Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
toplogical entropy of general tent map Measure theoretic entropy of General Tent maps
The linked question made me wonder how to calculate the topological entropy of a general tent map.
Let $I=[0,1]$ and $\alpha \in ( 0,1)$. Define $T: I \rightarrow I$ by
$T(x)= x/\alpha$ for $x \in [0,\alpha]$ and $(1-x)/(1-\alpha)$ f... | Since the peak of the skewed tent touches ocurrs at $(\alpha, 1)$, the pre-image of this point will consist of two points, therefore its second iterate will have four laps. But now we can apply this argument to both these new points (only because of the surjectivity of the function ensured by the peak of the tent "touc... | {
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"url": "https://math.stackexchange.com/questions/760717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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A question about a continuous function that satisfies the property $\forall x\in\mathbb{R},\exists xI got this question:
Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function that satisfies the property: forall $x\in\mathbb{R}$ there exists $y \in\mathbb{R}$ such that $x < y$ and $f(x)<f(y)$
I was able to prove (har... | Assume that there is an $a$ such that $f(a) > 5$. Let $\epsilon = f(a) - 5 > 0$. If
$$
\lim_{x\to \infty} f(x) = 5
$$
then you would have an $N$ such that if $x \geq N$ then $\lvert f(x) - 5 \rvert < \epsilon / 2$.
Now $f$ being continuous on $[a, N]$ where $f(a) > f(N)$, $f$ attains a maximum at $x_0$ on $[a, N]$. T... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Zonal averaging of partial derivatives Given a quantity $Q(x,y,t)$, the zonal average operator $[Q] = \frac{1}{2\pi}\int_0^{2\pi} Q\:\mathrm{d} \lambda$, and zonal anomaly $Q^\star$ such that $Q = [Q] + Q^\star$, my text book says that zonally averaging
$$\frac{\partial Q}{\partial t} + \frac{\partial}{\partial x}(uQ) ... | I figured it out. It's clear that the zonally averaged $\partial/\partial x$ is zero. The terms involving $\partial/\partial y$ come from expanding $vQ = ([v] + v^\star)([Q] + Q^\star)$, then noticing that $[v^\star [Q]] = [[v] Q^\star] = 0$. I also had to convince myself that $[\partial Q/\partial t] = \partial [Q]... | {
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"timestamp": "2023-03-29T00:00:00",
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Is this linear functional bounded? Find the norm. $$\ell^2\ni (x_n)\rightarrow2x_{1}+28x_2+35 x_{3}$$
I think it can be bounded:
$$|2x_{1}+28x_2+35 x_{3}| \le |2x_{1}|+|28x_2|+|35 x_{3}| \le 65 (\sum_{n=0}^{\infty}|x_n|^2)^{1/2}$$
But I can't find norm.
| What is the norm of the linear functional $L(x)$? It is the smallest constant $M$ such that
$$|L(x)| \leq M ||x||_2$$
holds for every $x \in \ell^2$.
Okay, so now note that your linear functional is of the form $\langle a, x \rangle$, so apply the Cauchy inequality to get
$$|L(x)| = |\langle a, x \rangle| \leq ||a||_2 ... | {
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A characterization for subgroups. Let $G$ be a group and $a_0,a_1,...,a_n\in G$ and
$$A=\{a_0,a_1,...,a_n\}$$
and
$$(\forall m\le n)(\forall i\le m)(a_{i}a_{m-i}\in A)$$
Is $A$ a subgroup of $G$? How if $G$ is abelian?
| Let $n = 1$, $G = \mathbb{Z}/4\mathbb{Z}$.
Now let $A = \{0,1\}$. Now note that each of the sums $0+0,0+1,1+0$ are in $A$.
Hence $A$ satisfies the given condition but is not a subgroup. (Notice that
the sum $2 = 1+1$ doesn't have to be in $A$ since that would correspond to
$a_1 + a_1$ which would require $i = 1,m = 2$ ... | {
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Graph Theory Question Related to Domination number. Let G be a graph whose diameter is at least 3. Prove that the domination number of the complement of G is at most 2.
I know that since the diameter of G is at least 3, the diameter of the complement of G is at most 3. However, this doesn't seem to be enough to prove ... | Damned if I know. Let me try and follow my nose here. The diameter of $G$ is at least $3$, what does that mean? It means there are two vertices $u,v$ in $G$ such that $\operatorname d(u,v)\ge3$. And we want to show that the complement $\bar G$ has a dominating set containing at most $2$ vertices. Hmm. Maybe I can show ... | {
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"timestamp": "2023-03-29T00:00:00",
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Evaluate the line integral $\int_C \ x^2 dx+(x+y)dy \ $
Evaluate the line integral $$\int_C \ x^2 dx+(x+y)dy \ $$
where $C$ is the path of the right triangle with vertices $(0,0), (4,0)$ and $(0,10)$ starts from the origin and goes to $(4,0)$ then to $(0,10)$ then back to the origin.
I did this problem but the ans... | The line integrations along each leg look like
$$ (0,0) \ \rightarrow \ (4,0) \ : \ \ dy \ = \ 0 \ \ \Rightarrow \ \ \int_0^4 \ x^2 \ dx \ \ ; $$
$$ (4,0) \ \rightarrow \ (0,10) \ : \ \ y \ = \ 10 - \frac{10}{4}x \ \ \Rightarrow \ \ \int_4^0 \ x^2 \ dx \ + \ (x \ + \ [10 - \frac{10}{4}x]) \ (- \frac{10}{4} \ dx) $$
[n... | {
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Verifying whether a quotient ring is indeed a ring. Take $$\frac{\Bbb{R[x]}}{\langle x^2+1\rangle}$$ This is a ring. In this quotient ring, the product of equivalence classes $[a+bx]$ and $[c+dx]$ is another equivalence class, as a ring is closed under multiplication.
$[a+bx]=\{a+bx,a+bx+(x^2+1),a+bx+2(x^2+1),\dots\}$... | To be honest, if you are trying to show this is indeed a ring, there are indirect methods that are far easier. For example:
Consider the evaluation homomorphism $ev_i:\mathbb{R}[x] \rightarrow \mathbb{C}$ defined as follows:
$$f(x) \mapsto f(i)$$
This homomorphism is indeed surjective since, given any $(a + bi) \in \m... | {
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Why normal approximation to binomial distribution uses np> 5 as a condition I was reading about normal approximation to binomial distribution and I dunno how it works for cases when you say for example p is equal to 0.3 where p is probability of success.
On most websites it is written that normal approximation to binom... | So I did some experiments. I think np>5 condition is not correct at all. It depends on Excess Kurtosis value for a given binomial distribution. If it is Mesokurtic then approximation will give accurate results.
Check following table
for n=11 and p=0.5 kurtosis will be around 0.18. That is platykurtic and so I don't th... | {
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Let $A$ be the set of all $4$ digit numbers $a_1a_2a_3a_4$ such that $a_1 < a_2 < a_3 < a_4$, then what is $n(A)$ equal to? How can you solve this problem relatively quickly using combinatorics? I found it really interesting.
Let $A$ be the set of all $4$ digit numbers $a_1a_2a_3a_4$ such that $a_1 < a_2 < a_3 < a_4$, ... | $7,6,5,4,3,2$ in second positions give $6,5,4,3,2,1$ cases respectively of the trees below in which for $7$ one has one case, for $6$ three, for $5$ six, for $4$ ten, for $3$ fifteen, and for $2$ twenty one cases (these are the number of endpoints in each tree). This gives finally
$$6\cdot1+5\cdot3+4\cdot6+3\cdot10... | {
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$\mathbb{R}\text{P}^{n-1}$ is not retract of $\mathbb{R}\text{P}^n$ I have to solve the following:
Show that $\mathbb{R}\text{P}^{n-1}$ is not retract of $\mathbb{R}\text{P}^n$ for $n\geq 2$.
I have done this with knowledge of homotopy-groups, by showing that $\mathbb{Z}$ cannot factor through $\mathbb{0}$ or $\mathbb... | One can use the fact that $H^*(\mathbb R P^n, \mathbb Z/2) \cong \mathbb Z/2[x]/x^{n+1}$ as a graded commutative ring, where $x$ is in degree one. The inclusion $\mathbb R P^{n-1} \to \mathbb RP^n$ induces a map of graded rings $\mathbb Z/2[x]/x^{n+1} \to \mathbb Z/2[x]/x^n$. By considering fundamental groups or using ... | {
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application of the sampling distribution of x the GBAs of all students enrolled at a large university have an approximately normal ditribution with a mean of 3.02 and a standard deviation of 0.29 ..find the probability that the mean GBA of a random sample of 20 student selected from this university is ,
a)3.10 or high... | Imagine taking a random sample of $20$ students. Let random variables $X_1,X_2,\dots, X_{20}$ be their GBAs, and let $Y=\frac{1}{20}(X_1+X_2+\cdots+X_{20})$ be the sample mean.
