Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
For each even number n greater than 2, there exists a 3-regular graph with n nodes. I am having a hard time of understanding a proof in the book, "Introduction to the Theory of Computation, Third Edition (international)", In page 21.
PROOF
Let $n$ be an even number greater than 2. Construct graph $G = (V,E)$
with n no... |
\begin{eqnarray*}
E = \{ (i, i + 1) | \text{for } i= 0 \cdots n − 2 \} \cup \{ (n − 1, 0) \} \\
\cup \color{red}{ \{(i, i + n/2) | \text{for } i=0 \cdots n/2 − 1 \} }.
\end{eqnarray*}
| {
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"timestamp": "2023-03-29T00:00:00",
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Inverse function of an integral function I am trying to find the inverse function of the following function , or prove that it does not have one , but I can't do either of those things.
$$ y(τ)= \int_{τ-1}^{τ+1} \cos(\frac{πt}{8})\,x(t) \,dt $$ where $x(t)$ is a function with the same domain as $y(t)$. Can someone help... | After evaluating the integral, we can see $y(t)$ is constant $\forall \ t$. Therefore, it has no inverse.
| {
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Does this tricky trigonometric series converge? My question was to know if the following series converges
$$
\sum_{n \geq 0}^{ }\cos\left(\frac{\pi}{4}\left(7+4\sqrt{3}\right)^n\right)
$$
I may have found a ( weird ) way to do it, but I would like to know how you, you would be doing this. Is there specific theorem that... | $$ (7 + \sqrt {48})^n + (7 - \sqrt {48})^n = (7 + \sqrt {48})^n + \left( \frac{1}{7 + \sqrt {48}}\right)^n $$
is always an integer. Indeed, always an integer $m$ such that $m \equiv 2 \pmod 4.$ Including negative $n,$ the values are
$$ ..., 2702, 194, 14, 2, 14, 194, 2702,... $$
such that
$$ a_{n+1} + a_{n-1} = 14 ... | {
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How to find all real and complex solutions to the polynomial $r(t) = t^9 − 1$ Thanks for the detailed answers everyone. As i understand it, this class of problem is standard and can be solved with De Moivre's theorem, even if I haven't informed myself about that yet.
Otherwise, I would just state that my knowledge of ... | .Your method can be improved considerably.
$$
t^9-1 = (t^3)^3 - 1^3 = (t^3-1)(t^6 + t^3+1) = (t-1)(t^2 + t + 1)(t^6 + t^3 + 1)
$$
At this stage, we can see that one of the roots is $t = 1$. Two others are obtained by solving for the quadratic $t^2+t+1 = 0$, easily done via the usual formula.
EDIT : The last term is $... | {
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Eigenvalue of a special random matrix? Given a $n \times n$ symmetric random matrix whose diagonal elements are $1$, and the elements of $k\times k, k<n$ leading principal sub-matrix are $1$. All other values are i.i.d. uniformly randomly drawn from $[0,1]$.
For example, it could look like $\left( {\begin{array}{*{20}... | Let $A$ be your matrix.
If $e$ is the vector of all $1$'s, the greatest eigenvalue of $A$ is at least
$\dfrac{e^T A e}{e^T e} =\dfrac{e^T A e}{n}$. Now $e^TAe$ is the sum of the entries of $A$, which $\ge n^2/2$ with probability $> 1/2$ (since the sum of the entries that are not fixed at $1$ has a symmetric distribu... | {
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Problem on Principle of Inclusion-Exclusion How many integers $1, 2,....., 11000$ are invertible modulo $880$?
$880$ can be rewritten as $2^4\cdot5\cdot11$.
So I am supposed to find the number of integers in this range that have $2$, $5$ or $11$ as a divisor and then subtract that value from $11000$.
So If I divide $1... | Denote the set of integers which are divisible by $n$ in the range $\{1,2,\cdots,11000\}$ by $N(n)$. We wish to find $|N(2)\cup N(5)\cup N(11)|$. We can use inclusion exclusion.
$$|N(2)\cup N(5)\cup N(11)|=|N(2)|+|N(5)|+|N(11)|-|N(2)\cap N(5)|-|N(2)\cap N(11)|-|N(5)\cap N(11)|+|N(2)\cap N(5)\cap N(11)|$$
As stated in t... | {
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prove the root of $(z-2018)^{2n}+(z+2018)^{2n}=0$ is purely imaginary Prove: if $z \in \mathbb{C}$ satisfies $(z-2018)^{2n}+(z+2018)^{2n}=0$,then $z=bi$ for some $b \in \mathbb{R} (b \neq 0)$.
The method i can think of is use the binomial theorem to get:
$$\sum_{k=0}^{2n} \binom{2n}{k}z^{2n-k}(-2018)^{k}=-\sum_{k=0}^{2... | If $(z-2018)^{2n}+(z+2018)^{2n}=0$, then $|z-2018|^2=|z+2018|^2$.
It is your turn to show:
$|z-2018|^2=|z+2018|^2 \iff z+ \overline{z}=0 \iff Re(z)=0.$
| {
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How to solve linear congruences as a newbie? Please, before referring to another problem on the site or giving a link, I would like to say I read most of them such as: This, This, and many others but due to the (subtle) difference in my question, I'm having a really hard time to apply the methods used in other question... | HINT.-As a newbie you could act as follows:
First at all, note that $17$ is invertible modulo $100$ because it is not divisible by $2$ nor by $5$ so $17b \equiv 1 (mod\ 100)$ has solution.
Since $17n=17+17\cdots+17$ (n times), you could add successively $17$ plus $17$ until you get a number of the form $100x+1$ (this ... | {
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$\int_0^1\int_0^1\binom{\text{something}}{\text{something}}\binom{\text{something}}{\text{something}}\cdot \text{something }dxdy$ with closed-form I've calculated an approximation of integrals like than $$\int_0^1\int_0^1\binom{f(x)}{f(y)}\binom{f(y)}{f(x)}dxdy\tag{1}$$ for simple functions $f(x)$. I don't know if some... | Well, first of all you can realise that:
$$\binom{\text{f}\left(x\right)}{\text{f}\left(y\right)}\cdot\binom{\text{f}\left(y\right)}{\text{f}\left(x\right)}=\frac{\sin\left(\pi\cdot\left(\text{f}\left(x\right)-\text{f}\left(y\right)\right)\right)}{\pi\cdot\left(\text{f}\left(x\right)-\text{f}\left(y\right)\right)}=$$
$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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"answer_id": 0
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A question about elements of sets If $x$ is a set and $y\in x$, will it imply $y$ is a set?
Can we prove it using just axioms of Set theory and formal proof system?
If we add this as axiom in Axiom of Set theory, will new axiom system be inconsistent because of Lowernheim Skolem theory? (as there will not any countable... | You seem to be assuming that the language of ZF includes a special sort for "sets" - so that there is a distinction between "set variables" and "general variables." This is a feature of some set theories with urelements, but not of ZF: there is only one kind of "object" (syntactically speaking, only one kind of variabl... | {
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Flip side of Feynman's trick for Integration If I differentiate the integral:
$$\int_{-a+2}^{a-2} \ (a-x) \, da$$
then I get 4 - 2 a.
1) Is it possible to get back to integral in the form $ \int_{-a+2}^{a-2} \ (a-x) \, da$?
The application would be to find a way to use the 'flip side of Feynman's trick' described on pa... | $x$ is a bound (also called dummy) variable. You can use any other variable except $a$
$$
\int_{-a+2}^{a-2}\ (a-x)\,dx =
\int_{-a+2}^{a-2}\ (a-z)\,dz = \cdots
$$
In any case, your integral only depends of $a$.
Edit:
$$
\frac{d}{da}\int_0^1\frac{x^a - 1}{\ln(x)}\,dx\,=
\int_0^1 x^a\,dx
$$
Are you mixing Leibniz rule (th... | {
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Solving a complex quadratic-like equation Let $a,b,c,d$ are given non-zero complex numbers (i.e., constants). Then is it true that the equation
$$a|z|^2+b z+c\bar{z}+d=0$$
which is equivalent to $$|z|^2+b'z+c'\bar{z}+d'=0$$
will always have a (at least one) solution for $z$? Or there is some necessary and sufficient c... | The equation wil not always have solutions, as pointed out already.
The following gives a necessary condition for solutions to exist. Consider WLOG the case $\,a=1\,$, then taking the complex conjugates on both sides gives $\,|z|^2+\bar b \bar z+ \bar c z+\bar d=0\,$. Eliminating $\bar z\,$ between the latter and the o... | {
"language": "en",
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How were the values of this trigonmetric ratio determined? I'm reading a book that is pretty spartan about definitions. How did the book come up with the length of the sides of this triangle?
