Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Cesaro limit of analytic functions Let $f_n$ be a uniformly bounded sequence of analytic functions on $\Omega\subset\mathbb C$.
If $f_n(z)\to f(z)$ forall $z\in\Omega$, then by the Montel's theorem I know that the convergence is uniform on compact sets, hence $f$ is an analytic function.
Assume instead only that
$$ \fr... | Yes, The sequence $T_n (z) =\frac{1}{n} \sum_{j=1}^n f_j (z)$ is also uniformly bounded sequence of analytic functions.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Help understanding Weyl's proof of Heisenberg's Inequality https://www.math.unl.edu/~scohn1/8423/heisenberg.pdf
Can someone help me understand how they go from line to line in this proof? I'm confused by it. The proof is on the last page. Specifically how do they go from $$\left(\int \left\vert xf^*(x)f'(x)\right\vert\... | For the first question, note that $\frac{1}{2}\frac{d}{dx}|f|^2 = ff'$. For the second question, use the Plancherel Theorem.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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is there a relation between the fact that the derivative of x^2 is 2x and that the difference between 1,4,9,16, ... is 3, 5, 7, 9, ...? Is there a relation between the fact that the derivative of x^2 is 2x and that the difference between 1,4,9,16, ... is 3, 5, 7, 9, ...?
And why is the difference always 2?
I think the... | Yes. The derivative of $x^2$ tells you how fast the function $x^2$ increases. The second derivative of $x^2$ tells you how fast the derivative of $x^2$ increases. The second derivative is 2, meaning that the speed at which the rate of growth of $x^2$ increases is fixed. Therefore, the difference of the difference of co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1560173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Power series expansion of $x\ln(\sqrt{4+x^2}-x)$
Find $a_n $ where $x \ln(\sqrt{4+x^2}-x) =\sum_{n=0}^{\infty} a_nx^n$.
I know that I must find power series expansion of $\ln(\sqrt{4+x^2})$ but it doesn't help. Can anyone give me a hint? many thanks
| The factor $x$ can be momentarily disregarded; consider $f(x)=\ln(\sqrt{4+x^2}-x)$ and note that
$$
f'(x)=\frac{\dfrac{x}{\sqrt{4+x^2}}-1}{\sqrt{4+x^2}-x}=-\frac{1}{\sqrt{4+x^2}}=-\frac{1}{2}(1+(x/2)^2)^{-1/2}
$$
The Taylor development of the derivative can be written down. Integrate and multiply by $x$.
$$
f'(x)=-\fra... | {
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Mean of overcooking time This question came up this week when I had to put my rice in the microwave for a third time.
Suppose the perfect cooking time for a meal is given by a random variable $X$ with values in seconds. Now suppose a quick check allows to determine if the food is :
*
*Uncooked,
*Perfectly cooked,
... | This is not a complete answer, but it is too long to fit in the comments. (No reference here to a similar sentence by a chap named Fermat.)
Let $\{ t_n \}_{n=1}^N$ be the sequence of times at which you plan to stop and check, where $N$ may be finite or infinite. In your question, for instance, this is defined recursive... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Parametrization of a cylinder that is parallel to x axis The answer is no it does not matter.
The surface is $y^2+z^2=4$, I parametrized it so:
$\mathbf r=x \mathbf i +2\cos\theta \mathbf j + 2\sin\theta \mathbf k$
But Pauls Outline works through the problem with the j and k components switched. Does it matter which wa... | It really doesn't matter. Thinking of a similar problem in two dimensions, both
$$\cos\theta\,\mathbf{i} + \sin\theta\,\mathbf{j}\text{ and }
\sin\theta\,\mathbf{i} + \cos\theta\,\mathbf{j}$$
parametrize the unit circle, but in different ways.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Indefinite Integral $\int\frac{ \sin^3 2x-x \cos^3 x}{\cos^2 x}~dx $ How can I calculate $\displaystyle \int\frac{ \sin^3 2x-x \cos^3 x}{\cos^2 x} dx$? Help me I'm quite blur and how should I start. A little show on working are gladly appreciated.
| Hint:
Expand and arrive at (which should prove easier)
$$ \int 8 \sin^3(x) \cos(x)~dx-\int x \cos(x) ~dx$$
| {
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Normed Vector Space Let $V$ be a real normed vector space.
Suppose that $A$ is an open set from $V$. Show that the set $\frac12 A = \left\{ \frac12 x \, : \, x \in A \right\}$ is also open.
Let $V$ be a complex vector space. A norm on $V$ is a function $|| \cdot || : V \to R$ that satisfies the following three conditio... | Let $x \in \dfrac12A$, then $2x \in A$. Since $A$ is open, $\exists\epsilon>0$ such that the open neighbourhood $B(2x,\epsilon)\subseteq A \iff B(x,\dfrac\epsilon 2) \subseteq \dfrac12A$, so $\dfrac12A$ is open, where the open ball $B(x,a)$ denotes the set $\{y \in V:\lVert y-x\rVert<a\}$ for any $a\in \Bbb R$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1560716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sketching $y=a^x-a^{2x}, a>0$ When I need to sketch: $y=a^x-a^{2x}, a>0$, do I need to graph two functions?
One when $0<a<1$ and the other when $a>1$?
I didn't get any difference in the ascending descending intervals.
Because $y'=lna(a^x-2a^{2x})$ so y'=0
when $x=log_a0.5$ and $y''(log_a0.5)<0$
So it has a max point.
... | Yes, the behavior of the function differs in the cases $0<a<1$ and $a>1$. To see why, consider the function
$$
f_a(x)=a^x-a^{2x}
$$
and note that
$$
f_{1/a}(x)=f_a(-x)
$$
For instance, the graph you get for $a=1/2$ is symmetric with respect to the $y$-axis to the graph for $a=2$.
However, after noting this, you can avo... | {
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maximum Difference between two zeros What is the maximum difference between the two consecutive zeros of the solutions of $y''+(1+x)y=0$ on $0\leq x<+\infty$? I have applied the Strum's comparison theorem (by comparison with $y''+y=0$), and I obtained this maximum as $2\pi$ but I think it may be $\pi$. Is it necessaril... | Equation $$y''+y=0$$
has a general solution
$$y=a\cos t + b\sin t$$
Which obtain zero at
$$t=\mathrm{atan2}\left(\pm\frac{a}{\sqrt{a^2+b^2}},\mp\frac{b}{\sqrt{a^2+b^2}}\right)=\begin{cases}
\arctan\frac{b}{\pm a}&\pm a>0\\
-\pi+\arctan\frac{b}{a}&\pm a<0,\mp b<0\\
\pi+\arctan\frac{b}{a}&\pm a<0,\mp b>0\\
\end{cases}$$... | {
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Obtaining the Fundamental Polygon of $\mathbb{R}P^2$ On this page, Wikipedia shows, under the "Examples" heading, the fundamental polygons of the Sphere and the Real Projective Plane. Can we obtain the latter diagram from the former? I thought that this might be possible since $\mathbb{R}P^2$ is defined as the quotient... | It suffices to convince yourself the diagram below represents the antipodal map, with the shaded triangle as fundamental domain.
On the subject, there's a nice way to visualize antipodal identification of a cuboctahedron (a polyhedral model of a sphere) yielding a tetrahemihexahedron (a polyhedral model of a real proj... | {
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Multiple eigenvectors for an eigenvalue and how to know We have the following matrix:
$$\begin{pmatrix}
0 & -1 & 0 \\
4 & 4 & 0 \\
2 & 1 & 2
\end{pmatrix} $$
This matrix has characteristic equation $- \lambda ^3 + 6 \lambda ^2 - 12 \lambda + 8$, which gives us the eigenvalue $\lambda = 2$. However, this eig... | Yes let $v=(x,y,z)^T$ an eigenvector for the given matrix $A$ associated to $2$ then we have
$$Av=2v$$
and if we solve the system of equations we find that $v$ is a linear combination of the two vectors given in your question.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1561112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Linear Algebra - seemingly incorrect result when looking for a basis
For the following matrix $$ A = \begin{pmatrix}
3 & 1 & 0 & 0 \\
-2 & 0 & 0 & 0 \\
-2 & -2 & 1 & 0 \\
-9 & -9 & 0 & -3
\end{pmatrix} $$
Find a basis for the eigenspace $E_{\lambda}(A)$ of each eigen value.