Then $Y$ has (approximately) normal distribution, with mean $3.02$ and standard deviation $\dfrac{0.29}{\sqrt{20}}\approx 0.064846$. From here ... | {
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If G acts on X, show that there must be a fixed point for this action. Please help. Suppose that $G$ is a group of order $p^k$, where $p$ is prime and $k$ is a positive integer. Suppose that $X$ is a finite set and assume that $p$ does not divide the $|X|$. If $G$ acts on $X$, show that there must be a fixed point for ... | Put together facts you should know:
*
*A $G$-set $X$ is the disjoint union of its orbits, so $|X|$ is the sum of the sizes of the orbits.
*The size of an orbit divides $|G|$ by orbit-stabilizer.
*Fixed points correspond to orbits of size $1$.
| {
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Find the number of real solutions Let $$f(x)=\frac{1}{2}( |x-a|+|x-b|),$$ where $x$ is a real number ; no information is given on $a$ and $b$.
Study the differentiability of this function and determine how many real solutions does the equation $\mathbf{f(x)=m}$ have, where $m$ is a real number. The problem asks us not ... | Suppose first that $a<b$. Then $$f(x)=\frac{1}{2}\begin{cases}a+b-2x & x\le a \\ b-a & a<x<b \\ 2x-a-b & b\le x\end{cases}$$
Taking the derivative, we get $$f'(x)=\frac{1}{2}\begin{cases} -2 & x< a \\ 0 &a<x<b \\ 2 & b<x\end{cases}$$
You will note that there are discontinuities of $f'(x)$ at $a, b$; at these points $f... | {
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Does the alternating group $A_5$ contain a subgroup isomorphic to $\Bbb Z_{20}$? What are all the possible orders of elements in the group $A_5$? Does $A_5$ contain a subgroup isomorphic to $\Bbb Z_{20}$? How about $\Bbb Z_{10}$? How about $\Bbb Z_5$? Justify your answers.
I've found that the possible orders are 1, 2,... | Hint: a subgroup isomorphic to ${\mathbb Z}_n$ would be generated by an element of order $n$.
| {
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Show that there are exactly two lines through a point p outside the circle that are tangent to the circle C Let $C$ be a circle of radius $r$ in the plane. Let $p$ be a point in the plane that lies outside of $C$. Show that there are exactly two lines through $p$ that are tangent to $C$.
It is one of those questions t... | Method Using Calculus:
Say we are given any circle and any point outside of that circle. WLOG, we can translate the circle to be centered at the origin, and rotate our system so that the point is situated along the $y$-axis.
From here, we claim that we can hit that point with exactly two tangent lines to the circl... | {
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probability, please help on bayes question I dont need exact answer, I just need help to judge wether my following method is correct or not.
Question:A physician has 5 patients. There are treatments A and B. Physician gives treament A to 3 randomly selected patients, and B to the other 2. Suppose that treament A produc... | The setup is correct, we want
$$\Pr(X=0)\Pr(Y=0)+\Pr(X=1)\Pr(Y=1)+\Pr(X=2)\Pr(Y=2).$$
We have $\Pr(X=0)=(0.7)^3$ and $\Pr(Y=0)=(0.4)^2$.
For $\Pr(X=1)$, the right expression is $\binom{3}{1}(0.3)(0.7)^2$. Similarly, $\Pr(Y=1)=\binom{2}{1}(0.6)(0.4)$.
Finally, $\Pr(X=2)=\binom{3}{2}(0.3)^2(0.7)$ and $\Pr(Y=2)=(0.6)^2$.... | {
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$\lim_{n\to \infty}\left(1 - \frac {1}{n^2}\right)^n =?$ Can you give any idea regarding the evaluation of the following limit?
$\lim_{n\to \infty}\left(1 - \frac {1}{n^2}\right)^n$
We know that $\lim_{n\to \infty}\left(1 - \frac {1}{n}\right)^n = e^{-1}$, but how do I use that here?
| Here is a hint:
$\left(1 - \dfrac{1}{n^2}\right) = \left(1 - \dfrac{1}{n}\right)\left(1 + \dfrac{1}{n}\right)$
Also use the fact that $(ab)^n = a^n b^n$.
| {
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Continuity and simplification of a function I have a question to ask about a function.
Suppose a function $$f(x) = \frac{x^2 - x}{ x - 1},$$ we can simplify this function to be $f(x) = x$. Yet, we say that this function is discontinuous at $x = 1$ but after the simplification, we say that the function $f(x)$ is continu... | The original function $$f\left(x\right)=\frac{x^{2}-x}{x-1}$$ has $\mathbb{R}\backslash\left\{ 1\right\} $
as (maximal) domain and is continuous. It is not defined on $\left\{ 1\right\} $
and consequently statements like '$f$ is (dis)continuous at $1$' don't
make sense. It can only be (dis)continuous at points that bel... | {
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Proving $\arctan1 + \arctan\frac13 +\cdots+\arctan\frac1{n^2+n+1}=\arctan (n+1)$ by induction How to solve this problem using mathematical induction?
$$\arctan1 + \arctan\frac13 +\cdots+\arctan\frac1{n^2+n+1}=\arctan (n+1)$$
| Hint: $$\tan(a+b)=\frac{\tan a+\tan b}{1-\tan a\tan b}$$
In your case, for the base case $n=1$, $$\tan\left(\arctan 1+\arctan\frac{1}{3}\right)=\frac{\tan(\arctan 1)+\tan(\arctan\frac{1}{3})}{1-\tan(\arctan1)\tan(\arctan\frac{1}{3})}=\frac{1+\frac{1}{3}}{1-\frac{1}{3}}=2$$So $\arctan 1+\arctan\frac{1}{3}=\arctan 2$.
| {
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For which $a$ is a matrix $A$ diagonalizable? Say I have a matrix $A_a$ with
$$A_a:= \left(\begin{array}{c}
2 & a+1 & 0 \\
-a & -3a & -a \\
a & 3a+2 & a+2
\end{array}\right)$$
I was wondering if there was an easy way to determine for which $a$ the matrix would be diagonalizable.
I tried to determine its ei... | If you factor the characteristic polynomial, you find it is
$$
(2a^2-6a)-(a^2-7a+4)\lambda+(4-2a)\lambda^2-\lambda^3=
(2-\lambda)(\lambda^2+(2a-2)\lambda+a^2-3a)
$$
If you know this polynomial has three distinct roots, then the matrix is diagonalizable. A root can be repeated only if either
*
*the discriminant of $f... | {
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Show that $Z_{p^2} \oplus Z_{p^2}$ has exactly one subgroup isomorphic to $Z_p \oplus Z_p$ Show that $Z_{p^2} \oplus Z_{p^2}$ has exactly one subgroup isomorphic to $Z_p \oplus Z_p$
Attempt: $Z_p \oplus Z_p$ has $p^2-1$ elements of order $p$ . Hence, all non trivial elements of $Z_p \oplus Z_p$ are of order $p$. Number... | Unless I am mistaken, the idea of the question is as follows: You know that $G:= \mathbb{Z}_{p^2}\oplus \mathbb{Z}_{p^2}$ has one subgroup $H$ isomorphic to $\mathbb{Z}_p\oplus \mathbb{Z}_p$. The question is asking you to prove that this $H$ is the unique such subgroup.
So suppose $K$ is any other subgroup isomorphic t... | {
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Trigonometric problem: Elevation angle The elevation of the top of a tower $KT$ from a point $A$ is $27^\circ$. At another point $B$, $50$ meters nearer to the foot of the tower where $ABK$ is a straight line, the angle of elevation is $40^\circ$. Find the height of the tower $KT$.
| Consider the following diagram:
Looking at the outer (right-angled) triangle ($TAK$), and using trigonometry, we have:
$$(1) \tan(27)=\frac{h}{50+x}.$$
Looking at the inner (right-angled) triangle ($TBK$), and using trigonometry, we have:
$$(2) \tan(40)=\frac{h}{x}.$$
Now we've got a pair of simultaneous equations, $(... | {
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Show that $x_n \rightarrow 0$ Let $f:[0,1] \rightarrow \mathbb{R}$ continuous, such that $f(0)=0$
We set $x_n=\int_0^1{f(x^n)}dx$
Show that $x_n \rightarrow 0$
$$$$
The function $f$ is continuous at a closed interval $\Rightarrow $ $f$ is bounded $\Rightarrow \exists M>0: |f(x)| \leq M, \forall x \in [0,1]$
The functio... | Hints using the notation and stuff you already did:
For all $\;\epsilon>0\;$ there exists $\;K\in\Bbb N\;\;s.t.\;\;n>K\implies |f(x^n)|<\epsilon\;$ (why?) , so:
$$\left|\int\limits_0^1f(x^n)\,dx\right|\le\int\limits_0^1|f(x^n)|dx\le\int\limits_0^1\epsilon\,dx=\epsilon$$
| {
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Steps in the solution of Korteweg-deVries PDE In the following solution of the Korteweg-deVries PDE
$$
u_t + 6uu_x + u_{xxx} = 0 \qquad (3.1)
$$
I do not understand the second integration step and how they arrive at the expression for the differentials.
