that $P=I-yy^T$ projects $x$ onto $y_{\perp}$. So the minimization problem is equivalent to minimizing $xP^TAPx$. It’s easy to see that $M:=P^TAP$ is symmetric. So it will be the smallest eigenvector/eigenvalue of $M$. Can you finish from here? As a hint: $Px=x$ whenever $(x,y)=0$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If every polynomial in $k[x]$ has a root in $E$, is $E$ algebraically closed? If $E/k$ is algebraic and for all $f$ in $k[X]$, all roots of $f$ lie in $E$, then $E$ is algebraically closed.
The question is:
If $E/k$ is algebraic and for all $f$ in $k[X]$, at least one root of $f$ lies in $E$, then is $E$ algebraically ... | This is true, but it is not trivial. See Gilmer, A Note on the Algebraic Closure of a Field.
The OP asked for another reference in the comments. A google search reveals Richman A theorem of Gilmer and the canonical universal splitting ring, which apparently gives a constructive proof. In this Math Stackexchange answe... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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Proposition $8.3$ - Fundamental groups and covering spaces by Elon Lages Lima Preliminaries maybe important:
Let $M$ and $N$ be oriented manifolds with the same dimension and $f: M \longrightarrow N$ a local diffeomorphism. We say that $f$ is positive (with respect to the chosen orientations) when, for each $x \in M$, ... | I think you are correct. What is meant is that $\pi(x)=\pi(y)$. Therefore you can compute $\pi^\prime(y)^{-1}\circ\pi^\prime(x)$ because $\pi^\prime(y)$ and $\pi^\prime(x)$ both have value in $T_{\pi(x)}M/G$. Finally as it is said $\pi^\prime(y)^{-1}\circ\pi^\prime(x)=\alpha^{\prime}(x)$ which is positive so you can co... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is the equivalent Cauchy sequence in a topological group well-defined? I was reading chapter 10 of Atiyah where I met the notion of
equivalent Cauchy sequences for topological groups.
Atiyah does not explain the reason why equivalent Cauchy sequences indeed give a equivalence relation. I can manage to prove the re... | This is a standard theorem on topological groups, and follows from the continuity
of the group operation. I'll write that as addition, and assume the operation is commutative, but the argument works for non-commutative groups too.
As addition is continuous, then if $U$ is an open neighbourhood
of $0$ then $W=\{(a,b)\in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Condition for a quotient map to have compact image. Let $q:X\to X_{/\sim}$ be a quotient map for some relation $\sim$ on $X$. If there is a compact subspace $A\subset X$ such that every element of $X$ is in relation with some element of $A$, then $X_{/\sim}$ is compact, simply because the condition can be rewritten $q(... | This is only a partial answer. If $Y = X/\sim$ is compact Hausdorff and $q$ is a local homeomorphism (see e.g. https://en.wikipedia.org/wiki/Local_homeomorphism), then the answer is "yes". This covers the case when $\sim$ is generated by a covering space action (then $Y$ is a compact manifold and $q$ is a covering proj... | {
"language": "en",
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"source": "stackexchange",
"question_score": "4",
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ZFC and axiom of power set It seems that I do not understand thoroughly the axioms of ZFC.
I am thinking of "is really the axiom of power set independent from the other axioms, and if it is, how to prove that?".
In other words, how to prove that ZFC without the axiom of power set is not equal to ZFC?
| Consider the set $H(\kappa)$ consisting of sets $x$ which satisfy $|\text{tc}(x)| < \kappa$. As mentioned in this answer, this set $H(\kappa)$ is a model of all axioms of ZFC except power set. In fact, if one takes $\kappa$ to be a successor cardinal (such as $\aleph_1$), then one can verify that the axiom of power set... | {
"language": "en",
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Last element in list We have to find the last element not marked in a list after certain operations. Operations are performed until only one element is left.
Suppose I have a list which has certain elements 'marked' alternatively after a particular index $x$. For example, I begin with list: $[a, b, c, d, e, f', g, h',... | First eliminate all elements marked before the first step.
Then, from the remaining elements, notice that after iteration k, the remaining numbers are on the positions $m\cdot2^k+1$, $m=0,1,\ldots$
So if you have $n$ unmarked elements in the beginning, the index of the last one in the reduced array will be for $m=\left... | {
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Evaluate $\int_{0}^\frac{\pi}{2} \sqrt{1+\sin^2(x)}dx$ I feel like I'm very close, so I would only like a hint. I'm only using real methods with the main thing I'm trying to connect the integral to is the Beta function. With a bunch of substitutions, I have boiled the integral down to
$$\int_{0}^\frac{\pi}{2} \sqrt{1+\... | Hint 1:
$$\frac{\mathrm d}{\mathrm dx}\sin(x)=\cos(x)=\sqrt{1-\sin^2(x)}$$
Hint 2:
Conjugate the "numerator" so that the only radical is in the denominator.
Hint 3:
Perform a simple substitution so that you get a linear function inside the radical.
All steps shown:
$$\int_0^{\pi/2}\sqrt{1+\sin^2(x)}~\mathrm ... | {
"language": "en",
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Prove that $\{\cos x, \sin x, e^x, e^{-x}\}$ is a linearly independent subset of $C^\infty (\mathbb R)$ I am going to prove $\cos x, \sin x, e^x$ and $e^{-x}$ is a linearly independent subset of $C^\infty (\mathbb R)$, which is smooth functions.
first we have $a\cos x+b\sin x+ce^x+de^{-x}=0$, WTS that $a=b=c=d=0$.
Supp... | "Sometimes much larger" isn't a precise term, although it can be phrased more rigorously. For example, I suggest considering limits $\lim_{x\rightarrow\pm\infty}$. If $$ a \cos x + b\sin x+ c e^x + d e^{-x} =0$$
for all $x$, then $$ \lim_{x\rightarrow\pm\infty} (a \cos x + b\sin x+ c e^x + d e^{-x}) = 0$$
and you can... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Uniqueness property for the space of finite measure. Let $\mu$ be a finite measure on $\mathbb{R}$ satisfying,
$$\int_{\mathbb{R}}f(x)d\mu(x)=0,~\forall f\in C_c(\mathbb{R})$$
Then is it true that $\mu =0$?
