Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Find $\int\ln^{n} x$ .
Find $\int\ln^{n} x$ .
My observation
$\int\ln^{1} x= x(\ln x- 1)+ constant$
$\int\ln^{2} x= x(\ln^{2} x- 2\ln x+ 2)+ constant$
$\int\ln^{3} x= x(\ln^{3} x- 3\ln^{2} x+ 6\ln x- 6)+ constant$
$\int\ln^{4} x= x(\ln^{4} x- 4\ln^{3} x+ 12\ln^{2} x- 24\ln x+ 24)+ constant$
$$\ddots$$
We have
$$\ddot... | $$ \int \ln^{n} x dx = x \ln^{n}x - n\int\ln^{n-1}x dx + C $$
Please see:
https://www.youtube.com/watch?v=xE0Pp4I7PiA
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3307965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Limit Cycle at $r=1$? Consider the non-linear ODE
$$u''+(u^2+u'^2-1)u'+u=0.$$ Transforming this to polar coordinates:
$$r'=-(r^2-1)r\sin^2(\theta),$$
$$\theta'=-\sin(\theta)\cos(\theta)(r^2-1)-1.$$
If we consider an annulus (trapping region), $\frac{1}{2}<x^2+y^2<2,$ how do we deal with the $\sin^2(\theta)$ term?
We ... | Hint.
Putting in the form
$$
\dot u_1 = u_2\\
\dot u_2 = -(u_1^2+u_2^2-1)u_2 - u_1
$$
then
$$
\frac 12(u_1^2+u_2^2)' = -u_2^2(u_1^2+u_2^2-1)
$$
and the stream plot gives
NOTE
When $u_1^2+u_2^2 = 1$ then $\dot u_1 = u_2$ and also
$$
2u_1\dot u_1 +2u_2\dot u_2 = 0\Rightarrow \dot u_2 = - u_1
$$
and those points have the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3308068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Proving the following inequality using Mathematical Induction I need to prove the following inequality is true for n > 2.
$n^3$ > 2$n^2$ + 3
*
*Prove using base case n = 3:
$3^3$ > 2$(3)^2$ + 3
27 > 2(9) + 3
27 > 18 + 3
27 > 21 (true)
*Assume true for n = k:
$k^3$ > 2$k^2$ + 3
*Prove for n = k +1
$(k+1)^3$... | Now, $$(k+1)^3=k^3+3k^2+3k+1=2(k+1)^2+3+k^3+k^2-k-4>2(k+1)^2+3.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3308215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Let $U\neq \emptyset$ In $\mathbb{R}^n$ Is Open, Then $U$ Is Not Compact Prove: Let $U\neq \emptyset$ In $\mathbb{R}^n$ Is Open, Then $U$ Is Not Compact
How can I approach this? I know that in $\mathbb{R}^n$ we have that $U$ is comapct $\iff$ $U$ is closed and bounded
So we can start with assuming the $U$ is compact, b... | There are many good answers already, but here's another approach. First, we can solve this for $\mathbb{R}$. An open set $U \subset \mathbb{R}$ is the (countable) disjoint union of open intervals, $U = \sqcup_{j \geq 1}(a_j,b_j)$. We can assume $U$ bounded, otherwise it will not be compact from the get go. Thus all int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3308298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
Prove that a group of order 5 doesnt have any self-inverse member other than the identity member. Assuming you only know the most basic ideas about groups properties-
If group G contains 5 members, how do you prove that no member other than the identity member is self inverse?
*I rather have a clue on how to begin solv... | Assume that $a=a^{-1}$ and $a \neq e$. $G$ can be partioned in two disjoint sets: put $S = \{x \in G: x=x^{-1}\}$ and $T=\{x \in G: x \neq x^{-1}\}$, then $G = S \cup T$ and since $a, e \in S$, $|S| \geq 2$.
Note that $|T|$ is even (elements of $T$ come in pairs), which leaves us with $|T|=0$ or $2$ (if its cardinalit... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Argand formula and more for quaternions? Is it possible to define a similar form of Argand's formula but for quaternions?
In the sense
$$ \cos(nA)+i\cos(nB)+j\cos(nC)+k\cos(n) =(\cos(A)+i\cos(B)+j\cos(C)+k\cos(D))^{n}, $$
where $A, B, C, D$ are the angles of the quaternion with respect the axes $x,y,z,t.$
Also for a q... | Since @LordSharktheUnknown discussed the trigonometry, I'll answer your later questions. Do you want $w:=z_1/z_2$ to satisfy $z_1=z_2w$ or $z_1=wz_2$? It matters, which is why we don't usually write such expressions as $\frac{i+j}{k}$; you'd want to say $(i+j)k^{-1}$ or $k^{-1}(i+j)$ instead. (These are respectively $-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3308586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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How to show that $\sqrt[4]{13}$ is not in $\mathbb{Q}_{13}(\sqrt[4]{26})$ Approach
*
*I was able to show that $[\mathbb{Q}_{13}(\sqrt[4]{26}):\mathbb{Q}_{13}] = 4$ since $x^4 - 26$ in an irreducible polynomial in $\mathbb{Q}_{13}[x]$ (this can be shown by using Eisenstein's Criterion and the Gaussian Lemma). Therefo... | Note that $2^{(13^2-1)/4}=(2^6)^7$ is congruent to $-1$ mod $13$, thus $2$ is not a fourth root in a quadratic extension of $\mathbb{F}_{13}$.
Therefore, $X^4-2$ is irreducible in $\mathbb{F}_{13}$, thus in $\mathbb{Z}_{13}$ thus in $\mathbb{Q}_{13}$.
So $K=\mathbb{Q}_{13}[X]/(X^4-2)=\mathbb{Q}_{13}(2^{1/4})$ is a fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3308818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Show that $\lim_{x\to \frac{\pi}{2}} \frac{1}{\big(x-\frac{\pi}{2}\big)}+{\tan(x)}=0$.
Prove that
$$\lim_{x\to \frac{\pi}{2}} \frac{1}{\big(x-\frac{\pi}{2}\big)}+{\tan(x)}=0.$$
I'm not really sure how to proceed. I know that I should not try L'Hôpital's rule (tried that) but not sure how I would incorporate into th... | First note that
$$\lim_{x\to0}\frac{\sin(x)}{x}=1$$
$$\lim_{x\to0}\frac{x}{\sin(x)}=1$$
$$\lim_{x\to0}\frac{1-\cos(x)}{x}=0$$
Keeping these fundamental trigonometric limits in mind, we have
$$\lim_{x\to\frac{\pi}{2}} \frac{1}{\left(x-\frac{\pi}{2}\right)}+\tan(x)$$
$$=\lim_{t\to0} \frac{1}{t}-\cot(t)$$
$$=\lim_{t\to0}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3308938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Two Solutions for an ODE: $x' = x^{\frac45}$ Find two different solutions $x_1, x_2 : \mathbb{R} \to \mathbb{R}$ of
$$ \dot{x} = x^{\frac45}, \quad x(1) = 1. $$
This is a problem in a 60-minute exam, so it should be quite simple but still I'm failing.
I can get $x(t) = \left(\frac{t+4}{5} \right)^5$ via separation of ... | To get two different solutions, you need to have a violation of the Lipschitz condition, even the local one. This can be tested by looking for singularities of the derivative of the right side. You will find that there is such a singularity at $x=0$, and find further that the constant-zero function is a solution.
You ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3309178",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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double tours embedding of nonhamiltonian bicubic graphs Can Georges Graph
(or any other nonhamiltonian bicubic graph ) be embedded on an oriented surface of genus -2, i.e. a double torus?
If it helps, it would have $F=E+\chi-V=75-2-50=23$ faces...
| A double torus is a sphere with two handles and two holes, so its genus (which is the number of holes, see, for instance, p. 133 in “Chromatic Graph Theory” by Gary Chartrand and Ping Zhang, CRC Press, 2009) is $2$. My search at the House of Graphs of bicubic non-hamiltonian graphs of genus at most two provided the fol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3309358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why does combining these two linear diophantine equations yield solutions, when each individually has none? The two following linear diophantine equations have no solutions:
$$412x + 18y = 49$$
$$33x + 99y = 15$$
We can however combine them as $412x + 18y - 49=0, 33x + 49y - 15 = 0$ gives
$412x+33x+18y+49y-49-15=0 \to ... | You lost some information. You have $a=0$ and $b=0$ and (essentially) conclude that $a=b$, which is correct. But you've lost the information that each side equals $0$. You'd get the same conclusion if $a=5$ and $b=5$. Concluding $a=b$ is correct, but you've lost the $5$-ness.