First step is to find the ... | You're missing an eigenvalue! I think you've made a typo in your characteristic polynomial, writing $-\lambda^4$ instead of $\lambda^4$. The characteristic polynomial factorizes to $(x-2)(x+3)(x-1)^2$, hence the eigenvalues are $-3$, $2$, and $1$. The vector you have found is indeed a basis for the 2-eigenspace, and y... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Adding and multiplying piecewise functions How do I add and multiply two piecewise functions?
$$
f(x)=
\begin{cases}
x+3 &\text{if }x<2\\
\dfrac{x+13}{3} &\text{if }x>2
\end{cases}
$$
$$
g(x)=
\begin{cases}
x-3 &\text{if }x<3\\
x-5 &\text{if }x>3
\end{cases}
$$
| It helps to think about the fact that you can only add and multiply stuff that actually exists (i.e. is defined):
We can only add and multiply these functions in places where they both exist at the same time, namely:
$x<2\\
2<x<3\\
x>3$
The final step is to check what the functions equal at each of these segments, the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1561455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Easy way of memorizing or quickly deriving summation formulas My math professor recently told us that she wants us to be familiar with summation notation. She says we have to have it mastered because we are starting integration next week. She gave us a bunch of formulas to memorize. I know I can simply memorize the lis... | I would like to share the way I ended up remembering these formulas. Most of them are geometric ways of remembering these summation formulas. I still like Raymond Manzoni answer, so I will leave that as my accepted answer! He really helped me on my test.
$\sum \:_{n=a}^b\left(C\right)=C\cdot \:\left(b-a+1\right)$: Thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1561570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 2,
"answer_id": 1
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What would this homotopy look like? I found a very weird definition of null-homotopy.
Definition.
Let $M$ be a $C^k$ surface which is a subset of $\mathbb{R}^3$. Let $\mathscr{R}$ be an open connected subset of $M$
Let $\gamma$ be a closed curve whose image is the boundary of $\mathscr{R}$
Let $\sigma$ be any closed c... | This is almost certainly a typo. Indeed, as written, every $\sigma$ is "null-homotopic", via the map $\Gamma(t,s)=\sigma(t(1-s))$. It should additionally say that $\Gamma(L,s)=x_0$ for all $s$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1561692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $S=((x,y) \in \mathbb{R}^2 : e^x+e^y\le 100$ and $x+y\geq 0)$ is compact I want to use the fact that a subset of $R^2$ is compact if it is exactly closed and bounded. For the closed part, I'm defining the continuous functions $f(x,y)=e^x+e^y$ and $g(x,y)=x+y$ and I'm saying
$S=A\cap B$, where
$A=((x,y)\in ... | You have that $e^x\leq e^x+e^y\leq 100$ so $x\leq\ln(100)$, and $y\leq\ln(100)$ too. As $x\geq -y\geq -\ln(100)$ and $y\geq -x\geq -\ln(100)$ you get $S\subset[-\ln(100),\ln(100)]^2$ which is bounded so is $S$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1561777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Solve for $f(x)=f'(x)$ without previous knowledge.
Solve for $f(x)=f'(x)$ without previous knowledge.
I know it is obviously $f(x)=e^x$, but could you prove this without knowing $\frac d {dx}e^x=e^x$?
And does there exist a $g(x)=g'(x)$ but $g(x)\ne f(x)$?
| $$
f = \frac{df}{dx}
$$
$$
dx = \frac{df} f
$$
$$ x + \text{constant} = \log_e |f|
$$
$$
e^x \cdot\text{positive constant} = |f|
$$
$$
e^x \cdot\text{constant} = f.
$$
(One must check separately that $f=0$ is a solution.)
Suppose there were some other solution $g$ (maybe equal to $0$ at some points and not at others? ... | {
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Prove $a^2+b^2\geq \frac{c^2}{2}$ and friends if $a+b\geq c\geq0$ Sorry for my inequality spam, but I got to prepare for my exams today :( Here's another:
Problem:
Prove$$a^2+b^2\geq \frac{c^2}{2}$$$$a^4+b^4\geq \frac{c^4}{8}$$$$a^8+b^8\geq \frac{c^8}{128}$$ if $a+b\geq c\geq0$
Attempt:
Working backwards:
$$a^2+b^2\g... | Using the information you supplied, you could also just square both sides of the original inequality to obtain $a^2+2ab+b^2 \geq c^2$. Divide the inequality by 2 and subtract $ab$ to the other side. Given the AM-GM, we know that $\frac{a^2+b^2}{2} \geq ab $. Adding this inequality to the previous one we obtain $2\fra... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $\sigma$ takes squares to squares, conclude $a < b$ implies $\sigma a < \sigma b$. Prove that any $\sigma \in $ Aut$(\mathbb{R/\mathbb{Q}})$ takes squares to squares and takes positive reals to positive reals. Conclude that $a < b$ implies $\sigma a < \sigma b$ ,$\forall a,b
\in R$.
proof: Suppose $\sigma: a → b$, ... | Instead of splitting $\sigma(q+b-b)$ as $\sigma(q+b)-\sigma(b)$, split it as $\sigma(q-b)+\sigma(b)=-\sigma(b-q)+\sigma(b)$. Since $b>q$, $\sigma(b-q)>0$, so you get $\sigma(q)<\sigma(b)$.
(Actually, you can do this without introducing $q$ at all! Just notice that $\sigma(b)=\sigma(a+b-a)=\sigma(a)+\sigma(b-a)$, and ... | {
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"timestamp": "2023-03-29T00:00:00",
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Area bounded between the curve $y=x^2 - 4x$ and $y= 2x/(x-3)$ I've determined the intersects to be $x = 0, 2, 5$ and that $\frac{2x}{x-3}$, denoted as $f(x)$, is above $x(x-4)$, denoted as $g(x)$, so to find the area, I'll need to find the integral from $0$ to $2$ of $f(x) - g(x)$. But I've been stuck for a while playi... | Notice, the area bounded by the curves from $x=0$ to $x=2$, is given as $$\int_{0}^{2}\left(\frac{2x}{x-3}-(x^2-4x)\right)\ dx$$
$$=\int_{0}^{2}\left(\frac{2(x-3)+6}{x-3}-x^2+4x\right)\ dx$$
$$=\int_{0}^{2}\left(2+\frac{6}{x-3}-x^2+4x\right)\ dx$$
$$=\left(2x+6\ln|x-3|-\frac{x^3}{3}+2x^2\right)_{0}^{2}$$
$$=4-\frac{8}{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $|G|=30$ and $|Z(G)|=5$, what is the structure of $ G/Z(G)$? The question is:
If $|G|=30$ and $|Z(G)|=5$, what is the structure of $G/Z(G)$?
I don't know what do we mean by 'structure' asked in the question. Please help.
| $|G|=30$, $|Z(G)|=5$. $|G/Z(G)|=6$
We know that all groups of order 6 are isomorphic to $S_3$ or $\mathbb{Z} /6\mathbb{Z}$.
well known result:
If $G/Z(G)$ is cyclic, then $G$ is abelian.
If $G/Z(G) \cong \mathbb{Z} /6\mathbb{Z} $ ,then $G$ is abelian which is contradiction to $|Z(G)|=5$ . Therefore $G/Z(G) \cong S_3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1562231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Image of a circle under conformal map $1/z$ The image of a circle under conformal map $1/z$ should be a circle, but how to prove it (or how to find the relationship between the two circles)?
$z = x + iy = d + a\exp(i\theta)$, where $a$ is the radius of the circle and $d$ is a real number, represents a circle with radiu... | The real points $d \pm a$ lie on a diameter of your circle. The image points $1/(d \pm a)$ lie on a diameter of the image circle, so the image circle has center
$$
\frac{1}{2}\left[\frac{1}{d - a} + \frac{1}{d + a}\right]
= \frac{d}{d^{2} - a^{2}}
$$
and radius
$$
\frac{1}{2}\left|\frac{1}{d - a} - \frac{1}{d + a}\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1562344",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Prove that $( 1 + n^{-2}) ^n \to 1$. I need to prove that $\left(1 + \frac 1 {n^2} \right)^n \to 1$.
I tried to use Bernoulli's inequality, but that is not very useful since in the original sequence there is a plus sign. I then tried to use the Sandwich Theorem by finding two sequences which would make bounds for the o... | Use $e^x \geq 1 + x$ for all $x \in \mathbb{R}$. There are many ways to see this. The easiest, in my opinion, is to note that $y = x + 1$ is tangent to $y = e^x$ and $x \mapsto e^x$ is convex, then use definition of convexity.
It follows that
$$
1 \leq
\left(1 + \frac{1}{n^2}\right)^n \leq
\left(\exp\frac{1}{n^2}\right... | {
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"question_score": "2",
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How to maximize $\frac 1 {PC}+\frac 1 {PB}$? Assume $P$ is a point inside the angle $\hat A$. How to draw a line that intersects with lines $a$ and $b$ AND maximizes $\frac 1 {PC}+\frac 1 {PB}$ ?
Here is the picture:
EDIT:
The point $P$ is given. the only thing changeable is the angle of the line passing the point $P$... | I claim that $AP \perp BC$.
We set $X \in AB$ such that $AC \parallel PX$, and $Y \in AP$ such that $XY \parallel BC$.