We know that the result is true for $L^1(\mathbb{R})$, which is a subspace of the above.
Edit after the comments ... | For all $n=1,2,3,\dots$ there exists a non-negative function $f \in \mathrm{C}_{\mathrm{c}}(\mathbb{R})$ such that $f=1$ on $[-n,n]$. Therefore, for all $n=1,2,3,\dots$
\begin{equation}
\mu([-n,n]) \leq \int_\mathbb{R} f d \mu = 0.
\end{equation}
By countable subadditivity $\mu(\mathbb{R})=0$, and we are done.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find a basis and the dimension of the solution space $\textsf{W}$ $$\left\{\begin{align}
x + 3y + 2z = 0 \\
x + 5y + z = 0 \\
3x + 5y + 8z = 0 \\
\end{align}\right.$$
So if we represent this as an augmented matrix
$$\begin{pmatrix}
1 & 3 & 2 & 0 \\
1 & 5 & 1 & 0 \\
3 & 5 & 8 & 0 \\
\end{p... | It is not correct. How did you obtain these basis vectors?
By row reduction you reduced your system to
$$\begin{cases} x_1 + \frac72x_3 = 0 \\ x_2 - \frac12x_3 = 0\end{cases}$$
so $$\begin{bmatrix} x_1 \\ x_2 \\ x_3\end{bmatrix} = t\begin{bmatrix} -\frac72 \\ \frac12 \\ 1\end{bmatrix}, \quad\text{ for some } t \in \mat... | {
"language": "en",
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"source": "stackexchange",
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Compact subgroups of a p-adic field Definition: A p-adic field is a finite extension of $Q_p$.
Question: Let $E$ be a p-adic field, $G$ is a nontrivial additive compact subgroup of $E$, how to prove: $G$ is isomorphic to $Z_p^n$ for some positive integer $n$. This isomorphism is not only a topological group isomorphism... | $G$ is compact, thus closed in $E$. It is stable by multiplication by an element of $\mathbb{Z}$ (dense in $\mathbb{Z}_p$), thus is a $\mathbb{Z}_p$-submodule of $E$.
Note that there exists a finite $\mathbb{Q}_p$-base of the vector subspace $V$ spanned by $G$, with vectors $a_1, \ldots, a_n$.
Now, for each $v \in V$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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I can't find the Centre of Mass I am currently trying to find the centre of mass (COM) with a general coordinate R (radius of big circle) of a circle which is missing another circle, with half of the radius of the big circle (R/2). Half of this smaller circle is in the first quadrant, and half is in the fourth quadrant... | Your integral should be
$$\frac{8}{\pi R^2} \left(\int_{0}^{R} x\left(\sqrt{R^2 - x^2} - \sqrt{\frac{R^2}{4} - \left(x - \frac{R}{2}\right)^2}\right)dx\right) = \frac{8R}{3\pi} - \frac R2$$
and then you get
$$x_{\text{cm}} = \frac{\left(-\frac{4R}{3\pi}\right)\cdot \frac{R^2\pi}2 + \left(\frac{8R}{3\pi} - \frac R2\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3286744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding $C$ such that $C^TAC=$diag$(I_k,-I_l,O)$ (Diagonal form) Let $A=\begin{pmatrix} 1 & 2 & 2 & 0 \\ 2 & 1 & 0 & 2 \\ 2 & 0 & 1 & 2 \\ 0 & 2 & 2 & 1 \end{pmatrix} \in M_4(\mathbb{R})$
I want to find a matrix $C$, such that $C^TAC=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end... | You’re well on the way to a solution, having done the hard part: you’ve found a matrix $D$ such that $DAD^T$ is a diagonal matrix. Now you just have to massage this diagonal matrix the desired form. There are two things that you’ll need to do, in either order:
*
*Move the negative element on the diagonal down to th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Does $\mathbb{E}[X^6] < \infty$ imply $\mathbb{E}[X^4] < \infty$?
Let $X$ be a random variable.
Does $\mathbb{E}[X^6] < \infty$ imply $\mathbb{E}[X^4] < \infty$?
My tries
*
*I know this isn't rigorous but I thought that when there was a counterexample, the rv in question would have density so I could write
$$
\m... | Jensen's inequality in fact gives $E[Y^k] \ge E[Y]^k$ for any nonnegative random variable $Y$ and any $k \ge 1$ ($k$ does not have to be an integer). Now apply this with $Y = X^4$ and $k=6/4$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Which properties does the box topology conserve? The Wikipedia article on the Product topology has a wealth of examples of properties conserved by the product topology.
The following is a quote from the linked article:
"
Separation
*
*Every product of T0 spaces is T0
*Every product of T1 spaces is T1
*Every produ... | It preserves the separation axioms up to Tychonoff.
In the Handbook of Set-theoretic Topology there is a chapter by Scott S. Williams on box products with the theorem:
If $X_i, i \in I$ are non-discrete, Hausdorff completely regular spaces then for an infinite index set $I$ $\prod_{i \in I} X_i$ in the box topology (a... | {
"language": "en",
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Why is the diameter of a circumcircle of a triangle related to the law of sines? Please refer to the following image for clarity.
In this diagram I must find AC ( which is$\sqrt21$) in order to find the radius of the circle. I know that the radius can be found by dividing the diameter by $2$, and I know that the diame... | Let $ABC$ a triangle with acute angle $\gamma$, let $M$ be the center of the circumcircle and $r$ be its radius. By moving $C$ on the arc over $AB$ the angle $\gamma$ doesn't change; move it to the intersection $C'$ of $AM$ and the circle. From Thales the triangle $ABC'$ is right-angled with a right angle at $B$. He... | {
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"timestamp": "2023-03-29T00:00:00",
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$n^n<(n-1)^{n+1}$ for integer $n\ge5$
For natural number $n\ge5$, by mathematical induction or otherwise,
prove that $n^n<(n-1)^{n+1}$.
Actually I was trying to solve the problem that I posted.