Since you lost information, you have fe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3309453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Find the intersection of the the sets given If $aN= \{ ax:x\in N \}$ then $3N\cap 7N$ =
Options are
a)3N
b)7N
c)N
d)21N
I think the answer should be 3N as 7N would contain 3N and we have to find their intersection. But the answer is 21N, and I am not able to figure out the reason for that answer.
| Think about it this way:
$3N=\{n:n\in N\land 3\mid n\}$
$7N=\{n:n\in N\land 7\mid n\}$
For an two sets $A$ and $B$, such that $A=\{a:\varphi(a)\}$ and $B=\{b:\psi(b)\}$, the intersection is $A\cap B= \{c:\varphi(c)\land\psi(c)\}$
So...
$$3N\cap7N=\{n:n\in N\land (3\mid n)\land (7\mid n)\}$$
If $3$ divides $n$ and $7$ d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3309516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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n empty balls and an observer Consider $n$ empty circles in a row. An observer sits in one of the $n−1$
gaps between them. A random subset of r circles is colored black. Show that the expected number of empty circles the observer sees is at most $
\frac{2(n − r)}{(r + 1)}$. (The observer can see through empty circles b... | There is no inconsistency between the claimed result and the value you think you get in the case $r=1$, since the result says "at most", and $\frac{n-1}{2}$ is certainly at most $n-1$.
Also, $\frac{n-1}{2}$ isn't correct for $r=1$. For example, if $n=2$ the expected value is precisely $1$ (there is one uncoloured circl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3309785",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Diagonalising matrices over different fields examples Let say
$M=\left[ {\begin{array}{cc}
1 & 1 \\
1 & 0 \\
\end{array} } \right]$ and so its characteristic polynomial is $x^2-x-1$, which will be diagonalisable if the field chosen is $\mathbb{R}$ but not if the field is $\mathbb{Q}$.
How can I determine if suc... | You're right, if the matrix is diagonalizable over $\mathbb{C}$ or $\mathbb{R}$, this must not hold for a finite field. Look at your example. I reduce it here to the case $p = 2$. The elements of $\mathbb{F}_2$ are $0$ and $1$. But since
$$ 0^2 - 0 - 1 = 1 = 1^2 - 1 -1,$$
we see that the matrix has no eigenvalues in $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3309894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Relations between two applications of Catalan Numbers Say I am looking at how many different balanced parenthesis I can make. Then I look at how many ways n triangles can be made with a n+2 polygon. Because they both use Catalan numbers, I know they are bijective. What I am having trouble understanding is HOW they are ... | I suppose we are talking about convex polygons on $n+2$ vertices.
Index vertices sequentially, I will give them indices in the next way $0, 0, 1, 2, \ldots, n$
Algo $1$ (From triangulation to parenthesis):
*
*Mark $0-0$ edge as "seen"
*Start at vertex $1$ (at each vertex go over all the outgoing edges in a clockwis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3310008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find range of parameters for which a given curve is a geodesic I am working on the following problem;
Given parameterised surface $X(u,v)=(u \cdot \cos v,u \cdot
\sin v,v)$ determine for which values $\alpha$ the curve $\gamma_{\alpha}=(t \cdot \cos (\alpha t),t \cdot \sin(\alpha t),\alpha t)$ is a geodesic.
Accordin... | There are some mistakes on the LHS of your equations. The corrected equations are as follows:
$\displaystyle \frac{d(E \dot{u} + F \dot{v})}{dt} = \frac{1}{2}\left(E_u \dot{u}^2 + 2F_u \dot{u}\dot{v} + G_u \dot{v}^2 \right)$
and
$\displaystyle \frac{d(F \dot{u} + G \dot{v})}{dt} = \frac{1}{2}\left(E_v \dot{u}^2 + 2F_v ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3310109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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The confusion usage of atlas and maximal atlas I'm studying Loring Tu's An Introduction to Manifolds. In p.60 he said
Given a smooth manifold $M=(\underline{M},\Phi_{\text{maxi}})$, it is understood by people that there exist a maximal atlas $\Phi_{\text{maxi}})$ of that underlying set $\underline{M}$. However, what d... | Isn't required that the atlas in (ii) be maximal, but a such atlas can be extended to a maximal atlas preserving the property, namely the maximal atlas defining the smooth structure.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3310216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Derivative is just speed of change? In school we've been told that derivative of $x^2$ is $2x$.
Also I've read that derivative is simply a speed of value change.
So if
$$f(x)=x^2$$
then, using simple explanation, derivative of that function would be
$$f'(x)=f(x+1)-f(x).$$
Now if we take derivative when $x=2$ we will ge... | You are right, the derivative is a speed (or rate) of change. People are often familiar with the idea that the slope of a line is it's rate(speed) of change.
I'd just like to add to Marco's answer and tell you to pay particular attention to his very first equation:
$gradient =\frac{f(x + h) - f(x)}{(x+h) - x}$.
What ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3310297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 7,
"answer_id": 4
} |
How to solve this weird ODE? I came across this differential equation which I'm having trouble finding an analytic solution to:
$$\frac{dy}{dx}=\frac{A}{xy}+\frac{B}{(xy)^2}$$
I'm trying to solve for y. I have initial conditions as $x_0=0.02$ and $y_0=100000$, and A and B are known constants. I don't have a very heavy ... | Assume $A,B\neq0$ for the key case:
Hint:
Let $u=xy$ ,
Then $y=\dfrac{u}{x}$
$\dfrac{dy}{dx}=\dfrac{1}{x}\dfrac{du}{dx}-\dfrac{u}{x^2}$
$\therefore\dfrac{1}{x}\dfrac{du}{dx}-\dfrac{u}{x^2}=\dfrac{A}{u}+\dfrac{B}{u^2}$
$\dfrac{1}{x}\dfrac{du}{dx}=\dfrac{u}{x^2}+\dfrac{A}{u}+\dfrac{B}{u^2}$
$\dfrac{1}{x}\dfrac{du}{dx}=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3310458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Find Automorphism Group if the following graph (Picture below) I'm studying for an exam and found the following task concerning Automorphism Groups:
"Find the Automorphism $Aut(G)$ Group of the following Graph $G$."
On the outside, the graph looks like a regular pentagon, which would be an easy task. I figured out, th... | Let $I$ be the identity isomorphism, and let $M$ be the mirroring isomorphism. We want to determine whether there exist any other isomorphisms.
In order to do this, let's imagine that there exists an isomorphism that I'll denote $F_1$ which is unequal to either $I$ or $M$. Let's see what we can learn about $F_1$; maybe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3310574",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can I write $y$ only in terms of $x$ in the following equation? How would you write $y$ only in terms of $x$ in this equation?
$$x^2 + xy + y^2=100$$
| This is a quadratic equation in $y$:
$$y^2+xy+x^2-100=0$$
As such, the quadratic formula gives an expression for $y$ in $x$:
$$y=\frac{-x\pm\sqrt{x^2-4(x^2-100)}}2=\frac{-x\pm\sqrt{400-3x^2}}2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3310647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Fourier series with all coefficients $\frac1n$ The function with Fourier series given by
$$f(x)=\sum_{n=1}^\infty \frac{\cos{(nx)}+\sin{(nx)}}n$$
appears to be a curve with vertical asymptotes at $x=2\pi k$ where $k\in\mathbb{Z}$. Is there an elementary closed form for $f(x)$? Wolfram gives us
$$f(x)=-\frac12(1+i)(\ln{... | $$ f(x) = - \frac{\ln(2-2\cos(x))}{2} + \arctan\left(\frac{\sin(x)}{1-\cos(x)}\right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3310746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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differential equations - exponential growth and decay
The population $P$ of bacteria in an experiment grows according to the equation $\frac{dP}{dt}=kP$, where $k$ is a constant and $t$ is measured in hours. If the population of bacteria doubles every $24$ hours, what is the value of $k$?
I was given this problem a... | From
$\dfrac{dP}{dt} = kP, \tag 1$
assuming
$P \ne 0, \tag 2$
we deduce that
$\dfrac{1}{P}\dfrac{dP}{dt} = k; \tag 3$
we integrate 'twixt $t_0$ and $t$, assuming $P$ takes the value $P(t_0)$ at $t = t_0$:
$\ln P(t) - \ln P(t_0) = \displaystyle \int_{t_0}^t \dfrac{1}{P(s)}\dfrac{dP(s)}{ds} \; ds = \int_{t_0}^t k \; ds =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3310856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to get the value of $A + B ?$ I have this statement:
If $\frac{x+6}{x^2-x-6} = \frac{A}{x-3} + \frac{B}{x+2}$, what is the value of $A+B$ ?
My attempt was:
$\frac{x+6}{(x-3)(x+2)} = \frac{A(x+2) + B(x-3)}{(x-3)(x+2)}$
$x+6=(x+2)A + B(x-3)$: But from here, I don't know how to get $A + B$, any hint is appreciated.
| By your work the coefficient before $x$ it's $1$, which ends.
Also, $$\frac{x+6}{x^2-x-6}=\frac{x+6}{(x-3)(x+2)}=\frac{x-3+9}{(x-3)(x+2)}=$$
$$=\frac{1}{x+2}+\frac{9}{5}\left(\frac{1}{x-3}-\frac{1}{x+2}\right)=\frac{\frac{9}{5}}{x+3}+\frac{-\frac{4}{5}}{x+2},$$
which gives $A+B=1.$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
I'm not understanding combinations and counting The question:
An urn has 10 black balls numbered from 1 to 10, and 10 white balls numbered from 1 to 10. In how many ways can we choose 5 balls from the urn?