I understand the trig ratios once we have the lengths... but how were the lengths of $\sqrt{3}$, 1, and 2 determined? I think ... | A bit of a story.
In 8th grade (mid-1980s for me), my math teacher drilled us on five Pythagorean triplets:
*
*$1, 1, \sqrt{2}$
*$1, \sqrt{3}, 2$
*$3,4,5$
*$5,12,13$
*$8,15,17$
We learned to find the missing side(s) by pattern-matching for these triangles. (We had a dozen triangles, and we were given four minu... | {
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Spivak Calculus 3rd Edition Chapter 1 Problem 4 (iii) I'm not sure on arriving at the solution to $5-x^2 < -2$ . I've got:
$5-x^2 < -2$
$-x^2 < -7$
$x^2 < 7$
$\sqrt x^2 < \sqrt 7$
$x < \sqrt 7$
But the actual solution is $x > \sqrt7$ or $x < -\sqrt7$
Can someone point me in the right direction on this, thanks.
| Note that
$$
-x^2 < -7\iff x^2> 7\iff|x|>\sqrt{7}.
$$
In the first step multiplying both sides of the inequality by $-1$ "flips" the inequality sign.
| {
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Sum of Big O - which one is it? On the wikipedia page, we have the following property:
If $f_1(x) = O(g_1(x))$ and $f_2(x) = O(g_2(x))$ then $f_1(x) + f_2(x) = O(|g_1(x)| + |g_2(x)|)$.
But in my textbook, I also see the following sum property:
If $f_1(x) = O(g_1(x))$ and $f_2(x) = O(g_2(x))$ then $f_1(x) + f_2(x) = O(... | They are equivalent (assuming that $g_1,g_2$ are assumed non-negative in your textbook). Note that
$$
\max(a,b)\leq a+b \leq 2\max(a,b)
$$
for any $a,b\geq 0$, and that the constant $2$ can be "hidden" in the $O(\cdot)$ notation.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "14",
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Finding Range from Domain In the book Thomas' Calculus,in exercise section i got one question e.g. Find the domain and Range of G(t) = $\frac{2}{t^2-16}$.
Ans: Domain is (-∞,-4) U (-4,4) U (4,∞)..I understand this.Let us discuss how they find Range
*
*t<-4 as (-∞,-4)
=> -t>4 Multiply by -1
=> $(-t)^2$ > $4^2$ ... | The solutions for $1$ and $2$ are incomplete and for $3$ is just wrong.
In $1$ and $2$, surely $\frac{2}{x^2-16}\gt 0$. However, it says nowhere that each of those values is reached by substituting some $x$.
In $3$, it should say $-4\lt x\lt 4$, and then $-16\le x^2-16\lt 0$, but then $-\frac{2}{16}\ge\frac{2}{x^2-16}$... | {
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How to integrate $\int_{0}^{t}{\frac{\cos u}{\cosh^2 u}du}$? How to integrate $\int_{0}^{t}{\frac{\cos u}{\cosh^2 u}du}$?
I'm trying to use the integration by parts but it's impossible...
Is there an other way?
| If you are just interested in the limit as $t\to +\infty$ (namely $\frac{\pi}{2\sinh\frac{\pi}{2}}$), there are better ways (namely the Fourier transform), but you may notice that
$$\begin{eqnarray*}\int_{0}^{t}\frac{\cos u}{\cosh^2 u}\,du&=&\left[\cos(u)\tanh(u)\right]_{0}^{t}+\int_{0}^{t}\tanh(u)\sin(u)\,du\\&=&\cos(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2603477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does there exist any relationship between non-constant $N$-Exhaustible function and differentiability? I was trying to solve this question. If $f ◦f$ is differentiable, then $f ◦f ◦f$ is differentiable. While trying to find the counterexample. I come across the Dirichlet function. $f(x) = \begin{cases} 1 & x \in \mathb... | I shall construct a few examples to show one way it can be done.
$
\def\rr{\mathbb{R}}
\def\lfrac#1#2{{\large\frac{#1}{#2}}}
$
Let $f$ be the function on reals such that $f(x) = 0$ for every real $x \le 0$ and $f(x) = -\exp(\lfrac1x)$ for every real $x > 0$. Then $f$ is infinitely differentiable and $2$-exhaustible.
Us... | {
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Particular commutator matrix is strictly lower triangular, or at least annihilates last base vector Let $(e_1,e_2,\ldots,e_n)$ be the canonical basis of ${\mathbb C}^n$. Let $A$
be a $n\times n$ matrix such that $Ae_k=e_{k+1} (1\leq k \leq n-1$ (so everything in
$A$ is specified except for the last column). Let $B$ be ... | It's true when $charpoly(A)$ has only simple roots; otherwise it's false.
Assume $n=4$ and consider
$A=\begin{pmatrix}0&0&0&-1\\1&0&0&0\\0&1&0&2\\0&0&1&0\end{pmatrix},B=\begin{pmatrix}0&0&-1&0\\0&-1&0&-2\\-1&0&0&0\\0&0&0&1\end{pmatrix}$ where $charpoly(A)=charpoly(B)=(x-1)^2(x+1)^2$.
Then $C=\begin{pmatrix}0&1&0&1\\1&... | {
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Example of simple modules
Find a simple $\mathbb{Z}[1/2]$-module.
What would be an example and how would we think about this type of problems?
First I looked at the $\mathbb{Z}$-module $\mathbb{Z}/p$ localized at the multiplicative set $S=\{1,2,2^2, 2^3,\cdots\}$ which is isomorphic to $\mathbb{Z}[1/2] \otimes \mathb... | Follow from Qiaochu's comment.
Given any non zero $m\in M$, $Rm = M$ since $M$ is simple, so we have the surjective $R$-module map
$$f: R\rightarrow M$$
and $R/\ker(f) \cong M$ as $R$-module.
$\ker(f)$ must be a maximal ideal, or else it is contained in some other maximal ideal $I$, then we have a proper submodule o... | {
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Prove $f(x,y)$ is not differentiable in ($0,0)$ Prove $f(x,y)$ is not differentiable in ($0,0)$
$$f(x,y)= \begin{cases}\dfrac{|x|y}{\sqrt{x^2+y^2}}& \text{if } (x,y)\not =0\\ \\ 0&\text{if } (x,y)=0
\end{cases}
$$
I try prove this by existence of the limit.
Let $y=x$ and $x\not = 0$, then
$$f(x,y)=\frac{|x|x}{\sqrt{x^2... | If it were differentiable at $(0,0)$, the corresponding Jocobian is the partial derivatives at $(0,0)$, which you can compute the Jacobian is actually the zero map as well.
So you may try to get a contradiction with the existence of
\begin{align*}
\lim_{(x,y)\rightarrow(0,0)}\dfrac{1}{\sqrt{x^{2}+y^{2}}}\dfrac{|x|y}{... | {
"language": "en",
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Multiplying prime numbers If I multiply $13$ and $17$ to get $221$ I can only get $221$ by multiplying $13$ and $17$ (excluding $1$ and $221$) does the same rule apply to multiplying $3$ numbers? (excluding the use of $1$)
| So you have two distinct positive primes, let's call them $p$ and $q$. Then their product has precisely four divisors among the positive integers: 1, $p$, $q$ and $pq$ itself, and we verify that $1 \times pq = pq$.
In your example with $p = 13$ and $q = 17$ (or $p = 17$ and $q = 13$, if you prefer), we verify that $1 \... | {
"language": "en",
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"source": "stackexchange",
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GRE question - unintersection I'm struggling to visualize the following GRE problem:
The solution says that for $m$ and $n$ to intersect on the right, it has to be the case that $2x+3x>180$, after that we get $x>36$. But I don't understand why. I can see that if $m$ and $n$ are parallel then $2x+3x = 180$ though. Can ... | If $m$ and $n$ intersect to the right, then a triangle is formed, two of whose angles are the angles just to the right of those labelled $2x^\circ$ and $3x^\circ$. These angles are $180-2x$ degrees and $180-3x$ degrees respectively, and so $180-2x+180-3x<180$ since the sum of all three angles in the triangle is $180$ ... | {
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Prove that $ \frac{1}{1+n^2} < \ln(1+ \frac{1}{n} ) < \frac {1}{\sqrt{n} }$ For $n >0$ ,
Prove that $$ \frac{1}{1+n^2} < \ln(1+ \frac{1}{n} ) < \frac {1}{\sqrt{n}}$$
I really have no clue. I tried by working on $ n^2 + 1 > n > \sqrt{n} $ but it gives nothing.
Any idea?
| This inequality is correct only for $n >1$ and not for $n >0$ as asked by the person.