My Attempt:
Step I: Verify that when $n=5$, then the given inequality holds true;
$5^5\overset{?}{<}4^6\Rightarrow 3125<4... | I'd write $m=n-1$. Then the statement reduces to
$$\left(1+\frac1m\right)^m<\frac{m^2}{m+1}.$$
It's well-known that $(1+1/m)^m$ increases to $e$, but more naively,
$$\left(1+\frac1m\right)^m=1+1+\frac1{m^2}{m\choose 2}
+\frac1{m^3}{m\choose 3}+\cdots<1+1+\frac12+\frac16+\cdots<3.$$
But
$$\frac{m^2}{m+1}>\frac{m^2-1}{m+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3287469",
"timestamp": "2023-03-29T00:00:00",
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Verifying that a branching process is a Markov chain I would like to verify that the following model of a branching process creates a Markov chain - a Markov chain here defined as having the property that $\mathrm{P}\left\{\xi_{k+1}=a_{k+1} | \xi_{0}, \ldots, \xi_{k}\right\}=\mathrm{P}\left\{\xi_{k+1}=a_{k+1} | \xi_{k}... | I believe it is sufficient to check that for any sequence $i_1,...,i_{k+1 } $, $\mathrm{P}\left\{\xi_{k+1}\right.=i_{k+1} | \xi_{k}=i_{k}, \xi_{k-1}=i_{k-1}, \ldots \}=\mathrm{P}\left\{\xi_{k+1}=i_{k+1} | \xi_{k}=i_{k}\right\}$, considering that for a partition $\{A_m \}$ of $\{\xi_k = b_j \}$, $1_{\{\xi_k = b_j \}}... | {
"language": "en",
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$\qquad f(tx)=t^2f(x)\iff\left\langle \nabla f(x),x\right\rangle =2f(x)$ $f:\mathbb{R}^n\to\mathbb{R}$ is a differentiable function. How do I show that the following are equivalent:
(i) $\qquad f(tx)=t^2f(x)\quad\ \; \ \qquad \forall t\gt 0\land x\neq 0 $
(ii)$\qquad \left\langle \nabla f(x),x\right\rangle =2f(x) \qqua... | If (i) is true let $\phi(t) = f(tx)$ and so $\phi'(t) = \langle \nabla f (tx), x \rangle = 2 t f(tx)$. Setting $t=1$ gives the desired result.
If (ii) is true let $\eta(t) = {1 \over t^2} f(tx)$ and so
$\eta'(t) = {1 \over t^3} (\langle \nabla f (tx), tx \rangle-f(tx)) = 0$. Hence $\eta(t) = \eta(1)$ and we get the des... | {
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Baby rudin 2.34 Baby Rudin 2.34: Prove that a compact subset of a metric space is closed.
I think I have an alternative solution for Rudin 2.34. So can you check whether my steps are correct?
Let $p$ be a limit point of a set $K$. Take any neighborhood $V_s(p)$ in metric space $X$. This neighborhood must contain some e... | I agree with Siong Thye Goh that it's not a valid proof, but I'm going to rewrite it to give detail where it goes wrong.
Let $K$ be a compact subset of a metric space, and assume for the sake of contradiction that it is not closed. Then there exists a limit point $p\notin K$. We will use $p$ to construct an open cover,... | {
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Probability Exercise. Find a joint distribution. I have been working on some exercises for probability. There is a problem that I cannot even figure out where to start. So, here is the question.
*
*Let $T$ be drawn from a uniform distribution on the interval $\left[0, \,\sqrt{\,{2}\,}\, - 1\right]$.
*Accept $T$ wit... | OK, let's go by steps.
First, for $T$, there is some rejection sampling happening. Noticing that and recalling the Cauchy distribution, we see that $T$ is drawn from a truncated Cauchy distribution -- i.e. $T$ has the distribution of a standard Cauchy random variable, conditioned on lying in the interval $[0,\sqrt{2}-1... | {
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On the determinant of a Toeplitz-Hessenberg matrix I am having trouble proving that
$$\det
\begin{pmatrix}
\dfrac{1}{1!} & 1 & 0 & 0 & \cdots & 0 \\
\dfrac{1}{2!} & \dfrac{1}{1!} & 1 & 0 & \cdots & 0 \\
\dfrac{1}{3!} & \dfrac{1}{2!} & \dfrac{1}{1!} & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \ddots & \v... | Hints
Prove it by induction.
At each step, expand by minors along the top row.
At the end, think about the binomial theorem.
| {
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Finding the projection matrix onto a subspace $V$ of $\mathbb R^n$ given an orthonormal basis of $V$ Let $V\subset \mathbb R^n$ be spanned by an orthonormal basis $\{v_1,\dots, v_d\}$, with each vector represented by a column vector under the canonical basis of $\mathbb R^n$. How can I find a projection matrix $P:\math... | You can show that $P(v) = \sum\limits_{k=1}^d \langle v_k,v\rangle v_k$ is a formula for $P$ by noting that it works on an orthonormal basis for $\mathbb R^n$ extending $\{v_1,\ldots,v_d\}$, and you can use this to find the matrix entries $\langle P(e_i),e_j\rangle = \sum\limits_{k=1}^d\langle v_k,e_i\rangle\langle v_k... | {
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Is there a bijection between the set of prime ideals of norm $~ q~$, and the set of ring maps to $~F_q~$? OK, for each given ring morphism,
$\mathfrak{o}\rightarrow~F_{q}~$,
there exists another distinct arrow after composing with the Frobenius map (or a power of it).
So, if $q = p^{n}$, then there are n distinct endom... | Prime ideals of norm $q$ are in bijection with surjections $R \to \mathbb{F}_q$, up to postcomposition with Frobenius, for exactly the reason you state. I don't know whether your text has a typo or whether you have misread it, but it is not true that prime ideals of norm $q$ are in bijection with surjections $R \to \ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3288474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Function of a random variable is a random variable Consider the following two statements
Statement 1
If $X$ is a continuous random variable and $Y=g(X)$ is a function of
$X$, then $Y$ itself is a random variable.
Statement 2
A random variable is a function from a sample space S into the real
numbers.
From statement... | $X : \Omega \to \mathbb{R}$, i.e. to each $\omega$ you pair some real number $X(\omega)$.
Then, $Y$ given by $g(X)$ should be interpreted as pairing $g(X(\omega))$ to each $\omega$.