(There are more questions, which is why there's balls that are different colors and numbered. This is just one of ... | You have to choose $5$ balls from $20$ balls; so the order of the balls does not matter. There are $20×19×18×17×16 $ number of ways contains only $\frac{20×19×18×17×16 }{5!}$ different combinations of the balls. Although,if order of the drawn balls did matter,then yours would be the correct answer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3311004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Number of ways to split $N$ up into $k$ baskets such that different arrangements of the $k$ baskets are considered equivalent? I've been considering the problem of integer partitions and while there have been some answers for related questions, I haven't came across a solution for my following problem.
Suppose you hav... | I believe that the Stirling numbers of the second kind $\big\{{n\atop k}\big\}$ are what you need. This is the number of ways to partition $n$ labelled elements into $k$ unlabelled non-empty subsets.
EDIT: If we're trying to partition $n$ unlabelled elements into $k$ subsets, then the function we're actually looking to... | {
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"timestamp": "2023-03-29T00:00:00",
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Are the $\Bbb S^2\times \Bbb R^2$ and $\Bbb R^2\times \Bbb S^2$ homeomorphic? Are the $\Bbb S^2\times \Bbb R^2$ and $\Bbb R^2\times \Bbb S^2$ homeomorphic? I know that the answer is certainly yes but what is confused me is the following:
*
*$\Bbb R^2\times \Bbb S^2$: Consider a plane and attach a $2$-sphere to each ... | As you said, the spaces are homeomorphic. You imagine a product $X \times Y$ in two different ways:
*
*A copy of $Y$ attached at each point of $X$.
*A copy of $X$ attached at each point of $Y$.
The copies $\{x\} \times Y$ are pairwise disjoint, and I guess you imagine them as "isolated bags hanging on string". Howe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3311276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Points A,B,C (fix) and X (variable) such that |AX| + |BX| = |CX| Let $A,B,C$ (fix) points of the plane. Where are points $X$ (variable) in the plane with $|AX| + |BX| = |CX|$ ?
It seems $X$ need to lie on the arc under $AB$ of the circumscribed circle of the triangle $ABC$.
How can I prove that?
Note: There is no Trap... | From the equation for $X$ we can get
$$4 |A-X|^2 |B-X|^2 = (|C-X|^2 - |A-X|^2 - |B-X|^2)^2$$
which gives you a quartic polynomial in the coordinates of $X$. If this doesn't factor (which in general it won't), the locus of $X$ will not be an arc of a circle, nor any conic section.
Here's a picture of it in the case $A... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Program to find intersection of subgroups of free groups As the title says, I am working on examples for a research project I'm doing, and I need a way to efficiently calculate the intersection of subgroups of a free group (say, of rank 2). Are there any computer programs to do this, or any papers explaining how such a... | The algorithm is easy and well known. Let $A, B$ be finitely generated subgroups of a free group $F$. Construct the Stallings cores $U,V$ of these subgroups. These are labeled graphs whose labels-generators of $F$ and which have basepoints $u,v$. Then $A$ (resp. $B$) consists of all labels of reduced loops of $U$ (resp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3311562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
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Calculating 2 non-definite integrals Calculate the following:
A) $\int \sqrt{3x^4 +x^6 +9x^2} \, dx$
B) $\int \sqrt[3]{{\frac{1}{x^2 +1}}}\, dx$
A) I managed to write $\int x \sqrt{3x^2 +x^4 +9} \, dx$, but then I didn't know what to do because of the square root, even with integration by parts.
B) I tried substituting... | In the integral
$$\int x \sqrt{x^4+3x^2+9} \ dx$$
substitute $u=x^2$, $du=2xdx$ to get
$$\frac{1}{2} \int \sqrt{u^2+3u+9} \ du = \frac{1}{2} \int \sqrt{ \left( u+\frac{3}{2} \right)^2 + \frac{27}{4} } \ du$$ and to continue, substitute
$$u+\frac{3}{2}= \frac{3\sqrt 3}{2} \tan \theta$$
you will need to remember the inte... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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On proving that the geometric realization of $\Delta^n = \Delta(-,n)$ is homeomorphic to $|\Delta^n|$. I'm trying to prove that the geometric realization of $\Delta^n = \Delta(-,n) : \Delta^{op} \to \mathsf{Set}$ coincides with the geometric realization of $\Delta^n$ as a simplicial complex (I will note $|\Delta^n|$ on... | Just observe that there is a map $|X|\to |\Delta^n|$: each element of $X_k=\Delta(k,n)$ determines a map $|\Delta^k|\to|\Delta^n|$, and so these maps together over all values of $k$ give a map $\coprod_kX_k\times|\Delta^k|\to|\Delta^n|$. This map respects the equivalence relation $\sim$, essentially by definition (if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3311960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How is $\sin 90° = 1$ possible? How can two angles of a triangle be equal to $90°$? If two angles were $90°$, this would mean that the two sides would be parallel and the angle of the third side would be equal to 0. Thus, there would be only two vertices and this wouldn't be a triangle at all, ultimately making $\sin 9... | Consider polar coordinates, $(r \cos\theta, r \sin \theta)$ in a unit circle such that $r=1$. Then, we can see that, the mapping in the first quadrant inside the unit circle is just $(\cos \theta, \sin \theta)$. Now, consider a moving point $A$ starting from $\theta=0°$ to $\theta=90°$ on the circumference of the circl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3312092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Green's identity and gradient estimate After the proof of the Green's identity in the book "Han Q., Lin F. - Elliptic partial differential equations - AMS (1997)", they state at page 9:
We may employ the local version of the Green's identity to
get gradient estimates without using mean value property.
Suppose $u \in... | You barely have to treat it. You have already shown $$|\partial_{a_i} u(a)| \le \int_{r < |x| < R} |u(x)| \left| \Delta_x(\varphi(x)\partial_{a_i} \Gamma(a,x))\right|dx,$$ so $$\sup_{a \in B_{1/2}} |\partial_{a_i} u(a)| \le \sup_{a \in B_{1/2}}\int_{r < |x| < R} |u(x)| \left| \Delta_x(\varphi(x)\partial_{a_i} \Gamma(a,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3312241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to evaluate the following integral$\int_a^{pa}\frac{ax}{\sqrt{(a-x)(x-pa)}}dx$? Can anyone help me to evaluate the definite integral $\int_a^{pa}\frac{ax}{\sqrt{(a-x)(x-pa)}}dx$?I encountered this integral while doing a problem of particle dynamics in Ganguly Saha(Applied Mathematics).Can this integral be evaluated... | With the variable change $u=x/a-1$ and the shorthand $q=p-1$,
$$I=\int_a^{pa}\frac{ax}{\sqrt{(a-x)(x-pa)}}dx=a^2\int_0^{q}\frac{u+1}{\sqrt{u(q-u)}}du=a^2(I_1+I_2)$$
where, $I_1$ and $I_2$ are given below, integrated with the convenient substitution $u=q\sin^2\theta$.
$$I_1=\int_0^{q}\sqrt{\frac{u}{q-u}}du=q\int_0^{\pi/... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3312359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Partial Fractions: Why does this shortcut method work? Suppose I want to resolve $1/{(n(n+1))}$ into a sum of partial fractions. I solve this by letting $1/{(n(n+1))} = {a/n} + {b/(n+1)}$ and then solving for $a$ and $b$, which in this case gives $a=1$ and $b=-1$.
But I learnt about a shortcut method. It says suppose $... | Let's take your example. We have \begin{align}\frac{1}{n(n+1)(n+2)} = \frac{a}{n} + \frac{b}{n+1} + \frac{c}{n+2}&=\color{red}{\boxed{\frac an+\frac{b(n+2)+c(n+1)}{(n+1)(n+2)}\cdot\frac nn}}\quad(\text{group together}\,b,c)\\&=\color{blue}{\boxed{\frac b{n+1}+\frac{a(n+2)+cn}{n(n+2)}\cdot\frac{n+1}{n+1}}}\quad(\text{gr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3312450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Boyd & Vandenberghe, problem 4.55 — how to show that solution is Pareto optimal? In problem 4.55 of Boyd & Vandenberghe's Convex Optimization, the authors ask the following.
Show that in a multicriterion optimization problem, a unique solution of the scalar optimization problem
$$ \min. \max._{i =1,2\cdots q}F_i (x)
... | Assume $x^*$ is not Pareto optimal then wlog there exists $y \neq x^*$ such that $F_1(y) < F_1(x^*)$ and $F_i(y) \leq F_i(x^*)$ for all $i =2\cdots q$. Now, this implies
$$ \max_{i =1,2\cdots q}F_i (y) \leq \max_{i =1,2\cdots q}F_i (x^*) = \min_{x} \max_{i =1,2\cdots q}F_i (x),$$
by definition of $x^*$. Therefore, $y$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Given a knight on an infinite chess board that moves randomly, what's the expected number of distinct squares it reaches in 50 moves? I was asked this in an interview and wasn't sure how to frame the answer. Basically as in the question you have a knight on an infinite chess board and it chooses one of its valid 8 move... | My experiments in Mathematica give $40.06$ as the average number of distinct cells (I've done several tries for $100'000$ trials).