Now, we have $$\triangle AYX \sim \triangle APB$$ $$\triangle PYX \sim \triangle APC$$
This gives us $$\frac{XY}{PB}=\frac{AY}{AP}$$ $$\frac{XY}{PC}=\frac{YP}{AP}$$
Summing them, we have $$\frac{1}{PB}... | {
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"timestamp": "2023-03-29T00:00:00",
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A question on proof of the Zermelo's theorem "Every set is well-orderable." There exists some proof for the theorem. Some of them use Transfinite Recursion.
Some of them use same argument with the following.
Proof:(copied from proofwiki)
"Let $S$ be a set.
Let $\mathcal{P}(S)$ be the power set of $S$.
By the Axiom of C... | The process has to stop because of Hartogs theorem:
For every set $X$ there exists a well-ordered set $Y$ such that there is no injection from $Y$ into $X$.
Or, if you have the axiom of replacement, you can replace $Y$ by an ordinal and obtain the following:
For every set $X$ there exists a least ordinal $\alpha$ su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1562784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Use the change of variables to evaluate the double-integral $\iint_R4(x+y)e^{x-y}\;dA$
Use the change of variables indicated to evaluate the double-integral $\iint_R4(x+y)e^{x-y}\;dA$ where $R$ is the interior of the triangle whose vertices are $(-1,1)$, $(0,0)$, and $(1,1)$; $x=\frac{1}{2}(u+v)$ and $y=\frac{1}{2}(u-... | If I'm not mistaken:
Notice how
$$u = x +y \qquad \text{and} \qquad v = x-y.$$
Then notice how $v = \alpha$ implies parallell lines to the first bisector of the plane. This means $v = 0$ results in the first bisector.
Furthermore $u =0$ results in the second bisector.
All that is remaining is the straight line $y = 1$... | {
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"url": "https://math.stackexchange.com/questions/1562878",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Linear Algebra and Quadratic Equations I'm just wondering if Linear Algebra is concerned only with Linear equations? Can quadratic equations(or any higher power) also be considered under Linear Algebra? What does the term Linear stand for?
| Linear algebra very much included Quadratic equations. in fact. Polynomial equations of any degree. For example $ax^2+bx+c=0$ may be written as product of a row vector $(a, b, c)$ with a column vector $(x^2,x,1)^\top$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1563030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Monoids where $\operatorname{Hom}(M,M) \cong M$ What are some examples of monoids where $\operatorname{End}(M) \cong M$? Is there a nice characterization of such monoids? E.g., they will necessarily have a zero element, since $\forall x (x \mapsto 1_M)$ is a zero of $M^M$ (so no groups satisfy $G^G \cong G$).
Examples:... | Partial answer. If $M$ is a finite monoid such that $\operatorname{End}(M) \cong M$, then either $M = \{1\}$ or $M = \{0, 1\}$ with the usual multiplication of integers.
Proof. Let $G$ be the group of invertible elements of $M$. Since $M$ is finite, $M - G$ is an ideal of $M$. Let $E(M)$ be the set of idempotents of $M... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1563135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Proof limit $b^n n^{\alpha} \to 0$, when $n \to \infty$ and $\forall |b| < 1 $. $$\lim_{n \to \infty} b^n n ^{\alpha} = 0, \forall |b| < 1, \alpha > 0$$
I have proved it for cases when $\alpha \le 0$, but I am not sure where to go further. I have tried D'Alembert's test, but I believe that it does not work for $\alpha ... | Because the limit of $b^n$ is $0$, since $|b| < 1$ and $n$ approaches infinity
$$\lim_{n \to \infty} (b^n) (n^a) = \lim_{n \to \infty} (b^n) \cdot \lim_{n \to \infty}(n^a)$$
since limit of $b^n$ is $0$, the limit of the previous one is $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1563233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Expectation of normal distributions Let $X$ have a normal distribution with mean $µ$ and variance $σ^2$.
Find $E[X^3]$ (in terms of $µ$ and $σ^2$).
the pdf of this function is $$\frac{1}{σ\sqrt{2\pi}} e^{\frac{-(x-µ)^2}{2σ^2}}$$
so $$E[X^3] = \int_{-\infty}^\infty x^3\frac{1}{σ\sqrt{2\pi}} e^{\frac{-(x-µ)^2}{2σ^2}}... | Using the symmetry of $\frac{X-\mu}{\sigma}$, one can conclude that
$$
E\left(\frac{X-\mu}{\sigma}\right)^{2n+1}=0, \text{for}\quad n=0,1,2,...
$$
Hence, using $Var(X)=\sigma^2$ and $EX=\mu$, you can show that $E\left(\frac{X-\mu}{\sigma}\right)^{3}=\frac{EX^3-3\mu\sigma^2-\mu^3}{\sigma^3}=0$, thus
$$
EX^3 = 3\mu\sig... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Group Theory - Cartesian Product Given $G = C_3 \times C_2 \times C_4 \times C_5$ where $C_2 = \langle\{0,1\}, +2\rangle$; $C_3 = \langle\{0, 1, 2\}, +3\rangle$; and so on. Consider the group $(G, *)$, here the group operator $*$ is defined by the constituent groups in the Cartesian product.
i) Write an expression for... | A cartesian product $X \times Y$ consists of all ordered pairs $(x,y)$, where $x$ is in $X$ and $y \in Y$. So $x$ and $y$ don't have to be the same in any way. Similarly, $G = C_3 \times C_2 \times C_4 \times C_5$ should consist of all ordered quadruples $(t_1, t_2, t_3, t_4)$, where $t_1 \in C_3, t_2 \in C_2, t_3 \i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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If the difference of the function only dependent on the difference of input, can we say it's linear without assuming it's continuous? As state in the title, if for a function $$f(t+\tau)-f(t)=R(\tau), \forall t,\tau$$
$R(\tau)$ is just a general function showing the difference has nothing to do with $t$.
then I can pro... | Yes, measurability implies that $f$ is affine. Proof: Let $g(t) = f(t) - f(0)$. Then $g$ is additive, because $$ g(a + b) = f(a + b) - f(0) = f(a) + R(b) - f(0),$$ and $R(b) = f(0 + b) - f(0)$, so $g(a + b) = f(a) + f(b) - 2f(0) = g(a) + g(b)$. Furthermore, $g$ is measurable. Every additive, measurable function is line... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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To evaluate limit $\lim_{n \to \infty} (n+1)\int_0^1x^{n}f(x)dx$
Let $f:\Bbb{R} \rightarrow \Bbb{R}$ be a differentiable function whose derivative is continuous, then
$$\lim_{n \to \infty} \left((n+1)\int_0^1x^{n}f(x)dx\right).$$
I think I have to use L'Hop rule, but I don't see how.
| We can prove that if $f$ is continuous function ($f$ being differentiable function is not needed)
$$
\lim_{n \to \infty} (n+1)\int_0^1x^{n}f(x)dx=f(1)
$$
Since $f$ is continuous, $f$ is bounded on $[0,1]$. Also
$$
\forall \epsilon>0, \exists \delta>0, \forall x\in(1-\delta,1],\text{there is } |f(x)-f(1)|<\epsilon
$$
\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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If $f''+f'=f$ then $f\equiv 0$ Let , $f:\mathbb R\to \mathbb R$ be a two times continuously differentiable function that satisfies the differential equation $f''+f'=f$ for all $x\in [0,1]$. If $f(0)=f(1)=0$ then , which is correct ?
(A) There exists $(a,b)\subset [0,1]$such that $f(x)>0$ in $(a,b)$.
(B) There exists $... | The function, which is continuous on $[0,1]$, must have a maximum point $x$. In the maximum point (if $x\in(0,1)$) you have: $f'(x)=0$ and $f''(x)\le 0$. So the equation says $f(x) = f''(x) \le 0$. With a similar reasoning you find that on the minimum you have $f(x)\ge 0$. Hence the function is identically zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1563775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Does showing term-by-term differentiation of a series via uniform convergence imply that the series defines an infinitely differentiable function? If I show that
$$\sum 2^{n\alpha} \sin\left(\frac{x}{\alpha}\right)$$
converges uniformly by the M-test, for all x in $R$, then we may differentiate term-by-term everywhere.... | 1) This is not a power series.
2) It is not enough to show that the series converges uniformly with each summand being differentiable.
3) It is enough to show that the series of derivatives converges uniformly to show that the limit function is differentiable.