The first integration is clear to me, but in the second step, why... | Before the second integration, the whole equation is multiplied by $u_\xi$ (a standard trick, but perhaps it should have been explained in the text), and then the chain rule is used backwards.
And if $c_2=0$, then
$$
\frac12 \left( \frac{du}{d\xi} \right)^2 = -u^3 + \frac12 c u^2 = \frac12 u^2 (c-u)
,
$$
so
$$
\frac{du... | {
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Finding roots of cubing equation Find all roots of the following polynomial:
$$x^3 + x^2 + 1$$
| You can use the formulas of Cardano equation but not for sheep, because the form of the equation which is elected by the Cardano formulas is:
$$x^3+px+q=0$$
If the equation is of the form:
$$x^3+ax^2+bx+c=0$$
Your equation is the form of the above, therefore, by means of substitution $$x+\frac{a}{3}=y$$
acquired forms:... | {
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Euclidean Algorithm for Modular Inverse, with negative numbers I might be on to something quite simple which I'm failing to see, while calculating modular inverses.
For example, calculating 7x = 5 (mod 12)
Which is the same as saying: 7x - 5 = 12k
Which becomes: 7x - 12k = 5
And then I proceed using Euclidean Algorithm... | In a Bezout identity
$$
a⋅x+b⋅y=c
$$
you can exchange multiples of $a⋅b$ or even $lcm(a,b)=a'\cdot b=a\cdot b'$ between the terms on the left, so that
$$
a⋅(x-k⋅b')+b⋅(y+k⋅a')=c
$$
is also a correct identity. This slightly extends the reasoning on modular equivalences in the comment of Bill Dubuque.
| {
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Doubt on proof of Implicit function theorem
On The second part of the proof, where it's stated that V is open as it is the inverse image of the open set $V_0$ under the continuous mapping $y \rightarrow (0, y)$.
Let $\pi$ be this continuous mapping. Then, $\forall _{U_{open\text{ in }\mathbb{R}^n\times\mathbb{R}^p}} \... | So, I think I now understand what's happening.
$Dom(\pi)=V_1$ and yet we still have $\pi^{-1} (V_0)=V$ since, $V=\{y\in V_1|(0,y)\in V_0\}$ and $\pi(y)=(0,y)$.
Silly me!
| {
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"url": "https://math.stackexchange.com/questions/763553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Homeomorphism Compact Subsets Are there compact subsets $A,B \subset \mathbb{R^2}$ with $A$ not homeomorphic to $B$ but $A \times [0,1]$ homeomorphic to $B \times [0,1]$?
| Yes. Consider the two sets below.
Edit: Jon, let me explain in more detail why the two sets (call them $A$ and $B$, respectively) are not homeomorphic. I find in practice that showing two sets to not be homeomorphic is a bit cumbersome. I didn't fill out all of the details for that reason, leaving them to you; let me ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/763663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How many different messages can be transmitted in n microseconds using three different signals...
How many different messages can be transmitted in n microseconds using three different signals if one signal requires 1 microsecond for transmittal, the other two signals require 2 microseconds each for transmittal, and a... | i found a online solution in which it is showing An = An-1+An-2 because we have two choise here either we can send first signal which 1 second time or we can send another signal which takes two second time anyone of them we can send first so i think the value of A1 = 1 because in 1st second we can send a signal which ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Integrating the product of Poisson and exponential pdf So I'll spare the background as to why, but I'm trying to integrate the following:
$$\int_0^{\infty} \frac{e^{-(\lambda+\mu)t}(\lambda t)^n}{n!} dt$$
If you parameterize a Poisson w/ $\lambda$ and an exponential w/ $\mu$ and multiply their pdf's, you get the above.... | You can use the Laplace Transform. The transform of $t^n$ is $$\frac{n!}{s^{n+1}}$$ So you get, after some algebra, the quantity must equal $$\lambda^n {n!\over n!}\frac{1}{((\lambda +\mu))^{n+1}}=\frac{\lambda^n}{(\lambda+\mu)^{n+1}}$$ Can someone check this please? And evidently you've made a mistake somewhere becaus... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $r^n/n!$ converges where $n\ge r$ The answer is in the title of the question. I need to show it converges to 0 and $r>0$. I am sorry if this is a bad question, I'm having trouble explaining it. So essentially this Do the $\lim_{n\to inf}\frac{r^n}{n!}=0.$
| I asume that you want to prove:
$$\lim_n\displaystyle\frac{r^n}{n!}=0 $$
and $r>0$ is fixed.
Let $N$ be an integer number such that $N> r$. Then for $n>N$ the following holds:
$$\displaystyle\frac{r^n}{n!}=\displaystyle\frac{r}{1}\cdots\displaystyle\frac{r}{N-1}\displaystyle\frac{r}{N}\cdots\displaystyle\frac{r}{n}<\d... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that if the prime $p$ divides $|G|$, then $|X|$ is divisible by $p$. Question :
Let $p$ be a prime number that divides the order of the finite group $G$. Let $X$ = $\bigcup_{P \in Syl_p(G)}P$. Show that $|X|$ is divisible by $p$.
| Lemma Let $G$ be a finite group and $p$ be a prime divding $|G|$. Let $H$ be a $p$-subgroup of $G$ and $P \in Syl_p(G)$. Then $H \cap C_G(P)=H\cap Z(P)$.
Proof It is clear that $H \cap Z(P) \subseteq C_H(P)=H \cap C_G(P)$. Conversely, observe that $C_H(P)$ is a $p$-subgroup (it is a subgroup of $H$!) and it normalizes ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A factorization problem involving Fibonacci and Lucas Polynomials Consider a sequence of polynomial $\{w_n(x)|\, n\geq 0\}$ which are defined recursively by $w_n(x)=xw_{n-1}(x)+w_{n-2}(x)$. With $w_0(x)=0$ and $w_1(x)=1$, one gets the so-called Fibonacci polynomials $w_n(x)=F_n(x)$. With $w_0(x)=2$ and $w_1(x)=x$, one ... | The key observation is that, for $n$ odd, $L_{2n}(x) = L_{n}(x)^2 + 2$; this is straightforward to show by induction, as a subcase of the more general identity $L_{m+n}(x) = L_{m}(x)L_{n}(x) + (-1)^{m-n}L_{m-n}(x)$.
Now, your polynomial $F_{n}(z)^2 - L_{2n}(z) + 2$ reduces to $F_{n}(z)^2 - L_{n}(z)^2 = (F_{n}(z) - L_{... | {
"language": "en",
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Caccioppoli inequality Assume we have established the following version of Caccioppoli inequality
$$\int |\nabla u|^2 \psi^2 dA\leq C \int u^2 |\nabla \psi| ^2 dA$$
for $C^2(\mathbb C)$- smooth functions $u\geq 0$ with $\Delta u\geq 0$, and $\psi\in C_c^\infty (\mathbb C)$ (compactly supported, smooth) test functions. ... |
$\nabla \psi$ exists almost everywhere, and it is bounded, and supported on a finite measure set?
As written: not enough. On the right, $|\nabla \psi|^2$ must be the weak derivative of $\psi$; nothing short of it can control the oscillation of $\psi$. Bounded pointwise a.e. derivative need not be weak.
