Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Direct sum of compact operators is compact I have that $T_n$ are bounded operators on $H_n$ ($n\geq 1$) and that $\sup ||T_i||<\infty$. Define $T=\oplus T_n$ and $H=\oplus H_n$. I want to show that $T$ is compact iff $T_n$ is compact for all $n$ and $||T_n||\rightarrow 0$.
Here is what I have so far:
Assume that $T$ ... | Fix $\varepsilon>0$. Choose $n_0$ such that $\|T_n\|<\varepsilon$ if $n> n_0$. For each $n=1,\ldots,n_0$, there exists a finite-rank $S_n$ with $\|S_n-T_n\|<\varepsilon$. Put $S_n=0$ for $n>n_0$. Then $\bigoplus_1^{n_0}S_n$ is finite-rank and
$$
\|T-S\|=\sup\{\|S_n-T_n\|,\ n\in\mathbb N\} <\varepsilon.
$$
So $T$ is a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/681707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Showing if a function is injective or surjective problem $F : \Bbb{P}(X) \rightarrow \Bbb{P}(X) ; U \rightarrow (U-A) \cup (A-U)$
My intuition has been telling me that this function is bijective but I having the most difficult time trying to show this. Any help would be appreciated, thank you!!
edit: So far, I introduc... | Trying it directly seems rough, especially since it's hard to picture if $U_1 = U_2$ using a Venn Diagram. I'm not sure how familiar you are with this idea, but $(U - A) \cup (A-U)$ is also called the symmetric difference of $U$ and $A$. It is denoted $U \triangle A$. Using the associativity of $\triangle$ with your fu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/681798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Lower semicontinuous functional Consider the space $A=(C^1([0,1],\mathbb R),\|.\|_{L^\infty})$ norm and look at the functional
$$
\mathcal F: A\to\mathbb R_+, f\mapsto \int_{0}^1\left|f'(t)\right|~\mathrm dt.
$$
This functional is not continuous. My question: Is it lower semi-continuous?
And could there be a meaningful... | This functional is DEFINITELY continuous, when defined on $C^1[0,1]$, with its standard norm.
Let $f\in C^1[0,1]$, then
$$
\lvert Af\rvert=\Big\lvert\int_0^1 f'(t)\,dt\Big\rvert\le \int_0^1 \lvert f'\rvert\,dt
\le \int_0^1 \| f'\|_\infty\,dt=\| f'\|_\infty\le \| f\|_\infty+\| f'\|_\infty=\|f\|_{C^1[0,1]}.
$$
Is it lowe... | {
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If $m = g \circ f$ in a dagger category and $m$ is an isometry, is it possible that $f$ fails to be an isometry? Question. Suppose $m : A \rightarrow B$ is an isometry in a dagger category (by which I mean that $m^\dagger \circ m=\mathrm{id}_A$), and that we're given arrows $f : A \rightarrow Y$ and $g : Y \rightarrow ... | Counterexample: let $m$ be the identity and pick $f$ and $g$ any inverses that aren't daggers of each other. For example in the dagger category of complex matrices with conjugate transpose:
$$m=\begin{pmatrix}1&0\\
0&1\\
\end{pmatrix}$$
$$f=\begin{pmatrix}1/2&0\\
0&2\\
\end{pmatrix}$$
$$g=\begin{pmatrix}2&0\\
0&1/2\\
\... | {
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Asymptotic expansion of $\sum_{k=0}^{\infty} k^{1 - \lambda}(1 - \epsilon)^{k-1}$ I'm seeing a physics paper about percolation (http://arxiv.org/abs/cond-mat/0202259). In the paper the following asymptotic relation is used without derivation.
$$
\sum_{k=0}^{\infty} k P(k) (1 - \epsilon)^{k-1} \sim \left<k\right> - \lef... | I have not been able to see how is coming the $\epsilon^{\lambda - 2}$ term. However, and, may be, this could be a track $$
\sum_{k=0}^{\infty} k P(k) (1 - \epsilon)^{k-1}=\frac{c \Phi (1-\epsilon,\lambda -1,0)}{1-\epsilon}$$ when $P(k) = c k^{-\lambda}$ ($\Phi$ being the the Hurwitz-Lerch transcendent $\Phi(z,s,a)$ fu... | {
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"timestamp": "2023-03-29T00:00:00",
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Abstract Algebra - Permutations I'm asked to show that $(1,2,3) \in S_3$ generates a subgroup which is normal.
I know that I could show it explicitly but that would be tedious. I think it may have to do with the fact that $(1,2,3)$ generates all even permutations but I'm sure there's something I'm missing. Any help wou... | Actually, showing it directly would be relatively untedious as far as getting your hands dirty with group theory is concerned. There are only six elements in $S_3$, and $\langle(123)\rangle$ has three elements so it only has two cosets (where did I get "two" from?), so there isn't much work involved. I urge you to do t... | {
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"timestamp": "2023-03-29T00:00:00",
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Proof of closed form of recursively defined sequence Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be defined by $f(1) = 5, f(2) = 13$, and for $n \ge 2, f(n) = 2f(n - 2) + f(n - 1)$. Prove that $f(n) = 3\cdot 2^n + (-1)^n$ for all $n \in N$
So far I've proved that $f(n)$ is true when $n = 1, 2$.
For $k \ge 3$, assume that... | Hint $\ $ Let $\,S g(n) = g(n\!+\!1)$ be the shift operator. $(S-2)(2^n) = 0 = (S+1)(-1^n)$ so their product $(S-2)(S+1) = S^2\!-S-2$ kills $\, f_n = c\,2^n + d (-1)^n\,$ for any $\,c,d\,$ independent of $\,n.$ Therefore we deduce $\, 0 = (S^2\!-S-2)f_n = f_{n+2} - f_{n} - 2f_n,\ $ i.e $\ f_{n+2} = f_{n+1} + 2 f_n.$
Re... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Finding Number of Automorphisms of Z8? I'm trying to find the number of automorphisms of Z8. When I google around, I find stuff like:
There are 4 since 1 can be carried into any of the 4 generators.
The problem hint tells me to make use of the fact that, if G is a cyclic group with generator a and f: G-->G' is an iso... | Note that $1$ generates $\mathbb{Z}_{8}$ as a group, so any group morphism $\varphi:\mathbb{Z}_8\rightarrow\mathbb{Z}_8$ is determined by $\varphi(1)$. Furthermore, if $\varphi$ is an automorphism, then $\varphi(1)$ generates $\mathbb{Z}_8$. The possible generators of $\mathbb{Z}_8$ are $1,3,5,7$. It then remains to ch... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/682286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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solve the differential equation by integrating directly I am trying to solve a differential equation and I don't know how to solve it when it comes to integrating directly. I'd like to know how to do this so I can start doing other problems. Thanks in advance.
Solve the differential equation by integrating directly
$${... | $$y'(t)=4t+\frac4{(t+1)^2}\implies y(t)=2t^2+\frac{4t}{t+1}+y(0)$$
$$y'(t)=\frac{4t+4}{(t+1)^2}\implies y'(t)=\frac{4}{t+1}\implies y(t)=4\log(t+1)+y(0)$$
The solution of the second version on the interval $(-\infty,-1)$ would be
$$
y(t)=4\log|t+1|+y(-2).
$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Find the required digit What is the digit at the $50^{th}$ place from left of $(\sqrt50 +7)^{50}$
I thought of binomial expansion but it was way too lengthy. Can anyone suggest any other way?
| Note that $a=7+\sqrt{50}$ and $b=7-\sqrt{50}$ satisfy $a+b=14$ and $ab=-1$. Note that $a\gt 14$ so that $a^n \gt 10^n$ (by some margin) so that $a^{50}\gt 10^{50}$ and $b^{50}\lt 10^{-50}$
Note that if $Y_n=a^n+b^n$ we have $Y_n=14Y_{n-1}+Y_{n-2}$ so the $Y_r$ are integers. ($a$ and $b$ satisfy $x^2-14x-1=0$, $Y_0=2, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Smallest next real number after an integer This might be a silly question, but is it possible at all for n.00000...[infinite zeros]...1 to be the next real number after n? If not, why not?
Firstly, I know (I think) that $$\lim_{x\to \infty} \frac{1}{10^x} = 0$$ but I'm not talking about taking its limit. Surely I'm not... | You can define a number like this one if you consider hyperreal numbers: and it would be written $n + \epsilon$, which is greater than n but less than any other number greater than n.
Indeed, if you could define n.00000...[infinite zeros]...1 , then you should be able to define n.00000...[infinite zeros]...2 and many o... | {
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How do I prove an inequality of a strict concave function? The function $f(x)$ is strict concave, strict increasing, and $f(0) = 0$. $a, b \in \mathbf R \; and \; a < b$, how can I get that $\frac{a}{b} < \frac{f(a)}{f(b)}$?
Thank you!