In other words $Y : \Omega \to \mathbb{R}$ and is just $g \circ X$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3288595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Goedel's representability of simple recursive sets I'm referring to Goedel's theorem as exposed here:
https://plato.stanford.edu/entries/goedel-incompleteness/
The formal system in question is named Q and is a first order formalization of natural numbers with addition and multiplication operations. A set S of natural n... | Thanks to the answer of @Noah Schweber and the description of the $\beta$ function in wikipedia here is a possibile way to write the requested formula:
$$
rem(a,b)=c \colon \qquad (c<b) \land \exists n\colon a=b\cdot n + c\\
\beta(a,b,i) = c\colon \qquad rem(a,1+(i+1)\cdot b) = y\\
a_i = c \colon \qquad \beta(a,b,i) = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3288715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Understanding the definition of an ideal of a semigroup Liapin wrote: Two-sided ideals are also all the possible unions of principal two-sided ideals. I get this as:
*
*If $I$ is an ideal of $S$ then for $a_1...a_k\in S$ we have $I= \cup(a_i)$.
*Also, I think $a_i\in S$ can be chosen from $D$-classes' representat... | Let $S$ be a semigroup and let $S^1$ be the semigroup equal to $S$ if $S$ is a monoid and equal to $S \cup \{1\}$, where $1$ is a new identity element, otherwise.
This notation is useful in the context of two-sided ideals because if $X$ is a subset of $S$, then the two-sided ideal generated by $X$ is $S^1XS^1$.
In part... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3288819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Why presheaves are generalized objects? While self studying category theory (Yoneda lemma), I came across the statement that for any category $\mathsf{C}$ the functor category $\mathsf{Fun}(\mathsf{C}^{op}, \mathsf{Set})$ represents generalized objects of $\mathsf{C}.$
Here generalized means bunch of objects of $\math... | The previous answers are very good, but I also like to always keep in mind a simple example when working with presheaves, to get a feel for what all this means.
Luckily, we have a very simple and intuitive category of presheaves to wrok with. Consider the category $\mathbb{G}$, whose objects are $[0]$ and $[1]$, and wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3288940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
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Vector field with constant length Is it correct, that for some pseudo-Riemannian manifold $M$, $X \in \mathfrak{X}(M)$, if $g(X,X)=1$, then the integral curves for $X$ are geodesics?
I have the following explanation: Let $\gamma$ be a curve, s.t. $\gamma'(t)=X_{\gamma(t)}$. Then
$g(\gamma', \nabla_{\gamma'} \gamma')=\g... | $g(\gamma',\nabla_{\gamma'}\gamma')=0$ means that the velocity and acceleration vectors are orthogonal, this does not necessarily implies $\nabla_{\gamma'}\gamma'=0$. A simple example is the punctured plane $\mathbb R^2\setminus\{0\}$. In polar coordinates, take the smooth vector field $X=(-\sin\theta,\cos\theta)$. The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3289040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Is there a polynomial $p(x)$ with integer coefficients such that $p(2013)=1789$ and $p(1515)=1830$?
Problem:
Is there a polynomial $p(x)$ with integer coefficients such that
$p(2013)=1789$ and $p(1515)=1830$?
My attempt:
After ruling out polynomials of degree 1, 2 and 3, and with further inspection, it appeared ... | $p(2013)-p(1515)$ is divisible by $2013-1515=498,$ but $1789-1830=-41$ is not.
It follows from the following reasoning.
Let $p(x)=a_0x^n+a_1x^{n-1}+...+a_n,$ where $a_i\in\mathbb Z$.
Thus, $$p(m)-p(k)=a_0(m^n-k^n)+a_1(m^{n-1}-k^{n-1})+...+a_{n-1}(m-k)=$$
$$=(m-k)(a_0(m^{n-1}+m^{n-2}k+...+k^{n-1})+a_1(m^{n-2}+...+k^{n-2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3289174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove that if A is an upper triangular matrix with distinct values on the main diagonal, then A is diagonalizable. I know that for a matrix to be diagonalizable, the eigenvectors of its eigenvalues must be linearly independent. However, I am unable to prove the theorem in the title.
| Given the matrix $A$ is a $k × k$ upper triangular matrix with distinct diagonal entries, $a_1, ~a_2, \cdots, ~a_k$.
The determinant of an upper triangular matrix is the product of its diagonal entries.
So $$f(t)= \det(A-tI)=(a_1-t)(a_2-t)\cdots(a_k-t) $$
Setting that to $~0~$, your $~k~$ eigenvalues are all distinct... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3289268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Stably trivial vector bundles over a torus It is not difficult to see that there are non-trivial stably trivial bundles of rank $2$ for a closed surface $\Sigma$ of genus $\neq 1$ using that $T \Sigma$ is non-trivial, but what happens in the genus $1$ case? Are there stably trivial real vector bundle over $T^2$ which a... | I solved the question, I'll leave the answer for reference.