But, I see th... | {
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$\displaystyle\Big(1-\frac{t}{n}\Big)^n$ is strictly increasing for $n>N$ and $t>0$
Show that $\exists N\in\mathbb N$ such that, $\displaystyle\Big(1-\frac{t}{n}\Big)^n$ is strictly increasing for $n>N$
$(n\in\mathbb N, t>0)$
Bernoulli Inequality didn't help me
I did;
$\displaystyle\frac{\Big(1-\frac{t}{n+1}\Big)^{n+... | As long as $n\gt t$, apply Bernoulli:
$$
\begin{align}
\frac{\left(1-\frac t{n+1}\right)^{n+1}}{\left(1-\frac tn\right)^n}
&=\left(1+\frac t{(n+1)(n-t)}\right)^{n+1}\left(1-\frac tn\right)\\
&\ge\left(1+\frac t{n-t}\right)\left(1-\frac tn\right)\\[9pt]
&=1
\end{align}
$$
| {
"language": "en",
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Convergence of $\sum^\infty_{n=1}\frac {\sqrt[m]{n!}}{\sqrt[k]{(2n)!}}$
Does the following series converges ? $$\displaystyle\sum^\infty_{n=1}\frac {\sqrt[m]{n!}}{\sqrt[k]{(2n)!}} \ \text{for} \ \ k,m\in \mathbb N$$
I tried the ratio test:
$ \displaystyle\lim_{n\to\infty}\frac {\sqrt[m]{(n+1)!}}{\sqrt[k]{(2n+2)!}}\c... | If you look at the ratio you have and rewrite it slightly, you get
$$n^{\frac{1}{m} - \frac{2}{k}}\frac{\left(1+\frac{1}{n}\right)^{(k-m)/(mk)}}{2^{2/k}\left(1+\frac{1}{2n}\right)^{1/k}}.$$
The fraction converges to
$$\frac{1}{2^{2/k}} < 1,$$
so it depends on the behaviour of $n^{1/m - 2/k}$. If $\frac{1}{m} > \frac{2}... | {
"language": "en",
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A convex function has a lower bound? Suppose that $f=f(x)$ is strictly convex for $x\in\mathbb{R}$, i.e. there exists $\epsilon>0$ such that $f''(x)\geq\epsilon>0$ for $x\in\mathbb{R}$. Does there exist $\delta>0$ such that $f(x)\geq \delta$ for $x\in\mathbb{R}$?
| A function satisfying the condition given in this question is called "strongly convex".
There is a more general definition of strong convexity which applies to functions that may not be differentiable: A function $f:\mathbb R^N \to \bar{\mathbb R}$ is strongly convex (with parameter $\epsilon > 0$) if and only if the... | {
"language": "en",
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Help understanding Recursive algorithm question We have a function that is defined recursively by $f(0)=f_0$, $f(1)=f_1$ and $f(n+2) = f(n)+f(n+1)$ for $n\geq0$
For $n\geq0$, let $c(n)$ be the total number of additions for calculating
$f(n)$ using $f_0$ and $f_1 $ as input with $c(0) = 0$ and $c(1) = 0$. For $n \geq... | In reality, only n-1 additions are needed. The only formula that we can use is f(n+2) = f (n+1) + f (n) for n >= 0. If f0 and f1 are given, the only value this formula allows us to calculate is
f2 = f1 + f0 (applying the formula with n = 0)
Now with f2 available as well, the only value the formula allows us to calcul... | {
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Approximate $\sqrt{e}$ by hand I have seen this question many times as an example of provoking creativity. I wonder how many ways there are to approximate $\sqrt{e}$ by hand as accurately as possible.
The obvious way I can think of is to use Taylor expansion.
Thanks
| And here is another answer. There is a known continued fraction expansion for $e^{1/n}$. Continued fraction sequences converge quickly (although with so many 1s, this particular continued fraction converges on the slower end of things). The downside is that you can't use the $n$th convergent to quickly find the $n+1$st... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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Limit of $f'(x) e^{-f(x)}$ Let $f$ be a real function verifying $f''\geq C>0$, where C is a constant.
Do we have : $\lim_{x\to +\infty}f'(x) e^{-f(x)}=0$ ?
| Hint: transform the limit to an indeterminate form such as $[\frac{0}{0}]$ or $[\frac{\infty}{\infty}]$, and apply De L'Hopital's Rule:
as you mentioned in a comment $\lim_{x\to +\infty}f'(x)=+\infty$ and $\lim_{x\to +\infty}f(x)=+\infty$, so $\lim_{x\to +\infty}f'(x) e^{-f(x)}$ would be an indeterminate form such as $... | {
"language": "en",
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"source": "stackexchange",
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Partial Derivatives using ChainRule Can any one please explain the second step:-
Step1:
$$\frac{\partial }{\partial x}\left[(1-x^2)\frac{\partial u}{\partial x}\right]+\frac{\partial }{\partial y}\left[y^2\frac{\partial u}{\partial y}\right]=0$$
Step2:
$$L.H.S.=-2x\frac{\partial u}{\partial x}+(1-x^2)\frac{\partial ^2u... | Are you sure about the +1 after step 2? I think, that 1 is disappearing after derivation.
If you derive $-x^2\frac{\partial U}{\partial x}$ then you have to use the product rule.
$$u=-x^2$$
$$v=\frac{\partial U}{\partial x}$$
$$(u \cdot v)'=u'\cdot v+u\cdot v'$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Trace norm of Hermitian matrix Let $A\in L(H)$ some Hermitian matrix, where $H$ is some finite dimensional Hilbertspace.
I want to show $$\left\|A\right\|_{tr} = \max_{U\in U(H)}|\text{tr}(UA)| \ \ \ (*)$$
where U is unitary, and $\left\|\cdot \right\|_{tr}$ is the trace-norm which is given by
$\left\|A\right\|_{tr}=... | $\langle i|U|i\rangle$ are the diagonal entries of $U$ in your orthogonal basis. They could certainly be different from $0$ and $1$, but you are sure that they are less than $1$ in absolute value:
$$
|\langle i|U|i\rangle|\leq\|U\|\,\|i\rangle\|^2=\|U\|=1.
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Equi integrability and weak convergence of measures Let $f_n$ be a sequence of functions in $L^1(K, m ; \mathbb{C})$, $K$ metric compact and $m$ a Radon measure on $K$. Assume that $\| f_n \|_1 \leq 1$.
From what I understand, there is a subsequence converging weakly in $L^1$ if and only if the $f_n$ are equi integrabl... |
there is a subsequence converging weakly in $L^1$ if and only if the $f_n$ are equi integrable
Not true as stated. One can interlace a convergent sequence with a non-equi-integrable sequence; the result will still have a convergent subsequence. What is true (and what you probably meant) is that equi-integrability is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/764975",
"timestamp": "2023-03-29T00:00:00",
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Find a vector $t \in \{x,y,z\}$ with base $\{u, v, w\}$ I don't know how to find a vector $\vec t$ that will suffice the condition:
$\vec t \in \{x,y,z\}$ with bases $\{u, v, w\}$
the given vectors are:
$$
\begin{array}{rcrrrrrl} u &=& [ & -3, & -1, & 1, & -2 & ] \\ v &=& [ & 2, & -3, & -2, & -2 & ] \\ w &=& [ & 1, & 1... | I am not sure, whether I have understood your problem correctly. If you have to say whether $x$, $y$ or $z$ are in $\operatorname{span}\{u,v,w\}$ you have to solve the three linear systems:
*
*$x = \alpha u + \beta v + \gamma w$
*$y = \alpha u + \beta v + \gamma w$
*$z = \alpha u + \beta v + \gamma w$
If there i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Expected value for games where you can replay? Lets just say there's a game where you roll one fair die. If you roll a 1 or 2, you pay 1. If you roll a 3 or 4, you win 2. If you roll a 5 or 6 you roll again until you get a 1, 2, 3, or 4.
How much are you expected to win? I can't figure out how to think about this.
... | Note that your expected gain given that you first rolled a $5$ or $6$ is the same as your expected gain initially… you just get to start over.
Using linearity of expectation, then, you can write your expected gain as
$$
E[G]=\sum_{i=1}^{6}E[G\;\vert\;X_1=i]\cdot P[X_1=1]=-\frac{1}{3}+\frac{2}{3}+\frac{1}{3}E[G]=\frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/765154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding number of roots of a polynomial in p-adic integers $\mathbb{Z}_{p}$ The problem is to find the number roots of $x^3+25x^2+x-9 $ in $\mathbb{Z}_{p}$ for p=2,3,5,7.
I read this equivalent to having a root mod $p^{k}~\forall k\geq 1$.
By Newton's lemma I can get whether there is at least one root.