I'm counting the initial square though, because it makes the program more simple.
I wouldn't know how to approach this problem theoretically, but it's simple enough to do tests. Knight's mo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3312820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can one prove this property of integrals?$\int_0^bf(x)(b-x)\,dx=\int_0^b\left(\int _0^xf(t)\,dt\right)\,dx$ $$\int_0^bf(x)(b-x)\,dx=\int_0^b\left(\int _0^xf(t)\,dt\right)\,dx$$
I can't understand how this property can be proven but it has held true for everything I have tried. How do you even approach this? I have ... | \begin{align}
& \int_0^b\left(\int _0^xf(t)\,dt\right)\,dx \\[10pt]
= {} & \iint\limits_{(t,x)\,:\,0\,<\,t\,<\,x\,<\,b} f(t)\, d(t,x) \\[10pt]
= {} & \int_0^b \left( \int_t^b f(t) \, dx \right) dt \\[10pt]
= {} & \int_0^b f(t)(b-t) \, dt.
\end{align}
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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proof of some statement about finite field extension I would like to have some explanation for the following statement
Let $K$ be an algebraically closed field of characteristic $p>0$, and $K((t))$, the field of Laurent series with coefficients in $K$. The Galois group of the polynomial $X^{p^n}-X=t^{-1}$ is isomorphi... | Say that $\alpha$ is a root of your polynomial $X^{p^n}-X-t^{-1}=0.$ Then it is obvious that if $a \in \mathbb{F}_{p^n},$ that $\alpha+a$ is a root as well, since $(\alpha+a)^{p^n}= \alpha^{p^n}+a^{p^n} = \alpha^{p^n}+a.$ So the Galois group is as claimed.
There are finite extensions with Galois groups isomorphic to ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Solutions for inequality $\frac{1}{x} + \frac{1}{1-x} \gt 0$ How to find all real solutions for $$\frac{1}{x} + \frac{1}{1-x} \gt 0$$
I came up with $\frac{1}{x(1-x)} \gt 0$ implies $x(1-x)>0$ and finally ended with $0<x<1$ but the answer provided was $0<x<1$ or $x>1$. I tried sample values for $x>1$ but they don't sat... | It should be $\frac{1}{x(1-x)} \gt 0$ thus, $x(1-x)>0$ and so, $0 < x < 1$ is the only solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3313178",
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"source": "stackexchange",
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What series should I use to compare? Direct comparison test Use direct comparison test to prove if the following series converge or not.
A) $\sum_{n=0}^\infty \frac{1}{3^n -1}$
B) $\sum_{n=0}^\infty\frac{1}{\sqrt{n+2}}$
In A) I wrote $3^n -1<3^n$ so $\frac{1}{3^n -1}>\frac{1}{3^n}$, but that is useless because $\frac{... | You have the right idea for A). Try to compare it with a geometric series. How about using/proving the inequality $2^n \leq 3^n - 1$?
Let me give a hint for B):
We have $$\frac{1}{\sqrt{n+2}} \geq \frac{1}{\sqrt{n+n}} = \frac{1}{\sqrt{2}}\frac{1}{\sqrt{n}}$$ for all $n \geq 2$. Does this help you?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Combinatorics: Partnerships Problem Overcounting My textbook presents the following story proof for partnership counting:
Let's use a story proof to show that
$$\dfrac{(2n)!}{2^n \cdot n !} = (2n - 1)(2n - 3) \dots 3 \cdot 1$$
Story proof: We will show that both sides count the number of ways to break $2n$ people into... |
This overcounts by a factor of $n! \cdot 2^n$ since the order of pairs doesn't matter, nor does the order within each pair.
We assume at the start that the pairs formed are ordered, both within the pair (A, B not the same as B, A) and between the pairs (this can be interpreted as the pairs standing side-by-side, in a... | {
"language": "en",
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Definition of 'product' for ordered pairs In 'Mathematics Form and Function' ch.2, section 4, 'Integers' by Saunders Mac Lane (p.50 in the 96 edition) I came across the following definitions of sum and product for ordered pairs:
(m, n) + (m', n') = (m + m', n + n')
(m, n) (m', n') = (mm' + nn', mn' + m')
My understandi... | The word ordered in ordered pair $(a,b)$ does not mean $a<b$
It simply means that $a$ is the first element and $b$ is the second one.
The difference between the set and ordered pair is that $\{a,b\}=\{b,a\}$ but $(a ,b)\ne (b,a)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3313504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the series $\sum_{n=1}^{\infty} \frac{e^{-n^2 x}}{n}$? Following Passare: How to compute $\sum 1/n^2$ by solving triangles
I tried the following
$$
\int_0^{\infty}\frac{e^{-nx}}{n^2} dx = \frac{1}{n^3}
$$
So we can write (with some help of Wolfram Alpha)
$$
\sum_{n=1}^{\infty} \int_0^{\infty}\frac{e^{-nx}}{n^2... | Let's replace:
$$x=y^2/4$$
Then we have:
$$e^{-n^2 y^2/4}= \frac{1}{\sqrt{\pi}} \int_{-\infty}^\infty e^{-t^2+i n y t} dt$$
Thus, provided the integral exists, we should have:
$$g(y)=\sum_{n=1}^\infty \frac{e^{-n^2 y^2/4}}{n}=-\frac{1}{\sqrt{\pi}} \int_{-\infty}^\infty e^{-t^2} \log \left(1-e^{i y t} \right) dt$$
Extr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3313634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Rank of a Differential This a mostly a sanity check sort of thing. I am working with the map given by
$$F:M_{2\times 2}(\mathbb{R})\rightarrow S_{2\times 2}(\mathbb{R}):A\mapsto A^t J A$$
where J is the matrix
$$J = \begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}.$$
I am trying to show that the set $S = \{A : F(A) = J\}$... | Ok, I just wanted to add a post showing another way of answering this question that is more in line with the structure posted in the question itself. This is NOT an elegant solution like the one provided above by Ted Shifrin (Thank you again!), but I think it might be useful for some people seeing this later to have bo... | {
"language": "en",
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"source": "stackexchange",
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Find the maximum possible perimeter of a right triangle
The ratio between the perimeter of a right triangle and its area is 2:3. The sides of the triangle are integers. Find the maximum possible perimeter of the triangle.
If the sides of the triangle are $A$, $B$ and $C$ (the hypotenuse), I have deduced that:
$$A+B+... | There are naturals $m$ and $n$ such that $m>n$, $a=m^2-n^2$, $b=2mn$ and $c=m^2+n^2$.
See here: https://en.wikipedia.org/wiki/Pythagorean_triple
Thus, $$\frac{m^2-n^2+2mn+m^2+n^2}{mn(m^2-n^2)}=\frac{2}{3}.$$
Can you end it now?
I got $56$ as the answer.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convergence integral given weak convergence of measures and functions/random variables Let $X\subset\mathbb{R}^d$ be compact.
Given sequences of real-vauled random variables $f_n\to f$
and positive radon measures $\mu_n\to \mu$ both converging weakly for $n\to\infty$.
Under which further conditions can we deduce that ... | Claim: the conclusion holds for every sequence $(\mu_n)$ converging weakly to $\mu$ iff $f_n \to f$ uniformly.
Since $\mu$ is Radon and $X$ is compact, $\mu$ is a finite measure. Since $\mu_n(X) \to \mu(X)$ it follows that $sup_n \mu_n(X)<\infty$. So, if $f_n \to f$ uniformly then $\int f_n d\mu_n -\int fd\mu_n \to 0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3314036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Inverse of a structured matrix of sines. Suppose I have a matrix $P$ defined by
$$
P =\begin{pmatrix}
\sin(\frac{\pi}{n+1}) & \sin(\frac{2\pi}{n+1}) & \cdots & \sin(\frac{n\pi}{n+1}) \\
\sin(\frac{2\pi}{n+1}) & \sin(\frac{4\pi}{n+1}) & \cdots & \\
\vdots & & \ddots & \\
\sin(\frac{n\pi}{n+1}) & \cdots & & \sin(\frac{n^... | I managed to solve it. It was a bit of a headache but here is my proof.
For $k, l \in \mathbb{N}$ and $n \in \mathbb{N}$ consider the sum
\begin{equation}
A_{kl} = \sum_{j=1}^n \sin \left(\frac{k \pi}{n+1}j\right) \sin \left(\frac{l \pi}{n+1}j\right).
\end{equation}
Claim:
\begin{equation}
A_{kl} = \frac{n+1}{2}\delta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3314129",
"timestamp": "2023-03-29T00:00:00",
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Need help with concept of expectation! I have been reading blogs on Expectation. I am quite familiar with what it is but still don't understand what do we infer from it, what does it tell us about the experiment?
For eg: expected number of coin flips for two consecutive heads is 6. So what does this "6" say...are these... | Intuitively, the expectation is the average of a random variable for an infinite number of drawings. E.g. with a fair coin, exactly $\dfrac12$ heads. It is the "real" average.