The answer is yes for the first question and no for the second. Consider the tangent bundle of $T^2 \times S^2$, which is trivial. $T^2 \times S^2$ contains tori of nonzero self-intersection number. Pick one such torus, say $Y$, and let $E \rightarrow Y$ be th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Expected time until there are two students
Students arrive at a help centre according to a rate $r$ Poisson process. When there are n ≥ 1
students in the centre, the first one to leave does so at a random $Exp (2r)$ time. Suppose that there are presently no students in the centre. What is the expected time until the... | Let $T_n$ be the interarrival times, $S_n$ the service times, and $$\tau = \inf\{t>0:X(t)=2\} $$ where $X(t)$ is the number of customers in the system at time $t$. Then
\begin{align}
\mathbb E[\tau] &= \mathbb E[T_1] + \mathbb E[T_1]\mathbb P(T_1<S_1) + (\mathbb E[S_1]+\mathbb E[\tau])\mathbb P(S_1<T_1)\\
&= \frac1r + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3289508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Logarithmic inequality (looking for a better solution) $1+\sqrt{17-\log_{x}{2}} \cdot \log_{2}{x^7} \geq \log_{2}{x^{27}}$
Let $t = \log_{2}{x}$. Then we get (taking account of the fact that $x>0$ and $x \ne 1$
$$1+\sqrt{17-\frac{1}{t}}\cdot 7t \geq 27t$$ or
$$\sqrt{17-\frac{1}{t}} \cdot 7t \geq 27t-1 \tag{1}$$
It's cl... | As you wrote, we have to have
$$t< 0\qquad\text{or}\qquad t\ge\frac{1}{17}\tag1$$
Now, we have
$$\begin{align}&1+\sqrt{17-\frac{1}{t}}\cdot 7t \geq 27t
\\\\&\iff 7t\sqrt{17-\frac{1}{t}} \geq 17t-1+10t
\\\\&\iff 7t\sqrt{17-\frac{1}{t}}\geq t\left(17-\frac 1t\right)+10t
\\\\&\iff t\left(17-\frac 1t\right)-7t\sqrt{17-\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3289791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Nonisomorphic Groups with the Same Order and Exponent I am trying to find two nonisomorphic finite abelian groups with the same order and exponent. I've tried solving this problem for a fews days, but I have had no luck. I tried looking for pairs of such groups of the form $\Bbb{Z}_{mn}$ and $\Bbb{Z}_m \oplus \Bbb{Z}_n... | Let $\mathbf{V}=\{(1),(12)(34),(13)(24),(14)(23)\}$ be the four-group, and let $\Gamma_{4}=\langle i\rangle=\{1,i,-1,-i\}$ be the multiplicative cyclic group of fourth roots of unity, where $i^{2}=-1.$ Then these are the two abelian groups with the same order. We will see they are not isomorphic.
If there were an isomo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3289931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Exercise showing that $\nabla f(r) = \frac{\mathrm{d} f}{\mathrm{d}r}\cdot \frac{\underline{r}}{r} $ Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function. Let
$$\underline{r}(x,y,z) := \begin{pmatrix}x\\y\\z\end{pmatrix}$$
be a vector field in Cartesian coordinates. The length $r$ of the vector $\u... | I'm going to use the notation $\vec{r} = (x, y, z)$. We are ultimately trying to find $\nabla f(r) = \nabla f(g(\vec{r}))$ where $g(r) = \|\vec{r}\|$. Now we know from the chain rule that $\nabla f(g(\vec{r}))=\frac{df}{dg(\vec{r})} \nabla g(\vec{r})$, which gives the desired answer.
When remembering the chain rule, i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3290035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Growth of the gradient of $f(x+y) \leq f(x) f(y)$ Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}_{\geq 0}$ be a radial continuous function and $C^2$ on $\mathbb{R}^3 \setminus \{0\}$ which satisfies the following functional inequality
$$ f(x+y) \leq f(x) f(y) $$
Does there exist constants $c,d$ such that for all $x\in \ma... | The answer to my question can essentially be found in the answer of another question here Derivatives of functions satisfying Euler-ish inequality $f(x+y)\le f(x)f(y)$.
Many thanks to @PhoemueX and @MaximilianJanisch (in particular for pointing out that we can actually smooth it without any problems).
Let me just give ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3290169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Solve $ab + cd = -1; ac + bd = -1; ad + bc = -1$ over the integers I am trying to solve this problem:
Solve the system of equations
\begin{align}
\begin{cases}
ab + cd = -1 \\
ac + bd = -1 \\
ad + bc = -1
\end{cases}
\end{align}
for the integers $a$, $b$, $c$ and $d$.
I have found that the first equation gives $d... | Hint:
Squaring the equations gives
\begin{align}
\begin{cases}
a^2b^2 +2abcd+ c^2d^2 = 1 \\
a^2c^2 +2abcd +b^2d^2 = 1 \\
a^2d^2 + 2abcd+b^2c^2 = 1
\end{cases}
\end{align}
Subtracting them yields
\begin{align}
\begin{cases}
(a^2-b^2)(c^2-d^2)=0 \\
(a^2-c^2)(b^2-d^2)=0 \\
(a^2-d^2)(b^2-c^2)=0
\end{cases}
\end{align}
Now ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3290290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Analytic continuation of Riemann zeta $\zeta(s)$ from the complex $\mathbb{C}$ to quaternion $\mathbb{H}$? One way to define Riemann zeta function is by the analytic continuation of
$$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$$
for the domain $Re(s)>1$ to the f... | $\Bbb{H}$ is just a sub-algebra of $M_2(\Bbb{C})$.
For $A \in M_n(\Bbb{C})$ use the Jordan normal form to obtain $A = P J P^{-1} = P (D+N)P^{-1}$ where $D$ is diagonal and $DN=ND$ and $N^n = 0$. Let $f(s) = (s-1)\zeta(s)= \sum_{k=0}^\infty c_k s^k$ which is entire then $$P^{-1} f(A)P =f(D+N)=\sum_{k=0}^\infty c_k (D+N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3290380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Geometric or Trigonometric way to fit a circle to series of points I'm working on a computer vision project where I'm trying to detect a moving colored circle (doughnut actually). After some work I'm able to get it working pretty well except for these two edge cases. My results are always a small set of approximately 5... | One way to fit points to a circle is to do linear regression on the equation in the form $x^2 + a x + y^2 + b y + c = 0$. Writing this as
$(x+a/2)^2 + (y+b/2)^2 = (a^2 + b^2)/4 - c$, this corresponds to a circle of centre $(-a/2, -b/2)$ and radius $\sqrt{(a^2+b^2)/4 - c}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3290512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
a proof of that $~C_c(X)~$ is dense in $~L^p~$ in Rudin's RCA I'm reading theorem $3.14$ in Rudin's RCA. The assertion is here:
For $1\leq p\leq \infty$, $~C_c(X)$ is dense in $L^p(\mu)$. Note that $C_c(X)$ denotes the class of continuous complex functions that support is compact and the measure $\mu$ has the properti... | If $s$ is a simple function, say $f= \sum\limits_{k=1}^{n} c_kI_{A_k}$ then $\sup |f|=\|f\|_{\infty}$ provided none of the sets $A_k$ has measure $0$. In this proof we can assume that each $A_k$ has positive measure.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3290656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Integrating a scalar function on a manifold So I have the following action in Minkowski spacetime $(M, \eta)$:
$
S[\phi] = \int \eta^{\alpha \beta}(\partial_{\alpha} \phi)(\partial_{\beta} \phi)\sqrt{-\eta}d^2x
$
Now, I have the following two charts $(x^{\alpha})$ and $(\xi^{\alpha})$ related as :
$
(x^{\alpha}(\xi^0,\... | I ended up finding the answer, I will leave the comment I wrote on my MSc. thesis for someone else to learn. If some experienced mathematician finds this answer incorrect let me know.