Any suggestions ... | 1)Using Newton's polygon method
2)Hensel's lemma
| {
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What are the best sites to get caught up on Calculus? I'm going back to college this summer and will be taking engineering statistics and calculus based physics. I dropped out of college about 4 years ago and took calculus 1-3 before leaving. I'm worried I have forgotten all of my calculus and won't be able to perfor... | All 3 mentioned, I find great for different things. Khan Academy for developing some insight (if you have forgotten the overarching idea), PatrickJMT for lots of examples of how to actually do questions, and Paul's Online Notes are also great for reading.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find the minimum of $(x(1+y)+y(1+z)+z(1+x))/\sqrt{xyz}$ over positive integers $x,y,z$ Let $x,y,z$ be positive integers.The least value
$$\frac{x(1+y)+y(1+z)+z(1+x)}{(xyz)^{\frac 12}}$$
I tried sum using arithmetic-geometric means inequality (seems promising as the denominator is similar to geometric mean of $x,y,z$).... | Notice
$$\frac{x(1+y)+y(1+z)+z(1+x)}{(xyz)^{\frac 12}} = \frac{x + xy + y + yz + z + zx}{(xyz)^{1/2}} \geq \frac{ 6 (x^3y^3z^3)^{1/6}}{(xyz)^{1/2}} = 6$$
By the AM-GM inequality
| {
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"timestamp": "2023-03-29T00:00:00",
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Last digits, numbers Can anyone please help me?
1) Find the last digit of $7^{12345}$
2) Find the last 2 digits of $3^{3^{2014}}$.
Attempt: 1)
By just setting the powers of $7$ we have $7^1 = 7$, $7^2=49$, $7^3=343$, $7^4 = 2401$, $7^5 = 16807$, $7^6 = 117649$, $\dots$
After the power of $4$, the last digits will rep... | Let $$3^{2014}-1=2x$$
Your number is : $3^{2x+1}=3\cdot3^{2x}=3(10-1)^{x}=3(-1+10)^{x}$
Using binomial theorem and neglecting powers of $10$ greater than $2$ as we want only last $2$ digits.
$$3(-1^{x}+x(-1)^{x-1}10)$$
Writing $3$ as $4-1$ in first expression will lead you to believe that $2x$ is divisible by $4$. Henc... | {
"language": "en",
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Beautiful Theorems and what constitutes as beautiful I often hear the phrase "mathematical beauty". That a proof or formula or theorem is beautiful. and I do agree I was awestruck when I first saw Euler's formula, connecting 3 seemingly unrelated branches of mathematics in a single formula $e^{i\pi}=-1$
But beauty is a... | If $M=M^2$ is a smooth compact $2$-dimensional Riemannian manifold with (smooth) boundary $\partial M$, $K$ denotes it's Gauß-curvature, $k_g$ the geodesic curvature of it's boundary und $\chi(M)$ the Euler-Characteristic, then the theorem of Gauß-Bonnnet states that
$$\int_M K dA + \int_{\partial M}k_g ds = 2\pi \chi(... | {
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"timestamp": "2023-03-29T00:00:00",
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can set $\mathcal{A}$ be written as union of countable set which are rare sets $\mathcal{A}$=the set of all fnite sequences in $l_1$
$l_1$: the space of sequences of $x_n$ s.t. : $\sum^\infty_1 |x_n|<\infty$
$A_n$ is rare set if the interior of the closure is empty,${Int}\bar A_n=\emptyset$,
is it possible that: $\mat... | Try $A_n$ = the set of all sequences of length $n$. It is clear that the union of $A_n$ is $\mathcal A$. It can also be shown that $A_n$ is closed and has an empty interior.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is the determinant differentiable? I was wondering, given an $n \times n$ square matrix, let function $\det : \left(a_1,a_2,\ldots,a_{n^2}\right) \to \textbf{R}$ give the determinant, where $a_{k}$'s are the entries of the $n \times n$ matrix.
*
*Is this function (determinant) a differentiable kind?
*If so, is the ... | If you would not know that the determinant is a polynomial in the entries of the matrix you may know that it is, if considered as a function of the columns (or rows) of the matrix, mulitilinear, hence $C^{\infty}$ as a function of the columns. Since the matrices depend smoothly on their entries they also depend smoothl... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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What can we say about the one point compactification of a Suslin tree? As a continuation to this question:
The Alexandroff one point compactification of $(X,\tau)$, is a space $X \cup \{a\}$, where open neighborhoods of $X$ are $\{ U : U \in \tau\} \cup \{ V \cup \{a\}: V \in \tau $ and $V^C$ is closed and compact in ... | From this answer we have that if $X = \{ \infty \} \cup T$ is the one-point compactification of an Aronszajn tree $T$ with the tree topology, then the sets of the form $$U ( s_1 , \ldots , s_n ) = \{ \infty \} \cup {\textstyle \bigcap_{i \leq n}} \{ x \in T : x \not\leq_T s_i \}$$ where $s_1 , \ldots , s_n \in T$ form ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$-5| 2+4x | = -32(x+3/4)- | x | + 1$ This was my attempt:
$$-5| 2+4x | = -32\left(x+\frac34\right)- | x | + 1\\
\implies|2+4x|=\frac{-32x-24- | x | + 1}{-5}\\
\implies2+4x=\pm \frac{-32x+-24- x + 1}{-5}\\
\implies4x=\pm \frac{-33x+-23 }{-5}-2\\
\implies-20x=\pm (-33x-23)+10\\
\implies-20x= -33x-13\text{ or} -20x=33x+3... | Hint:
There are two equations here. When you have absolute value, you have to have one equation where you use the original and another where the stuff inside the absolute value signs is negative. So try solving the equation where the stuff in the absolute value signs are negative.
| {
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Prove that if $\dim X'<\infty$ then $\dim X<\infty$ I have to prove that $\dim X'<\infty$ then $\dim X<\infty$ where $X$ is a normed vector space and $X'$ is a space of all linear and continuous functionals from $X$.
How can I prove this? I always try to figure out sth by myself before posting here, but this time I hav... | Here's another approach. Again, we let $X'$ be the continuous dual.
Well, if $\dim X' < \infty$ then $\dim X'' = \dim X' < \infty$. But then note
$X \subset X''$ (using the canonical embedding) so $X$ is then finite dimensional!
| {
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Modeling a greatest integer function I'm trying to model a function that resembles a greatest integer function. The domain is from [0, $\infty$). The inputs from 0 to 1.5 (non-inclusive) need to be mapped to an output of 0, and 1.5 to $\infty$ mapped to 1. But, I'm trying to not use a piecewise function. Is it possible... | Try this function,
$$\frac{|x-1.5|}{2x-3}+\frac12$$
Although this is very artificial, it works.
| {
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Is the $\sum_{n=1}^{\infty} \frac{(2n+1)^{1/2}}{n^2}$ convergent or divergent? For this question I am not really sure which test to use to determine this. I was thinking the comparison or limit comparison test but it doesn't seem to be working. I was wondering what the steps are to figure this out, and if it is the com... | We have
$$\frac{\sqrt{2n+1}}{n^2}\sim_\infty\sqrt2\frac{1}{n^{3/2}}$$
hence the given series is convergent by the asymptotic comparison with a convergent Riemann series.
| {
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Examples of properties not preserved under homomorphism An isomorphism indicates that two structures are the same, using different names for the elements. Therefore it's obvious that every (algebraic) property of the first structure must be present in the second.
However, homomorphisms only indicate that the two struct... | An image of an algebraic object is equivalently a quotient in the most elementary cases. Taking a quotient is an identification process, so a general class of properties not preserved under images are those relating to uniqueness of solutions of equations.
For instance, in any free abelian group a linear equation with... | {
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Are there complex solutions for $z^3=\bar z$ I'm asked to solve $z^3=\bar z$. I got $z=0, 1, -1$. Are there any complex solutions $a+bi$ to this though?
| Writing $z = re^{it}$, we have that
$$r^3 e^{3it} = re^{-it}$$
Taking absolute values, we find that $r^3 = r$, so that $r = 0$ or $r = \pm 1$. In the second two cases, we get that
$$e^{3it} = e^{-it} \implies e^{4it} = 1$$
It follows that $4t$ is an integer multiple of $2\pi$, so there are corresponding complex solutio... | {
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Is this a valid method of finding magnitude of complex fraction If I have a complex fraction $\dfrac{a+bi}{c+di}$ and I want the magnitude, then will it be $\left|\dfrac{a+bi}{c+di}\right|=\dfrac{|a+bi|}{|c+di|}$?
Scratch that ... I just found the answer on another page; however, I'm still unclear why it's true?
| You can make use of complex exponents.
$$\dfrac{a+\mathrm{i} \ b}{c+\mathrm{i} \ d}=\frac{\rho_1e^{\mathrm{i} \varphi_1}}{\rho_2e^{\mathrm{i} \varphi_2}}=\frac{\rho_1}{\rho_2}e^{\mathrm{i}(\varphi_1-\varphi_2)}$$
where $\rho_1=\sqrt{a^2+b^2}, \rho_2=\sqrt{c^2+d^2}$ are the magnitudes and $\varphi_1=\arg\{a+\mathrm{i} \... | {
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probility, placing balls, covariance Can you please help to see where I did wrong?