For $n >1$, we know that $$\tag1 \frac{1}{n}<\frac{1}{\sqrt n}$$ and $$\tag2 \frac{1}{(1+n^2)} < \frac{1}{(1+n)}$$
Also, $\tag3\frac1{n+1}<\ln\left(1+\tfrac1n\right)<\frac1n, \forall n>0$
So the overall inequality $$\tag4\frac{1}{(1+n... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
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"answer_id": 1
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Determining a $4\times4$ matrix knowing $3$ of its $4$ eigenvectors and eigenvalues The precise question given goes as follows;
The matrix 'A' $\in \Bbb R^{4 \times4}$ has eigenvectors $u_1, u_2, u_3, u_4$ where
$u_1 = \begin{pmatrix}
1 \\
-1 \\
1 \\
1 \\
\end{pmatrix}
, u_2 = \begin{pmatrix}
... | The answer could be arbitrary unless $w$ is a combination of the given eigenvectors (if it's not then an arbitrary value of $Aw$ defines completely the linear map since you know the images of a linear basis).
One approach will be to find this combination -- it is
$$
w = 16 u_1 + 20 u_2 + 11 u_3.
$$
From this you comp... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Levi-Civita-connection of an embedded submanifold is induced by the orthogonal projection of the Levi-Civita-connection of the original manifold Let $(M, g)$ be a Riemannian manifold with Levi-Civita-connection $\nabla$, and let $N \subseteq M$ be an embedded submanifold with a $g$-induced Riemannian metric $h$. I now ... | A possible way to prove this is to remember that the LC connection is the unique torsion free connexion for which the metric tensor is parallel.
The fact that your formula gives a connexion is obvious.
To check that it is torsion free, note that $g_N(\tilde \nabla _X Y, Z)= g(\nabla _X Y, Z)$ for every triple of tang... | {
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$\int \ x\sqrt{1-x^2}\,dx$, by the substitution $x= \cos t$ I have tried to determine $\int \ x\sqrt{1-x^2}\,dx$ using trigonometric
formula by the substition $x=\cos t$ I have got :
$$-\int \cos t \sin^2 t\,dt\tag{1}$$ for $\sin t > 0$ and $$ \int \cos t \sin^2 t\,dt\tag{2}$$ for $\sin t <0 $.
But both $(1)$ and $(2... | From $\int \cos t \sin^2 t dt$ you can use $u=\sin t, du=\cos t \ dt$ to get $\int u^2 du=\frac {u^3}3+C$ and backsubstitute.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2605115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How to I find the Taylor series of $\ln {\frac{|1-x|}{1+x^2}}$?
The expression is $$\ln {\frac{|1-x|}{1+x^2}}$$
I'm told there's an easy way to do it to get the first 2 non-zero term but I ended up differentiating this answer several times and got a very long answer that is not correct.
What I did in specific:
*
*s... | So you have $$f(x)=\ln {\frac{|1-x|}{1+x^2}},$$ that means
$$f'(x)=-\frac1{1-x}-\frac{2x}{1+x^2}=(-1-x-x^2-\ldots)-(2x-2x^3\pm\ldots)=-1-3x-x^2+\ldots$$
Integration gives $$f(x)=-x-\frac32x^2-\frac13x^3+\ldots,$$
since $f(0)=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2605214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Find the integral $\int\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}\,dx.$ The question is $$\int\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}\,dx.$$
I have tried to multiply both numerator and denominator by $1-\sqrt{x}$ but can't proceed any further, help!
| I think its useful to learn some standard substitution you can use for such kind of problem.
for this case instead of using $x = \cos^2\theta$ I'm gonna try using $x=\cos^22\theta$ $$x=\cos^22\theta \Rightarrow dx=\left(-4\cos 2\theta \sin 2\theta \right)d\theta$$
so we have $$\int\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2605444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 5
} |
Why is substitution valid in Intergals? So i am a little confused why the u-substitution is valid. I do not have a problem when the integral is in the form $\int f(g(x))g'(x)dx=F(g(x))$ clearly then we can do $u=g(x)$ find $F(u)$ and do $F(g(x))$. However, things are not always so simple. Take the function $\int{\sqrt{... | Actually you have that form in this case too, it is a little hidden. You can write
$$\int x^5\sqrt{1+x^2}dx=\int xx^4\sqrt{1+x^2}dx=\int\frac{2}{2}xx^4\sqrt{1+x^2}dx=\frac{1}{2}\int2xx^4\sqrt{1+x^2}dx.$$
Now you can notice that if you derive $1+x^2$ you get $2x$. So you try to make your integral easier by letting $1+x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2605668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Steps to simplify this boolean expression How do you simplify: ~A*B+A*~C+B*~C to A * ~C + B * ~A
I tried the distributive law but I end up going in circles.
| This equivalence is well known and called the Consensus Theorem.
It can be proven as follows:
$$A'B + AC' + BC' \overset{Adjacency}{=}A'B + AC' + ABC' +A'BC' \overset{Absorption (2x)}{=}A'B + AC' $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2605746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Trouble computing $\int_0^\pi e^{ix} dx$
I am trying to compute the integral of $\int_0^\pi e^{ix} dx$ but get the wrong answer. My calculations are $$
\begin{eqnarray}
\int_0^\pi e^{ix} dx &=& (1/i) \int_0^\pi e^{ix} \cdot i \cdot dx = (1/i) \Bigl[ e^{ix} \Bigr]_0^{\pi} \\
&=& (1/i) \Bigl[ e^{i\cdot \pi} - e^{i\cdot ... | your answer is correct just see that $-i =\frac{1}{i}$. Whereas it could be simpler to write
$$e^{ix} = \cos x+i\sin x$$
Then $$\int_0^\pi e^{ix} dx = \int_0^\pi \cos x dx+i\int_0^\pi \sin x dx \\ =\left[\sin x\right]_0^\pi+i\left[-\cos x\right]_0^\pi=2i$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2605851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
For which natural numbers are $\phi(n)=2$? I found this exercise in Beachy and Blair: Abstract algebra:
Find all natural numbers $n$ such that $\varphi(n)=2$, where $\varphi(n)$ means the totient function.
My try:
$\varphi(n)=2$ if $n=3,4,6$ and I think that no other numbers have this property. So assume $n>7$.
Case ... | For every prime $p\geq 4$ and ever natural number $k$ you have $p^k-p^{k-1}>2$. Since $φ(ab)=φ(a)φ(b)$ if $(a,b)=1$ this means that if you have a prime number $p\geq4$ which $p|n$ then $φ(n)>2$. Therefore only $2$ and $3$ can divide $n$. But $2^{k_1}-2^{k_1-1}=2^{k_1-1}$ and $3^{k_2}-3^{k_2-1}\geq2$ for $k_2\geq2$ the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2606016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Principal Ideal Ring which is not Integral In Atiyah & McDonald: Commutative Algebra the Principal Ideal Domain is a principal ideal ring which is also an integral domain.
I tried but couldn't find examples of commutative rings with identity that have the property that every ideal is generated by a single element but a... | In general, if $R$ is a PID, then every quotient of $R$ is a PIR.
Ironically, $\Bbb Z_6$ is such a ring, because it is $\Bbb Z / (6)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2606121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Why does $\alpha = -\alpha$, where $\alpha \in F = \{0, 1, \alpha, \alpha^2\}$? $F$ is a field with 4 elements, $\{0, 1, \alpha, \alpha^2\}$, where $\alpha \neq 0$ and $\alpha \neq 1$
This is the setup for a previous exam paper question. The question is of little importance, as the solution isn't too difficult for me t... | As mentioned in the comments, $\alpha = - \alpha$ follows at once because a field with $4$ elements has characteristic $2$ and so $2x=0$ for all $x$. This follows from Lagrange's theorem applied to the additive group of the field. Indeed, $0 = 4 \cdot 1 = (2 \cdot 1)(2 \cdot 1)$ implies $2 \cdot 1
=0$, because a field ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2606248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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If $|ax^2+bx+c|\le \frac12$ for all $|x|\le1$, then $|ax^2+bx+c|\le x^2-\frac12$ for all $|x|\ge1$
Prove that if $|ax^2+bx+c|\le \frac12$ for all $|x|\le1$ then $|ax^2+bx+c|\le x^2-\frac12$ for all $|x|\ge1$.
My attempts:
Let $f(x)=ax^2+bx+c$
I know that
1) if $f(a)<0$ and $f(b)>0$ then exist $x_0\in[a;b]$ then $f(x_... | Since $$f(-1)=a-b+c,\quad f(0)=c,\quad f(1)=a+b+c$$ we can write
$$a=\frac{f(1)+f(-1)-2f(0)}{2},\quad b=\frac{f(1)-f(-1)}{2},\quad c=f(0)$$
Suppose here that there exists a real number $p$ such that
$$|p|\ge 1\qquad\text{and}\qquad |ap^2+bp+c|\gt p^2-\frac 12$$
Then,
$$\begin{align}p^2-\frac 12&\lt\left|\frac{f(1)+f(-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2606384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Prove that two kernels are equal Let $V$ be a finite dimensional linear space.