Oh sorry I forgot to mention that $f:\mathbf R_{+} \rightarrow \mathbf R_{+}$ is co... | $\frac{f(a)}{f(b)}-\frac{a}{b}=\frac{bf(a)-af(b)}{bf(b)}$. Now, $bf(b)>0\forall b\in \mathbb{R}$ since $f(\cdot)$ is increasing and $f(0)=0$. Now, $$bf(a)-af(b)=(b-a)f(a)-a(f(b)-f(a))$$ For concave functions, if $f$ is differentiable, $\forall x,y\in \mathbb{R}$ $$f(y)\le f(x)+(y-x)f'(x)$$ So, $$f(b)-f(a)<(b-a)f'(a)\\
... | {
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"timestamp": "2023-03-29T00:00:00",
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Combinatorial proofs of the identity $(a+b)^2 = a^2 +b^2 +2ab$ The question I have is to give a combinatorial proof of the identity $(a+b)^2 = a^2 +b^2 +2ab$.
I understand the concept of combinatorial proofs but am having some trouble getting started with this problem, any help would be appreciated.
| Hint. You have $a$ different blue shirts and $b$ different pink shirts. In how many ways can you choose one shirt to wear today and one to wear tomorrow?
| {
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Basic B-Spline basis function question I am studying the basic recursion formula for generating B-Spline basis functions N(i,j) of a given degree from the basis for the lower degree, and puzzling at the magic.
In particular what I am having a hard time getting through my head is why the new functions obtained have one ... | Splines satisfy many more identities than the Cox-de Boor relations. As an example, the cardinal B-splines $B_n(x)$ of degree $n-1$ and support $[0,n]$ also satisfy
$$B_{n+1}'(x)=B_n(x)-B_n(x-1).$$
This relation directly shows that smoothness increases with degree.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A way of finding $x \in \mathbb{Z}$, congruency I'm trying to find all $x \in \mathbb{Z}$ that satisfies this equation
$$3x \equiv 1 \pmod 6$$
I tried using trial and error, but couldn't find a suitable number for x.
I know that the $\mbox{gcd}$ is $3$.
How would I approach this?
| A solution mod $\,6\,$ remains a solution mod $\,3,\,$ yielding $\ 0\equiv 3x\equiv 1\pmod 3,\,$ contradiction.
| {
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If $a$ and $b$ are odd then $a^2+b^2$ is not a perfect square
Prove if $a$ and $b$ are odd then $a^2+b^2$ is not a perfect square.
We have been learning proof by contradiction and were told to use the Euclidean Algorithm.
I have tried it both as written and by contradiction and can't seem to get anywhere.
| Let $a=2m+1$ and $b=2n+1$.
Assume $a^2+b^2=k^2$. Then:
$$(2m+1)^2+(2n+1)^2=k^2 \iff \\
4(m^2+m+n^2+n)+2=k^2 \iff \\
4(m^2+m+n^2+n)+2=(2r)^2 \iff \\
2(m^2+m+n^2+n)+1=2r^2 \iff \\
2s+1=2r^2,$$
which is a contradiction. Hence, the assumption $a^2+b^2=k^2$ is false.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculate possible intersection point of two lines
I'd like to know the $I$ coordinates $[x, y]$. $I$ is the possible intersection point of line $|AB|$ with $|XY|$ and $|X'Y'|$. Values that are known:
*
*Angle $AIX$ and $AIX'$
*Coordinates of points $A$, $B$, $X$, $Y$, $X'$, $Y'$
I have no idea how to calculate ... | Given the coordinates $(x,y)$ of your four points $A,B,X,Y$, the parametrized equations for the two continuation lines are
$$\vec{r}_1(t)
=\begin{bmatrix}x_A \\ y_A \end{bmatrix}
+t\cdot\begin{bmatrix}x_B-x_A \\ y_B-y_A \end{bmatrix},$$
$$\vec{r}_2(s)
=\begin{bmatrix}x_X \\ y_X \end{bmatrix}
+s\cdot\begin{bmatrix}x_Y-x... | {
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Determining Linear Independence and Linear Dependence? I understand that when solving for a linear dep/independent matrix, you can take the determinant of the matrix and if it is zero, then it is linearly dependent. However, how can I go about doing this for something like two $3\times 1$ vectors?
Example:
$$\begin{bma... | Define a matrix $A\in\mathbb{R}^{3\times 2}$ by setting $A:=\begin{bmatrix}-6 & -1 \\ -1 & -5 \\ -7 & 4 \end{bmatrix} $.
By performing the row operations on the matrix $A$, we can show that the null space $\mathcal{N}(A)=\left\{\begin{bmatrix}0 \\ 0\end{bmatrix}\right\}$.
Since the null space of $A$ contains only the z... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Stability of a feedback system Take the following feedback system:
$\dot{x} = (\theta - k_1) x - k_2 x^3$
Now my book says:
For $\theta > k_1$, the equilibrium $x = 0$ is unstable. I wonder why...
Furthermore my book indicates that it is easy to see that $x(t)$ will converge to one of the two new equilibria $\pm \sqrt... | Question 1: From stability theory it is known, that
a fixed point $x^*$ of $\dot{x}=f(x)$ is stable $\Leftrightarrow$ all eigenvalues of the jacobian $f'(x^*)$ have negative real parts.
For $f(x)=(\theta - k_1) x - k_2 x^3$ and $x^*=0$ this restricts to $\theta<k_1$
Question 2: You can use the same argument.
For $... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Differentiating this implicit expression? I am given:
$$\tfrac{1}{4}(x+y)^2 + \tfrac{1}{9}(x - y)^2 = 1$$
Using the chain rule, and factoring out $y'$, I'm left with:
$$y' \left(\tfrac{1}{2}(x+y) - \tfrac{2}{9}(x-y)\right) = 0$$
Now I need to isolate $y'$ but I'm not sure how.
Should I do:
$$y' = \frac{1}{\tfrac{1}{2}... | Your first step is wrong. It should be $$\frac{1}{2}(1+y')(x+y)+\frac{2}{9}(1-y')(x-y)=0$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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The Hedgehog space is locally compact We'll consider The Hedgehog space, and use the definition from Wikipedia (link to the current revision):
Let $\kappa$ a cardinal number, the $\kappa$-hedgehog space is formed by taking the disjoint union of $\kappa$ real unit intervals identified at the origin. Each unit interval i... | I don't think this space is locally compact. Indeed the whole space isn't compact (when $\kappa$ is infinite), as the open cover $$\big(B((2/3)_{\alpha},1/2)\big)_{\alpha\in\kappa}\cup\big(B(0,1/2)\big)$$
clearly admits no finite subcover. If the space were locally compact, $0$ would admit a compact neighborhood $K$ wi... | {
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"timestamp": "2023-03-29T00:00:00",
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To maximise my chance of winning one prize should I put all my entries in a single draw? Every week there's a prize draw. It's free to enter using the code from a soup tin lid. You can enter as many times as you like during the week until Monday's draw and then it starts all over.
The prizes are experiences and the val... | If there are lots of entries (so the number $n$ of your entries doesn't much change the probability of a number winning) and the probability of one ticket winning is $p$ so that $np$ is small then:
If you enter all at once, the probability of winning is $np$
If you enter in $n$ separate weeks the probability of winning... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove two graphs are isomorphic if and only if their complements are isomorphic? How this can be proved that two graphs $G_1$ and $G_2$ are isomorphic iff their complements are isomorphic?
| Let graph $G$ be isomorphic to $H$, and let $\overline G$, $\overline H$ denote their complements.
Since $G$ is isomorphic to $H$, then there exists a bijection $f: V(G) \to V(H)$, such that $uv \in E(G)$ if and only if $f(u)f(v) \in E(H)$. -> [this should be edge set]
Equivalently, there exists a bijection $f: V(G) \t... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Vector question regarding plane and area How do i find the area enclosed within the $3$ points when the plane intersects the $x, y$ and $z$ axis and
The plane is $ax+by+cz=1$
Need help on solving this as it has been bothering me for some time
| Find the points where the plane intersects the $x$, $y$, and $z$ axis. Then find the distances between these three points and use Heron's formula to find the area of the trianlge defined by them.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proof of a bijection to the set of subsets? For part of a proof I wanted to show that $f: \{1,2\} \to \mathcal{S}(X)$ is a bijection, where $\mathcal{S}(X)$ is the set of subsets of $X$, which in this case I know to be $\{\emptyset , X\}$. So I define $f$ as $f(1) = \emptyset$ and $f(2) = X$. But then $f(1) = \emptyset... | You are confusing between $f^{-1}(a)=\{x\mid f(x)=a\}$ and $f^{-1}(a)=\{x\mid f(x)\in a\}$. The latter is sometimes written as $f^{-1}[a]$ to avoid this sort of confusion when $f(x)$ is a set itself.