More precisely, if each $f_n : I \to \Bbb{R}$ is $C^1$, if ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Finding the extremal point of a function I have to find and classify all extremal points of a few problems and I am finding it difficult as I don't know how to even start. I can't find any examples of what to do. So if someone could help with one question then I can use that to figure out the rest. One of the questions... | In general, when having real functions $ f: \mathbb{R}^m \rightarrow \mathbb{R}^n $, which are continuous and twice differentiable, to find their extreme values, you may follow the following strategy:
*
*Find the gradient of $f$, and study its critical points (i.e. points where $\nabla f =0$). Note, it may happen t... | {
"language": "en",
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Finding the sum of n terms $S_n$ starting from sigma $k=0$ $$\sum_{k=0}^{n} ((4k-3)\cdot 2^k)+4=(2^{n+3}+4)n-7\cdot2^{n+1}+15$$
How? I've tried everything but i don't see it. Any equivalent solutions are also welcome, thanks.
| You can exploit a telescoping sum:
$$
\sum_{k=0}^n\bigl((k+1)2^{k+1}-k2^k\bigr)=(n+1)2^{n+1},
$$
because you also have:
$$
\sum_{k=0}^n\bigl((k+1)2^{k+1}-k2^k\bigr)=
\sum_{k=0}^n\bigl(k2^k-2^{k+1}\bigr)
$$
so that
$$
\sum_{k=0}^n k2^k=
\sum_{k=0}^n\bigl((k+1)2^{k+1}-k2^k\bigr)+\sum_{k=0}^n2^{k+1}
=(n+1)2^{n+1}+2(2^{n+... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Series' convergence - making my ideas formal
Find the collection of all $x \in \mathbb{R}$ for which the series $\displaystyle \sum_{n=1}^\infty (3^n + n)\cdot x^n$ converges.
My first step was the use the ratio test:
$$ \lim_{n \to \infty} \dfrac{(3^{n+1}+n+1) \cdot |x|^{n+1}}{(3n+n) \cdot |x|^n} = \lim_{n \to \inf... | The series $\displaystyle \sum_{n=1}^\infty (3^n + n)\cdot x^n$ converges if $|x|<\frac{1}{3}$
Indeed $$(3^n + n)\cdot x^n\sim_{\infty} 3^n \cdot x^n$$ but this is a geometric series that converges only if $|x|<\frac {1}{3}$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Exercise 1.2.19 of Hatcher's Algebraic Topology I've been trying to prove exercise $1.2.19$ of Hatcher's algebraic topology:
Show that the subspace of $\mathbb{R}^3$ that is the union of the spheres of radius $\frac{1}{n}$ and center $(\frac{1}{n}, 0, 0)$ for $n = 1, 2, ···$ is simply-connected.
I was thinking of app... | Let $S_n$ be the sphere of radius $1/n$, and let $U_n$ be an open neighborhood of the origin $(0,0,0)$ within $S_n$. Then we can cover the space $X$ by $V_n=(\bigcup_{n=1}^\infty U_n)\cup S_n$ for $n=1,2,\cdots$. Since $V_n$ are path-connected and have trivial fundamental groups, the intersection of any two or more $V_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 1
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There are 31 houses on north street numbered from 1 to 57. Show at least two of them have consecutive numbers. I thought to use the pigeon hole principle but besides that not sure how to solve.
| There is a unique subset with the maximum number of non-adjacent houses: {1,3,5,...,57}. The number of houses (29) in this set is less than 30.
This observation solves the problem, but it also indicates that with 30 houses there are at least 2 adjacencies. With only 1 adjacency $(i-1,i)$ you could add $+1$ to all the... | {
"language": "en",
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Find $\lim_{x\to0}\frac{\sin\left(1-\frac{\sin(x)}{x}\right)}{x^2}$. Is my approach correct? Find:
$$
L = \lim_{x\to0}\frac{\sin\left(1-\frac{\sin(x)}{x}\right)}{x^2}
$$
My approach:
Because of the fact that the above limit is evaluated as $\frac{0}{0}$, we might want to try the De L' Hospital rule, but that would lead... | By L' Hospital anyway:
$$\frac{\sin\left(1-\frac{\sin(x)}{x}\right)}{x^2}$$ yields
$$\cos\left(1-\frac{\sin(x)}x\right)\frac{\sin(x)-x\cos(x)}{2x^3}.$$
The first factor has limit $1$ and can be ignored.
Then with L'Hospital again:
$$\frac{x\sin(x)}{6x^2},$$
which clearly tends to $\dfrac16$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1564729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Cauchy sequence in $L^1$ Space I am learning about the $L^1$ space (the complete Riemann integrable functions) and I am not used to using $\epsilon, \delta$ in these type of problems yet. Here is my attempt. Below I want to that $f_n$ is a Cauchy sequence.
Let $\epsilon > 0$ be given.
$$
\int |f_n - f_m| = \int_{1... | Let $f_n = x^{-1/2}\chi_{[1/n, 1]}$.
Let $\epsilon >0$. Choose $N>\frac{4}{\epsilon^2}$ such that when $n,m > N$, then
$$\begin{align}\int |f_n - f_m| &= \int |x^{-1/2}\chi_{[1/n, 1]}-x^{-1/2}\chi_{[1/m, 1]}| \\
&= \int_{1/n}^{1/m} x^{-1/2} dx \\
&= 2 \cdot \left[\left(\frac{1}{m}\right)^{1/2} - \left(\frac{1}{n}\right... | {
"language": "en",
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Math induction problem with large numbers I am trying to figure out how to prove $17^{200} - 1$ is a multiple of $10$. I am talking simple algebra stuff once everything is set in place.
I have to use mathematical induction.
I figure I need to split $17^{200}$ into something like $(17^{40})^5 - 1$ and have it as $n = 17... | Since it seems a bit strange to use induction to solve for particular case ($17^{200} - 1$), and it seems from your question that you want to see this solved via an inductive proof, let's use induction to solve a somewhat more general problem, and recover this particular example as a special case.
So, let's make the co... | {
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"source": "stackexchange",
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If $\lambda$ is an eigenvalue of $A$, determine eigenvalues of $A^2$ and $A^3$ Also what is an eigenvalue of $A^n$?
I know you can square the eigenvalues, etc... but how do I prove this for a general case?
| If $p(x)$ is a rational function defined on the spectrum of $A$ then the spectrum $$\sigma(p(A))=p(\sigma(A))$$
The spectrum being the set with all the eigenvalues or in a more precise way $\sigma(A)$={ $\lambda I-A$ is not invertible}.
Such definition is because this theorem comes from operator theory where in princi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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the first player to win three games in a row or a total of four games wins. In a competition between players X and Y, the first player to win three games in a row or a total
of four games wins.
a. How many ways can the competition be played in total?
b. How many ways can the c... | Suppose four games are played and one player wins them all. WWWW. There are 2 ways that can happen. (Either one player wins all four or the other player does.
Suppose five games are played and one player winning 4 and losing one. LWWW or WLWW or WWLW. The winner can't lose the last game (because then the winner wo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Correlation Coefficient I am trying to understand the following equation for Correlation Coefficient:
$r = \frac{\sum_{i=1}^{n}(x_i-\bar x)(y_i-\bar y)}{\sqrt(\sum_{i=1}^{n}(x_i - \bar x)^2\sum_{i=1}^{n}(y_i-y)^2)}$
Can someone dissect this equation and provide reasoning as to why this equation does what it does, pro... | Let $\mathbf{z}$ be a vector in $n$-dimensional space (think in two-dimensions, $n=2$, if that is easier). In terms of a given coordinate system with unit vectors $(\mathbf{e}_1, \mathbf{e}_2, \dots, \mathbf{e}_n$), the vector can be expressed as the sum of its components
$$
\mathbf{z} = z_1\mathbf{e}_1 + z_2\mathbf... | {
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$H$ is a finite abelian of order $k$, also $q \in Z^{+}$ with $(k,q)=1$. $\phi:H \rightarrow H$ by $\phi(h)=h^q,\forall h \in H$ belongs to $Aut(G)$. $H$ is a finite abelian group of order $k$, also $q \in Z^{+}$ with $(k,q)=1$. $\phi:H \rightarrow H$ by $\phi(h)=h^q,\forall h \in H$ belongs to $Aut(G)$.
What I did so ... | The condition that $H$ is abelian is used in proving that $H$ is a homomorphism.
Since $(k,q) = 1$, $\exists x,y \in \Bbb Z$ such that $qx + ky = 1$.
If $\phi(h_{1})=\phi(h_{2})$, then $h_{1}^q=h_{2}^q$.
Since $|H| = k$, $h_1^k = h_2^k = e$.
$$h_1 = h_1^{qx+ky}=h_1^{qx}=h_2^{qx} = h_2^{qx+ky}=h_2.$$
Similarly, surjecti... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Multiplicity of 0 eigenvalue of directed graph Laplacian matrix I am looking for a result (if it exists) for directed graphs relating the multiplicity of 0 eigenvalues of the directed Laplacian matrix.
Consider a directed graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ and define the in-degree Laplacian as $L_{in}(\mathc... | I'm a few years late but I think tst conjecture is correct, according to https://arxiv.org/pdf/2002.02605.pdf (which also cites other references).
They use the notion of Reach :
Definition 2.3 :
i) Let $i\in V$. The reachable set $R(i)$ consists of all $j\in V$ such that there exists a path from $i$ to $j$
ii) A ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Sum of all values of $b$ if the difference between the largest and smallest values of the function $f(x)=x^2-2bx+1$ in the segment $[0,1]$ is $4$ Find sum of all possible values of the parameter $b$ if the difference between the largest and smallest values of the function $f(x)=x^2-2bx+1$ in the segment $[0,1]$ is $4$.... | Since $f(x) = (x-b)^2 + (1 - b^2)$, the vertex will be at $(b, 1-b^2)$.