"Here is where the standard abuse of notation shines with malice. Expressions above naively suggest the actions are everything but equal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3290868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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"Almost" in the kernel Is there a notion of "almost kernel" of a matrix? Namely, a vector $v$ is "almost" in the kernel of the matrix if it is mapped "close" to 0. And thus the "almost kernel" of a matrix is a subspace (this would need to be verified) of all vectors that are mapped closed to zero.
I guess one could de... | You can consider for fixed $\epsilon>0$ the set
$$
C:=\{v: \ \|Av\|\le \epsilon \|v\|\}.
$$
This has some nice properties: it is a closed and convex cone. Also $v\in C$ implies $-v\in C$.
Such constructions are useful in constrained optimization and second-order sufficient optimality conditions. Second-order necessary... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3290957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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mathematical Induction (algebra) assume there is a function like $f:A→B$ which is injective, why it means $\left|A\right|\le\left|B\right|$ or in another way why a function like $g:B→A$
stands for $\left|B\right|\le\left|A\right|$
my problem has been clearly explained here:
assume a set like $A=[1,2,3,4]$ and $B=[1,... | In old books you can find a difference between function and aplication.
*
*A function $f\colon A\to B$ has as domain a subset of $A$.
*An application $f\colon A \to B$ has as domain all $A$
for people, in general , those means the same.
If you want to speak strictly, the intuition behind injectivity is:
Y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3291105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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How the roots are calculated for this function?
It says that the roots of $$f(x) = x^{-1} \sin(x^{-1}\log(x))$$ are defined as
$$1 > a_{1} > a_{2} > \cdots > 0$$
where $a_{i} = \exp(-b_{i})$ and $b_{i}$ is the unique solution to the equation $b \exp(b) - i\pi = 0$, $1 < b < \infty$.
I am wondering how are formulas of... | Because the $x_i>0$, we can find values $b_i$ such that $x_i = \exp(-b_i)$. Plug this into $f(x)$ to get
\begin{align}
f(\exp(-b_i)) &= \exp(-b_i)^{-1}\sin{(\exp{(-b_i)}^{-1}\log{(\exp{(-b_i)})})}\\
&= \exp(b_i)\sin(-b_i\exp{b_i})
\end{align}
Setting this equal to $0$ and using the fact that $\exp(b_i)>0$ we get
$$\sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3291282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\sup \{ \varepsilon x:\, x\in A\}=\varepsilon \sup A$ Since this is an introductory course to real analysis I am looking for the most simple and direct proof. Based mostly on definitions.
Let $\varepsilon$ be a positive real number. If $A$ is a non-empty bounded subset of $\mathbb{R}$ and $B = \{\varepsilo... | If $S$ is a subset of $\mathbb{R}$ with a minimum, $m$, then $\epsilon m \leqslant \epsilon x$ for all $x \in S$, i.e. the set $\epsilon S$ has the minimum $\epsilon m$.
If $S$ is the set of all upper bounds of $A$, then $\epsilon S$ is the set of all upper bounds of $\epsilon A$, and the result follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3291350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
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Let $X \subseteq \Bbb Q^2$. Suppose each continuous function $f:X \to \Bbb R^2$ is bounded. Then $X$ is finite.
True or false: Let $X \subseteq \Bbb Q^2$. Suppose each continuous function $f:X \to \Bbb R^2$ is bounded. Then $X$ is finite.
Now it will be compact for sure just by using distance function.
Now what can w... | Hint: Consider $X = \{(1,0), (1/2,0), (1/3,0), ..., (0,0)\}$. What can we say about the behaviour of $f(x)$ as $x\to (0,0)$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3291490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Cannot find the p.d.f. with jacobian transformation. Suppose $X$ and $Y$ are continuous random variable with joint probability density function
$$f(x,y)=
\begin{cases}
12xy(1-x)&0<x<1,0<y<1\\
0&\text{for other } x
\end{cases}.
$$
If $Z_1=X^2Y$, determine the probability density function of $Z_1$.
Because of the p.d.f. ... | $Z_1$ and $Z_2$ cannot take all values between $0$ and $1$. There is an extra inequality they have to satisfy: $Z_1 =X^{2}Y <X^{2}=Z_2^{2}$. So the joint density vanishes if $z_1 >z_2^{2}$. Now see if yo get the density of $Z_1$ correctly.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3291566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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How to prove $ \frac{\ln^k(1+x)}{k!}=\sum_{n=k}^\infty(-1)^{n-k} \begin{bmatrix} n \\ k \end{bmatrix}\frac{x^n}{n!}$
Prove the following formula involving Stirling numbers of the first kind:
$$\frac{\ln^k(1+x)}{k!}=\sum_{n=k}^\infty(-1)^{n-k} \begin{bmatrix} n \\ k \end{bmatrix}\frac{x^n}{n!}$$
where $\begin{bmat... | As said within the comment section it is sufficient to visit the Wikipedia Page and therefore no need to invoke some kind of Harmonic Numbers here. Within the subsection Generating Functions we eventually find the following paragraph:
A variety of identities may be derived by maniplulating the generating function:
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3291701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Calculate $\phi * \phi$
Show that $$\phi * \phi(x)=\begin{cases} >0 , \operatorname{if} x\in(0,4\pi)\\
= 0, \operatorname{otherwise} \end{cases}$$
where $\phi: \mathbb R \to \mathbb R$ and $\phi(t)=\begin{cases} 1-\cos{(t)} , \operatorname{if} t\in[0,2\pi]\\
0, \operatorname{otherwise} \end{cases}$
My idea:
$$\phi ... | Since $\;\phi=0\;$ outside $\;[0,2\pi]\;$ , we get:
$$\phi * \phi(x)=\int_{\mathbb R} \phi(x-y)\phi(y)dy=\int_0^{4\pi}\left(1-\cos(x-y)\right)\left(1-\cos y\right)\,dy=$$