So $f^{-1}(\varnothing)=\{1\}$ and $f^{-1}[\varnothing]=\varnothing$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/683948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Factoring a hard polynomial This might seem like a basic question but I want a systematic way to factor the following polynomial:
$$n^4+6n^3+11n^2+6n+1.$$
I know the answer but I am having a difficult time factoring this polynomial properly. (It should be $(n^2 + 3n + 1)^2$). Thank you and have a great day!
| There's another approach: check that for, say $n=\pm 1,\, 0,\,\pm 2$ your polynomial is a perfect square:$$\begin{cases}25,&n=1\\1,&n=-1\\1,&n=0\\1,&n=-2\\121,&n=2.\end{cases}$$ Therefore, in five points a polynomial of fourth degree is a perfect square - you can hope to find a polynomial of the second degree that sat... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that a set of vectors spans $\Bbb R^3$? Let $ S = \{ (1,1,0), (0,1,1), (1,0,1) \} \subset \Bbb R^3 .$
a) Show that S spans $\Bbb R^3$
b) Show that S is a basis for $\Bbb R^3 $
I cannot use the rank-dimension method for (a). Is it possible to show via combination of 3 vectors? What would the final equations be like... | Let $$v\in \Bbb R^3$$ Then v = (v1,v2,v3). Let s1 = (1,1,0), s2 = (0,1,1), s3 = (1,0,1).
So by row reducing the matrix (s1 s2 s3 v), we get that $$v = ((v_1+v_2-v_3)/2)*s_1 + ((-v_1+v_2+v_3)/2)*s2 + ((v_1-v_2+v_3)/2)*s_3$$
And since v was chosen arbitrarily this means that $$S\ \text{spans}\ \Bbb R^3$$
Now all you nee... | {
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"answer_id": 0
} |
weighted sum of two i.i.d. random variables Suppose we know that $X_1$ and $X_2$ are two independently and identically distributed random variables. The distribution of $X_i$ ($i=1,2$) is $P$, and we have some constraints on $P$ that $$\mathbb{E} X_1 = 0$$ (zero-mean) and $$\mathbb{E} X_1^2 = 1$$ (variance is normalize... | If you let $a$ and $\omega_1 + \omega_2 =: c$ be fixed and, for notational convenience, let $0 < \omega_1 =: t$, then
\begin{align}
f(\omega_1,\omega_2,a)
= f(t)
&= \sup_{P \in \mathcal{D}}P[t X_1 + (c-t)X_2 \geq a] \\
&= \sup_{P \in \mathcal{D}} \int P\left[X_1 \geq \left(1-\frac{c}{t}\right)X_2 +
\frac{a}{t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/684167",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Rewrite $f(x) = 3 \sin (\pi x) + 3\sqrt{3} \cos (\pi x)$ in the form $f(x) = A \sin (Kx+D)$ I got a question like that said "Rewrite $f(x) = 3 \sin (\pi x) + 3\sqrt{3} \cos (\pi x)$ in the form $f(x) = A \sin (Kx+D)$".
I'm inclined to think that since the periods are the same ($2$), that the amplitudes will just add up... | The standard way to combine such functions is to use the R Formula ( http://www.oocities.org/maths9233/Trigonometry/RFormula.html) :
Specifically, if we wanna combine your function into :
$$a\sin{\theta} + b\cos{\theta} = R\sin{(\theta + \alpha)}$$
Then it is possible to compute $R,\alpha$ :
$$R = \sqrt{a^2 + b^2}, \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/684248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Tychonoff vs. Hilbert Let $(\mathscr H_n,\langle\cdot,\cdot\rangle_n)_{n\in\mathbb N}$ be a sequence of Hilbert spaces. Let $$\mathscr H\equiv\bigoplus_{n\in\mathbb N}\mathscr H_n\equiv\left\{(h_n)_{n=1}^{\infty}\,\Bigg|\,h_n\in\mathscr H_n\,\forall n\in\mathbb N,\,\sum_{n\in\mathbb N}\|h_n\|_n^2<\infty\right\}$$ denot... | If a product of compact sets lies within the direct sum, it will always be compact.
The space in question is metrisable, so it's enough to check sequential compactness, and thanks to completeness, it's not hard to do that using the standard diagonal argument.
This is not a consequence of Tychonoff's theorem, however, a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/684399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
chain rule using trig functions So I have the following:
$$
y = cos(a^3 + x^3)
$$
This is what I got.
$$
y' = \cos(a^3 + x^3) \ ( -sin(a^3 + x^3) ) \ ( 3a^2 + 3x^2 )
$$
I'm not sure what to do after this? Would this be the final answer?
| $$d[\cos(a^3+x^3)]=-\sin(a^3+x^3)d(a^3+x^3)=-\sin(a^3+x^3)3x^2dx$$
assuming $a$ is a constant and $x$ is the variable. If you want to explicitly use the chain rule then let
$$u=a^3+x^3,\frac{du}{dx}=3x^2$$
$$y=\cos u,\frac{dy}{du}=-\sin u$$
$$\frac{dy}{dx}=\frac{du}{dx}\times\frac{dy}{du}=-3x^2\sin(a^3+x^3)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/684488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Show $1+\cosθ+\cos(2θ)+\cdots+\cos(nθ)=\frac{1}{2}+\frac{\sin[(n+1/2)θ]}{2\sin(θ/2)}$ Show
$$1+\cosθ+\cos(2θ)+\cdots+\cos(nθ)=\frac12+\frac{\sin\left(\left(n+\frac12\right)θ\right)}{2\sin\left(\frac\theta2\right)}$$
I want to use De Moivre's formula and $$1+z+z^2+\cdots+z^n=\frac{z^{n+1}-1}{z-1}.$$ I set $z=x+yi$, bu... | Hint.
$$
\operatorname{Re}(1+e^{i\theta}+\cdots +e^{ni\theta})=\operatorname{Re}\frac{e^{(n+1)i\theta}-1}{e^{i\theta}-1}
$$
where $e^{i\theta}=\cos\theta+i\sin\theta$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/684603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Summation of factorials modulo ten I have read that$$\sum\limits_{i=1}^n i!\equiv3\;(\text{mod }10),\quad n> 3.$$
Why is the sum constant, and why is it $3$?
| Think about what you are summing:
$$1+2+6+24+120+720+\dots = 33 + 120 + 720 + \dots$$
Taking mod $10$ of the sum, you can see that $33$ gives $3$, can you see that all other sumands are divisible by $10$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/684668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Cut locus that is a geodesic Can we characterize surfaces $S$, for which cut locus $C_p(S)$ with respect to a point $p$ on $S$, is itself a geodesic between the points it passes through? This holds for example for a cylinder and therefore surfaces isometric to it which do have a cut locus. I am looking for more example... | If the manifold is compact a topological classification can be obtained as follows:
First observe that the cut locus of a closed connected manifold is connected, as noted in Jason De Vitos comment. In fact for $p \in S$ and $q \in S \setminus \{p\}$ consider a minimal geodesic $\gamma$ from $p$ to $q$. Mapping $q$ to $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/684821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Different basis over $\Bbb{R}$ and $\Bbb{C}$ V is a finite dimensional vector space over $\Bbb{C}$ and
{v$_1$,...,v$_n$}
be a basis of V.
Show {v$_1$,iv$_1$,...,v$_n$,iv$_n$} is a basis of V over $\Bbb{R}$ and conclude:
dim$_{\Bbb{R}}$V=2dim$_{\Bbb{C}}$V.
I have proved this is true for the case V = $\Bbb{C}^2$ using ... | Let $\alpha_1,\beta_1,\alpha_2,\beta_2,\ldots,\alpha_n,\beta_n\in\Bbb R$ such that
$$\alpha_1 v_1+\beta_1 i v_1+\cdots+\alpha_n+\beta_n i v_n=0$$
so with $z_i=\alpha_i+i\beta_i$ we have
$$z_1v_1+\cdots+z_n v_n=0\Rightarrow z_i=0 \;\forall i$$
since $(v_1,\ldots,v_n)$ are linearly independant, moreover, it's clear that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/684917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
What are the ideals in ${\Bbb C}[x,y]$ that contain $f_1,f_2\in{\Bbb C}[x,y]$? This question is based on an exercise in Artin's Algebra:
Which ideals in the polynomial ring $R:={\Bbb C}[x,y]$ contain $f_1=x^2+y^2-5$ and $f_2=xy-2$?
Using Hilbert's (weak) nullstellensatz, one can identify all the maximal ideals of $R$... | Here's a procedure that works more generally: the ideal $I := (x^2 + y^2 - 5, xy - 2)$ has height $2$ in $\mathbb{C}[x,y]$. To see this, note that $x^2 + y^2 - 5$ (or $xy - 2$) is irreducible over $\mathbb{C}$ (e.g. by Eisenstein), hence generates a height $1$ prime ideal in $\mathbb{C}[x,y]$, which does not contain th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/684984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Independent Event Complements I have the following homework assignment that I've already finished, but am confused on whether I've gotten right/wrong, and was hoping someone could help explain so I understand the problem better.