There are 10 balls, and each ball to be place in bin 1 and bin 2. Each ball is placed indepedently. Let X be the number of balls in bin 1 and Y be the number of balls in bin 2. Compute Cov(X,Y).
My attempt:
write X=X1+X2+-------+X10, whe... | Since each ball can go into bin 1 or bin 2, mutually exclusively and exhaustively, it's a binomial distribution:
$\operatorname{P}(X=x) = \dbinom{10}{x} \dfrac{1}{2^{10}} \\ \quad = \dfrac{10!}{x!(10-x)! 2^{10}}$
The expected value is thus:
$\operatorname{E}[X] = \sum\limits_{x=0}^{10} x\cdot\operatorname{P}(X=x) \\ \q... | {
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Showing that $38^n+31$ is prime I was reading a question in one of the previous pages, in searching for a proof I stumble across what seem like a contradiction. All I want is for someone to provide the missing link in my argument.
The question
Find the least $n$ for which $38^n+31$ is prime.
My attempt at a proof
I... | For $38^x+31$, it is useful to note that the small primes divide $38^x+1$, as $x$ is odd or double-odd. So, eg $3 \mid 38^x+1 $ for odd x. $5 \mid 3d^x + 1$ for $x=2 \pmod 4$. So after removing 3 and 5, one is left with $4 \mid x$.
Running through the output of factor for $38^{4x}+31$, gives some rather difficult... | {
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How $\tan{\frac{A}{2}}\tan{\frac{B}{2}}=\frac{1}{2}$,then find $\angle C$ In $\Delta ABC$, if $$\tan{\dfrac{A}{2}}\tan{\dfrac{B}{2}}=\dfrac{1}{2}\\\sin{\dfrac{A}{2}}\sin{\dfrac{B}{2}}\sin{\dfrac{C}{2}}=\dfrac{1}{10}$$
Find the $\angle C$
My try: since
$$2\sin{\dfrac{A}{2}}\sin{\dfrac{B}{2}}=\cos{\dfrac{A}{2}}\cos{\dfra... | Denote $~~a=\tan\dfrac{A}{2}$, $~~b=\tan\dfrac{B}{2}$ $($let $a\le b$$)$.
$\sin\dfrac{A}{2}\sin\dfrac{B}{2}\sin\dfrac{C}{2}=\dfrac{1}{10}$;
$\sqrt{\dfrac{1-\cos A}{2}} \cdot \sqrt{\dfrac{1-\cos B}{2}} \cdot \sqrt{\dfrac{1-\cos C}{2}} = \dfrac{1}{10}$;
$(1-\cos A) (1-\cos B)(1-\cos C) = \dfrac{2}{25}$;
$(1-\cos A) ~ (... | {
"language": "en",
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relation between singular values and eigenvalue How is this inequality proved
$\sigma_{min}(A) \leq \min_{i}|\lambda_i|\leq\max_{i}|\lambda_i| \leq \sigma_{max}(A) $
where $\sigma$ are the singular values and $\lambda $ are the eigen values of a matrix A
in the book i am reading (Matrix computations, Golub), it says ... | The point is that if $T=(t_{ij})\in\mathbb{C}^{n\times n}$ is triangular, then
$$\tag{$❀$}
\sigma_{\min}(T)\leq\min_i|t_{ii}|\leq\max_i|t_{ii}|\leq\sigma_{\max}(T).
$$
In other words,
$$
\sigma_{\min}(T)\leq|t_{ii}|\leq\sigma_{\max}(T), \quad i=1,\ldots,n.
$$
To see this, consider the vector $e_i$ (the $i$th column of ... | {
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Linear Algebra, geometric multiplicity I have a matrix and the question says I that I have an eigenvalue of 0.
The question asks me to find the geometric multiplicity of that eigenvalue. I know the answer is 4. I just don't understand how it is 4 since this matrix can be reduced to just one row of 1 1 1 1 1and th... | The geometric multiplicity of an eigenvalue $\lambda$ for the $n\times n$ matrix $A$ is, by definition, the dimension of the subspace
$$
E_A(\lambda)=\{v\in K^n:Av=\lambda v\}
$$
where $K$ is the base field, in your case probably $\mathbb{R}$ or $\mathbb{C}$, and the elements of $K^n$ are column vectors.
This subspace ... | {
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Cases of Partial Fraction Decomposition How many cases are there in integration using partial fractions?
| If I understood your question correctly, I would say there are $5$ cases.
Assume you have a rational function $\dfrac{p(x)}{q(x)}$, where the degree of $q(x)$ exceeds the degree of $p(x)$.
Case $1$: $q(x)$ is a product of distinct linear factors
Example: Consider $q(x)=\dfrac{x}{(x+3)(x-1)}$
Case $2$: $q(x)$ is a produ... | {
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A Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues - Proof Strategy [Lay P397 Thm 7.1.3c] Would someone please explain the proof strategy at Need verification - Prove a Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues? I brook the algebra so I'm not asking about for... | Regarding 1:
Intuitively, it comes down to the fact that we need to prove the fact that $\lambda \in \mathbb{R}$ using the facts that $A = A^*$ and $Av = \lambda v$ for some vector $v \neq 0$. In order to bring $A^*$ into play, we have to take the Hermitian conjugate of both sides at some point.
As for multiplying by... | {
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Sturm Liouville with periodic boundary conditions Background and motivation: I'm given the boundary value problem:
$$y''(x)+2y(x)=-f(x)$$
subject $y(0)=y(2\pi)$ and $y \, '(0)=y \, '(2\pi)$.
EDIT: These were not given to be zero !! Maybe this helps...
The text (Nagle Saff and Snider, end of Chapter 11 technical writ... | I will write $a$ for $\sqrt{2}$ for simplicity.
The general solution of the homogenous equation $y''+2y=0$ has the form
$$y(x)=C\cos(ax)+D\sin(ax)$$
Using the variation of parameters method to find a particular solution of the non homogenous problem, we have to determine $C$ and $D$ with
$$
\eqalign{C'\cos(ax)+D'\sin(a... | {
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prove that $P(X=0) \leq \frac{Var(X)}{E(X^{2})}$. Let $X$ be a random variable taking integral nonnegative values, let $E [X^2]$
denote the expectation of its square, and let $Var [X]$ denote its variance. Prove
that $P(X=0) \leq \frac{Var(X)}{E(X^{2})}$.
I try to use this theorem which will be prove... | Since $\text{Var}[X] \geq 0$ (this is a well-known fact), $E[X^2] - (E[X])^2 \geq 0$ and the inequality desired follows.
| {
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If $\sum_{n=1}^\infty a_n$ converges and $\sum_{n=1}^\infty \frac{\sqrt a_n}{n^p}$ diverges, then p $\in$ {?} Let {$a_n$} be a sequence of non-negative real numbers such that the series $$\sum_{n=1}^\infty a_n$$ is convergent.
If p is a real number such that the series $$\sum_{n=1}^\infty \frac{\sqrt a_n}{n^p}$$
diverg... | Seems like we'll have to make use of the A.M. $\ge$ G.M. inequality here, so that we have $$\frac {\sqrt{a_n}} {n^p} \leq a_n + \frac{1}{n^{2p}}$$
Now, this diverges for p $\le$ $\frac12$. So that must be the required answer.
Can anyone show how to do this using Cauchy-Schwarz inequality? It is simple I reckon.:)
| {
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Prove that this segment bisects another
The circle touches the trapezoid $GFEC$ at the points $C$, $D$ and $E$. The point $A$ is the center of the circle. The rest of the information can be seen in the diagrams below. What we have to prove is that $FI=HI$.
I've added some diagrams below. The first one simply shows wh... | Let $s$ the line defined by $FE$, $t$ the line defined by $CD$ And $J$ the point of intersection between $s$ and $t$. See the following figure:
Hints:
Note that $\angle GDC = \angle GCD = \angle FDJ = \angle DJF$.
Therefore $FJ=ED$.
But $FD=FE$ (Why?).
Note that $\triangle JFI \sim \triangle JEC$, hence ...
| {
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Why is $\lim\limits_{N\to\infty}x^{N+1}=0$, where $|x|<1$? How is this done?