How do you prove that there exists such an m that:
$$\text{ker} (T^m) = \text{ker} (T^{m+1}).$$
I have managed to prove using induction that for each $m≥1$,
$\text{ker} (T^m) ⊆ ker (T^{m+1})$.
So, I have a unidirectional inclusion.
Instead ... | Consider $\{\dim\ker(T^k):k\ge0\}$; this set of natural numbers is bounded by $\dim V$, hence there is an $m$ so that $\dim\ker(T^m)$ is maximal.
Now you know that $\ker(T^{m})\subseteq\ker(T^{m+1})$, which implies
$$
\dim\ker(T^{m})\le\dim\ker(T^{m+1})
$$
By maximality of $\dim\ker(T^{m})$, you infer the two dimension... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2606460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Finding $Im(T)$ and $Ker(T)$ of the following linear transformation
Let $$T:\mathbb{R}^4\to\mathbb{R}^3$$
$$T(x,y,z,w)=(x-y+z-w,x+y,z+w)$$
I need to find $\operatorname{Ker}(T),\operatorname{Im}(T)$ and the basis of them and to show if $T$ is is one-to-one and if it onto $\mathbb{R}^3$
I'm having hard time finding $... | You have found that $$A=\begin{pmatrix}1&-1&1&-1\\ 1&1&0&0\\ 0&0&1&1\end{pmatrix}...\to\begin{pmatrix}1&-1&1&-1\\ 0&2&-1&1\\ 0&0&1&1\end{pmatrix}$$Note that the Row Space gives you the rank of $A$ and in your last matrix you have three linearly independent vectors.
That implies the $ Rank(A)=3$ which is the dimension ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2606752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Somewhat shady way of solving a problem from Baby Rudin The problem statement is: If $\mathbb{R}^n$ is the (countably) infinite union of closed sets, show at least one of those closed sets has non empty interior.
My shady way of solving this is noting that a closed set without interior is a boundary (i.e. of some open ... | Lebesgue measure of a boundary (of an open set) need not be zero. Not even in $\mathbb R^1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2607002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Non-standard axioms + ZF and rest of math I've never taken a formal course of Set Theory, but I've been wondering about this for some time now.
Are non-standard axioms, like $\mathbb{V}=\mathbb{L}$ and axioms about large cardinals and any other you can think of (which are independent of $ZFC$) used outside pure Set The... | Grothendieck universes, developed by Grothendieck for use in algebraic geometry, and which have applications in category theory, are equivalent to strongly inaccessible cardinals.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2607089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Clarificaiton on barycentric coordinates This is related to ray tracing (which I learnt and then forgot).
Given a triangle in 3D $\widehat{ABC}$, where $A,B,C$ are the points of the triangle
And a parametric line described by $(O,\vec v)$ (origin and direction vector)
We find the point $P$ as the intersection of the l... | The intuition depends on the method you use to perform the computation. If you're comfortable with algebraic areas, I'm pretty sure you can still interpret negative coefficients as (algebraic) areas of the appropriate triangles. Note that in that case, you have to be careful with the permutation you use, since $\text{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2607325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Book/Online Video Lectures/Notes Recommendation for Analysis(topics mentioned) I am going to start a first course in Analysis soon in university this semester (in around a week).
Can anyone please recommend me good books/online notes or video lectures that can help me in studying analysis? I'll be studying the followi... | I personally recommend studying analysis through 'Understanding Analysis' by Stephen Abbott. I found it to be a fantastic book, with the treatment rigourous and suitable very much for beginners.
Also see this earlier MSE question.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2607608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Solving the heat equation with robin boundary conditions I have a coupled non-dimensional diffusion system in $v(z,\tau)$, formulated by the following equations
\begin{align}
\frac{\partial v}{\partial \tau} &= \Delta\frac{\partial^2 v}{\partial z^2},
%
\qquad &\text{for}\ z\in[0,1],\ \tau>0 \\
%%%
\frac{\partial v}{\... | The equation in $Z$ is
$$
-Z'' = \lambda^2 Z, \\
Z'(0)-EZ(0)=0 \\
Z'(1)+DZ(1)=0.
$$
If $Z_1$ is a solution for $\lambda_1$ and $Z_2$ is a solution for $\lambda_2$, then
\begin{align}
(\lambda_2^2-\lambda_1^2)\int_{0}^{1}Z_1(z)Z_2(z)dz & = \int_{0}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2607742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Given a local field is the maximal unramifield extension always finite? I was just wondering given a local field complete with non-archimedean valuation, is the maximal unramified extension always finite or could it be infinite? Any comments are appreciated!
| For a finite extension of $\Bbb Q_p$ the maximal unramified extension
is infinite. It is generated by adjoining the $n$-th roots of unity
for all $n$ coprime to $p$. The Galois group is the cyclic profinite
group $\hat{\Bbb Z}$, and is naturally isomorphic to the Galois
group of the algebraic closure of $\Bbb F_p$. Thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2607918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding DNA sequences of length $3$ In the following question I am trying to determine how many DNA sequences of length $3$ that have no $C$'s at all or have no T's in the first position.
Below are my workings,
So there are $4$ DNA letters, $A,T,C,G$
Considering how many DNA sequences of length $3$ that have no $C'$s,... | Your answer is correct for each separate question -- you are using the product rule correctly.
If the question imposed both restrictions simultaneously, you should proceed much in the same manner as you did before: how many possibilities are there for the first position? What about the second? And so on. There is no '... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2608001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Let $G$ be a cyclic (or not) group of order $n$ and let $k$ be an integer relatively prime to $n$. Prove that the map $x \mapsto x^k$ is surjective. Let $G$ be a cyclic group of order $n$ and let $k$ be an integer relatively prime to $n$. Prove that the map $x \mapsto x^k$ is surjective. Use Lagrange's Theorem (Exercis... | The difference is how you get this property: $g^{n}=1$ for any $g\in G$. (You used this property in your last step: $y^{n}=1$.)
For the cyclic group case, $G=\langle t\rangle$. Let $g\in G$. Then $g=t^{m}$. So $g^{n}=t^{mn}=(t^{n})^{m}=1$.
For the finite group case, this is proved as a corollary of Lagrange's Theorem.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2608094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Compute the limit without L'Hospital's rule I'm not sure how to handle the trig functions with different arguments when computing this limit using L'Hospital's rule.
$$\lim_{x \rightarrow 0} \frac {x^2\cos(\frac {1} {x})} {\sin(x)}.$$
I have come up with the correct numerical answer via a different method, but am unsu... | $$
\frac {x^2\cos\frac 1 x} {\sin x} = x\cdot \frac x {\sin x} \cdot \cos\frac 1 x
$$
Now use the fact that $-1 \le \cos \frac 1 x \le 1$ and $x\to0$ and one further fact not mentioned in your question:
$$
\frac x {\sin x} \to 1 \text{ as } x\to0.
$$
Without that last fact or something else other than what's in your q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2608201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
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What is $\lim\limits_{n \to 0} \frac{d}{dx} \frac{1}{n} x^n$? A limit that I find rather intriguing is $\lim\limits_{n \to 0} \frac{d}{dx} \frac{1}{n} x^n$. Following the usual rules for differentiation of polynomials, this would be $\lim\limits_{n \to 0} \frac {nx^{n-1}}{n} = x^{-1}$. It seems unlikely that this is a... | There is no mistake in your reasoning. What you may find confusing is that you started from something that seems to have nothing to do with $\ln(x)$. Except in the process you took $n$ to tend to $0$ at some point. So to be fair let's see what happens when you try to do that before taking the derivative. You will see t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2608368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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What does it mean to say that a function of an arbitrary norm is "continuous with respect to the 1-norm"? I am trying to understand a proof of the equivalence of all norms. Many of the proofs start by showing that two arbitrary norms $|| x ||_p$ and $|| x ||_q$ are equivalent to $||x||_1$ and thus the equivalence relat... | Assume that $\|\cdot\|_{\beta}\leq c\|\cdot\|_{1}$, for the function $f:(X,\|\cdot\|_{1})\rightarrow{\bf{R}}$, $f:x\rightarrow\|x\|_{\beta}$, if $x_{n}\rightarrow x$ in $\|\cdot\|_{1}$, then $\|x_{n}-x\|_{1}\rightarrow 0$, apply Squeeze Theorem to the inequality $\|x_{n}-x\|_{\beta}\leq c\|x_{n}-x\|_{1}$, we get $\|x_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2608508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is this proof correct? (Proof Theory) Is this proof correct?