An oil exploration company currently has two active projects, one in Asia and the other in... | (a) This question belongs to "conditional model"
But since A and B are independent
You may take directly p(B not)=0.3
Reason: independent means
P(A .B)=P(A).P(B)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/685090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Need help with a proof concerning zero-free holomorphic functions. Suppose $f(z)$ is holomorphic and zero-free in a simply connected domain, and that $\exists g(z)$ for which $f(z) =$ exp$(g(z))$.
The question I am answering is the following:
Let $t\neq 0$ be a complex number. Prove that $\exists h(z)$ holomorphic such... | Nudge: Remember that for real numbers $r>0$, you can define $r^t = \exp(t\ln r)$. Maybe you can do something similar for holomorphic functions?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/685174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Tricky limit involving sine I'm trying to evaluate
$$\text{lim}_{(x,y) \rightarrow (0,0)} \frac{x^4 + \text{sin}^2(y^2)}{x^4+y^4}.$$
I'm pretty sure that the limit exists and is $1$; at the very least, you get that if you approach $(0,0)$ along the lines $x=0$ and $y=0$ and $x=y$. But I can't seem to figure out how to... | Consider
$$1-{x^4+\sin^2(y^2)\over x^4+y4}={y^4-\sin^2(y^2)\over x^4+y^4}={y^4\over x^4+y^4}\left(1-{\sin^2(y^2)\over y^4} \right)$$
Now
$$\left|{y^4\over x^4+y^4}\right|\le1$$
for all $(x,y)\not=(0,0)$ and
$$\lim_{y\to0}{\sin^2(y^2)\over y^4}=\lim_{u\to0}\left({\sin u\over u}\right)^2=1$$
That should take care of thin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/685264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Proving that an uncountable set has an uncountable subset whose complement is uncountable. How does one prove that an uncountable set has an uncountable subset whose complement is uncountable. I know it needs the axiom of choice but I've never worked with it, so I can't figure out how to use. Here is my attempt (which ... | Your idea is generally correct.
Using the axiom of choice, $|X|=|X|+|X|$, so there is a bijection between $X$ and $X\times\{0,1\}$. Clearly the latter can be partitioned into two uncountable sets, $X\times\{0\}$ and $X\times\{1\}$.
Therefore $X$ can be partitioned to two uncountable disjoint sets.
Indeed you need the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/685349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Trying to get a bound on the tail of the series for $\zeta(2)$ $$\frac{\pi^2}{6} = \zeta(2) = \sum_{k=1}^\infty \frac{1}{k^2}$$ I hope we agree.
Now how do I get a grip on the tail end $\sum_{k \geq N} \frac{1}{k^2}$ which is the tail end which goes to zero?
I want to show that $\sqrt{x}\cdot \mathrm{tailend}$ is bound... | Use either the "Euler-Maclaurin summation formula" or "Abel summation formula" applied to the function $f(x) = 1/x^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/685435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 3
} |
Why must metric tensor be invertible? The metric can be written as a matrix, but why must this matrix be invertible? At the points where the matrix is singular, why is the metric not defined?
| This worried me at one time as well. The way I thought about it was by working at a fixed point and using the Gram-Schmidt process for inner products on the coordinate basis $\partial_1,...,\partial_n$ to produce an orthonormal basis $e_1,...,e_n$. It's a standard and easy fact that the matrices that represent these bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/685544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Closed form for derivative $\frac{d}{d\beta}\,{_2F_1}\left(\frac13,\,\beta;\,\frac43;\,\frac89\right)\Big|_{\beta=\frac56}$ As far as I know, there is no general way to evaluate derivatives of hypergeometric functions with respect to their parameters in a closed form, but for some particular cases it may be possible. I... | Using Euler-type integral representation for the Gauss's function:
$$
{}_2F_1(a,b; c; z) = \frac{\Gamma(c)}{\Gamma(b) \Gamma(c-b)} \int_0^1 u^{b-1} (1-u)^{c-b-1} (1-z u)^{-a} \mathrm{d}u
$$
for $c = \tfrac{4}{3}$ and $b=\tfrac{1}{3}$, differentiating with respect to $a$ at $a=\tfrac{5}{6}$:
$$
\left.\frac{\mathrm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/685644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
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What steps are taken to make this complex expression equal this? How would you show that $$\sum_{n=1}^{\infty}p^n\cos(nx)=\frac{1}{2}\left(\frac{1-p^2}{1-2p\cos(x)+p^2}-1\right)$$ when $p$ is positive, real, and $p<1$?
| Since $0<p<1$, we have
\begin{eqnarray}
\sum_{n=1}^\infty p^n\cos(nx)&=&\Re\sum_{n=1}^\infty p^ne^{inx}=\Re\sum_{n=1}^\infty(pe^{ix})^n=\Re\frac{pe^{ix}}{1-pe^{ix}}=\Re\frac{pe^{ix}(1-pe^{-ix})}{|1-pe^{ix}|^2}\\
&=&p\Re\frac{-p+\cos x+i\sin x}{|1-p\cos x-ip\sin x|^2}=p\frac{-p+\cos x}{(1-p\cos x)^2+p^2\sin^2x}\\
&=&p\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/685705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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How find this value $\left|\frac{z_{1}z_{2}+z_{1}z_{3}+z_{2}z_{3}}{z_{1}+z_{2}+z_{3}}\right|$ let three complex $z_{1},z_{2},z_{3}$ such
$$z_{1}+z_{2}+z_{3}\neq 0,|z_{1}|=|z_{2}|=|z_{3}|=1$$
Find this value
$$\left|\dfrac{z_{1}z_{2}+z_{1}z_{3}+z_{2}z_{3}}{z_{1}+z_{2}+z_{3}}\right|$$
My idea:if $z_{1},z_{2},z_{3}$ is r... | Note that
$$\lvert z_2z_3+z_3z_1+z_1z_2\rvert=\lvert z_1z_2z_3\rvert\cdot\lvert z_1^{-1}+z_2^{-1}+z_3^{-1}\rvert=\lvert\overline z_1+\overline z_2+\overline z_3\rvert=\lvert z_1+z_2+z_3\rvert$$
Can you point out the reason of each equality?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/685795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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$f(x) = \arccos {\frac{1-x^2}{1+x^2}}$; f'(0+), f'(0-)? $f(x) = \arccos {\frac{1-x^2}{1+x^2}}$
$f'(x) = 2/(1+x^2)$,
but I see graphic, and it is true only for x>=0.
For x<=0 => $f'(x) = -2/(1+x^2)$
How can I deduce the second formula or proof that it is.
| Method $\#1:$
Let $\displaystyle\arccos\frac{1-x^2}{1+x^2}=y$
$\displaystyle\implies(1) \cos y=\frac{1-x^2}{1+x^2}\ \ \ \ (i)$
Using the definition of Principal values,
$\displaystyle \implies(2)0\le y\le\pi \implies 0\le\frac y2\le\frac\pi2\implies \tan\frac y2\ge0$
Applying Componendo and dividendo on $(i),$
$\displa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/685900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $\operatorname{tr}(A+B)>\operatorname{tr}(A)$, does it hold that $\operatorname{tr}((A+B)^k)>\operatorname{tr}(A^k)$ for all $k\geq 1$ I wonder whether the following holds and if so how it could be proved:
Let $A, B$ be (non-commuting) positive semi-definite matrices,
If $\operatorname{tr}(A+B)>\operatorname{tr}(A)$... | Very partial answer:
It is true for k=2:
$tr((A+B)^2)=tr(A^2+AB+BA+B^2)=tr(A^2)+tr(AB)+tr(BA)+tr(B^2)=tr(A^2)+2tr(AB)+tr(B^2)$
by linearity of trace and by the fact that $tr(AB)=tr(BA)$. Furthermore, as B is psd, $tr(B^2)\geq 0$, and $tr(AB)\geq 0$ (see A.U. Kennington, Power concavity and boundary value problems, Indi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/685985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Does X = Y in distribution and X being Y-measurable imply Y is X-measurable? Suppose $X,Y$ are random variables taking values in some Borel space, $X \overset {d}{=} Y$, and $X$ is $Y$-measurable.
It follows from the fact that $X$ is $Y$-measurable that there exists a measurable $f$ such that $X = f(Y)$ a.s.
Is it the ... | No.
Let $Y$ be uniformly distributed on $[-1,1]$, let $f(x) = 2|x|-1$ and let $X = f(Y)$. Then $X$ is also uniformly distributed on $[-1,1]$, but $Y$ is not $X$-measurable, since, intuitively, by looking at $X$ you cannot tell the sign of $Y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/686165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Inner-product question Let $V$ be $\mathbf{R}^2$ equipped with usual inner product, and $v$ be a nonzero vector. $S_v(u)= u- 2 \frac{\langle u,v\rangle}{\langle v,v\rangle } v$ and $\Phi$ be a non-empty set of unit vectors in $\mathbf{R}^2$ such that $S_v(u) \in \Phi$ and $2\langle u,v\rangle\in\mathbf{Z}$ for any $u,v... | First, note that if $u,v \in \mathbb{S}^1$, so is S_v(u).