Why is $\lim\limits_{N\to\infty}x^{N+1}=0$, where $-1<x<1$?
| A relatively "low-tech" way to see the limit must be zero (assuming the limit exists) is to call the limit $L$ and note that
$$
L = \lim_{N \to \infty} x^{N+1}
= \lim_{N \to \infty} (x \cdot x^{N})
= x \lim_{N \to \infty} x^{N}
= xL.
$$
Subtracting and factoring, $(1 - x)L = 0$. Since $x \neq 1$ by hypothesis, it... | {
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value of summation of $2^i\cdot i$ I'm trying to calculate the value of
$$2^0\cdot0 + 2^1\cdot1 + 2^2\cdot2 + .... 2^n\cdot n$$
I figured this would be summation $2^i \cdot i$ from $i = 0$ to $n$. But iI'm unable to calculate its value.
I have tried searching online but haven't been able to find a formula or any pro... | WolframAlpha claims that
$$
\sum_{n=0}^{N}2^{n}n=2+2^{N+1}(N-1)
$$
We can verify this using induction. Clearly it holds for $N=0$. Now assuming the formula holds for $N$, we have
$$
\sum_{n=0}^{N+1}2^{n}n=2+2^{N+1}(N-1)+2^{N+1}(N+1)=2+2^{n+1}(2*N)=2+2^{N+2}(N+1-1)
$$
And so the formula holds for all $N\in\{0,1,2,\ldots... | {
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What kind of matrix is it that when multiplied with its transpose produces the identity? If $A^TA = I$, where $A$ is a lower triangular matrix, does that mean $A$ has to be an identity matrix (and nothing else)? In general, which kind of matrix $A$ must be for that equality to hold?
| OK, edits complete!
Setting lower triangularity aside for the moment, take any $m$ mutually orthonormal vectors $\vec n_1, \vec n_2, \ldots, \vec n_m \in \Bbb R^m$ so that $\langle \vec n_j, \vec n_k \rangle = \delta_{jk}$, and turn them into a matrix $N$ by using them as columns, so we may write $N$ in columnar form ... | {
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Evaluate the integral of sec(2x + 1) dx I got $\ln|\sec(2x +1) + \tan(2x+1)| + \text C$ as an answer. I saw that the integral of $\sec x$ is $\ln|\sec x + \tan x| + \text C$. But I feel I may have left something out because that was too easy.
| $$\int\sec({2x+1})\,dx=\frac{1}{2}\int\sec({2x+1})\,d(2x+1)$$
Can you take it from here?
| {
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Rows of orthogonal matrix from an orthonormal basis of $R^n$ The question is:
Let $U$ be an $n \times n$ orthogonal matrix. Show that the rows of U form an orthonormal basis of $\mathbb R ^n $.
So far I have stated: Since $U$ is orthogonal its column vectors are linearly independent and by the Invertible Matrix theorem... | Hint: This is long (and can be very much shortened), but I think it will be a good learning curve for you to make it long. It became this long, because we start with the way you have understood orthogonal matrices (as seen in comments). I am reproducing it here "A set is orthogonal if each pair of distinct vectors with... | {
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Branch of logarithm which is real when z>0 I am familiar with the complex logarithm and its branches, but still this confuses me. I read this in a textbook:
"For complex $z\neq 0, log(z)$ denotes that branch of the logarithm which is real when $z > 0$.
What does this mean?
| Let $\hbox{Log} z=\ln|z|+i\hbox{Arg}z$ be the principal branch of the logarithm,
that corresponds to a cut along the negative real numbers. Now, consider any branch of $log$ that is defined on a connected neighborhood of $\Bbb{R}^+=(0,+\infty)$. Clearly,
$x\mapsto \varphi(x)=log(x)-\hbox{Log}(x)$ is a continuous funct... | {
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If $2xf ' (x) - f(x) = 0$ find $f$ So $2xf '(x) - f(x) = 0$ and we know that $f(1) =1$. So I actually need to find the integral of $2xf'(x) - f(x)$.
Thanks.
| $$\frac{f'(x)}{f(x)}=\frac{1}{2x} \Rightarrow \ln|f(x)|=\frac{1}{2} \ln{x} +c \Rightarrow \ln|f(x)|= \ln{{x}^{\frac{1}{2}}} +c \Rightarrow f(x)= \pm c_1 \sqrt{x} \Rightarrow f(x)=C \sqrt{x}$$
$$f(1)=1 \Rightarrow 1= C $$
So $$f(x)=\sqrt{x}$$
| {
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How can I Prove that H=G or K=G Let subgroups $H,K \subseteq G$ such that $G=H \cup K$
Prove $H=G$ or $K=G$
| In the special case when $G$ is a finite group, it is also possible to use a counting argument. (This is less general and a bit more complicated than the other answers, but I find the method interesting.)
Suppose $H$ and $K$ are both proper subgroups of $G$. Then $|H|$ and $|K|$ are proper divisors of $|G|$. Therefore,... | {
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Manipulating inequalities and probabilities We have the Chebychev's inequality $$\mathrm{P}\left(|X - \mu| \ge k\sigma\right) \le \frac{1}{k^{2}}$$ which tells us $\frac{1}{k^{2}}$ is the upper bound. How do we find the lower bound of this probability using this information?
| The lower bound is zero. Most of the time, you can construct examples that hit these bounds exactly. E.g., a uniform distribution has support on $\mu \pm \sigma \sqrt{3}$, so it places zero probability for $k\ge \sqrt{3}$. But Chebyshev's inequality does not know about it, and that's not its job, anyway. On the other b... | {
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series convergence test with parameter As part of a bigger proof I reached the series $\sum {1 \over n^\alpha}$. $\alpha \in \mathbb{R}$
Obviously, the convergence depends on the value of $\alpha$.
I already know the harmonic series diverges while $\sum {1\over n^2}$ converges.
How to solve the general case?
| Some notes:
*
*Because $\sum \tfrac 1n$ diverges you can prove the divergence for $\alpha \le 1$ with the limit comparison test
*Because $\sum \tfrac 1{n^2}$ converges you can prove the convergence for $\alpha \ge 2$ with the limit comparison test
*Concerning an article I have read, you can use the Cauchy condensa... | {
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Hyperbolic Fixed Point Let $f:M\rightarrow M$ be a $C^{1}$-class diffeomorphism . Let $x\in M$ be a fixed point.
I've been looking for a while on Internet for a proof of the following fact, but i couldn't find :
$\lbrace x\rbrace$ is a hyperbolic set for $f$ if and only if $x$ is a hyperbolic fixed point.
The definiti... | This is simply a matter of definition: a hyperbolic fixed point is defined to be a point $x$ such that $\{x\}$ is a hyperbolic set. See for example the wikipedia entry on hyperbolic sets where they use the term "hyperbolic equilibrium point" instead of "hyperbolic point".
| {
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Find $b$ such that $0 \le b < 101$ and $2^{987654321} \equiv b \pmod {101}$. Find the unique integer $b$ with $0 \le b < 101$ satisfying $2^{987654321} \equiv b \pmod {101}$.
I am having trouble just starting this problem. This is a home work problem that is going to be on my test and I just can not think of what to do... | Hint: $101$ is prime. Use Fermat's Little Theorem.
| {
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"source": "stackexchange",
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"answer_count": 1,
"answer_id": 0
} |
lifting a product of commutators of standard generators on 2-manifolds I have a problem with understand the proof http://www.ams.org/journals/proc/1972-032-01/S0002-9939-1972-0295352-2/S0002-9939-1972-0295352-2.pdf
I don't understand this part: "(...) we can easily construct $p: \tilde{F} \rightarrow F$ to be a six she... | Justin Malestein and I give a very explicit construction of such a cover in the proof of Lemma 2.1 of our paper
Malestein, Justin, Putman, Andrew;
On the self-intersections of curves deep in the lower central series of a surface group.
Geom. Dedicata 149 (2010), 73–84.
which is available here. We actually construct an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/769420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Bayes theorem application A professor gives a true-false exam consisting of thirty T-F questions. The questions whose answers are “true” are randomly distributed among the thirty questions. The professor thinks that 3/4 of the class are serious, and have correctly mastered the material, and that the probability of a co... | Let $A$ be the event the student is "serious" and $B$ the event she answers $2$ randomly chosen questions correctly. We want $\Pr(A|B)$, which is $\frac{\Pr(A\cap B)}{\Pr(B)}$.