Also $f(1) = 2-2b$ and $f(0) = 1$.
When the vertex is at or to the left of $x=0$, then $f(x)$ is increasing on $[0, 1]$. So
$$ \max_{x \in [0,1]}f(x) - \min_{x \in [0,1]}f(x) = f(1) - f(0) = 1-2b$$
and $1-2b=4$ when $b = -\frac 32$
When the vertex ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Group of order $255$ is cyclic
Let $G$ a group and its order is $255$.
Prove that $G$ is cyclic.
I easily demonstrated that the group has only one $17$-Sylow subgroup $P$ that is normal in $G$ and it's cyclic since it is of a prime order. Then $G/P$ is also cyclic since a group of order $15$ is cyclic. Then $G$ can... | Yes I think it is correct. Because of the orders of all elements must divide to the order of the group and gcd(15,17)=1 you can prove that all the group is generated by only one element (by the same way you prove that a group of order 15 is cyclic)
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Series of functions convergence: $a_n \sin\left(\frac 1{tn}\right),\, t\geq 2/\pi$ Problem: If the partial sums of $\sum_{n=1}^{\infty} a_n$ are bounded, then prove that $\sum_{n=1}^{\infty} a_n \sin\left(\frac{1}{tn}\right)$ converges for $t\geq\frac{2}{\pi}$.
I'm trying to use Dirichlet's Test because the partial sum... | Hint: Show that if $\pi/2 \ge x_1 \ge x_2 \ge \cdots \to 0,$ then $\sin x_n$ decreases to $0.$ Apply this to $x_n = 1/(tn).$
| {
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Prove the uniqueness of the Markov distribution Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. I want to prove the uniqueness of a probability distribution under which a process $(X_n)_{n\geq0}$ is a Markov chain with transition probabilities $P$ and initial distribution $\mu_0$.
As an hint I have that I... | The phrase "the sigma algebra of the probability space where my Markov chain is defined" is pretty vague. Specifically, you need to use the $\sigma$-algebra generated by the process, i.e., ${\cal F}=\sigma(X_0,X_1,\dots).$ A generating set $\cal D$ for $\cal F$ is the collection of all sets of the form $$\bigcap_{j=0}^... | {
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Galois Field to bits, implementation is fine, but the mathematics is not. I am working on a hardware implementation of the SIMON cipher and the key expansion is based on GF(2). The original paper is here, https://eprint.iacr.org/2013/404 I have successfully created the hardware, but the fact that I cannot grasp the m... | Of course, after writing this up, I came up with a resolution. Via Matlab:
W=[0 1 0 0 0; 0 0 1 0 0; 1 0 0 1 0; 0 0 0 0 1; 1 0 0 0 0 ]
A_o=[0;0;0;0;1];
for i=1:31
A_o=W*A_o;
A_o=mod(A_o,2)
end
The result is that the MSB matches the bitstream.
| {
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If $a_{n+1}=\frac {a_n^2+5} {a_{n-1}}$ then $a_{n+1}=Sa_n+Ta_{n-1}$ for some $S,T\in \Bbb Z$. Question
Let $$a_{n+1}:=\frac {a_n^2+5} {a_{n-1}},\, a_0=2,a_1=3$$
Prove that there exists integers $S,T$ such that $a_{n+1}=Sa_n+Ta_{n-1}$.
Attempt
I calculated the first few values of $a_n$: $a_2=7,a_3=18, a_4=47$ so I'd ... | The result is $a_{n + 1} = 3 a_n - a_{n - 1}$.
HINT: Let $a_{k + 1} = a_k + b_k$. Then, by the original equation, we get $$a_{n - 1} + b_{n - 1} + b_n = \frac {(a_{n - 1} + b_{n - 1})^2 + 5} {a_{n - 1}} = a_{n - 1} + 2 b_{n - 1} + \frac {b_{n - 1}^2 + 5} {a_{n - 1}}.$$ Hence, $$b_n = b_{n - 1} + \frac {b_{n - 1}^2 + 5}... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Combinatorics - how many possible solutions are there for: $|x_1| + x_2+x_3 = 16$ How many possible solutions are there for this equation:
$|x_1| + x_2+x_3 = 16$ ; $x_1 \in Z$ $x_2,x_3 \in N$
I know it's a simple combinatorics question but I'm still having trouble figuring it out.
Since $x_1$ is an integer, there could... | If $x_1 = 0$ then there are $15$ pairs for $x_2, x_3$.
If $x_1 = \pm 1$ then there are $14$ pairs.
If $x_1 = \pm 2$ then there are $13$ pairs.
if $x_1 = \pm 13$ then there are $2$ pairs.
if $x_1 = \pm 14$ then there is $1$ pair.
So it's a stars and bars problem for the second part. Each bin contains at least one of th... | {
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$A_4$ is not the direct product of its sylow subgroups? This is an exercise in Hungerford. I've tried proving this as follows (forgetting for the moment that in $A_4$, $n_2=1$ and $n_3=4$):
By Sylow III, $n_2=1$ or $3$.
If $n_2=3$, the number of elements of order $2$ in $A_4$ is $n_2 (2-1)=3(2-1)=3.$ Then the number of... | There are many incorrect statements in your proof.
Anyway, this is beside the point as you can prove this statement quickly without counting Sylow subgroups, or doing any other hard analysis:
Assume $A_4$ is a direct product of its Sylow subgroups. The Sylow 2-subgroups of $A_4$ have order 4, hence are abelian. Simila... | {
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Few group theory questions I am trying to solve the following;
First, given $G$ is a group and $H$ a subgroup of $G$, what can we say about the relation $a \cong b$ if $b^{-1}a \in H$
I can show that it is reflexive as the identity is always in the subgroup.
if $a \cong b$ then $b^{-1}a \in H$ and so $(b^{-1}a)^{-1}=a... | *
*Correct, commutativity is not needed and not used in your proof.
*The group operation is rather addition.
Hint: Where can the equivalence class $[1]_{n\Bbb Z}$ be mapped by a homomorphism to $\Bbb Z$?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$\lim\limits_{n\to\infty}$ $(1+ {2\over n})^n$ = $e^2$ If I set N=$2\over n$, the equation becomes $(1+ N)^{2\over N}$, which I can take the natural log of and work down until I reach $e^2$. However, my teacher wants me to use subsequences, starting with $x_n$ = $(1+ {1\over n})^n = e$, to prove this and I am stuck.
| You could rewrite this as
$$\lim_{n \to \infty}\left(1+\frac{1}{n/2}\right)^n = \lim_{n \to \infty}\left(\left(1+\frac{1}{n/2}\right)^{n/2}\right)^2 = \lim_{n\to\infty} (x_n)^2$$
where $x_n=\left(1+\frac{1}{n/2}\right)^{n/2}$. Now as $n \to \infty$, $\frac{n}{2} \to \infty$ as well. Hence $\lim_{n\to\infty} x_n = e$. ... | {
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Fake proof that there don't exist complicated numbers So there's this false proof going around that I can't seem to find now that says that complicated numbers don't exist. So let me explain what it's about (I've added some technical details of my own, but the idea is the same).
Definition A number $a \in \mathbb{N}$ i... | This paradox is called the Berry paradox. The problem is that you haven't been specific enough about what it means to express a number using ASCII characters. Once you fix the meaning of "express," the first proof becomes a proof by contradiction which shows that "the first complicated number" is not a well-defined exp... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove the set $\{x \in \mathbb{R} | f(x) > 0\}$ is open Let $f: \mathbb{R} \to \mathbb{R}$ be continuous. I need to use the definition to prove that the set where $f(x) > 0$ is open. I think it's true that a set defined by $f(x)$ is closed if it contains its limit points. I'm not sure what do with the proof however.
| Approach 1: pick a point $x_0$ in $\{x \mid f(x)>0\}$. By the definition of continuity, some small neighborhood around $x_0$, say, $(x_0-\delta,x_0+\delta)$ is mapped by $f$ to a $(f(x)/2, 3f(x)/2) \subset (0,\infty)$. Thus, $(x_0-\delta,x_0+\delta) \subset \{x \mid f(x)>0\}$.
Approach 2: equivalently we can show the c... | {
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Martingale related question Let $S_n = Z_1 + · · · + Z_n$ where ${Z_k}$ are i.i.d. $N (0, 1)$ variables. Find constants $c_n$ such that $M_n = e^{S_n+c_n}$ is a martingale. How can I show using martingale theory that there exists a finite random variable $M_{\infty}$ such that $M_n → M_{\infty}$ a.s.
| $M_n$ is nonnegative. Hence $\lim M_n$ exists a.s.