Then $2<u,v> \in \mathbb{Z}$ means $2cos(\theta) \in \mathbb{Z}$, where theta is the angle between $u$ and $v$. Then $\theta=0, \pi/3,\pi/2$ or $\pi$, and you 're done!!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/686247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Is $L = \{(x,y,z) | x+y=z\}$ a regular language? Suppose $x,y,z$ are coded as decimal or their binary representations in an appropriate DFA. Is $L$ regular?
My intuition tells me that the answer is no, because there are infinitely many combinations such that $x+y=z$ and a DFA must contain a finite amount of memory. Is ... | This is a funny question, and the answer is: it depends.
First, if $x$, $y$ and $z$ are given sequentially, then pumping lemma implies that triplet $\langle 1(0^n),0,1(0^m)\rangle$ would have to be accepted for multiple values of $m$ and $n$, not necessarily equal.
On the other hand, if the numbers are given simultaneo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/686352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Tricky question about differentiability at the origin Let $f: \mathbb{R}^2 \to \mathbb{R} $ be given as
$$
f(x,y) =
\begin{cases}
y, & \text{if }\text{ $x^2 = y $} \\
0, & \text{if }\text{ $x^2 \neq y $}
\end{cases}
$$
Is this function differentiable at $(0,0)$ ?
| Yeah. It is easy to show that $f$ admits partial derivatives at $(0,0)$, both equal to $0$, for example $$\frac{\partial f}{\partial x}(0,0)=\lim_{x\to 0}\frac{f(x,0)-f(0,0)}{x}=0$$
To show that $\lim_{(x,y)\to 0}\frac{f(x,y)}{||(x,y)||}=0$ note that $$\Bigg|\frac{f(x,y)}{||(x,y)||}\Bigg|\leq \frac{x^2}{||(x,y)||}.$$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/686407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Finding system with infinitely many solutions The question asks to find equation for which the system has infinitely many solutions.
The system is:
\begin{cases}
-cx + 3y + 2z = 8\\
x + z = 2\\
3x + 3y + az = b
\end{cases}
How should I approach questions like this?
I tried taking it to row... | You can do row reduction; consider the matrix
\begin{align}
\left[\begin{array}{ccc|c}
-c & 3 & 2 & 8 \\
1 & 0 & 1 & 2 \\
3 & 3 & a & b
\end{array}\right]
&\to
\left[\begin{array}{ccc|c}
1 & 0 & 1 & 2 \\
-c & 3 & 2 & 8 \\
3 & 3 & a & b
\end{array}\right]\quad\text{swap 1 and 2}\\
&\to
\left[\begin{array}{ccc|c}
1 & 0 &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/686489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Simplifying solutions I am given the differential equation $$\frac{dz}{dx} = m(c_{1}-z)(c_{2}-z)^{\frac{1}{2}}, z(0) =0$$ and have arrived at a solution: $$z(x) = c_{2} - (c_{1}-c_{2})\tan^{2}{\left[\arctan{\left(\frac{\sqrt{c_{2}}}{\sqrt{c_{1}-c_{2}}}\right)} - \frac{mx}{2}\sqrt{c_{1}-c_{2}}\right]}.$$ I was wondering... | Using the identity
$$
\tan(\arctan(u)-v) = \frac{u - \tan(v)}{1 + u \tan(v)}
$$
I was able to simplify your answer to
$${{{ c_1}\,\tan \left({{\sqrt{{ c_1}-{ c_2}}\,m\,x}\over{2
}}\right)\,\left(2\,{ c_2}\,\tan \left({{\sqrt{{ c_1}-
{ c_2}}\,m\,x}\over{2}}\right)-{ c_1}\,\tan \left({{\sqrt{
{ c_1}-{ c_2}}\,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/686578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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List of Common or Useful Limits of Sequences and Series There are many sequences or series which come up frequently, and it's good to have a directory of the most commonly used or useful ones. I'll start out with some. Proofs are not required.
$$\begin{align}
\sum_{n=0}^{\infty} \frac1{n!} = e
\\ \lim_{n \to \infty} \l... | My opinion.
The most useful series is the geometric series, in both its finite and infinite forms:
$$
1 + x + x^2 + \cdots + x^n = \frac{1 - x^{n+1}}{1-x} \quad (x \neq 1)
$$
and
$$
1 + x + x^2 + \cdots = \frac{1 }{1-x} \quad (|x| < 1)
$$
You can derive many others from it by substituting values and by formal manip... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/686665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Expressing $\cos(x)^6$ as a linear combination of $\cos(kx)$'s Let $$(\cos^6(x)) = m\cos(6x)+n\cos(5x)+o\cos(4x)+p\cos(3x)+q\cos(2x)+r\cos(x)+a.$$
What is the value of $a$?
| $\newcommand{\+}{^{\dagger}}%
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\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\down}{\downarrow}%
... | {
"language": "en",
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Concerning the sequence $\Big(\dfrac {\tan n}{n}\Big) $ Is the sequence $\Big(\dfrac {\tan n}{n}\Big) $ convergent ? If not convergent , is it properly divergent i.e. tends to
either $+\infty$ or $-\infty$ ? ( Owing to $\tan (n+1)= \dfrac {\tan n + \tan 1}{1- \tan1 \tan n}$ and the non-covergence of
$\Big (\tan n \Bi... | Since $\pi/2$ is irrational, a theorem of Scott says there exist infinitely many pairs of positive integers $(n,m)$ with $n$ and $m$ odd such that $\left| \dfrac{\pi}{2} - \dfrac{n}{m} \right| < \dfrac{1}{m^2}$. For such $m$ and $n$
we have $|\cos(n)| < |m \pi/2 - n| < 1/m$ and thus $|\tan(n)|/n > k$ for suitable nonze... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/686841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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the rank of a linear transformation Let $V$ be vector space consisting of all continuous real-valued functions defined on the closed interval $[0,1]$ (over the field of real numbers) and $T$ be linear transformation from $V$ to $V$ defined by
$$(Tf)(x) = \int_0 ^1 (3x^3 y - 5x^4 y^2) f(y)\,\mathrm dy$$
Why is $\operato... | Note that $(Tf)(x) = 3x^3 \int_0^1 y f(x)dy - 5 x^4 \int_0^1 y^2 f(y)dy$, hence
$Tf \in \operatorname{sp} \{ x \mapsto x^3, x \mapsto x^4 \}$, so $\dim {\cal R} T \le 2$.
If we choose $f(x) = {2\over 3} -x$, we see $(Tf)(x) = {5 \over 36} x^4$, and if we choose $f(x) = {4\over 3} -x$, we see that $(Tf)(x) ={1 \over 24}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculating $ x= \sqrt{4+\sqrt{4-\sqrt{ 4+\dots}}} $ If $ x= \sqrt{4+\sqrt{4-\sqrt{ 4+\sqrt{4-\sqrt{ 4+\sqrt{4-\dots}}}}}} $
then find value of 2x-1
I tried the usual strategy of squaring and substituting the rest of series by x again but could not solve.
| I assume you mean
$$ x=\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-\sqrt{4\pm\ldots}}}}}$$
so that
$$ \begin{align}x&=\sqrt{4+\sqrt{4-x}}\\
x^2&=4+\sqrt{4-x}\\
(x^2-4)^2&=4-x\\
0 &= x^4-8x^2+x+12= (x^2-x-3)(x^2+x-4)\end{align}$$
Since clearly $x\ge \sqrt 4=2$, the second factor is $x^2+x-4\ge2>0$, which leaves us with the posit... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find an equation of the tangent line to the curve $y = x\cos(x)$ at the point $(\pi, -\pi)$ I concluded that the equation is
$$(y + \pi) = (\cos(x) + x-\sin(x)) (x - \pi)$$
1) Is this correct so far? Wolfram doesn't seem to process this correctly.
2) How would I expand this to get it in $y$-intercept form? I know I ca... | Your derivative, which we need for slope, is close, but $y'$ should be $$y' =\underbrace{(1)}_{\frac d{dx}(x)}\cdot(\cos x) + (x)\underbrace{( -\sin x)}_{\frac d{dx}( \cos x)}= \cos x -x\sin x$$ Now, for slope itself, we evaluate $y'(\pi) = \cos (\pi) - \pi\sin(\pi) = -1 - 0 = -1$.
That gives you the equation of the... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A three-way duel (probability puzzle) This puzzle is taken from Mathematical Puzzles: A Connoisseur's Collection [Peter Winkler]. I don't understand the solution.