$$=\int_0^{4\pi}\left(1-\cos y-\cos(x-y)+\cos y\,\cos(x-y)\right)\,dy=4\pi-\overbrace{\left.\sin y\right|_0^{4\pi}+\left.\sin(x-y)\right|_0^{4\pi}}^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3291830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find values of $x$ so that the matrix is invertible Find values of $x$ so that the matrix is invertible
$$A=\begin{pmatrix}
x & 0 & x \\
x & 2 & 1 \\
2x & 0 & 2x \\
\end{pmatrix}$$
I know that a matrix is invertible if determinant is not $0$, but I don't know how to find the $x$ values. I feel is a tricky questio... | Note that $\forall x, R_3=2R_1\implies Rank(A)<3\implies \det(A)=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3291925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Rewriting infinity as a limit to infinity (in terms of Fourier series) Put informally: When writing down the complex Fourier series of a function, is it proper to write
$$\displaystyle\sum_{n=-\infty}^\infty \tag*{(1)}$$
or
$$\displaystyle\lim_{k\to\infty}\displaystyle\sum_{n=-k}^k? \tag*{(2)}$$
From what I've seen, I ... | The symbolic form $$\sum_{n=-\infty}^\infty \dfrac{1}{z+n}$$ means
$$
\lim_{a,b\to\infty}\sum_{n=-a}^b \dfrac{1}{z+n},
$$
that is, the limit should not depend on the path to infinity that the pair $(a,b)$ takes in the grid $\Bbb N\times\Bbb N$. In the given example, this is not the case as the one-sided series are harm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3292044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solving the diophantine equation $x^3+y^3 = z^6+3$ I've the following problem:
Show that the congruence $x^3+y^3 \equiv z^6+3\pmod{7}$ has no solutions. Hence find all integer solutions if any to $x^3+y^3-z^6-3 = 0.$
We can rearrange the first equation to $z^6 \equiv (x^3+y^3-3) \mod{7}$. But $z^6\equiv 1\mod{7}$ so... | Any solution for $x^3+y^3-z^6-3=0$ is also a solution mod $7$. Therefore there are no solutions to the original equation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3292195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Use Ito lemma to solve some SDE Take B$_t$ to be a Brownian motion and Z$_t$ = e$^{\int_{0}^{t}g(s,w)dB_s-\frac{1}{2}\int_{0}^{t}g^2(s,w)ds}$, how can we apply Ito lemma to get SDE of Z$_t$?
I set Z$_t$ = f(t, B$_s$), then by Ito lemma, it follows that
dZ$_t$ = f$_t$dt + f$_{x}$dB$_s$ + $\frac{1}{2}$f$_{xx}$dt, but ... | You should write $Z_t=e^{Y_t}$ and notice that you know the SDE of $Y_t$ as it is provided in integral form. Then you can apply Ito's lemma on the exponential function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3292328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Convexity implies $ \frac{\varphi(c)-\varphi(c-h)}{h} \leq \frac{\varphi(c+h)-\varphi(c)}{h}$?, $h>0 $ Suppose $\varphi $ is a convex function on the real line. I wonder if the following is true? For $h>0 $
$\frac{\varphi(c)-\varphi(c-h)}{h} \leq \frac{\varphi(c+h)-\varphi(c)}{h}$
This seems like a trivial fact that... | May be, you could just use Taylor expansions around $h=0$. This would give
$$\text{lhs}=\frac{\varphi(c)-\varphi(c-h)}{h}=\varphi '(c)-\frac{1}{2} h \varphi ''(c)+\frac{1}{6} h^2 \varphi
^{(3)}(c)-\frac{1}{24} h^3 \varphi ^{(4)}(c)+O\left(h^4\right)$$
$$\text{rhs}= \frac{\varphi(c+h)-\varphi(c)}{h}=\varphi '(c)+\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3292583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Merging two functions So I have two fuctions like this :-
$f(x) = (x/5)^2$
and $g(x) = \sqrt{(x/5)}$
and a third fuction as a combination of both
$ h(x) = \Biggl[ { }^{ x\; \lt \; 5 : \; f(x) }_{ x \;\ge \; 5: \; g(x)}\Biggr] $
When I put $x =5$ in the first function I get $f(5) = 1$ and in the second one I get $g(5) ... | $$h(x) = \left ( \frac 1 {10} \left (x - 5 - \sqrt {\strut (x - 5)^2} \right ) + 1 \right )^2 \sqrt {\frac 1 {10} \left (x - 5 + \sqrt {\strut (x - 5)^2} \right ) + 1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3292680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Can anyone give a real life example to illustrate why does the principal axis that has the maximal variance retain the most information? One job in PCA is to maximize variance, because the
principal axis that has the maximal variance retain the most information.
Why is that? How to understand this in a easy or concre... | This could be quite difficult to be seen in a real life example since it is a theoretical or abstract assumption.
When you do PCA you want to project data in a low dimensional space. This projection obviously will loss information so you want to retain the most information. As the PCA assumption, the eigen-values of th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3292809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Which method to solve this differential equation? $$x'=(x+t+1)^2$$
I need to solve this differential equation but do not know how. We cannot use separation of variables so my only guess here would be to use an integrating factor but how would I find that?
EDIT:
the official answer is $x(t) = −t − 1 + \tan(t + C)
$
EDIT... | Substitute $$x+t+1=u$$ then $$x'+1=u'$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3292875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Sort of positive matrix Hy i want to find a block real or complex matrix of the form $\begin{pmatrix}C&B&C\\B&B&A\\C&A&A\end{pmatrix} $ such that it will be positive semi-definite but not such $A=B=C.$ All blocks are of same size and hermitian.
I couldn't find such a matrix in dimension three so that's why i am asking... | In the case that $A, B, C \in \mathbb{R}$, you can use the Sylvester's criterion for positive semi-definiteness. In short, this means any square sub-matrix along the diagonal must have a non-negative determinant. There are two cases we must consider:
*
*$A = 0$. Sylvester's Criterion tells us that $A, B, C \geq 0$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3293013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Probability of getting exactly one pair of aces, knowing there's one ace in the draw? We draw 5 cards in a 52 cards game. What is the probability of getting exactly one pair of aces, knowing there's one ace in the draw ?