From Probability with Martingales:
| {
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Double integral over annulus
Evaluate $$\iint_D\ e^{-x^2 - y^2}\ dA,$$
where $D$ is annulus $a \le x^2 + y^2 \le b$
My understanding is it involves polar coordinates but I don't understand how to convert it.
| Since $r^2 = x^2 + y^2$ (do you see why?), we're considering the annulus $\sqrt{a} \le r \le \sqrt{b}$. Then the integral is
$$\iint_D\ e^{-r^2}\ dA,$$
The area element is $dA = r\ dr\ d\theta$ in polar coordinates. Now what do you make the limits of integration?
| {
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Show $f$ is constant zero function on $[a, b]$ if $\int_a^x f(t) dt = \int_x^b f(t) dt \forall x \in [a, b].$ I've started by trying to use the Fundamental Theorem of Calculus and have that
$$\int_a^x f(t) dt - \int_x^b f(t) dt = 0$$
$$= F(x) - F(a) - (F(b) - F(x))$$
$$0 = 2F(x) - F(a) - F(b)$$
but am not sure how to p... | Just differentiate both sides to get:
$$f(x)=-f(x)$$
for all $x$, then $f(x)=0$ for all $x$.
Recall that the fundamental theorem of calculus tells you that for $f(x)$ continuous:
$$\frac{d}{dx}\int_a^x f(t) dt=f(x)$$
Then note that:
$$\int_x^b f(t) dt=-\int_b^x f(t) dt$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Describe $\text{Jac}(n\mathbb Z)$ in terms of the prime factorization of $n$ Let $I$ be an ideal of the commutative ring $R$ and define
$\mathrm{Jac}(I)$ to be the intersection of all maximal ideals of $R$ that contain $I$.
Let $n > 1$ be an integer. Describe $\text{Jac}(n\mathbb Z)$ in terms of the prime factorizat... | Hints :
*
*In $\mathbb Z$, an ideal $I\ne ${$0$} is a prime ideal if and only if it is a maximal ideal, if and only if $I=p\mathbb Z$ for some prime number $p$.
*An ideal $I=m\mathbb Z$ in $\mathbb Z$ contains $n\mathbb Z$, if and only if $m|n$
*Jac $n\mathbb Z=m\mathbb Z$, where $m$ is the product of the prime ... | {
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How to take an integral using half angle trigonometric substitution. So i have this question which is asking to take the integral using a predefined trigonometric substitution which is $$u=\tan\frac{x}{2}$$
and the integral equation is $$\int\frac{\sin x\ dx}{(6\cos x-2)(3-2\sin x)}$$ How would i go on about this probl... | We have $$x=2arctan(u)$$, that gives $$dx=\frac{2}{u^2+1}du$$
Furthermore, we have $$cos(x)=\frac{1-u^2}{1+u^2}$$
and $$sin(x)=\frac{2u}{u^2+1}$$
Inserting those terms, you get a rational function of $u$. Try it from here.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is weak topology considered weak while others are strong. I know for a banach space $\mathbf{X}$ that the norm $||\cdot||$ produces a topology for the space. I also know this is considered "strong". I also understand that the dual, $\mathbf{X}^\ast$, is the set of all continuous linear functionals. I also understan... | We speak of weak/strong topologies and also of coarse/fine topologies. Either metaphor relates to the same relationship.
A topology on a set $X$ is a subset $\tau$ of $\mathfrak{P}X$ - the power set of $X$. To qualify as a topology the collection $\tau$ must satisfy the usual axioms.
If $\sigma$ is another topology on ... | {
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Optimization of Rectangle A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing \$50 per foot and on the other three sides by a metal fence costing \$10 per foot. If the area is 24 square feet, what are the dimensions of the garden that minimize the cost? (Let x= side with bri... | Let $C(x)$ be the the cost of covering the garden of a side $x$. Since $y=24/x$ (by area), $C(x)=50x+10x+2(10\frac{24}{x})=60x+\frac{480}{x}$. Since $0<x<240$, our restriction will be $x\in(0,240)$. To find for critical number/s,
$$C'(x)=60-\frac{480}{x^2}=0\ (or\ undefined)$$
$$x^2=\frac{480}{60}=8$$
$$\therefore x=2\... | {
"language": "en",
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Why is this Conditional Density Function correct? This answered question shows how to solve the problem but I still do not understand how to get the conditional density function, i.e.
${"}$Let $Z=X+Y$, then the density $f_{X,Z}$ of $(X,Z)$ is defined by $f_{X,Z}(x,z)=f_X(x)f_Y(z-x)$ because $X$ and $Y$ are independent ... | The general formula for conditional density is
$$
f_{X\mid Z}(x\mid z) = {f_{X,Z}(x,z)\over f_Z(x)}.\tag1
$$
Your error is in replacing the numerator in (1) with
$$f_{X,Z}(x,z)=f_X(x)f_Z(z),
$$
which is true only if $X$ and $Z$ are independent.
What you should do is replace the numerator in (1) with
$$
f_{X,Z}(x,z)=f_X... | {
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Prove that $e^\frac{x+y}{2} < \frac{1}{2}(e^x+e^y)$ for $x\neq y$ Should I use the taylor series expansion of the exponential function?
| In general if $f''>0$ and $x>y$ then there exist $a\in ([x+y]/2,x)$ and $b\in (y,[x+y]/2)$ and $c\in (a.b)$ such that $$[f(x)+f(y)]/2 -f([x+y]/2)=$$ $$[f(x)-f([x+y]/2)]/2-[f([x+y]/2)-f(y)]/2=$$ $$=[(x-y)f'(a)]/4-[(x-y)f'(b)]/4=(x-y)^2f''(c)/4>0.$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 1
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Finding the value of Inverse Trigonometric functions beyond their Real Domain I wanted to ask how can we calculate the values of the inverse of trigonometric functions beyond their domain of definition, for example $\arcsin{2}$ beyond its domain of $-\frac{\pi}{2}<x<\frac{\pi}{2}$.
I tried to use the Euler form but did... | @Neelesh Vij, he makes an interesting point. I will do my best to explain.
We have as follows:$$\sin(z)=x, z=\arcsin(x)$$$$e^{iz}=\cos(z)+i\sin(z)$$$$e^{-iz} =\cos(-z)+i\sin(-z)=\cos(z)-i\sin(z)$$And we are trying to solve for an unknown z, given x.
Made clear, the last line has the equation $e^{-iz}=\cos(z)-i\sin(z)$... | {
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Trig : With point $(X,Y)$ and a end point $(x_2,y_2)$, work out the 2 angles So imagine your arm, you have a shoulder point be $X,Y (0,0)$ you have a angles off this, then you have a elbow point $(x_1,y_1)$ and another angles which ends at your hand.
If you know the arm length (4 and 4), you know your starting point $(... | Draw a line from the origin to the hand, called the distance line, then you have (based on your image, I assumed $y = -2$):
$$
\begin{eqnarray}
l &=& 4 & \textrm{length of an arm segment} \\
d &=& \sqrt{x^2 + y^2} = 2\sqrt{10} & \textrm{distance from the origin to the hand} \\
\tan \theta_1 &=& \frac{y}{x} = -\frac{1}{... | {
"language": "en",
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Is it possible to simplify $3\sin(z)\cos^2(z)-\sin^3(z)$? I have to write $\sin(3z)$ only with $\sin$'s and $\cos$'s. By using the additions-theorem I obtained $3\sin(z)\cos^2(z)-\sin^3(z)$ and I don't know If I can continue simplifying it.
| It's not completely clear what "simplify" means in this context -- very often, two equivalent trigonometric formulas can be equally "simple" -- but one thing you can do is replace $\cos^2(z)$ with $1-\sin^2(z)$. Then you'll have
$$3\sin(z)(1-\sin^2(z))-\sin^3(z)$$
$$3\sin(z)-3\sin^3(z)-\sin^3(z)$$
$$3\sin(z)-4\sin^3(z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1568045",
"timestamp": "2023-03-29T00:00:00",
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Triangle of numbers sum What is the total of all these numbers?
1
1 2
1 2 3
1 2 3 4
1 2 3 4 5
. . . . . .
1 2 ........N
The answer should be a single binomial coefficient. It looks pretty easy, but I cannot com... | The sum of the $k$-th row is $\sum_{h=1}^k h=\frac{k(k+1)}{2}$.