Alice, Bob, and Carol arrange a three-way duel. Alice is a poor shot, hitting her target only 1/3 of the time on average. Bob is better, hitting his target 2... | Arno proved that if Alice, Bob and Carol want to survive, they will go to a slatemate, so as Daniel V points out, the goal should be to win the duel and not to survive. But assuming that the goal is to win the duel, if Alice and Bob shoot in the air, why Carol could not shoot in the air too? Because she will have to sh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/687272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 6,
"answer_id": 3
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Why do you need to use the chain rule in differentiation of ln? I understand application of chain rule in the differentiation of a random function $(x^2+3)^3$.
But, why do you need to use chain-rule when differentiating something like $\ln(2x-1)$; why won't it just be $\displaystyle\frac 1{2x-1}$? Please help.
| Let us first call
$$y = \ln(2x + 1)$$
Physicists and engineers use the simpler Leibniz calculus to calculate the differential quotient $\frac{dy}{dx}$ instead of using those pesky Newtonian fluxion dots ($\dot{y}$) or French apostrophes ($y'$) .. :-)
We substitute
$$u = 2x + 1$$
and get
$$y = \ln(u)$$
and
$$\frac{dy}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/687337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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What is the number of real roots of $(\log x)^2- \lfloor\log x\rfloor-2=0$ $\lfloor\,\cdot\,\rfloor$ represents the greatest integer Question : What is the number of real roots of $(\log x)^2- \lfloor\log x\rfloor-2=0$. $\lfloor\,\cdot\,\rfloor$ represents the greatest integer function less than or equal to $x$.
I kno... | Since $ [\log x] \leq \log x $
we have $(\log x)^2-\log x -2 \leq 0$
This is equivalent to $-1 \leq \log x \leq 2$
When $-1 \leq \log x \leq 0, [\log x ] =-1$ so that $\log x =\pm 1$
If we see that $\log x =1$ is not in the specified range. Hence $\log x =-1$ and $x =\frac{1}{10}$
When $0 \leq \log x < 1$ , $[\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/687437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to ensure Topological Correctness Question:
I read through an enormous amount of material on topology and knot-theory in wikipedia, but I still am stuck at the following fundamental problem:
Given two representations of closed curves, how do you establish their "linkedness"?
So in a really simple example, given t... | Choose a generic 2-plane in 3-space and project your link onto it. Then use the idea in http://en.wikipedia.org/wiki/Linking_number#Computing_the_linking_number.
To make it computationally feasible, you might have to approximate your link by a sufficiently close polygonal curve.
(This answers what I think is your main ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/687525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Determinant of long exact sequence Let the following be a long exact sequence of free $A$-modules of finite rank:
$$0\to F_1\to F_2\to F_3\to...\to F_n\to0$$
I want to show that $\otimes_{i=1}^n (\det F_i)^{-1^{i}} \cong A$, where the notation $^{-1}$ means taking the dual.
My attempt was to break this into SES's like
... | The determinant can be determined for every finitely generated projective module, because these are precisely the locally free modules of finite rank (which doesn't have to be constant, but it is locally constant, and on each constant piece we take the corresponding exterior power). It is additive on short exact sequen... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can someone help me understand Cramer's Rule? I'm taking notes for my class and they define cramers rule and afterwards give us an example problem.
\begin{align*}
x_1 + 2\,x_2 =& 2\\
-x_1 + 4\,x_2 =& 1
\end{align*}
They compute
$$\det(a_1(b)) = \begin{vmatrix}2&2\\ 1&4\end{vmatrix}$$
and then they compute
$$\det (a... | No, the matrices are correct.
For Cramer's rule, you replace the column corresponding to the variable with the column of numbers on the other side of the equals sign. As Amzoti pointed out, to get to the final answer, you divide both $\det(a_1)$ and $\det(a_2)$ by the same determinant of the matrix of the coefficients... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$\sin^2 \alpha + \sin^2 \beta - \cos \gamma < M$ given that the sum of the angles is $\pi$ Question: Find the least real value of $M$ such that the following inequality holds:
$$\sin^2 \alpha + \sin^2 \beta - \cos \gamma < M$$
Given that $\alpha, \beta, \gamma \in \mathbb{R}^+$, $\alpha + \beta + \gamma = \pi$
My attem... | It should be a lot easier to look at the function:
$$\sin^2(x)+ \sin^2(y)-\cos(\pi - x - y)$$
And note it is symmetric when interchanging $x$ and $y$, and noting that comparing it's derivatives to zero leads to $\sin(2x)=\sin(2y)$. Thus $x=y+n\pi$.
Now find the maximum value of the function:
$$\sin^2(x)+\sin^2(x+n \pi)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How do I represent such a transformation? Let's say I have a 2d rectangle defined by $ [0,x_0] \times [0,y_0]$. Now lets say I cut out the middle rectangle $[\frac{1}{3} x_0, \frac{2}{3} x_0] \times [\frac{1}{3} y_0, \frac{2}{3} y_0]$. Now suppose I take the hyperreal extension of this rectangle. I then "fill" back up ... | As far as looking for hyperreal approaches to constructing the carpet, which is how I understood your idea, I would suggest looking first at hyperreal approaches to constructing nowhere differentiable functions. This was dealt recently in a paper by McGaffey here: http://arxiv.org/abs/1306.6900
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/688048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Composite with a zero arrow Why any composite with a zero arrow must itself be a zero arrow? I interpret this as
$a \rightarrow z \rightarrow b \rightarrow c = a \rightarrow z \rightarrow c$
(the zero arrow in the composite is $a \rightarrow z \rightarrow b$)
| I assume $z$ is a zero object. That is $z$ is initial and terminal. Being initial, for all object $c$ there is a unique arrow $z \to c$, so for any arrow $b \to c$,
$$ (z \to b \to c) = (z \to c) .$$
Being terminal, for all object $a$ there is a unique arrow $a \to z$. Composing (on the left) by this arrow for some $a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The unitary implementation of $*$-isomorphism of $B(H)$ Is it possible to construct $*$-isomorphism of (factor von Neumann) algebra $B(H)$ which is not unitary implementable?
| Let $\theta:B(H)\to B(H)$ be a $*$-automorphism. Fix an orthonormal basis $\{\xi_j\}$ of $H$, and write $E_{jj}$ for the corresponding rank-one projections, i.e. $E_{jj}\xi=\langle\xi,\xi_j\rangle\,\xi_j$. We can expand $\{E_{jj}\}_j$ to a system of matrix units $\{E_{kj}\}_{k,j}$, where $E_{kj}\xi=\langle\xi,\xi_j\ran... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Product of inverse matrices $ (AB)^{-1}$ I am unsure how to go about doing this inverse product problem:
The question says to find the value of each matrix expression where A and B are the invertible 3 x 3 matrices such that
$$A^{-1} = \left(\begin{array}{ccc}1& 2& 3\\ 2& 0& 1\\ 1& 1& -1\end{array}\right)
$$ and
$$B^... | Note that the matrix multiplication is not commutative, i.e, you'll not always have: $AB = BA$.
Now, say the matrix $A$ has the inverse $A^{-1}$ (i.e $A \cdot A^{-1} = A^{-1}\cdot A = I$); and $B^{-1}$ is the inverse of $B$ (i.e $B\cdot B^{-1} = B^{-1} \cdot B = I$).
Claim
$B^{-1}A^{-1}$ is the inverse of $AB$. So basi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "41",
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"answer_id": 2
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Rounding up to $3$ significant figures when adding If $a,\ b$ and $c$ are real numbers and you are required to find $a + b + c$ to $3$ significant figures, to how many significant figures could $a,\ b$ and $c$ be rounded up to to give the result?
| In general, if you want $\mathrm{round}_3(a+b+c) = \mathrm{round}_{3+k}(a)+\mathrm{round}_{3+k}(b)+\mathrm{round}_{3+k}(c)$ you should make $k$ as large as you possibly can.
A simple example is $a=499001,b=499.001,c=0.499001$.
*
*If you don't do any rounding before addition you get $\mathrm{round}_3(a+b+c) = \mathr... | {
"language": "en",
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Is torsion subgroup of elliptic curve birationally invariant? It's probably a very basic question: Having two birationally equivalent elliptic curves over $\mathbb{Q}$ - is the torsion subgroup unchanged under the birational equivalence?
| The group structure of the torsion subgroup may be the same, but the group law may look very different! I find the following example to be interesting and related to your question.