I know that the answer is $\frac{\binom{4}{2} \binom{48}{3}}{\binom{52}{5}-\binom{48}{5}}$, but I d... | It looks like you assume when the guaranteed ace is gonna appear - this is not the same. It can be drawn at any of the five draws.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3293128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How do we know $\sin$ and $\cos$ are the only solutions to $y'' = -y$? According to Wikipedia, one way of defining the sine and cosine functions is as the solutions to the differential equation $y'' = -y$.
How do we know that sin and cos (and linear combinations of them, to include $y=e^{ix}$) are the only solutions to... | Starting from $y'' = -y$, we can add $y$ to form
$$y'' + y = 0$$
This is a homogeneous second order linear differential equation which we can simplify by writing the characteristic polynomial as
$$r^2 + 1 = 0$$
or
$$r = \pm i$$
which are distinct roots. The general solution can be written as
$$y(x) = c_1\sin(x) + c_2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3293359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 9,
"answer_id": 2
} |
Isomorphism, Homeomorphism and the necessity of proving individual invariants I was wondering if there was a way of avoiding having to prove individual invariants of isomorphism and/or homeomorphism are, in fact, invariants. Consider homeomorphisms. We have to prove that compactness is a topological property. The wikip... | The fact that something is preserved under isomorphism (in whatever category) is the formal way of saying that it is "an inherent property of your structure".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3293469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Could you please recommend some open courses on Real Analysis, mainly about Lebesgue measure and Lebesgue Integral? I am a Chinese student and will be enrolled in the course about Lebesgue integral. I want some open courses to help preview the course. A little bit content about functional analysis is okay.
Thank you!
| You may find this useful for real analysis, and this for functional analysis.
Moreover, you can find more video courses here about some other topics in mathematics.
If Chinese is OK to you, this course from Shanghai Jiao Tong University is acessible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3293571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Confused with a system of equations with three variables that has infinitely many solutions I'm studying High School Algebra and it had this question:
Solve the system by equations:
\begin{align*} x + y - z &= \,0 \\ 2x + 4y - 2z &= 6 \\ 3x +
6y - 3z &= \,9 \end{align*}
The solution was:
infinitely many so... | Hint: Dividing the second equation by $2$ and the third by $3$ we get
$$x+y-z=0$$
$$x+2y-z=3$$
$$x+2y-z=3$$
the second and the third equation are the same.
Multiplying the first equation by $-1$ and adding to the second we get $y=3$.
Plgging this into the first and second equation we get
$$x-z=-3$$
$$x-z=-3$$ so we obt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3293662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Algebra and Combinatorics books for Mathematical Olympiads Could you kindly point out to me, some good contest-preparation book's to develop theory and problem solving skills in Algebra?
It would be good if the book is less of theory and more of problems. I have "Principles and techniques in Combinatorics", but I woul... | 'Problem Solving Strategies' by Arthur Engel is a classic for math olympiad training at a fairly advanced level. Contains hundreds of problems that take one line to state, two lines to solve with basic means and are still difficult enough to humble olympiad veterans.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3293811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Probability of hitting the target for the nth time on the mth throw A boy is throwing stones at a target. Probability of him hitting the target is $\frac{1}{2}$ . Then the probability of him hitting the target for the 5th time in 10 throws is ?
The answer given is 1/2 .
Now ,the reason given is that no matter how man... | In probability theory this is called a negative binomial distribution. What it does is answer the question of what is the probability that you will have $n$ successes after $m$ attempts. Technically this notation is wrong, but I modified it to match your question.
For example, you could say that in the event of a Best... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3293907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Calculate the value of $I(9)/I(3)$ when $I(m)=\int_{0}^{\pi}\ln\left(1-2m\cos(x)+m^2\right)\,dx$ We are given that $$I(m)=\int_{0}^{\pi} \ln\left(1-2m\cos (x)+m^2\right)\,dx.$$
I could see that there weren't any standard techniques to calculate this integral directly so I concluded that there must be some kind of reduc... | \begin{align*}
2I(m)&=\int_{0}^{\pi} \ln\left(1-2m\cos (x)+m^2\right)+\ln\left(1+2m\cos (x)+m^2\right)\,dx\\
&=\int_{0}^{\pi}\ln\left(1-2m^2\cos 2x+m^4\right)\, dx\\
&=\frac{1}{2}\int_{0}^{\color{red}{2\pi}}\ln\left(1-2m^2\cos t+m^4\right)\, dt && (\text{let }t=2x)
\end{align*}
Now use the fact that if $f(2a-x)=f(x)$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3294030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Unital commutative Banach algebra $A$, $A/\operatorname{radical}(A)$ has no quasinilpotent elements I am trying to show that for a unital commutative Banach algebra $A$, $A/\operatorname{radical}(A)$ has no quasinilpotent elements (where $\operatorname{rad}(A)=\{x\in A: x \,\text{quasinilpotent}\}$). I know that $\oper... | Assume $x\in A$ such that $x+\operatorname{rad}(A)$ is nilpotent in $A \operatorname{rad}(A)$. This implies that for all $\varepsilon > 0$ there is an $n\in \mathbb N$ and $r\in \operatorname{rad}(A)$ for which $|x^n-r|\leq\varepsilon^n$.
$r$ is quasi-nilpotent so there exists $M\in\mathbb N$ s.t. for all $m>M$, $|r^m|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3294138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find function $f'(x)$ such that its domain $D'=\mathbb{R}$, $f'(x)=f(x)$ $\forall x\in D$ and $f'(x)$ is continuous. Let $f(x)=\frac{^3\sqrt{x^3+3x^2+7}}{x+2}$.
I was asked to find $f'(x)$ such that
$a)$ the domain $D'$ of $f'(x)$ is $\mathbb{R}$,
$b)$ $f(x)=f'(x)$ $\forall x\in D$, with $D$ being domain of $f(x).$
$... | We have
$$\lim_{x\to-2^+} \frac{\sqrt[3]{x^3+3x^2+7}}{x+2} = \frac{\sqrt[3]{11}}{0^+} = +\infty$$
so $f$ cannot be extended to a continuous function on $\mathbb{R} \to \mathbb{R}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3294231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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