Use C rule to prove $$\vdash\exists xC(x)\to\exists x(B(x)\lor C(x))$$
Proof:
By hypothesis, $\exists xC(x)$
By the C rule, $C(c)$
By $C\vdash B\lor C,C(c)\lor B(c)$
By $\exists-introduction, \exists x(B(x)\lor C(x))$
By Deduction theorem, $\vdash\exists xC... | Yes. The steps and justifications are okay, and indeed clearly what is required.
$$\begin{split}\exists x~B(x)&\vdash \exists x~B(x)&\textsf{Assumption} \\\exists x~B(x)&\vdash B(c) & \textsf{C-rule / Existential elimination}\\ B(c)&\vdash B(c)\vee C(c) & \textsf{Disjunction introduction}\\B(c)\vee C(c)&\vdash \exist... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2608670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
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Statements on function with finite integral over $[0, \inf[$
Let $f: [0, \infty[\to[0, \infty[$ be a continuous function such that:
$$\int_0^\infty f(x) dx < \infty$$
Which of the following statements are true?
*
*The sequence $\{f(n)\}_{n\in\mathbb{N}} $ is bounded.
*$f(n) \to 0$ as $n\to \infty $
*The se... | $a)$ In the words of angryavian (who deleted his/her answer), consider as a counterexample a function which has infinitely many, thin spikes, progressively taller spikes and is zero elsewhere. The height of the spikes is unbounded, but the width at the bottom of each spike could be made small enough to make the area un... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2608779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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Divisibility property of product two elements in abelian group Let $G$ be a finite abelian group and $d=o(ab), \ m=o(a), \ n=o(b)$.
Show that $d\mid \frac{mn}{\text{gcd}(m,n)}$ and $\frac{mn}{\text{gcd}(m,n)^2}\mid d$.
In particular, if $m$ and $n$ are coprime then order of product is multiplicative.
Proof: $(ab)^{\te... | If $ab$ has order $d$ in abelian group, then $(ab)^d=a^db^d=e$ so $a^d=b^{-d}$. Now write $(m, n)=ms+nt$ for some integers $s$ and $t$. So $$a^{(m, n)d}=a^{msd}\cdot a^{ntd}=b^{-ntd}=e.$$ Therefore $m|(m, n)d$. Similarly $n|(m, n)d$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2608919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Provide a bijection between power set of natural numbers and the Cantor set in $[0,1]$
Question: I am trying to prove that Cantor Middle Third Set $C$ is uncountable, by establishing a bijection $f$ from $C$ onto the power set of $\mathbb{N}.$
My attempt: We know that every element of Cantor set $C$ has a ternary rep... | Here's another proof:
First, note that we can create a bijection between $C$ and all the decimals written using only 0s and 1s by switching all the 2s to 1s. Now consider all those decimals in base 2. Those are precisely all the rational numbers between 0 and 1, so this is a bijection between $C$ and $[0,1]$, which has... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2609018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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If a continuous function is bigger than another at a point, then it is so in an entire neighbourhood...proof? Let $f,g: \mathbb{R}\to \mathbb{R}$, both be continuous at zero. If $f(0)>g(0)$, show that there exists positive $\delta$ such that for all $y,z \in (-\delta , \delta)$, $f(y)>g(z)$.
Attempt at a proof:
I used ... | Choose any number $k$ such that $f(0)>k>g(0) $. Such a number $k$ exists because reals are dense. Now $f(0)>k$ and $f$ is continuous at $0$ so if we take a small neighborhood of $0$ values of $f$ in this neighborhood can be ensured to be closer to $f(0)$ than to $k$ and thus all these values will be greater than $k$. S... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2609111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 4
} |
The dimension of the intersection of two vector subspaces We're given: $$ V = Span\left\{ \begin{bmatrix} 2 \\ 2 \\ 2 \\ 1 \end{bmatrix}, \ \begin{bmatrix} 2 \\ 1 \\ 1 \\ 0 \end{bmatrix}, \ \begin{bmatrix} 5 \\ 4 \\ 1 \\ 1 \end{bmatrix}
\right\} \ \ \ \mathrm{and} \ \ \ W = Span \left\{\begin{bmatrix}1\\-3\\2\\1 \en... | At the first find the $\mathrm{dim}(V\cup W^{\bot})$.
And we know that $\mathrm{dim}(V\cup W^{\bot})=\mathrm{dim}(V)+\mathrm{dim}(W^{\bot})-\mathrm{dim}(V\cap W^{\bot})$
Finding the dimension of $(V\cup W^{\bot})$ is also an easy work.
For above one form a matrix $A$ like following one:
$$\begin{bmatrix} 2 \\ 2 \\ 2 \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2609215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to solve a partial differential equation with 3 variables? I have just learned the Characteristic Method with 2 variables to solve Partial diferential équations... I would like to know how to solve the next partial diferential equation with 3 variables
$$
\frac{df}{dx}+ Q(z_1)\frac{df}{dz_2}+ Q(z_2)\frac{df}{d z_2}... | $$
\frac{df}{dx}+ Q(z_1)\frac{df}{dz_1}+ Q(z_2)\frac{df}{d z_2}=P(x,z_1,z_2)f
$$
$$
\frac{dx}{1}= \frac{dz_1}{Q(z_1)}= \frac{dz_2}{Q(z_2)}=\frac{df}{P(x,z_1,z_2)f}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2609364",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Understanding theorem 9.12 in Rudin's PMA $9.11$ Definition Suppose $E$ is an open set in $R^n,$ f maps $E$ into $R^m,$ and x $\in E.$ If there exists a linear transformation $\mathbf{A}$ of $R^n$ into $R^m$ such that
$$\lim_{h\to 0}\frac{\left|\mathbf{f(x +h)-f(x)-Ah}\right|}{|\mathbf{h}|}=0,\tag{14}$$ then we say $\m... | By Rubin's question is very important in all of calculus:
[derivatives] are the unique transformation of input (displacement) that provides for a linear approximation that goes to zero [faster than the input does] -- his proof is general enough to cover single variable calculus as well as any other mapping $R^m$ to $R^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2609454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Show that $e^{xy}+y=x-1$ is an implicit solution to the differential equation $\frac{dy}{dx} = \frac{e^{-xy}-y}{e^{-xy}+x}$ I began by using implicit differentiation on $e^{xy}+y=x-1$.
From that I got:
$$\left(y+x\frac{dy}{dx}\right)e^{xy}+\frac{dy}{dx}=1$$
Then using algebra I got to the point where I had this equat... | we have $$e^{xy}y+e^{xy}y'x+y'=1$$so we get
$$y'(e^{xy}x+1)=1-ye^{x}$$
$$y'=\frac{1-ye^{xy}}{1+x^{xy}}$$
multiplying numerator and denominator by $$e^{-xy}$$
we get
$$y'=\frac{e^{-xy}-y}{e^{-xy}+x}$$
this is what we want to prove
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2609579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Elliptic Curve and Differential Form Determine Weierstrass Equation I am reading Fermat's Last Theorem by Diamond, Darmon and Taylor and they state:
"An elliptic curve E over a field F is a proper smooth curve over F of genus
one with a distinguished F-rational point. If $E/F$ is an elliptic curve and if $\omega$ is a ... | In my opinion, the statement, that an elliptic curve together with a non-zero holomorphic differential form determines a Weierstrass equation, unfolds its full meaning when considering a family of elliptic curves $E\to S$ over a base scheme $S$ which is not necessarily a field. In this case the sheaf $\Omega_{E/S}$ doe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2609788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Equivalence of weak convergences Let $X$ be a normed space. Let $(x_n)$ be a sequence in $X$. We say that $x_n\to x\in X$ weakly as $n\to \infty$ if $\ell(x_n)\to \ell(x)$ as $n\to \infty$ for all $\ell\in X^*$.
I found a note, where it says in a remark (with no explanation) that, if $X$ were Hilbert space, then $x_n\... | Yes, it remains to show that $\phi(z)$ is actually a bounded map: $|\phi(z)(x)|=|\left<x,z\right>|\leq\|x\|\|z\|$, so $\|\phi(z)\|\leq\|z\|<\infty$, so $\phi(z)\in X^{\ast}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2609890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Finding matrix linear transformation Question:
Find the $3 \times 3$ matrix $ A$, associated with the linear transformation that projects vectors in $\mathbb{R^3}$ (orthogonally) onto the plane $x+y+z=0$.
I was given this question just as a review question for my class. I took linear algebra over 2 years ago, so my me... | All you have to do is to find where the transformation sends the
basic unit vectors
$e_1=\pmatrix{1\\0\\0}$, $e_2$ and $e_3$ to, since these will be
the columns of your transformation matrix.