Let $E:y^2=x^3+1$ with zero at $[0,1,0]$, and consider $E':y^2=x^3+1$ where we now declare zero to be $[2,3,1]$. Then, $E$ and $E'$ are cle... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Whether derivative of $\ln(x)$ is $\frac{1}{x}$ for $x>0$ only? Whether derivative of $\ln(x)$ is $\frac{1}{x}$ for $x>0$ only? Can't we write $$\frac{d}{dx} {\ln|x|} = \frac{1}{x} $$ so that we can get the corresponding integration formula for $\frac{1}{x}$ easily as $$\ln|x|$$
I have gone thorough this but it discuss... | Since $|x|=\sqrt{x^2}$ we have $\bigl(\ln|x|\bigr)'=\bigr(\ln(\sqrt{x^2})\bigr)'=\dfrac{1}{\sqrt{x^2}}\dfrac{1}{2\sqrt{x^2}}\cdot2x=\dfrac{x}{|x|^2}=\dfrac{x}{x^2}=\dfrac{1}{x}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/688607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Elliptic curve over $\mathbb{Q}$ cannot have $\mathbb{Z}_4\times\mathbb{Z}_4$ as a subgroup Show that an elliptic curve over $\mathbb{Q}$ cannot have $\mathbb{Z}_4\times\mathbb{Z}_4$ as a subgroup.
We've been told that for this problem, we are not allowed to use Mazur's Theorem. Unfortunately that is the only way I can... | Hint: the existence of the Weil pairing.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Number of reflection symmetries of a basketball Excerpt from John Horton Conway, The Symmetries of Things, pg. 12.
Basketballs have two planes of reflective symmetry, as do tennis balls.
I read this sentence and it immediately struck me as incorrect: from my understanding of the pattern of lines on a basketball, the... | In the basketball I hold in my hands just now, there really are just two planes of symmetry. The plane perpendicular to the two great circles is not a symmetry. This is because the lines which are not great circles intersect one of the great circles near one of the poles, but the other great circle near the other pole.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Textbook for graduate number theory I am attending a graduate number theory, the professor did not assign any textbook. The materials are somewhere along the advanced/algebraic level such as Ring of Gaussian Integers, Quadratic Number Fields and especially about Euclidean Domain. Any suggestion to textbooks that I can ... | a good book is Problems in Algebraic Number Theory by Murty. It overs all of those things, and more and is 'problem-orientated,' so you do most of the work!
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $(\cos \alpha + i \sin \alpha )^n = 1$ then $(\cos \alpha - i \sin \alpha )^n = 1$ Prove that if $(\cos \alpha + i \sin \alpha )^n = 1$ then $(\cos \alpha - i \sin \alpha )^n = 1$.
What should I use? De Moivre's formula? Exponential form? I tried, but It doesn't work.
| Yet another approach: Since $(\cos\alpha+i\sin\alpha)^n=1,$ then $$(\cos\alpha-i\sin\alpha)^n=(\cos\alpha-i\sin\alpha)^n(\cos\alpha+i\sin\alpha)^n=\bigl((\cos\alpha-i\sin\alpha)(\cos\alpha+i\sin\alpha)\bigr)^n$$ Now, expand $(\cos\alpha-i\sin\alpha)(\cos\alpha+i\sin\alpha)$. What can we do then?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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complement of compact set is connected Let A be a compact subset of R, the real numbers. Prove that the complement of A in the complex numbers C is connected.
My thoughts: If A is compact then it is contained in a finite union. So if it's complement in C was disconnected it would imply C was disconnected-contradicton
| My thoughts:
Whenever possible, I prefer dealing with path-connected spaces to connected spaces, because I can more easily visualize it. If $A \subset \mathbb{R}$ is compact, then there's an $R > 0$ such that $A \subset [-R, R]$. Now if we have two points $z, w \in \mathbb{C} \setminus A$, then we have a couple of ea... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $p$ is an odd prime and $a$ is a positive integer not divisible by p, then the congruence has either no solution or 2 incongruent solutions My question is as follows:
Show that if $p$ is an odd prime and $a$ is a positive integer not divisible by p, then the congruence $x^2 \equiv a \pmod{p}$ has either no solution... | Hint $\ $ prime $\,p\mid(x-b)(x+b)\,\Rightarrow\, p\mid x-b\ \,$ or $\,\ p\mid x+b\,$ by uniqueness of prime factorizations (or some equivalent, e.g. Euclid's Lemma).
Alternatively if $\,c\not\equiv \pm b\,$ is a root then $\,(x-b)(x+b) \equiv (x-c)(x+c)$ contra polynomials over a field have unique prime factorization... | {
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Class divided into 5, probability of 2 people in the same team This might be a very simple question, but: a class of 25 students is divided into 5 teams of 5 each. What is the probability of student X and Y being in the same team?
is it just 4/25?
| Imagine that our heroes, A and B, are assigned to teams, in that order, with the rest being assigned later.
Whatever team A is assigned to, there are $4$ empty spots on that team. The probability B is given one of these spots is $\frac{4}{24}$.
Or else we can do more elaborate counting. Imagine the teams are labelled ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/689216",
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Interesting and unexpected applications of $\pi$ $\text{What are some interesting cases of $\pi$ appearing in situations that do not seem geometric?}$
Ever since I saw the identity $$\displaystyle \sum_{n = 1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$
and the generalization of $\zeta (2k)$, my perception of $\pi$ has ... | Too long for a comment:
What are some interesting cases of $\pi$ appearing in situations that are not geometric ?
None! :-) You did well to add “do not seem” in the title! ;-)
All $\zeta(2k)$ are bounded sums of squares, are they not ? And the equation of the circle, $x^2+y^2=$ $=r^2$, also represents a bounded sum... | {
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"answer_id": 6
} |
Correspondence as a graph of a multifunction Suppose I'd like to say that a projection of $R\subset X\times Y$ on $X$ is the whole $X$. That is, $R$ is a graph of a certain multifunction, or equivalently it is a left-total relation. I do remember seeing somewhere the term correspondence being used exactly for such purp... | It is standard terminology in mathematical economics. See for example, Aliprantis & Border 2007 Infinite Dimensional Analysis: A Hitchhiker's Guide. Terminology varies as to whether any subset of $X\times Y$ is a correspondence or whether the projectiont to $X$ has to be surjective (a "nonempty-valued correspondence").... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/689350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Using discriminants to find order of extension Any hints on how to show $[G:H]^{2}=\frac{disc(H)}{disc(G)}$,where G,H are free abelian groups of rank n and $H\subset G\subset K$,where K is a number field?
Alternative formulation, how to relate $[R:Z[a]]$ and disc(R),disc(Z[a])?
thanks
| I assume that what you mean is something to this effect:
If $\mathcal{O}\subseteq\mathcal{O}'$ are orders of $K$ (for some number field $K$) then $[\mathcal{O}':\mathcal{O}]\text{disc}(\mathcal{O}')=\text{disc}(\mathcal{O})$. This follows immediately from the following theorem of algebra:
Theorem: Let $R$ be a PID, an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/689443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Should I divide this permutation problem into cases or are there any quicker methods? I have got an idea for the second question but I think my approach is too long and I would like to ask whether there are any other quicker methods?
Eight cards are selected with replacement from a standard pack of 52 playing cards, wi... | There are $\binom{8}{3}$ ways to choose the places in the sequence of $8$ cards that the picture cards will occupy. For every such choice, the places can be filled in $12^3$ ways. So this part of the job can be done in $\binom{8}{3}\cdot 12^3$ ways.
For every way of dealing with the picture cards, there are $\binom{5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/689612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What does this line in Lang's "Algebra" mean? Lang's "Algebra" says the following:
Let $S$ be a set. A mapping $S\times S\to S$ is sometimes called a law of composition (of $S$ into itself).
I always thought $S\times S\to S$ implied a binary operation on two elements of $S$, and $S$ being closed on that binary operat... | The phrase “law of composition” is a direct translation from the French loi de composition (usually also interne is added). See Bourbaki, Éléments de Mathématique.
It's just a name and has nothing to do, in general, with function composition. Indeed, Lang says that sometimes a map $S\times S\to S$ is called a law of co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/689670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Properties of the cofinite topology on an uncountable set
Let $X$ be an uncountable set and let $\mathcal T = \{U \subseteq X : U = \varnothing\text{ or }U^c \text{ is finite} \}$.
Then is topological space $(X,\mathcal T)$
*
*separable?
*Hausdorff?
*second-countable (has a countable basis)?
*first-countable (... | Hints:
*
*Show that every infinite set is dense; in particular, the countably infinite sets. (Fix an infinite $A \subseteq X$, and show that $U \cap A \neq \varnothing$ for every nonempty open $U \subseteq X$.)
*Show that any two nonempty open sets have nonempty intersection.