Recalling that $\sum\limits_{k=1}^n k^2=\frac{n(n+1)(2n+1)}{6}$, we have that the sum of the numbers in the triangle is $$\sum_{k=1}^n \left(\frac{k^2}{2}+\frac{k}{2}\right)=\frac{n(n+1)(2n+1)}{12}+\frac{n(n+1)}{4}=\\=\frac{n(n+1)(2n+4)}{12}=\frac{n(n+1)(n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1568118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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the function of a complex variable $\int\limits_{0}^{\pi}\frac{cos^2{\varphi}}{2-sin^2{\varphi}}d\varphi$
I think
$e^{i\varphi}=z$ $\to d\varphi=\frac{dz}{iz}$
$cos\varphi=\frac{z^2+1}{2z}$
$sin\varphi=\frac{z^2-1}{2iz}$
$\oint\limits_{|z|=1}^{}\frac{(z^2+1)^2}{iz(z^4+6z^2+1)}dz$
and then get 4 roots that are not good
... | You see, upper limit is only $\pi$. Do substitution after doubling of the argument:
$$\int\limits_0^{\pi}\dfrac{\cos^2 \phi}{2-\sin^2\phi}d\phi=\int\limits_0^{\pi}\dfrac{2\cos^2 \phi}{2+2\cos^2\phi}d\phi=\int\limits_0^{\pi}\dfrac{1+\cos 2\phi}{3+\cos2\phi}d\phi,$$
$$\quad e^{2i\phi}=z,\quad d\phi=\dfrac{dz}{2iz},\quad ... | {
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Finding a vector field Give a formula $F=M(x,y)i + N(x,y)j$ for the vector field in the plane that has the property that F points toward the origin with magnitude inversely proportional to the square of the distance from (x,y) to the origin. (The field is not defined at (0,0)).
I first found the norm of the vector $|F|... | $n$ should be so that $||n||= 1$
| {
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What is a formal proof for why there are exactly $\frac{p-1}{2}$ elements that are not quadratic residues? If $p$ is prime, I was trying to understand why $Z^*_N$ has $\frac{p-1}{2}$ non-quadratic elements exactly.
The current proof that I have is sort of more of an intuitive argument (or at least it doesn't feel rigor... | Your argument, as well as I can follow it, would seem to extend to a proof that $\mathbb{Z}_N^*$ has $\phi(N)/2$ non-residues for any $N$, not just when $N$ is prime. But that's not true in general: If $N=15$, for example, there are $2$ quadratic residues and $6$ non-residues, not $4$ of each. So what seems to be mi... | {
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Connecting the formula for the surface area of a sphere to differential forms I have a formula for computing the area of the surface of any object in $\Bbb R^3$, namely given some parametrization $\Phi(s,t)$, I take the cross product of the partial derivative with respect to each variable and norm it.
This looks suspic... | Here's the calculus method:
Begin with parametrization of a sphere with radius R.
$$S=\big(R\sin(\phi)\cos(\theta),R\sin(\phi)\sin(\theta), R\cos(\phi)\big)$$
For $\theta\in[0, 2\pi], \phi\in[0, \pi]$.
The equation for the surface area $A$ of a parametric surface $S$ is:
$$A=\iint_D \big|\,S_\phi \times S_\theta\big|\,... | {
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Express the following complex numbers in standard form $$\left(\frac{\sqrt 3}{2}+\frac{i}{2}\right)^{25}$$
I know that you have to put it in the form $\cos\theta+i\sin \theta$ but I'm not sure how to go about it.
| When you want to raise a number to an integer power, the polar form is very handy. When in doubt, draw a picture-plot $\frac {\sqrt 3}2+\frac i2$. You should notice that the radius is about $1$. Do you know how to compute the radius and find it is exactly $1$? Then you should know that $(re^{i\theta})^n=r^ne^{in\thet... | {
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Quantifier: "For all sets" I've seen the following statement a few times:
"Let $A$ be a set, then $\emptyset\subseteq A$".
Or, written 'more formally':
$$
\forall A\,\, \emptyset\subseteq A
$$
My doubt is: I've always seen the quantifier $\forall x$ to mean "for all $x$ elements of some set $S$". However, when talk... | $\forall A$ means "for all $A$" or more completely "for all objects $A$ that the theory we are working in is talking about". If the theory is set theory (say, we started off by writing down the axioms of ZFC), then our $A$ is a set; when working in other theories it may instead mean that $A$ is a natural number or that... | {
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Find the limit of $\dfrac{e^{x^2}}{xe^x}$ Please help me find this limit. I have already tried to use L'Hopitals rule and logarithmic differentiation but it turns into a quadratic.
| I will assume you mean to evaluate the limit as $x\to\infty$.
Note, the expression is actually $\dfrac{e^{x^2-x}}{x}$. Now, we know for large $y$, the inequality $e^y\geq 1+y$ is true.
So noting that our expression is always non-negative, and putting $y=x^2-x$ in the above inequality, $\dfrac{e^{x^2-x}}{x}\geq \dfrac{1... | {
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"question_score": "1",
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"answer_id": 1
} |
Let $f(x),g(x),h(x)$ are three differentiable functions satisfying:
Let $f(x),g(x),h(x)$ are three differentiable functions satisfying:
$\int (f(x)+g(x))dx=\frac{x^3}3+C_1, \int(f(x)-g(x))dx=x^2-\frac{x^3}{3}+C_2$, $\int\frac{f(x)}{h(x)}dx=-\frac1x+C_3$.
$(1)$ The value of $\int (f(x)+g(x)+h(x))dx$ is equal to... | For the second part, $x=-1,0,1$ are not differentiable as the limit from left and right are not equal.
For the third part, you have $x^0,x^{\pm1},x^{\pm2},...,x^{\pm15}$ so $\Sigma n=15\implies n=5$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1568913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
group actions on spheres Let $\mathbb{Z}/2$ act on the $m$-sphere $S^m$ freely and properly discontinuously. If the action is not trivial, can we conclude that the action is homotopy equivalent to the antipodal action? That is, the following diagram
commutes up to homotopy?
Can we generalize this to $S^1$-actions on ... | If $\Bbb Z/2$ acts freely on $S^n$, then acting by the nontrivial element of $\Bbb Z/2$ one gets a map $f:S^n \to S^n$ which has no fixed point.
Thus, $f(x) \neq x$ for all $x \in S^n$. The homotopy $$F(x, t) = \frac{(1-t)x - tf(x)}{|(1-t)x - tf(x)|}$$
between $f$ and the antipodal map $-\text{id}$ is well-defined. Thu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1569093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Write $\frac {1}{1+z^2}$ as a power series centered at $z_0=1$ I'm trying to solve a question where I need to write $\frac {1}{1+z^2} $ as a power series centered at $z_0=1$
I'm not allowed to use taylor expansion. So my first thought was to rewrite the function in a form where I can apply the basic identity:
$$ \frac{... | $$
\begin{align}
\frac1{1+z^2}
&=\frac1{2+2w+w^2}\tag1\\[9pt]
&=\frac1{2i}\frac1{1-i+w}-\frac1{2i}\frac1{1+i+w}\tag2\\[6pt]
&=\frac{1-i}4\frac1{1+\frac{1+i}2w}+\frac{1+i}4\frac1{1+\frac{1-i}2w}\tag3\\
&=\sum_{k=0}^\infty\frac{(-1)^k}4\left[\left(\frac{1+i}2\right)^{k-1}+\left(\frac{1-i}2\right)^{k-1}\right](z-1)^k\tag4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1569168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Kernel in Modern Algebra What is a Kernel and how can it be describe in the real world and how can it be defined well and precisely. Tried asking my professor and he just tell us it is just abstract idea.
| The kernel of a linear map ($T$) between two vector spaces $U$ and $V$ are those elements ($v$) of $V$ such that $T(v)=0$. It is defined analogously for other structures.
You can see it as a measure of how far is the map from being one to one.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1569270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 1
} |
Probability :Knock Out Tournament Of Ranked Players
Thirty-two players ranked 1 to 32 are playing in a knockout
tournament. Assume that in every match between any two players, the
better-ranked player wins, the probability that ranked 1 and ranked 2
players are winner and runner up respectively, is ?
I dont und... | If you picture the usual tree diagram for a tournament bracket, you'll see that the 1 and 2 seeds will meet in the final if and only if they start on opposite sides of the bracket. By symmetry, it doesn't matter where you position the 1 seed (so you may as well place her say at the top left). This leaves $31$ possibl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1569352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Does the sequence $f_n(x)={1+x^n\over 2+x^n}$ converge uniformly on $[0,1)$? Does the sequence of functions $$f_n(x)={1+x^n\over 2+x^n}$$
converge uniformly to ${1\over 2}$ on the interval $0\le x\lt1$?
I don't think so because $\sup\left|f_n(x)-{1\over2}\right|=1$. If the interval is $[0,a]$ where $0\lt a\lt1$, then... | You're right, it is not uniformly convergent. And given $\varepsilon>0$, you can even find how badly $N(x)$ degenerates as $x$ approaches 1, where $N(x)\in\mathbb{N}$ is such that $|f_n(x)-0.5|<\varepsilon,\forall n>N(x)$. Notice that
$$
f_n(x) = 1 - \frac{1}{2+x^n}
$$
so
$$
\left|f_n(x)-\frac{1}{2}\right| = \frac{x^n}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1569571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
(Perhaps) An Easy Combinatorics Problem: (Perhaps not so easy...)