The Law of Large Numbers tells that the average computed on an increasing number of drawings does converge to the expectation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3314338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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How to solve this word problem using graph theory? Suppose that there is a global network of one hundred airports, and that between each pair of airports there is a direct connection. In connection with the cuts the governments of various countries want to eliminate connections as far as possible. It must however still... | Hint: Consider one really big airport and 99 small ones.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3314435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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I perform 5 "independent" card draws from a deck w/ replacement. All drawn cards are queens, what is the probability of queen of spades drawn twice? So, drawing a queen of spades on the first draw has probability $\frac{1}{52}$.
As there is replacement involved the next card, assuming the card drawn is again, the queen... | Pretending that the deck has only the four queens is great advice.
Let's assume that you need exactly two queen of spades.
Choose which two of the five draws you get your queen of spades. How many ways?
How many ways can you get three non-queen of spades in the other three slots?
Divide by the total number of ways to d... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why do we want lots of $0$’s in a matrix? I am working thru Axler’s “Linear Algebra Done Right.” I am okay with the math, but I’m losing sight of the forest.
Like all books on linear algebra, there’s a lot of time and energy spent on finding, proving existence of, and interpreting matrices with “lots of zeros” - tha... | There area a whole theory about SPARSE MATRICES (matrices most of the elements are zero) which are used in computing science and numerical analysis. You can find many papers on this subject if you are interested in knowing more about it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3314632",
"timestamp": "2023-03-29T00:00:00",
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In the exapnsion of $(1+x+x^3+x^4)^{10}$, find the coefficient of $x^4$ In the exapnsion of $(1+x+x^3+x^4)^{10}$, find the coefficient of $x^4$.
What's the strategy to approach such problems. Writing expansion seems tedious here.
| Referring to Jack Crawford's comment above, there are three possible ways to get $x^4$:
$$\underbrace{1\times1\times\cdots\times 1}_{9}\times x^4\\
\underbrace{1\times1\times\cdots\times 1}_{8}\times x\times x^3\\
\underbrace{1\times1\times\cdots\times 1}_{6}\times x\times x\times x \times x$$
The first is the combinat... | {
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Prove that $ I_{R} (A) $ is the "biggest" subring in $R$ in which $A$ is ideal.
Let $A$ be subring of ring $R$. Let $ I_{R} (A) = \{ x \in R \colon xa, ax \in A, \forall a \in A \} $. Prove that $ I_{R} (A) $ is the maximal (biggest) subring of $R$ in which $A$ is ideal.
I easily proved that $I_{R} (A) $ is subring o... | Let $A\subseteq S\subseteq R$ with $S$ a ring and suppose $A$ is an ideal of $S$. If $x\in S$, then $xa$, $ax\in A$ for all $a\in A$. Therefore $x\in I_R(A)$. So $S\subseteq
I_R(A)$.
| {
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"url": "https://math.stackexchange.com/questions/3314970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Exercise in differential geometry using Gauss-Bonnet For a positive real number $r$ let $M_{r}$ be the regular surface
$M_{r}=\{(x,y,z) \mid x^2+y^2=z<r^2, x>0,y>0\}$.
Let $K$ denote the Gaussian curvature of $M_{r}$. Determine
$\int_{M_{r}}KdA$ and $\lim_{r \rightarrow \infty}\int_{M_{r}}KdA$.
Solution;
Using Gauss-... | As a complement to the other answer here is a solution that uses Gauss-Bonnet:
Let $S_{r}=\{(x,y,z) \mid x^2+y^2=z\leq r^2\}$. By symmetry $\int_{M_{r}}K\;dA=\frac 14\int_{S_{r}}K\;dA$.
Since $S_{r}$ is a compact two-dimensional Riemannian manifold by Gauss-Bonnet
$$\int_{S_{r}} K\;dA+\int_{\partial S_{r}}k_g\;ds=2\pi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find the average rate of change of $g(t)=t^2+3t+1$ on the interval [0,a] I am working on a textbook question "Find the average rate of change of $g(t)=t^2+3t+1$ on the interval $[0,a]$". The solution provided, along with the steps in between is:
Avg. rate of change:
= $\frac{g(a)-g(0)}{a-0}$
= $\frac{(a^2+3a+1)-(0^2+3... | For any number $a, a-0=a$. Subtracting $0$ makes no change.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3315185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is this limit for the sum of remainders correct? I found the following problem:
$$\lim_{n\to\infty}\left(\frac{1}{n^2}\sum_{m=1}^{n}n\space\text{mod}\space m\right)$$
and decided to give it a go. I got an answer of $1-\frac{\pi^2}{12}$, but I am not sure if this is correct. It was posted with no source/context, and I w... | Using $$n\bmod m = n-m\left\lfloor \frac{n}{m}\right\rfloor$$ you get:
$$\sum_{m=1}^{n} n\bmod m = n^2-\sum_{m=1}^{n} m\left\lfloor \frac{n}{m}\right\rfloor$$
or:
$$\frac{1}{n^2}\sum_{m=1}^{n} n\bmod m = 1-\frac{1}{n}\sum_{m=1}^{n} \frac{m}{n}\left\lfloor \frac{n}{m}\right\rfloor$$
But $\frac{1}{n}\sum_{m=1}^{n} \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3315285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Domain of $7^{\log_7(x^2-4x+5)}$
If $$7^{\log_7(x^2-4x+5)}=x-1$$ then $x$ may have values...
My attempt: $$x^2-4x+5=x-1$$
So, $$x^2-5x+6=0$$
So, $$x=2,3$$
To check the domain of log, $$x^2-4x+5>0$$ i.e., $$(x-2+i)(x-2-i)>0$$ That gives me, $x<2-i$ and $x>2+i$. Is this a valid way of writing domain here? If No, how ... | Note that $$x^2-4x+5 = (x-2)^2 +1 >0 $$ for all x, so there is no problem with logarithm and we have
$$7^{log_7(x^2-4x+5)}=x^2-4x+5$$
Therefore $$ 7^{log_7(x^2-4x+5)}=x-1 \iff x^2-5x+6=0$$
Thus the solutions are $x=2$ and $x=3$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Modular arithmetic $(2n+1)x \equiv -7 \pmod 9$ Find a solution $(2n+1)x \equiv -7 \pmod 9$
I’m sure this is trivial but I still have doubts about it.
I know the equation has solution for certain $n \in \mathbb {Z}$. Actually I have tried a few and got a similar results (with Diophantine equations ). I wonder if there’s... | Since $2n + 1$ 'cycles through' the modulo $9$ residues, the problem is reduced to solving
$$\tag 1 x'x \equiv 2 \pmod 9$$
This is equivalent to $x'x = 9k +2$ and we need only look for solutions
$$ 0 \le x' \lt 9 \text{ and } 0 \le x \lt 9$$
We represent both $x'$ and $x$ in $\text{base-}3$ format,
$$\tag 2 x' = a' + ... | {
"language": "en",
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Find every $n$: $n^2 + 340 = m^2$ Let $n$, $m \in N$.
The problem asks to find every natural number $ n $ such that:
$ n^2 + 340 = m^2 $
I tried to solve the equation like this:
$ n^2 - m^2 = 340 $
$ (n + m)(n - m) = 2^2 * 5 * 17 $
I listed all possible pairs of dividers of 340:
$(1, 340), (2, 170), (4, 85), (5, 68), (... | I think the way you did it is the best way.
A worse way would be that if we let $m = n + k$ then $n^2 + 2nk + k^2 = m^2$ so $2nk + k^2 = 340$. Clearly $k$ is even so if $k=2k'$ then $nk' + k'^2 = 85$. And $k'(n+k') = 85$ so for $k' =1....\sqrt{85};k|85=5*17$ or in other words for $k'=1, 5$ we get $n=84,12$ and $k=2... | {
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Does there exist an area-preserving map from the hyperbolic plane to the Euclidean plane? Fairly simple question: does there exist an area-preserving map from the hyperbolic plane to the Euclidean plane?
If not, does there exist an area-preserving map from an arbitrarily large subset of the hyperbolic plane, to an arbi... | There does in fact exist an area-preserving map, as demonstrated in this video at 11:20: the Lambert azimuthal equal-area projection.
The idea is that you take polar coordinates of the hyperbolic plane and map them to polar coordinates of the euclidean plane via a map $(r, \theta) \mapsto (f(r), \theta)$ where $f$ is c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3315718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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How to find value of $k$ and $p$ Im so confused How can i solve this problem
the given below is:
1.Find the $k$; $2k+1,3k+4,7k+6$ in geometric
2.Find the $k$; $k-3,k+2,k+3$ in geometric
3.Find the $p$; $p+7,3p+9,p+3$ in arithmetic
Thanks for the solution
any answer is highly appreciated.
| Hint: for three consecutive terms $a,b,c$ in a geometric progression, we must have $ac=b^2$, and if they are instead in an arithmetic progression we must have $a+c=2b$. Thus you can formulate a quadratic equation for the geometric problems and a linear one for the arithmetic problems.
| {
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Can anyone help with geometry (area with an unknown length) question? I would really appreciate it.