A normal vector to the plane $x+y+z=0$ is $n=\pmatrix{1\\1\\1}$
so $e_1$ will be sent to $e_1+\alpha n=\pmatrix{1+\alpha\\\alpha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2610117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Consider the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$
Let $ABCD$ be a quadrilateral circumscribing the ellipse $$\frac{x^2}{25}+\frac{y^2}{16}=1.$$ Let $S$ be
one of its focii, then what is the sum of the angles $\angle{ASB}$ and $\angle{CSD}$?
My try:
I have done this for the extreme case of quadrilateral, taking... | Here, I am not very sure about where ABCD are. I am assuming A is at the top left, B at the top right, C at the bottom right, and D at the bottom left. If this interpretation of the question is wrong, please correct me.
We know that the ellipse is "lying down", with a width of $2*5=10$ and a height of $2*4=8$. The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2610215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Pigeonhole principle: prove that a class of 21 has at least 11 male or 11 female students. Here is the problem in full with no other special restrictions:
"If there are 21 students in a class, show that at least 11 must be male or female."
| Let us proof by contradiction. By assuming that there are only 10 boys and 10 girls, we attempt to not fulfill the requirement. However, there are 21 people, so there must be one more boy or girl, meaning that there would be 11 boys or 11 girls.
To answer the comment, we also proof by contradiction. Let us say there... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2610377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Identify $\operatorname{co}(\{e_n:n\in\mathbb N\})$ and $\overline{\operatorname{co}}(\{e_n : n\in\mathbb N\})$ in $c_0$ and $\ell^p$ The question is:
identify $\operatorname{co} (\{e_n:n\in\mathbb N\})$ and $\overline{\operatorname{co}}(\{e_n : n\in\mathbb N\})$ in $c_0$ and $\ell^p$ where $1\leq p\leq \infty$.
N... | The answer is $\{\{a_j\}:a_j \geq 0$ and $\sum a_j =1\}$. This set is contained in $c_0$ and every $l^{p}$ and sequences in this set can be approximated by similar elements of $c_{00}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2610487",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What does it mean by standard coordinates on $R^n$
Here what does "standard coordinates" mean? Is it just the identity map of $R^n$? Or is it an arbitrary element of the standard smooth structure on $R^n$? The text is somewhat confusing. Could anyone please clarify it?
| By "standard coordinates" on $\mathbb R^n$ it must be meant that points are identified by $(x_1,x_2,\ldots,x_n)$.
This corresponds to representing points in terms of the standard (ordered) basis for $\mathbb R^n$ as a vector space, $\mathbf e_1,\ldots,\mathbf e_n$, where $\mathbf e_k$ is the vector whose $k$th coordina... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2610667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Computing the convolution of $f(x)=\gamma1_{(\alpha,\alpha+\beta)}(x)$ If I have a top hat function
$$
f(x)=\begin{cases}
\gamma & \quad \text{if } \alpha<x<\alpha+\beta\\
0 & \quad \text{otherwise }
\end{cases}
$$
and I convolute it with itself:
$$
f*f(x)=\int^{\alpha+\beta}_{\alpha}\gamma f(x-t)\ d... | Answer:
$$f*f(x)=\begin{cases}
\gamma^2(\beta- |x-2\alpha-\beta|) & \quad \text{if } |x-2\alpha-\beta|< \beta\\
0 & \quad \text{if } |x-2\alpha-\beta|\ge \beta
\end{cases}$$
see the details below
-----------------------------------------------------------------------------
Let $x\in \Bbb R$ for $t\in (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2610751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
What does "maximum value" of a set of random variables mean? In our statistical mechanics lecture the professor said something along the lines of:
If we have some independent random variables $x_1,x_2,x_3,...,x_n$ having identical distributions:
Suppose $M_{n}=\text{max}(x_1,x_2,x_3,...)$, then we say that
probabili... | Yes, it is the random variable with the highest value.
For your second question, consider $n$ real numbers $x_1, \dots, x_n$. Suppose the largest of these is less than some value $x$. Then it follows that $x_1, \dots, x_n$ must all be less than $x$ as well. By independence
$$\mathbb{P}(M_n < x) = \mathbb{P}(X_1 < x, X_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2610848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 2
} |
Find the argument of complex number
What is the argument of $z = (1+\cos 2a)+i(\sin 2a)$ if $\pi/2<a<3\pi/2 $?
After using the formula of $\sin2a$ and $\cos2a$, I am getting the argument as $a$ when $\pi/2<a<\pi$ and $a-2\pi$ when $\pi<a<3\pi/2$ but both the answers are incorrect
My approach
Answer given in book
| I am giving a different approach here, less formal but with the intention to see what is going on and why the argument cannot be $a$, as the OP understandably suggested. Without a bit "digging" into the numbers, it is understandable that an average reader (like me!) wouldn't immediately follow the last to lines of the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2610960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
simplification of $\cos\left( \frac{1}{2}\arcsin (x)\right)$ I know this identity :Simplifying $\cos(\arcsin x)$? but how I can simplify
$$\cos\left( \frac{1}{2}\arcsin (x)\right)$$
if you have any idea. It could also be a sinus instead of a cosinus. When I do it I loop on something...
| Let $c=\cos\left(\frac12\arcsin x\right)$ and let $s=\sin\left(\frac12\arcsin x\right)$. Then $2cs=\sin(\arcsin x)=x$ and $c^2+s^2=1$. Can you take it from here?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2611102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Proving formula to find area of triangle in coordinate geometry. Given 3 points, $A$, $B$ and $C$ in anti clockwise order, I have to find the area of the $\triangle ABC$. The formula is area $=\frac{1}{2}(A_x*B_y+B_x*C_y+C_x*A_y-A_y*B_x-B_y*C_x-C_y*A_x)$. Here $A_x$ is the $x$ coordinate of point $A$, and $A_y$ the $... | Start with something more basic: The area of a triangle with vertices $P=(x_1,y_2)$, $Q=(x_2,y_0)$ and $O=(0,0)$ is the absolute value of $$a(\triangle{OPQ}) = \frac12\begin{vmatrix}x_1&x_2\\y_1&y_2\end{vmatrix}.$$ You can find several proofs that the above determinant gives the area of the parallelogram with sides $OP... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2611421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Whether this vector norm proposition is true? $ \vert x_i \vert \ge \vert y_i \vert \Rightarrow \Vert X \Vert \ge \Vert Y \Vert$ Whether this vector norm proposition is true?
$ \vert x_i \vert \ge \vert y_i \vert \Rightarrow \Vert X \Vert \ge \Vert Y \Vert$
where $X, Y \in \mathbb R ^n$
Is it true for all kinds of nor... | Define a norm on $\Bbb R^2$ by $$||(x,y)||=|x|+|x-y|.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2611566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Changing modulus in modular arithmetic Is it true that
$$a\equiv b\pmod{m}\implies\frac{a}{n}\equiv\frac{b}{n} \pmod{\frac{m}{n}},$$ where $a, b, m, n, \frac{a}{n}, \frac{b}{n}, \frac{m}{n}\in\mathbb{N}$? If so, how do I prove it?
| $$a\equiv b\pmod m$$
means
$$\frac{a-b}m\in\Bbb Z.$$
$$\frac{a}n\equiv \frac bn\pmod{\frac mn}$$
means
$$\frac{a/n-b/n}{m/n}\in\Bbb Z.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2611739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 1
} |
Need a hint regarding this question... In how many ways can we select three vertices from a regular polygon having $2n+1$ sides ($n>0$) such that the resulting triangle contains the centre of the polygon?
| HINT: When we fix a vertex of the $(2n+1)$-gon, we can assume that there are $n$ vertices on left of the fixed vertex and $n$ vertices on the right. In order for the resulting triangle to contain the center of the polygon, we need to choose three vertices such that when we fix every vertex one by one, we should have:
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2611888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Pure imaginary numbers : $ ( j^{2n} - j^n ) \in i\Bbb R $? Considering the complex number $j$ such that $$ j = \frac{-1}{2} + i\frac{\sqrt3}{2} $$
Prove that $ \forall n \in \Bbb Z : $
$$ ( j^{2n} - j^n ) \in i\Bbb R $$
( $i\Bbb R$ being the set of pure imaginary numbers)
| Hint:
Note that using Euler's formula, we can write $$ j = \cos \frac{2\pi}3 + i\sin \frac{2\pi}3 = e^{\dfrac{2\pi i} 3}$$ which is a complex root of unity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2612053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Solve $ \int_\frac{-π}{3}^{\frac{π}{3}}\frac{\cos^2(x)-\sin^2(x)}{\cos^2(x)}dx$ I came across this question in my textbook and have been trying to solve it for a while but I seem to have made a mistake somewhere.
$$ \int_\frac{-π}{3}^{\frac{π}{3}}\frac{\cos^2(x)-\sin^2(x)}{\cos^2(x)}dx$$ and here is what I did. First I... | $\begin{align} J=\int_\frac{-π}{3}^{\frac{π}{3}}\frac{\cos^2(x)-\sin^2(x)}{\cos^2(x)}dx=2\int_0^{\frac{π}{3}}\frac{\cos^2(x)-\sin^2(x)}{\cos^2(x)}dx\end{align}$
Observe that for $x\in [0;\frac{\pi}{3}]$,
$\begin{align} \frac{\cos^2(x)-\sin^2(x)}{\cos^2 x}&=\frac{\cos^2 x(1-\tan^2 x)}{\cos^2 x}\\
&=\frac{1-\tan^2 x}{1+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2612141",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Are mathematical relations intrinsically transitive? Here's the question:
Let there be a set A = {1,2,3}.