Still have some difficulty with problems like this.
Suppose I have Box with various buttons inside. For the sake of example, let the box contain: $$[ 4 \textrm{ Red}, 5 \textrm{ Blue}, 7 \textrm{ Green} ]$$ How many distinct tuples of, say, 5 buttons ca... | If Im not wrong I see 5 different values for red (from 0 to 4), 6 different values for blue and 7 different values for green.
You want that they sum exactly 5 (or starting from 1 instead of 0, 8). This is the coefficient of $x^5$ of this generating function:
$$f(x)=(1+x+x^2+x^3+x^4)(1+x+x^2+x^3+x^4+x^5)(1+x+x^2+x^3+x^4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1569671",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
the space of lipschitz function is complete with respect to some norm Let $V$ be the space of real valued lipschitz functions over $[a,b]$,we define:
$M_f=sup_{x\neq y}\frac{|f(x)-f(y)|}{|x-y|}$
and lipschitz norm:
$||f||_{Lip}=|f(a)|+M_f$
prove that $V$ with lipschitz norm is a complete normed vector space.
it is easy... | You should start from a hypothetical Cauchy sequence of functions (Cauchy in the sense of the newly defined norm). Prove that the values of that function at the left end $a$ converge as real numbers, and then prove that the functions converge pointwise at any $x\in[a,b].$ This allows you to write $f(x)$ for the pointwi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1569788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Determine whether a matrix is othrogonal I need to determine if the following matrix is orthogonal
$A = \frac{1}{\sqrt{2}} \begin{pmatrix}
1 & 1 \\
0 & 0 \\
1 & -1
\end{pmatrix}$
Here is what I did:
$u \cdot v = (\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}) + (\frac{1}{\sqrt{2}} \cdot -\frac{1}{\sqrt{2}}) = 0$
$||u||... | Your matrix satisfies $A^TA=I_2$, that is the columns are orthonormal. Such matrices are useful when you want to define the polar decomposition of a $m\times n$ matrix $M$ with $m\geq n$ and $rank(M)=n$. The decomposition is $M=US$ where $U$ is a $m\times n$ matrix with orthonormal columns and $S$ is a $n\times n$ SDP ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1569900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Problem with $SL(2)$ isometric action on a compact homogeneous space Let $G=SL(2,\mathbb{R})$, fix any left-invariant Riemannian metric $g$ on $G$. Let $\Gamma$ be a cocompact discrete subgroup of $G$ and $X=G/\Gamma$. Because $\Gamma$ acts by isometries $g$ descends to $X$ (call it $g'$) and the action of $G$ on $X$ m... | Suppose that $G$ is endowed with a bi invariant metric, it induces on its Lie algebra $sl(2)$ a scalar product invariant by the adjoint representation. This implies that the image of $SL(2)$ by the adjoint representation is conjugated to a subgroup of $SO(3)$ and this is not true since the adjoint representation has un... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1570023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How many ways to roll a die seven times How many ways to roll a die seven times and obtain a sequence of outcomes with three 1s, two 5s, and two 6s?
Ans: When I was thinking of a way to decompose the problem I first thought that if a die is getting rolled seven times then each roll is independent of the other thus: ... | You want to know in how many ways you can get $1155666$ in any order.
But there are $7!$ possible permutations of these numbers, except that some of the numbers are equal, and you need to compensate for that.
Since there are two $1$'s, two $5$'s, and three $6$'s, I've counted the same permutation $3!2!2!$ times, which ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1570131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Why does $\sqrt{x^2}$ seem to equal $x$ and not $|x|$ when you multiply the exponents? I understand that $\sqrt{x^2} = |x|$ because the principal square root is positive.
But since $\sqrt x = x^{\frac{1}{2}}$ shouldn't $\sqrt{x^2} = (x^2)^{\frac{1}{2}} = x^{\frac{2}{2}} = x$ because of the exponents multiplying togethe... | First of all, $\sqrt{a}\sqrt{b} = \sqrt{ab}$ only when $a$ and $b$ are non-negative. $\sqrt{a}$ and $a^{\frac{1}{2}}$ are not the same function, as Cameron Williams points out in the comments above. Secondly, we can break your square of square root up. For positive values, $\sqrt{x^2} = x$, as one would expect. However... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1570221",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 2
} |
Finding the geometric series of the fraction I am confused as to how to turn a fraction into a sum using geometric series.
I have $\frac{z+2}{(z-1)(z-4)}=\frac{2}{z-4}+\frac{-1}{z-1}$
I do not know how I turn the last 2 fractions into geometric series and write them as sum. Can someone please help me?
| You may have previously seen that
$$\sum_{n=0}^\infty z^n =\frac{1}{1-z}, \;\;|z|<1$$
Then you may use $\frac{2}{z-4}=-\frac{2}{4}\cdot\frac{1}{1-z/4}$ and $\frac{-1}{z-1}=\frac{1}{1-z}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1570301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Statistics: Deriving a Joint Probability Function From a Definition of Other PDF's Here's a particular question I'm trying to understand from the lecture notes.
It says:
Assume that $Y$ denotes the number of bacteria per cubic
centimeter in a particular liquid and that $Y$ has a Poisson
distribution with paramete... | This follows from definition of conditional probability. If there are two events $A,B$, then the probability of their joint occurrence is $P(A\cap B)$, conditional probability of occurrence of $B$ given that event $A$ has taken place is written as $P(B\lvert A)$, which is defined as
\begin{equation}
P(B\lvert A)=\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1570383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
$p$-adic logarithm, $|\log_p(1 + x)|_p = |x|_p$? Define the $p$-adic logarithm$$\log_p(1 + x) = \sum_{i =1}^\infty (-1)^{i-1}x^i/i.$$How do I see that if $p > 2$ and $|x|_p < 1$, then $|\log_p(1 + x)|_p = |x|_p$?
| Here is an outline of a proof. First recall that the disc of convergence of $\log_p(1+x)$ is $D(0,1) = p\mathbb{Z}_p$. Let $v_p$ denote the $p$-adic valuation. Given $x \in p\mathbb{Z}_p$, then $v_p(x) \geq 1$.
($1$) Show that $v_p\left(\frac{x^n}{n}\right) > v_p(x)$ for all $n \geq 2$. Intuitively, this should hol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1570481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the length of common chord of two intersecting circles Let us consider two intersection circles
$x^2+y^2+2g_1x+2f_1y+c_1=0$
and $x^2+y^2+2g_2x+2f_2y+c=0$.
Then equation of common chord of the above two circles is
\begin{equation}2(g_1-g_2)x+2(f_1-f_2)y+c_1-c_2=0............(1)
\end{equation}
I want to find the... | Equation is unnecessary. The length is given by $2\sqrt{(C_1P)^2-(C_1M)^2}$ where $C_1P$ is radius of circle 1 and $C_1M$ is length of perpendicular from $C_1$ to common chord $PQ$ can you take it from there.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1570575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Separation of variables to derive the solution $u(x, t)$ How do I solve this question, what are the steps ?
Use the method of separation of variables to derive the solution $u(x, t)$ to the equation for a vibrating string
$$\frac{\partial ^2u}{\partial \:t^2}=9\:\frac{\partial ^2u}{\partial x^2} \hspace{4em} (0 < x < 4... | I advise you to go through the procedure outlined here (first hit on Google when you search for 'separation of variables'), which shows the method for the equation
\begin{equation}
\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}.
\end{equation}
Adopting it to your problem shouldn't be too hard... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1570759",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that $(\mu \times \mu)(G) = 0$ where $\mu$ is the Lebesgue measure using Fubini
Suppose $f: [0,1] \rightarrow [0,1]$ is a Borel measurable function. Suppose $G = \{(x,y)\mid f(x) = y\} \subset [0,1] \times [0,1]$. Use Fubini to prove that
$$
(\mu \times \mu)(G) = 0.
$$
What I've tried:
$$(\mu \times \mu)(G)... | Let $G_{x}=\lbrace y\in[0,1]:(x,y)\in G\rbrace$. Clearly, $\mu(G_{x})=0$ for all $x$ (even a.e. would be sufficient ). Now use the fact that
$$(\mu\times\mu)(G)=\int_{[0.1]}\mu(G_{x})d\mu(x)=0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1570856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
What is the greatest $4$-digit integer for which the sum of its proper divisors is $n-1$
What is the greatest $4$-digit integer $n$ for which the sum of its proper divisors is $n-1$?
I can't think of a way to solve this without guessing and checking.
Edit: It is easy to show that all powers of $2$ are such numbers. ... | I get $8192$. This is also $2^{13}$, which I'm sure is significant. I was lazy and programmed it. Its divisors are 1,2,4,8,16,32,64,128,256,512,1024,2048,4096.
using System;
class Program
{
static void Main()
{
int result = 9999;
for (int i = 9999; i >= 1000; i--)
{
if (SumOfDivisors(i) == i - 1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1570953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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