**Note - the problem I'm struggling with is how to calculate the area of APBQ (the last question)
Figure 1 on the right shows a right-angled
triangle ABC where AB ＝ 1 cm, AC ＝ 2 cm, and
angle BAC ＝ 90°. Triangle PAB is ... | Following on from your comment:
$\angle CAB = \tan^{-1} (2)$, and since $PA$ and $BC$ are parallel, $\angle PAB = \tan^{-1} (2)$ as well.
Now if you split $\Delta PAB$ in half where $M$ is the midpoint of $PB$, you will have $\sin PAM = \frac{PM}{PA} = \frac{PM}{1}$. This gives me a value of $PM = \sqrt{\frac{2}{5+\sqr... | {
"language": "en",
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"source": "stackexchange",
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Parallel system functioning problem I am currently solving the following problem about conditional probability:
"A parallel system functions whenever at least one of its components works. Consider a parallel system of n components, and suppose that each component works independently with probability $\frac{1}{2}$. Find... | Let's call event "whole parallel system works" as $W$ and event "first component works" as $W_1$. Our task is to find $P(W_1|W)$. Let's use conditional probability definition to expand it:
$$P(W_1|W) = \frac {P(W_1 \cap W)}{P(W)} = \frac {P(W | W_1)\cdot P(W_1)}{P(W)}$$
Now, $P(W_1) = \frac{1}{2}$. $P(W) = 1 - P(\bar W... | {
"language": "en",
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Why ReLU function is not differentiable at 0? I'm kind of rusty in calculus.
Why is the ReLU function not differentiable at $f(0)$?
$$
f(x) =
\begin{cases}
0 & \text{if $x \leq 0$} \\
x & \text{if $x > 0$}.
\end{cases}
$$
| If you look at $x > 0$, or the righthand derivative, $$\frac{df}{dx} = \frac{d}{dx} x = 1$$ for all $x$.
If you look at $x \le 0$, or the lefthand derivative, $$\frac{df}{dx} = \frac{d}{dx} 0 = 0$$ for all $x$.
Since $x = 0$ is the "break" point, the lefthand and righthand derivatives are not the same, and thus, the de... | {
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Analytic Continuation of Complex Function I am triyng to solve the following problem in Brown and Churchill's complex variables textbook.
Show that the function $f_2 (z) = 1/z^2$ ($z \neq 0$) is the analytic continuation of the function
\begin{align*}
f_1 (z) = \sum\limits_{n=0}^{\infty} (n+1)(z + 1)^n \ \ \ (|z+1... | We need to show that $1/z^2$ is the analytic continuation of $S$, so amongst other things we need to show they are equivalent on $D_1\cap D_2$.
Splitting the sum,
\begin{eqnarray}
S&=&\sum_{n=0}^\infty(z+1)^n+\sum_{n=0}^\infty n(z+1)^n\\
&=&\frac{1}{1-(z+1)}+(1+z)\frac{d}{dz}\sum_{n=0}^\infty (1+z)^n\\
&=&-\frac{1}{z}-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3316545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Why is $\sum_{k = 0}^{\infty}(-x^{2})^{k}$ not uniformly convergent on $(-1,1)$ but it is on $[-r,r] \subset (-1,1)$? Clarification Note: I'm aware there are similar postings about the same idea, but I'm attempting to develop understanding of the concept through my explanation of what is happening.
Why is $\sum_{k = 0}... | We will use the following result to prove that the series $\sum_{k=0}^\infty(-x^2)^k$ does not converge uniformly.
Result: Let $D$ be a subset of a metric space $(X,d)$, and a series of function $\sum{f_k}$ be uniformly convergent on $D$ to a function $f$. Let $x_0$ be a limit point of $D$ and $\lim_{x\rightarrow x_0... | {
"language": "en",
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What are bases, really? I'm taking a course in Linear Algebra right now, and am having a hard time wrapping my head around bases, especially since my prof didn't really explain them fully. I would really appreciate any insight you could give me as to what bases are! Also, can there can be multiple different bases for a... | While a bit late to the game, I thought another perspective might help.
Consider the following physical example. Now, without being too pedantic about definition, a basis for a vector space is much like a building block of a biological system. We can build a human body from a set of cells. That is, we can construct al... | {
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The name for equations/problems like "How many four-digit numbers have four different digits (without a leading $0$)?" I'm not very quick at these kinds of problems so I want to improve on them. But I don't know what topics to search for.
Should I be searching for significant figures + base arithmetic?
How many four-... | As you were told in the comments, the topic is called permutation. But, as you don't the whole set of digit numbers some may refer to that as an variation on some texts. I have read that a variation is a permutation of r elements from n. By this definition we could get the closed formula,
$$V(n,r)=\frac{n!}{(n-r)!}.$$
... | {
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Algorithm for computing algebraic numbers? (Why are algebraic numbers computable?) Suppose $b$ is algebraic over the rationals.
In other words:
$p(b) = 0$ for some polynomial where all the coefficients are rational.
I am told $b$ is computable. But why?
*
*Can I derive a polynomial from $p$ that I can evaluate to ge... | The answer is that iterative root-finding algorithms, like the Aberth method, exist to numerically find the roots of polynomials whose coefficients are themselves computable. Therefore if p(x) has rational coefficients, there exists a program that will produce b. So b is computable.
| {
"language": "en",
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Renting Vs Buying?? Define Function? A family has $100,000 in savings accounts. They seek financial advice to develop a ten-year housing strategy. The two options are: continue renting the apartment or take a bank loan to buy a property.
(a) Assume that they spend USD1900 per annum on renting the apartment and put thei... | At a) the family starts with $100,000$. This amount is compounded once. At the end of the year they pay $1900$. The net value after one year is therefore $NV_1=100,000\cdot 1.02-1900=100,100$. Now this value is compounded one year:
$(100,000\cdot 1.02-1900)\cdot 1.02=102,102$. Here we see that the rent is compounded as... | {
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$f$ convex $\iff f(y)\geq f'(x)(y-x)+f(x).$ I want to prove that $f$ is convex $\iff f(y)\geq f'(x)(y-x)+f(x)$.
The implication is fine, but I have difficulties with the converse implication. I tried to prove that $f'$ is increasing, but still, it doesn't work : Let $y>x$, then $$\frac{f(y)-f(x)}{y-x}\geq f'(x),$$
but ... | Graphically, this means that the tangent at any point of a convex function lies below the graph.
Convexity implies $ f(y)\geq f'(x)(y-x)+f(x)$
Now, take $0 < \theta < 1$, we know that for $x,y \in \operatorname{dom}(f)$, we also have that $x + \theta(y-x) \in \operatorname{dom}(f)$. Using the definition of convexity... | {
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Continuity of $x\sin\frac{1}{y}$ at $(x, 0)$ I need to check if function is continue in $(x,0)$
$$f(x,y) =
\left\{
\begin{array}{ll}
x\sin\frac{1}{y} & \mbox{if } y \ne 0 \\
0 & \mbox{if } y = 0
\end{array}
\right.$$
Can someone help me understand if I approached this correctly?
First, I check the following lim... | It's easier to use sequences: Fix $0\neq x\in \mathbb R$. There are sequences $(y_n)$ and $(z_n)$ such that $y_n\to 0$ and $z_n\to 0$ and such that $\sin(1/y_n)=1$ and $\sin(1/z_n)=-1$ (why?). Then, $\alpha_n=(x,y_n)\to (x,0)$ and $\beta_n=(x,z_n)\to (x,0)$ but $f(\alpha_n)\to x$ and $f(\beta_n)\to -x$ so $f$ is not co... | {
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If $f$ and $g$ are paths with $g$ constant, then $f'\simeq g'$ rel$\{1\}$ if an only if there exist a free homotopy $f'\simeq g'.$ Using Theorem 1.6 (http://www.ugr.es/~acegarra/Rotman.pdf page 17), show (using the notation of exercise 3.2 (page 40)) that if $f$ and $g$ are paths with $g$ constant, then $f'\simeq g'$ r... | You confuse $f,g : I \to X$ which are closed paths and $f',g' : S^1 \to X$. Thus your homotopy $H : I \times I \to X$ cannot be a homotopy from $f'$ to $g'$. So take a (free) homotopy $H : S^1 \times I \to X$ such that $H(z,0) = f'(z), H(z,1) = g'(z) = x_0$.
In Example 1.7 Rotman shows that if $f : S^n \to X$ is map ... | {
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Computation Of Integrals Computer the Integral: $$\int\frac{2x+1}{(x-1)(x-2)}dx$$
Now using partial fraction we can write $$\frac{2x+1}{(x-1)(x-2)}=\frac{A}{x-1}+\frac{B}{x-2}$$, So we get $$\frac{2x+1}{(x-1)(x-2)}=\frac{A(x-2)+B(x-1)}{(x-1)(x-2)}$$ Now for all $x$ not equal to $1, 2$ we can cancel out the denominator ... | The expression
(2x+1)/(x-1)(x-2). = A / ( x-1). + B/(x-2) is discontinuous at x= 1 and at x= 2 but identy for restall value of x.