Let relation R in set A be defined as R = {(1,2),(3,3)}
My textbook says that the relation is neither reflexive nor symmetric but transitive.
I was not quite sure of this so I rechecked the definition of a transit... | If there weren't any two pairs $(a_1,a_2)$ and $(a_2,a_3)$ both belonging to $R$, then the implication
$$(a_1,a_2)\in R \wedge (a_2,a_3) \in R \quad \implies \quad (a_1,a_3)\in R$$
would be true because of the antecedent being always false.
Note however that this is not the case, since you have one case to analize:
$$(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2612353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
A Problem on Beta distribution .
In this problem i know that $X\sim B(m,n)$ and $(1-X)\sim B(n,m)$
After putting values in $Y_i$ i got this $\dfrac{x^2}{1-x^2}$ I have not idea what do further . I am confused.
| It suffices to find the standard deviation of $Y_{i}$. To this end first define a random variable $W\sim\text{Gamma}(p)$ ($p>0$) if $W$ has density
$$
f_W(w)=\frac{1}{\Gamma(p)}w^{p-1}e^{-w}\quad (w>0).
$$
It is easy to see that $EW^d=\frac{\Gamma(p+d)}{\Gamma(p)}$ by the definition of the gamma function. Now $X_i\stac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2612461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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How many sets in the power set containing a given integer? Let $\mathcal{J}\equiv \{1,...,J\}$ and let $\mathcal{C}$ be the power set of $\mathcal{J}$ (with cardinality $2^{J}$).
Question: take any $j\in \mathcal{J}$. How many elements of $\mathcal{C}$ (sets) contain $j$?
For example: if $J=3$ and $j=1$, then $\mathcal... | Hint: each set containing $j$ can be paired nicely with a certain set not containing $j$ and vice versa, so the answer is "half of them".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2612578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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How do I use Maple to calculate the Christoffel Symbols of a Metric? I have been tasked with calculating all the non-vanishing Christoffel symbols (first kind) of a metric and have done these long-hand using the Lagrangian method and shown my working. However, for peace of mind I would like to run the metric through M... | I don't know much about the DifferentialGeometry package, but it seems to me you want to first use
DGsetup([t,x,y,z],M);
and then in defining g1 use dt &tensor dt etc. instead of (dt)^2 etc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2612696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Zero function implies zero polynomial. I'm trying to help someone with a problem in Apostol's book (Chapter 1 BTW, so before basically any calculus concepts are covered) at the moment and I'm stumped on a question.
I'm trying to prove that if $p$ is a polynomial of degree $n$, that is where
$$p(x) = a_0 + a_1x + \cdots... | Note that according to the Fundamental Theorem of Algebra a polynomial of degree $n$ has exactly $n$ roots.
Now your function has infinitely many zeros, therefore it can not be a polynomial of degree $n$ for any $n$.
Thus all the coefficients are zero which makes your function to be identically zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2612814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 11,
"answer_id": 1
} |
Confusion about the rephrase of Recursion Theorem From textbook A Course in Mathematical Analysis by Prof D. J. H. Garling, I'm confused about how he rephrases the Recursion Theorem.
First, he states the theorem:
Then he says:
Finally, he expresses the theorem in a more general term:
My question is: the author says ... | What you feel is exactly what the theorem says! It does NOT say that "there exists a unique mapping $f^n$ …". It says that
For each $\color{blue}{n}\in\mathbb{Z}^{+}$ there exists a unique mapping $f^{\color{blue}{n}}:A\to A$ …
So all in all there are many these mappings, just as you said: there's one for $n=0$, one ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2613010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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$\sum a_n$ and $\sum b_n$ converges. Prove $\sum a_nb_n$ converges. Let ${a_n}$ and ${b_n}$ sequences with positive terms, such that $\sum a_n$ and $\sum b_n$ converges. Prove that $\sum a_nb_n$ converges as well.
What I did:
$\sum a_nb_n \le \left(\sum a_n\right) \left(\sum b_n\right)$ and by comparison test we are do... | Since $\sum\limits_{k = 1}^\infty b_k$ converges, then $b_n \to 0 \ (n \to \infty)$. Note that $\{b_n\}$ is a sequence of positive numbers, thus there exists $N \in \mathbb{N}_+$ such that$$
0 < b_n < 1. \quad \forall n > N
$$
Therefore,$$
0 < \sum_{k = 1}^\infty a_k b_k = \sum_{k = 1}^N a_k b_k + \sum_{k = N + 1}^\inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2613213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
} |
Young Tableaux generating function The number of young tableaux of $n$ cells is known to satisfy the recurrence $a_{n+1} = a_{n} + na_{n-1}$. I am trying to find the generating function but I keep getting something dependent on $n$. Here's what I did so far:
Denote by $f(x) = \sum_{n\geq 1}a_nx^n$. We have $\sum_{n \g... | The first step is to look in the OEIS and see that this is sequence A000085. The sequence grows so fast that the o.g.f. has radius of convergence $0$. This strongly suggests looking at the e.g.f instead, which in the OEIS entry is given as $\exp(x+x^2/2).$ The question now is how to derive this e.g.f. from the recursio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2613328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Arc length of curve of intersection between cylinder and sphere
Given the sphere $x^2+y^2+z^2 = \frac{1}{8}$ and the cylinder $8x^2+10z^2=1$, find the arc length of the curve of intersection between the two.
I tried parametrizing the cylinder (the task specifies this as a hint). My attempt:
$$x(t) = \frac{1}{\sqrt{8}... | From $8x^2 + 10z^2 = 1$,you get $z^2 = \frac{1}{10}.(1-8x^2)$. Substitute this in the other equation $ x^2+y^2+z^2 = \frac{1}{8}$ you get
$$x^2 + 5y^2 = \frac{1}{8}$$
This is the curve of intersection, now parameterize this ellipse with
$x = \frac{1}{\sqrt{8}} sint$
and
$ y = \frac{1}{\sqrt{40}} cost$
$ z = \frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2613408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Show that if $\gamma$ is any isometry on $\mathbb{R}^n$, then so is $a\gamma(\dfrac{1}{a})$ Take $v \in \mathbb{R}$
and denote translation over $v$ as $\tau v$. Let a ∈ $\mathbb{R}$ with $a \neq 0$.
a) Verify that $a \tau_v \dfrac{1}{a} $ is again a translation
b) Show that if $\gamma$ is any isometry on $\mathbb{R}^n$... | These both answer easily by applying the definitions of things.
$\tau_v:x\mapsto x+v$ is a translation, and $\rho_a:x\mapsto ax$ would be a dilation.
So $a\tau_v\frac1a$ can be thought of as three things:
*
*1) $x\mapsto x/a$,
*2) $x/a\mapsto x/a+v$
*3) $x/a+v\mapsto a(x/a)+av=x+av$
Next, what is an isometry? H... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2613512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A recursive divisor function Question:
Function definition:
$$f(1)=1$$
$$f(p)=p$$ where $p$ is a prime, and
$$f(n)=\prod {f(d_n)}$$ where $d_n$ are the divisors of $n$ except $n$ itself.
End result:
The end result of the function is when all divisors have been reduced to primes or 1.