*If $\mathcal{B} \subseteq \mathcal{T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/689756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Heisenberg XXX spin model Let $\pi$ be the standard representation of $sl_2(\mathbb{C})$ on $\mathbb{C}^2$. Let $p_1,p_2,p_3$ the three Pauli matrices. Define $S^a:=\frac{1}{2}\pi(p_a)$. What does such matrices looks like?
| Using
In[6]:= s1 = -I/2 PauliMatrix[1]; s2 = -I/2 PauliMatrix[2];
s3 = -I/2 PauliMatrix[3];
We can verify that $S^1$, $S^2$ and $S^3$ satisfy commutation relations of $\mathfrak{sl}_2(\mathbb{C})$:
In[8]:= {s1.s2 - s2.s1 - s3, s3.s1 - s1.s3 - s2, s2.s3 - s3.s2 - s1}
Out[8]= {{{0, 0}, {0, 0}}, {{0, 0}, {0, 0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/689876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Existence of a certain subset of $\mathbb{R}$
To every real $x$ assign a finite set $\mathcal{A}(x)\subset \mathbb{R}$
where $x\not\in \mathcal{A}(x)$. Does there exist $\mathcal{W}\subset
\mathbb{R}$ such that:
$$1.\;\;\mathcal{W}\cap \mathcal{A}(\mathcal{W})=\varnothing\qquad
2.\;\;|\mathcal{W}|=|\mathbb{R}|$$
... | Let $\mathcal{Q}=\{[p,q]:\;p,q\in\mathbb{Q},\;p<q\}$. $\mathbb{Q}$ dense in $\mathbb{R}$ and $\mathcal{A}$ finite $\Rightarrow$ we may choose $\phi:\;\mathbb{R}\to\mathcal{Q}:$ $$(\text{i}):\;\;x\in\phi(x)\qquad (\text{ii}):\;\;\phi(x)\cap\mathcal{A}(x)=\varnothing$$
Since $|\mathcal{Q}|=|\mathbb{N}|$ there exists $I\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/689966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 0
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Show that the equation, $x^3+10x^2-100x+1729=0$ has at least one complex root $z$ such that $|z|>12.$ Show that the equation,
$x^3+10x^2-100x+1729=0$ has at least one complex root $z$ such that $|z|>12.$
| Let $~α_1,~α_2,~α_3~$ be roots of the equation. (These are complex numbers and existence is guaranteed by the Fundamental Theorem of Algebra). What do we already know about these roots? We know the sum of roots, sum of product of roots taken two at a time, and product of roots. These are expressible by the coefficients... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/690033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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how to find a matrix A given the solution? if we need,for example, to find a nonzero 3x3 matrix A such that we are given a 3x1 vector as a solution to Ax = 0. What is the general procedure we can follow to obtain such Matrix A?
Thank you :)
| Let's suppose that your vector $v$ is a column vector.
One of options is to look at the matrix $B:=vv^T$: it's a $3\times 3$ matrix, and $Av = \|v\|^2v$. Now we can look at the matrix $A:=(\|v\|^2Id-B)$: easy to check that $v$ belongs to its nullspace. We need to check that our $A$ is not zero; indeed, take any nonzer... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/690139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$" I have a question that says this:
Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$ according to the fundamental theorem of finitely generated abelian groups.
I would like to see how it is c... | I'm assuming that $\langle(1,1,1)\rangle$ means the subgroup generated by $(1,1,1)$, or in other words $\{(k\bmod 5,k\bmod 4,k\bmod 8)\mid k\in\mathbb Z\}$.
In that case we can see that each of the cosets that make up the quotient must contain an element of the form $(0,x,0)$. Namely, assume that $(a,b,c)$ is some elem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/690209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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What is $\Bbb R^{\times}$? [unit group, ring to "times" power] I'm doing some sheets for my Abstract Algebra class and I can't seem to remember the group defined as $\mathbb{R}^{\times}$. It's obviously some variation of $\mathbb{R}$ but I'm away from college on reading week so can't ask my tutor. If someone could clea... | The notation is often used on the form $\Bbb R^*$ i.e. with a star and it means $\Bbb R\setminus \{0\}$ and we have $(\Bbb R^*,\times)$ is a multiplicative group.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/690301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
All subgroups of index 2 Can we construct examples of non-abelian groups G (finite or infinite) such that for all of it's non-trivial subgroups has index 2 in G?.
| Not of finite order. Suppose $G$ has subgroup of index 2. Then order of $G$ is even, as it's a union of two cosets of such subgroup. By Cauchy's Theorem, $G$ has member of order two. If $|G|>4$, index of the subgroup generated by this element is greater than 2. If $G$ has no subgroups at all, it is cyclic (of prime or... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/690454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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$H_{n-1}(M;\mathbb{Z})$ is a free abelian group need help with this problem:
show that if $M$ is closed connected oriented n-manifold then $H_{n-1}(M;\mathbb{Z})$ is a free abelian group.
thanx.
| $$O \to Ext(H_{n-1}(M),\mathbb{Z}) \to H^n(M) \to Hom(H_n(M),\mathbb{Z}) \to 0$$
As the latter arrow is an isomorphism when M is closed, connected and orientable, it follows that $Ext(H_{n-1}(M),\mathbb{Z})=0$.
You just need to understand why it implies your $n-1$ torsion group $T_{n-1}$ is $0$...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/690857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Validating a PDE problem solution I have the following problem, which I have tried to solve myself and I would like someone to verify that my answer is valid. The problem is the following:
By separation of variables, derive the family
$$u_{mn}^{\pm}(x,y,z) = \sin(m\pi x)\cos(n\pi y)\exp(\pm\sqrt{m^2+n^2}\pi z)$$
of th... | The method is correct. At the end, to arrive at the given solution you choose some coefficients. A little remark: for completeness, you should also consider the case $b=0$.
This cases arises when, dealing with the boundary conditions for $Y(y)$, you write
$$Y^{'}(0)=C_4b=0, $$
$$Y^{'}(1)=-C_3b\sin b+C_4 b\cos b=0.$$
L... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/690973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Calculate the series: $\sum^\infty_{n=1}\frac{(-1)^n}{n^2}$ using dirichlet's theorem This question was in my exam:
Calculate the series: $$\sum^\infty_{n=1}\frac{(-1)^n}{n^2}$$.
I answered wrong and the teacher noted: "You should use dirichlet's theorem".
I know my question is a bit general,
but can you please explai... | By absolute convergence you can simply write:
$$\sum_{n=1}^{n}\frac{(-1)^n}{n^2}=\sum_{n \text{ even}}\frac{1}{n^2}-\sum_{n\text{ odd}}\frac{1}{n^2}=2\cdot\sum_{n \text{ even}}\frac{1}{n^2}-\sum_{n\geq 1}\frac{1}{n^2}=\frac{2}{4}\sum_{n\geq 1}\frac{1}{n^2}-\sum_{n\geq 1}\frac{1}{n^2}$$ $$=-\frac{1}{2}\sum_{n\geq 1}\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/691230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Localization at a maximal ideal and quotients.
If we have a commutative ring $R$ and a maximal ideal $m$, then is
$m/m^2$ isomorphic to $m_m/m^2_m$?
Thx.
| It is enough to show that $R/\mathfrak{m} \cong R_\mathfrak{m}/\mathfrak{m}_\mathfrak{m}$, since $\mathfrak{m}/\mathfrak{m}^2$ and $\mathfrak{m}_\mathfrak{m}/\mathfrak{m}_\mathfrak{m}^2$ are just the base changes of the $R$-module $\mathfrak{m}$ to these respective rings.
This is straightforward with universal properti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/691469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Is $\sum_{x=1}^n (3x^2+x+1) = n^3+2n^2+3n$? I wanna check if the following equation involving a sum is true or false? How do I solve this? Please help me.
$$ \sum_{x=1}^n (3x^2+x+1) = n^3+2n^2+3n$$
for all $n \in \{0,1,2,3, \dots\}$.
| Not quite. Note that it fails at $n=1$. A closed form expression for the sum is $n^3+2n^2+2n$.
Remark: Recall that $\sum_1^n k^2=\frac{n(n+1)(2n+1)}{6}$ and $\sum_1^n k=\frac{n(n+1)}{2}$. And of course $\sum_1^n 1=n$.
Or else you can prove the result directly by induction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/691566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
The limit of $|z|^2/z$ in the complex plane What is the limit of $|z|^2\over z$ in the complex plane at $z_0=0$?
This is how I do it: ${|z|^2\over z}={{x^2+y^2}\over {x+iy}} $, then along the real axis, and the imaginary axis, the limit approaches different value, namely $y/i$ and $x$, so the limit DNE. Is that correct... | Let $f(z)=|z|^2/z$ then
$$\lim_{z\to0}|f(z)|=|z|=0$$
so
$$\lim_{z\to0}f(z)=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/691685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Who generates the prime numbers for encryption? I was talking to a friend of mine yesterday about encryption. I was explaining RSA and how prime numbers are used - the product $N = pq$ is known to the public and used to encrypt, but to decrypt you need to know the primes $p$ and $q$ which you keep to yourself. The fact... | They are generated on the machine doing the encryption. Generating primes of a given size is fairly easy, and verifying that they are prime can be done much faster than trial division.
1024-bit RSA requires two 512-bit primes. On my (old) machine it takes about 34 milliseconds to generate a 512-bit prime (so generating... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/691797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
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