Once we write as 2x +1 = A (x-2) + B (x-1) since left hand side is continuous thus Right-hand side will be having same behaviour .hence we can put x= 2 and x= 1 to find A and B
| {
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"source": "stackexchange",
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"answer_id": 3
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$f(z)=\sqrt{1-z^2}$ pole at infinity Consider integrating $f(z)=\sqrt{1-z^2}$ with a branch cut of $[-1,1]$ around the following contour.
$\gamma_1:[-1,1]\to\mathbb{C}, t\mapsto t+\epsilon i$
$\gamma_2:[-\pi/2,\pi/2]\to\mathbb{C}, t \mapsto 1+\epsilon e^{-it}$
$\gamma_3:[-1,1]\to\mathbb{C}, t\mapsto -t-\epsilon i$
$\ga... | See this answer. You have implicitly used the condition that $f(x + i0) > 0$ for $-1 < x <1$ (otherwise you would get $I = -\pi$). With this condition, $f$ can be written as
$$f(z) = -i z \sqrt {1 - \frac 1 {z^2}},$$
where $\sqrt z$ is the principal value of the square root.
$\gamma$ goes around the origin clockwise, t... | {
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Find distribution function through moment generating function Suppose that the moment generating function $M_X$$(t)$ of a random variable $X$ is given by
$$ M_X(t)=\frac{e^t+e^{-t}}{6} + \frac 23 $$
I need to find the distribution function $F_X(x)$.
Until now, I have been given (in my lecture notes) that I can expres... | Hint: From the moment generating function we can determine the distribution of $X$, which is $P(X=1)=P(X=-1)=\frac16$, $P(X=0)=\frac23$. I believe that you can move on now.
| {
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Why is $\nabla \cdot u=0$? (If $u=v\cos(k\cdot x)$) Suppose $k\in \mathbb R^3$ and $v \in \mathbb S^2$ (2D sphere) and $k\cdot v=0$. Let $u=v\cos(k\cdot x)$, then why is $\nabla \cdot u=0$?
To calculate the divergence, I look at one of the components $\partial_{j} u_{j}$. $$\partial_{j} u_{j}=\partial_{j} v_{j}\cos(k_... | I like to avoid indices altogether whenever possible, and would use the product rule for divergence instead:
$$\nabla \cdot \left[f(x)v(x)\right] = \langle \nabla f(x), v(x)\rangle + f(x) \nabla \cdot v(x).$$
In particular if $v%$ is constant, the second term vanishes and you just get
$$\nabla \cdot u = v \cdot \nabla ... | {
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How dense are primes congruent to 1 and 3 (mod 4)? There are infinitely many primes of the form $4n+1$ and $4n+3$. In a given interval $[0,N]$ for a large enough $N$ do we expect to see the same number of primes congruent to $1$ and $3$ (mod 4)?
| Dirichlet theorem of primes in arithmetic progression says that the asymptotic density of primes of the form $4k+1$ and $4k+3$ are both equal to $\frac{x}{2\log x}$. However in the small scale we observe a phenomenon called Chebyshev bias where in the actual number of primes of the form $4k+3$ are slightly more than th... | {
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"source": "stackexchange",
"question_score": "3",
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Finding a basis for the Eisenstein space $\mathcal{E}_1(12,\chi)$ I am trying to find a basis of the Eisenstein space $\mathcal{E}_1(12,\chi)$ of modular forms of weight 1, level 12 and Dirichlet character
\begin{equation*}
\chi(m)=\genfrac{(}{)}{}{}{-12}{m}=\begin{cases}
1 ~~&\text{ if } m\equiv 1,\,7\pmod{1... | I don't know about how to generate that Eisenstein space, but from quadratic reciprocity for $O_K = Z[\frac{\sqrt{-3}+1}{2}]$ which is a PID we have
$$\sum_{a,b \ne (0,0)} |a+b \frac{\sqrt{-3}+1}{2}|^{-2s} = |O_K^\times| \zeta_K(s) =6 \zeta(s) L(s,(\frac{-3}{.}))=6 \zeta(s) L(s,(\frac{.}{12}))\\ =6 \sum_{n=1}^\infty ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3318140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$ab$ divides $a^2+b^2 \implies a=b$ Let $a$ and $b$ be two positive integers.
If $ab$ divides $a^2+b^2$ then $a=b$.
I can show that $a$ divides $b^2$ and $b$ divides $a^2$ but then I get stuck. Any ideas?
| Hint $\ n = \dfrac{a^2\!+b^2}{ab} = \dfrac{a}b + \dfrac{b}a =\, x+x^{-1}\,\overset{\large {\times\, x}}\Longrightarrow\,x^2-n\,x + 1 = 0$
By RRT = Rational Root Test $\ a/b\, =\, x\, = \pm 1.\,$ It is special case $\, j = 1 = k,\, c_1 = 0\,$ of below.
Generally applying RRT as above yields the degree $\,j+k\,$ homog... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3318249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Distance from any point in the plane to vertices of a triangle related to its sides I was writing a program calculating all possible configurations of $n$ random circles with random radius and center coordinate in the plane and met this problem. Being puzzled by it for quite a long time, I decided to have a try at thi... | $a',b',c'$ are the tripolar coordinates of $P$. MathWorld gives two equations $F$ by Euler, one of which is reproduced below (using $x,y,z$ for $a',b',c'$):
$$(a^2+b^2-c^2)(x^2y^2+c^2z^2)+(a^2-b^2+c^2)(b^2y^2+x^2z^2)+(-a^2+b^2+c^2)(a^2x^2+y^2+z^2)-(a^2x^4+b^2y^4+c^2z^4)-a^2b^2c^2=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3318361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Extending Pre-measure into two different measures. I am trying to find an example of an algebra $\mathcal{A}$, and a pre-measure $\mu_0$ such that, you can extend $\mu_0$ in the $\sigma-$algebra generated by $\mathcal{A}$ to two different measures.
By Caratheodory's extension theorem, you must have that the trivial ext... | On $[0,1)$, consider the algebra $\mathcal{A}$ of all finite unions of half-open intervals, of the form $\bigcup_{i=1}^n [x_i, x_{i+1})$. This generates the Borel $\sigma$-algebra. Consider the pre-measure $\mu_0$ which assigns measure $+\infty$ to every non-empty set in $\mathcal{A}$. Then one extension of $\mu_0$ t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3318441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the derivative of an improper triple integral I want to find the density function of the random variable $Y=X_1+X_2+X_3$, where the random variables $X_1, X_2$ and $X_3$ have a joint density function $$f_{X_1, X_2, X_3}(x_1, x_2, x_3) = (2\pi)^{-\frac{3}{2}}e^{-\frac{1}{2}(x_1^2+x_2^2+x_3^2)},\qquad -\infty<x_1, x... | Since no one answers it for 2 years and now I know where I went wrong, I'll post my answer here.
The short answer is:
(1) The bounds are incorrect.
(2) Once we fix that, the expression can be evaluated quite easily using Leibniz's rule and the Gaussian integral.
Let's consider the case with 2 random variables first, i.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3318517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving $\lim\limits_{n \to \infty} \frac{n^a}{c^n} = 0$ using L'Hôpital's Rule I am trying to prove $\displaystyle \lim_{n \to \infty} \frac{n^a}{c^n} = 0$ using L'Hôpital's Rule, but I'm stuck.
Here's what I have so far:
$$ \lim_{n \to \infty} \frac{n^a}{c^n} = \lim_{n \to \infty}\frac{an^{n-1}}{c^n \ln c} = \lim_{n ... | Hint If $c>1$, the limit is trivial for $a \leq 0$. For $a>0$ show instead that
$$\left( \lim\limits_{n \to \infty} \frac{n^a}{c^n} \right)^\frac{1}{a}=0$$
Then, raise both powers to $a$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3318614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Finding angle between two points on circle given cartesian coordinates of the points If I am given the coordinates of two points on a circle of radius'R', how can I find the angle between those two points, as well as the area of the arc between them(created on the circumference of the circle).
| If we take the dot product of the vectors from the center to these points then, $u\cdot v = |u||v|\cos \theta$ where theta is the angle you seek.
And since this is a circle $|u| = |v| = R$
If your circle is centered at the origin, and your points are $(x_1,y_1),(x_2,y_2)$
$\theta = \arccos \frac {x_1x_2 + y_1y_2}{R^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3318680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
} |
Methods for finding the minimum coordinate of an equation (any shortcuts?) I want to know if there's a quicker method to find the minimum coordinates of a quadratic equation other than the one described below.
Some background:
I was reviewing "Completing Squares and Inequalities" when I came to this inequality: $ y \l... | Derivative is also one of tool,
$\frac{dy}{dx}=2x-2 =0$
At $x=1$:
$\frac{d^2y}{dx^2} = 2 >0$
We get min. At $x=1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3318816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How to find number of words made using letters of word 'EQUATION' if order of vowels do not change Find number of words made using letters of word 'EQUATION' if order of vowels do not change.