Example:$$f(12)=f(2)f(3)f(4)f(6)=f... | I Found
$$f(a^n\cdot b^m) = a^{{(2^{m})}{(m-1)}} \cdot b^{{(2^{m-2})}{(m+1)}} \cdot
\prod _{j=1} ^{m} \prod _{i=1} ^{n-1} f(a^i\cdot b^{j})^{2^{m-j}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2613617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Difficulty in understanding cantor normal form Cantors normal form of x is defined as the following
$x = \omega^{a_1} n_1 + \dots + \omega^{a_k} n_k$, Where $x$ is an ordinal and where $\langle a_i \rangle$ is a strictly decreasing finite sequence of ordinals, $\langle n_i \rangle$ is a finite sequence of ordinals and ... | First, prove that the map $\alpha\mapsto\omega^\alpha $ is normal, that is, strictly increasing and continuous at limits. Use this to show that for any $\alpha $ there is a least $\beta $ such that $\alpha <\omega^\beta $, and that, if $\alpha\ne0$, then this least $\beta $ is a successor ordinal, say $\beta=\beta_0+1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2613718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to tell whether a left and right riemann sum are overestiamtes and underestimates? I know that in a positive and increasing function, the right riemann sum is an overestimate and the left is an underestimate, but what about if the function is negative and increasing like this? Which one would be an overestimate and... | It makes no difference whether the values of a function are positive or negative, if you always choose the smallest value of the function on each interval, the Riemann sum will be an underestimate. If you choose the largest value of the function on each interval, you will get an overestimate:
$$\sum_i \left(\min_{t_{i-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2613809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Faster way of determining the coefficient of a polynomial function? Question: Determine if the leading coefficient of the function "a", is positive or negative.
a) $$f(x)=(x-3)^2(x+1)(x+2)^3$$
In my notes I stated the sign of the leading coefficient without work but in order to get the answer now I had to expand the p... | You asked for the leading coefficient of $$f(x)=(x-3)^2(x+1)(x+2)^3$$
As you see this is a polynomial of degree $6$ therefore you want to know the coefficient of $x^6$
How many $x^6$ are there in the product?
Well there is only $1$ because you need to multiply the leading coefficients of each factor to get the leading ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2613967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
For Field extension, $[E :F]=1$ implies $E=F$ Suppose $E/F$ is a field extension then
$[E :F]=1$ implies $E=F$.
This sounds very trivial but i don't know how to formally write this.
| The expression “$E/F$ is a field extension” has some ambiguity.
Almost everybody (including you, I am sure) uses this expression to mean that $F$ and $E$ are fields with $F\subset E$. In this case, equality between $F$ and $E$ is equivalent to the degree being $1$, and with others’ hints, I’m sure you can prove it.
The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2614080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
converting a density function to a distribution function I get stuck converting density functions to distribution functions,
(i) the density function $f(y)$ is given by: y (for 0<=y<=1), 1 (for $1<y<=1.5$), and 0 elsewhere. What is $f(y)$?
I get: (a) $y^2/2$ (for $0<=y<=1$), and (b) $y$ (for $1<y<=1.5$)
But the answ... | It is customary to denote the cumulative distribution function of $Y$ as $F_Y(\cdot).$ Sometimes, it is helpful to use a neutral symbol (here $t$) for the variable of
integration.
(i) For $0 \le y < 1,$ we have $F_Y(y) = \int_0^y t\,dt = \frac 1 2 y^2.$
For $1 \le y < 1.5,$ we have
$F_Y(y) = \int_0^y f_X(t)\,dt = \int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2614312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Book Recommendation: Introduction to probability theory (including stochastic processes) I'm a first year undergraduate engineering student and we've got a course "Introduction to Probability Theory" which roughly covers the following topics:
addition, multiplication, marginal and conditional probability, joint
prob... | As a mathematics student I've had courses on probability theory and stochastic processes, and for both of those courses used the book Probability Theory and Random Processes by Grimmet and Stirzaker. The explanations were clear, and I remember the exercises in the book to be quite challenging. At least in my experience... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2614389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Calculating the integral $\int \sqrt{1+\sin x}\, dx$. I want to calculate the integral $\int \sqrt{1+\sin x}\, dx$.
I have done the following:
\begin{equation*}\int \sqrt{1+\sin x}\, dx=\int \sqrt{\frac{(1+\sin x)(1-\sin x)}{1-\sin x}}\, dx=\int \sqrt{\frac{1-\sin^2 x}{1-\sin x}}\, dx=\int \sqrt{\frac{\cos^2x}{1-\sin ... | As pointed out by other answers, you need to take signs into consideration. Indeed, starting from your computation we know that
$$ \int \sqrt{1+\sin x} \, dx = \int \frac{\left|\cos x\right|}{\sqrt{1-\sin x}} \, dx $$
Now let $I$ be an interval on which $\cos x$ has the constant sign $\epsilon \in \{1, -1\}$. That is, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2614504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Convergence of the sequence $ \sqrt {n-2\sqrt n} - \sqrt n $ Here's my attempt at proving it:
Given the sequence $$ a_n =\left( \sqrt {n-2\sqrt n} - \sqrt n\right)_{n\geq1} $$
To get rid of the square root in the numerator:
\begin{align}
\frac {\sqrt {n-2\sqrt n} - \sqrt n} 1 \cdot \frac {\sqrt {n-2\sqrt n} + \sqrt n}{... | It's perfect except for the justification that $2\sqrt {n}/n $ converges to $0 $. What you have written only proves that it is bounded above by $1 $.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2614626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Trying to calculate the series $\sum_{n=0}^{\infty}{{-1}\choose n}z^n$ I'm trying to calculate the series $\sum_{n=0}^{\infty}\binom{-1}{n}z^n$.
Here is what I have so far:
\begin{align*}
\sum_{n=0}^{\infty}\binom{-1}{n}z^n &= \sum_{n=0}^{\infty}\bigg(\frac{1}{n!}\cdot\prod_{j=0}^{n-1}(-1-j)\bigg)z^n \\&= \sum_{n=0}^{\... | Note that
$$
\binom{-1}{n}=(-1)^{n}\frac{n!}{n!}=(-1)^n.
$$
Hence
$$
\sum_{n=0}^\infty
\binom{-1}{n}z^{n}
=\sum_{n=0}^\infty(-z)^n=\frac{1}{1+z};\quad (|z|<1)
$$
by the geometric series.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2614854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Estimating the limit $x_{n+1} =x_n - x_{n}^{n+1} $ I wonder whether there is a general method for accurately estimating the limit of the sequence:
\begin{equation}
x_{n+1} = x_n - x_{n}^{n+1}, \forall x_1 \in (0,1)
\end{equation}
After showing that the limit exists, since $ x_n $ is decreasing and bounded, I managed to... | As you noted, $(x_n) $ is convergent as a decreasing positive sequence.
We should have
$$\lim_{n\to\infty}x_n^{n+1}=$$
$$\lim_{n\to\infty}e^{(n+1)\ln (x_n)}=0$$
for $A <0$ ans great enough $n,$
$$\ln (x_n)<\frac {A}{n+1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2614969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Open subset of Euclidean space can't have non-spherical manifold compactification. The title is not complete, since it would be too long. Consider the following statement:
Let $U \subset \mathbb{R}^n$ be open, connected and such that its one-point compactification is a manifold. Then, this compactification must be (ho... | I can imagine an ellementary approach only for the special case of $\mathbb{R^2} $ and $\mathbb{R}$.
For $\mathbb{R^2}$:We know that all the compact surfaces arise from adding to the sphere a finite amount of handles or Mobius-strips. In any case, if you remove a point from a compact surface which is something of the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2615185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Does the series $\sum 2^n \sin(\frac{\pi}{3^n})$ converge?
Check if $$\sum_{n = 1}^{\infty}2^n \sin\left(\frac{\pi}{3^n}\right)$$ converges.
I tried to solve this by using the ratio test - I have ended up with the following limit to evaluate:
$$\lim_{n \to \infty} \left(\frac{2\sin\left(\frac{\pi}{3 \cdot 3^n} \rig... | Same idea as Olivier express in a different way, you know that
$$
\sin\left(x\right)\underset{(0)}{=}x+o\left(x\right)
$$
Hence
$$
2^n\sin\left(\frac{\pi}{3^n}\right) \underset{(+\infty)}{\sim}\pi \left(\frac{2}{3}\right)^n
$$
What can you say about $\displaystyle \sum_{n \geq 0}\left(\frac{2}{3}\right)^n$ ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2615300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Generators of multiplicative group Let G be the multiplicative group generated by the complex number e^$(2πiθ)$, θ a real number. For what values of θ is G a finite group? What is its order in that case?
How would one proceed to solve this question?
| By definition, $G$ is cyclic with generator $e^{2\pi i \theta}$. Hence, the order of $G$ is equal to the order of the element $e^{2\pi i \theta}$, i.e. the smallest positive integer $n$ such that $e^{2\pi i \theta n} =1$ (in case such $n$ does not exist, the order is not finite). Now what do you know about the exponent... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2615417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Can I draw an acceleration vs. time graph from an acceleration vs. distance graph? An object with a known initial velocity, starting from the origin, moves along a line and its acceleration is graphed as a function of distance from the origin. I want to sketch $ x''(t) $ vs. $t$ given $ x''(t) $ = $ f(x(t))$. I will ca... | $$x'' = f(x)$$
$$x'x'' = f(x)x'$$
Here we can integrate with respect to $t$. $F$ is an antiderivative of $f$.
$$\frac12 (x')^2 = F(x)+C_1$$
$$(x')^2 = 2F(x)+C_2$$
You could plug in initial values for position and velocity, to solve for $C_2$. Also, the sign of initial velocity determines the sign of this square root:
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2615558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
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