My attempt:- since we do not have to change the order of the vowels hence,
_E_U_A_I_O_
we have $6$ places to fil... | You need to put Q, T and N into a blank eight-letter word, and fill in the remaining five places with the vowels in the prescribed order. There are $3!=6$ ways to order the consonants, and $\binom{6}{3}=56$ choices of three positions to put them in. So the answer is $6\times56=336$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3318935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Parametrisation of a surface in $\mathbb{R}^3$ I am trying to parameterise $M=\bigl\{(x,y,z) \mid e^z=\frac{\cos x}{\cos y}\bigr\}\subset \mathbb{R}^3$ where $x,y \in \bigl(-\frac{\pi}{2},\frac{\pi}{2}\bigr)$ in $(u,v)$ i.e $2$ variables but without any luck. Can someone see how that is supposed to be done?
| Since $z=\ln\frac{\cos x}{\cos y}$, we immediately get the parametrisation
$$\left(x,y,\ln\frac{\cos x}{\cos y}\right)\qquad x,y\in(-\pi/2,\pi/2)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3319048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Colors and corresponding numbers
The 10 colors Green, Blue, Violet, Red, Orange, Yellow, Cyan, Magenta, Fuchsia, Brown are associated to each of the numbers 1, 2,…, 10 but we don’t know which color corresponds to each number. In a large box there are infinite sealed envelopes, each containing one card of the above co... | I think this works:
11234 implies (1)(234)
25567 implies (1)(2)(34)(5)(67)
36889 implies (1)(2)(3)(4)(5)(6)(7)(8)(9)
and the only color you haven't seen is (10)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3319147",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
If $\sec A-\cos A=1$, then determine the value of $\tan^2\frac A2$ This is what I tried
$\sec A=\frac{1}{\cos A}$, so the equation becomes
$1-\cos^2A=\cos A$
If we solve the above quadratic equation, we the values of $\cos A$ as $\frac{-1\pm \sqrt5}{2}$
Therefore, $\tan\frac A2$ becomes
$$\sqrt \frac{3-\sqrt 5}{1+\sq... | Let $t=\tan^2\dfrac A2$. Then $\cos A=\dfrac{1-t}{1+t}$.
$\dfrac{1+t}{1-t}-\dfrac{1-t}{1+t}=1$
$4t=1-t^2$
$(t+2)^2=5$
$t=-2+\sqrt5$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3319274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Operator that has no fixed point Let $X = \{f \in C[0,1]; \|f\|_{\infty}\leq1, f(0)=0, f(1)=1\}$ be a subset of $C[0,1]$ and define the operator $T:X \rightarrow X$ by $Tf(t)=f(t^{2})$. Show that $T$ is continuous and has no fixed point.
Could you help me with the later question?
| Suppose $f(t)=f(t^2)$ for some $f\in C[0,1]$ such that $f(0)=0$ and $f(1)=1$. Then
$$f(x^{2^n})=f(x)$$
for all $n\geq 1$. As $f$ is continuous and $x^{2^n}$ converges to $0$ for all $x\in [0,1)$ for $n\to\infty$, we have that $f(0)=f(x)$ for all $x\in [0,1)$ which is a contradiction to continuity and $f(1)=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3319357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Coloring the grid points with three colors About half a year ago I posted a problem: "Coloring grid points with two colors" (The problem)
I found it really interesting, so I thought I make some research. Now I need help with my following question. I am thankful for every idea, hint and solution.
Let $S$ be a set of fin... | Yes, this can always be done.
Lemma. This can be done when every vertical and horizontal line with points on it contains exactly $3$ points.
Proof. In this case, all three points on a line must receive different colors.
We can think of this problem as a graph theory problem. Consider the bipartite graph with vertice... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3319447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why does unit group of valuation ring contains $K^{\times}$ Let $K$ be a number field, $K_v$ be a completion of $K$ by non-archimedean valuation $v$ of $K$ and $U_v$ be a unit group of valuation ring $R_v$.
Then why $K^{\times}\subset U_v$?(for almost all $v$)
This implies principal idele is idele.
Edit this is wrong
| Let $K = \Bbb{Q}$ then $$\Bbb{A_Q} = \{ a_\infty\prod_p a_p \in \Bbb{R} \times \prod_p \Bbb{Q}_p, \text{ for all but finitely many } p, a_p \in \Bbb{Z}_p\}$$
it is a commutative unital ring whose $1$ is $1_\infty \prod_p 1_p$ (with $1_p$ the $1$ of $\Bbb{Q}_p$) and
$$\Bbb{A_Q}^\times = \{ a \in \Bbb{A_Q}, \exists b \i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3319706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Maximizing $\frac{a^2+6b+1}{a^2+a}$, where $a=p+q+r=pqr$ and $ab=pq+qr+rp$ for positive reals $p$, $q$, $r$
Given $a$, $b$, $p$, $q$, $r \in\mathbb{R_{>0}}$ s.t.
$$\begin {cases}\phantom{b}a=p+q+r=pqr \\ab =pq+qr+rp\end{cases} $$
Find the maximum of $$\dfrac{a^2+6b+1}{a^2+a}$$
This question is terrifying that I ... | Here is an approach, which provides, so I think, a certain understanding of the "working domain", prior to the consideration of maximisation issue.
Your issue is to find values of $a$ and $b$ such that polynomial equation :
$$(x-p)(x-q)(x-r)=0 \ \ \iff$$
$$x^3-(p+q+r)x^2+(qr+pr+pq)x-pqr=0 \ \ \iff \tag{2}$$
$$x^3-ax^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3319843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Help with this inequality : $-1 \le \frac{1+x^2}{2x} \le 1$ I have to solve
$$
-1 \le \frac{1+x^2}{2x} \le 1
$$
My attempt at solution:
$$ -1 ≤ (1+x²)/2x ≤ 1 $$
$$ - 1 ≤ (1+x²)/2x \quad\text{and}\quad (1+x²)/2x ≤ 1 $$
$$ 0≤(1+x²)/2x + 1 \quad\text{and}\quad (1+x²)/2x - 1 ≤0 $$
i.e. $(x²+2x+1)/2x \quad\text{a... | This problem is equivalent to $$\left|\frac{1+x^2}{2x}\right|\le 1,$$ which gives $$(1+x^2)^2\le(2x)^2,$$ which gives $$(1-x)^2(1+x)^2\le 0.$$ Since LHS of last inequality can never be negative, it follows that the only solution will occur when LHS vanishes. Can you continue now?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3319993",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Prove that $T=\frac{2X}{\sqrt{Y}} \sim t_4$ where $X \sim N(0,1)$ and $Y \sim \chi^2_4$ Question. Let $X,Y$ be independent random variables. Suppose $X \sim N(0,1)$ and $Y \sim \chi^2_4$, use a bivariate transformation to show that $T=\frac{2X}{\sqrt{Y}} \sim t_4$.
Attempt. Use the bivariate transformation $T=\frac{2X}... | If $U,\,V$ are independent continuous random variables of respective pdfs $f_U,\,f_V$ with $V$ of support $[0,\,\infty)$, $W:=U/V$ has pdf $f_W(w):=\int_0^\infty vf_U(wv)f_V(v)dv$. With the choice$$f_U(u)=\frac{1}{\sqrt{2\pi}}\exp-\frac{u^2}{2},\,f_V(v)=\frac12 v^3\exp -\frac{v^2}{2}$$so $U\sim N(0,\,1),\,V\sim\chi_4$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3320125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find "A" in this equation. $$ \sqrt[3] {A-15√3} + \sqrt[3] {A+15√3} = 4 $$
Find "A" ?
The way of exponentiation took too much time, is there any easier method?
| If there exists a cubic polynomial of integer coefficients with roots:
$$0,\sqrt[3] {A-15√3} , \sqrt[3] {A+15√3} $$
then it will be of the form
$x^3-4x^2 + \sqrt[3]{A^2-675}x$
You just need to find $A$ such that $A^2-675$ is a perfect cube.
Trivially $A^2 = 676$ will do.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3320229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
constant evaluation when using differential equations. This is regards to constant evaluation when using differential equations.
*
*A solution is given to be:
$$y=(e^{2x}+e^x ) \ln(1+e^{-x} )-(c_1+1) e^x+(c_2-1) e^{2x}$$
*A simplified solution in an answer book is given as:
$$y=(e^{2x}+e^x ) \ln(1+e^{-x} )+(c_1... | Both solutions are correct and they are equivalent. The constants $C_1$ and $C_2$ are just place holders for numbers to be found from initial values and you may as well call them $-C_1-1$ or $C_2+1$
Once the initial values are given the constants are found and the final result is unique regardless of the notations for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3320339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Why must there be disjoint sets $A$ and $B$ such that $m^*(A \cup B) < m^*(A) + m^*(B)$? (Theorem 18, Royden) I am trying to follow Theorem 18 in Royden's Real Analysis book (fourth edition). It says the following:
Proof
The "preceding theorem" is Vitali's theorem which says that any set with positive outer measure co... | Any outer measure (like $m^*$) must satisfy countable subadditivity for all sets, in particular for $A$ and $B$, which is why $m^*(A\cup B)>m^*(A)+m^*(B)$ cannot occur. Regarding your first question, the equality for all sets means that the outer measure $m^*$ actually satisfies all the conditions of a measure, and mor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3320469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
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