We first calculate $\Pr(B)$. The event $B$ can happen in two ways: (i) the student is serious, and answers the two questions correctly and (ii... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/769579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Compute an improper integral. Suppose $A = [0,\infty) \times [0, \infty) $. Let $f(x,y) = (x+y)e^{-x-y} $. How can I find $ \int_A f $? I know since $f$ is continuous on $A$, then $\int _A f $ exists, Do I need to evaluate
$$ \int_{0}^{\infty} \int_{0}^{\infty} (x+y)e^{-x-y}\ dx\ dy\ ??$$
| HINT :
Rewrite:
$$ \int_{0}^{\infty} \int_{0}^{\infty} (x+y)e^{-x-y} dx dy =\int_{0}^{\infty} e^{-y} \int_{0}^{\infty} (x+y)e^{-x} dx dy\tag1
$$
Note that
$$
\int_0^\infty z^n e^{-z}\ dz=(n+1)!\tag2
$$
Use $(2)$ to evaluate $(1)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/769669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to count generators for a cyclic group Show that there are $\varphi (n)$ generators for a cyclic group $G$ of order $n$. Give their form explicitly. Here $\varphi (n)$ is the Euler's function. I don't know what to do, please help.
| Note that $G\cong\Bbb Z_n$, so we can suppose that $G=\{\bar0, \bar1,\ldots,\overline{n-1}\}$. I'll prove that these sentences are equivalent:
*
*$\bar a$ is a generator of $G$.
*For each integer $k$, $n$ divides $ak$ if and only if $n$ divides $k$
*$\gcd(a,n)=1$
$1\Rightarrow2$: Suppose that $\bar a$ generates ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/769790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Residues of $\frac{x^s}{s}\frac{\zeta'(s)}{\zeta\phantom{'}(s)}$ Going through a proof in Analyti number theory, the calculation
of the residues of
$$
f(s) = \frac{x^s}{s}\frac{\zeta'(s)}{\zeta\phantom{'}(s)}
$$
came up. I do have some experience with complex analysis so I tried
to compute the residues myself. So ... | Observe that poles and zeroes of $\zeta$ are all simple poles for its logarithmic derivative, hence the singularities of $s\longmapsto\frac{\zeta'(s)}{\zeta(s)}\frac{x^{s}}{s}$ lie all in halfplane $\{\sigma\leq1\}$.
Now, the poles of this function are: the pole of $\frac{x^{s}}{s}$ ($s=0$) which residue is
$$
Res\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/769888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If $a_n\to0,$ then $\sum a_n$ and $\sum (a_n + a_{n+1})$ converge/diverge together? Let $a_n$, a sequence suh that $\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=0$
and the series: $\sum a_n$, $\sum (a_n + a_{n+1})$
Prove/Disprove: The series converge/diverge together.
I'll be glad for an hint or a guidance.
| Hint: Consider $a_n = (-1)^n + 1/n^2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/769971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Differentiability of a Complex Function I'm just doubtful whether my proof(s) for showing functions are complex differentiable suffice as valid proofs.
e.g. let's take the classic example $f(z)=\bar{z}$.
You inevitably know from studying complex analysis that this function isn't differentiable for any $z \in \mathbb{... | Yes, the cauchy reimann equations are a necessary condition for complex differentiability.
So if they do not hold, then it is not complex differentiable. but it is good to know how to use the limits in case you are asked to specifically not use the equations.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/770096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How this absolutely convergent series property work? An absolutely cconvergent series may be multiplied with another absoultely convergent series. The limit of the product will be the product of the individual series limits.
How does it work?
| Consider absolutely convergent series $\sum a_i$ and $\sum b_i$, for $i \ge 0$. The product series is defined to be
$$
\sum_{n \ge 0} \left( \sum_{i=0}^n a_i b_{n-i} \right)
$$
Proof that this series is absolutely convergent
For any $N$, we have
\begin{align*}
\sum_{n = 0}^N \left| \sum_{i=0}^n a_i b_{n-i} \right|
&\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/770186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How do I solve this square root problem? I need to solve the following problem:
$$\frac{\sqrt{7+\sqrt{5}}}{\sqrt{7-\sqrt{5}}}=\,?$$
| \begin{align}
\frac{\sqrt{7+\sqrt{5}}}{\sqrt{7-\sqrt{5}}}&=\frac{\sqrt{7+\sqrt{5}}}{\sqrt{7-\sqrt{5}}}\cdot \frac{\sqrt{7+\sqrt{5}}}{\sqrt{7+\sqrt{5}}}\\
&=\frac{(\sqrt{7+\sqrt{5}})^2}{\sqrt{(7-\sqrt{5})(7+\sqrt{5})}}\\
&=\frac{7+\sqrt{5}}{\sqrt{7^2-(\sqrt{5})^2}}\\
&=\frac{7+\sqrt{5}}{\sqrt{49-5}}\\
&=\frac{7+\sqrt{5}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/770259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Use mathematical induction to prove that $(3^n+7^n)-2$ is divisible by 8 for all non-negative integers. Base step: $3^0 + 7^0 - 2 = 0$ and $8|0$
Suppose that $8|f(n)$, let's say $f(n)= (3^n+7^n)-2= 8k$
Then $f(n+1) = (3^{n+1}+7^{n+1})-2$
$(3*3^{n}+7*7^{n})-2$
This is the part I get stuck. Any help would be really appr... | $f(n)$ satisfies $f(n) = 11 f(n-1) - 31f(n-2) +21f(n-3)$
So by induction it's enough to check this is true for $n =0,1,2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/770344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Computing the Gaussian curvature of this surface $z=e^{(-1/2)(x^2+y^2)}$. Compute the Gaussian curvature of $z=e^{(-1/2)(x^2+y^2)}$. Sketch this surface and show where $K=0 $, $K>0$, and $K<0$.
So would the easiest way to do this question be to construct a parametrization $$\mathbf{x}(u,v)=(u, v, e^{-\frac{1}{2}(u^2+v^... | If $g$ is the first fundamental form, and $h$ is the second fundamental form. Then we know
$$K = \frac{ \det h}{ \det g } $$
Since you have a graph $z=f(x,y)$, it's straight forward to compute the fundamental forms by definition. I'll leave $g$ to you, but $h$ is given by
$$ h = f_{xx} dx^2 + 2 f_{xy} dxdy + f_{yy} dy^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/770524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Using L'Hospital's Rule to evaluate limit to infinity I'm given this problem and I'm not sure how to solve it. I was only ever given one example in class on using L'Hospital's rule like this, but it is very different from this particular problem. Can anyone please show me the steps to solve a problem like this?
Evaluat... | So the first step is to notice that the original equation can be simplified to
$$\lim_{x \rightarrow \infty }e^{\ln{\left( \left( 1+\frac{11}{x} \right) ^{\frac{x}{9}}\right)}}$$
Which can be made into: $$e^{\lim_{x \rightarrow \infty }\frac{x}{9}(\ln{ 1+\frac{11}{x}})}$$
So now we jsut need to solve for the exponent... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/770614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
How to compute time ordered Exponential? So say you have a matrix dependent on a variable t:
$A(t)$
How do you compute $e^{A(t)}$ ?
It seems Sylvester's formula, my standard method of computing matrix exponentials can't be applied here given the varying nature of the matrix and furthermore the fact that it may not alwa... | There seems to be some confusion here between two different problems: the first is the computation of a time-ordered (also called path-ordered) exponential, a matrix E(t',t) solving $\frac{d}{dt}E(t,0) = M(t)E(t,0) $ and the computation of the matrix exponential of a time dependent matrix $\exp(A(t))$.
Recall that the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/770679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
} |
Evaluate analytically...$\lim\limits_{x\to 0} \left(\frac{\sin(x)}{x}\right)^{\cot^2(x)}$ Can someone help? My professor likes to call these easy problems...Wolfram Alpha says the answer should be $e^{-1/6}$. Every time I do it I get something different. Help!
| $$\lim_{x \to 0} \left( \frac{\sin x}{x} \right) ^ {\cot^2{x}}$$
A simpler solution than l'Hopital's rule is to use Taylor expansions, as we are interested in the function as $x \to 0$. Note that $$\frac{\sin{x}}{x} = 1 - \frac{x^2}{6} + O(x^4)$$
Note that $\tan{x} = x + O(x^3)$, and thus $\cot^2{x} = 1/x^2 + O(x^{-4})... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/770759",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Inequality Problem of examination Show that if $n > 2$, then $(n!)^2 > n^n$.
I cannot find any way out to these.Please help.
I tried break $n!$ and then argue but failed.
| $$(n!)^2=1(n)\times2(n-1)\times3(n-2)\times \cdots n(1)$$
But
$$n\ge a+1 \implies a(n)\ge a(a+1)\implies (a+1)(n-a)\ge n$$
And equality is achieved iff $a=0$ or $a=n-1$. Since for $n>2$ we have $a=1\neq n-1$ in our product, we have the strict inequality $(n!)^2>n^n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/770950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Why does $\int^{ab}_{a} \frac{1}{x} dx = \int^{b}_{1} \frac{1}{t} dt$? I can't understand how the integral having limits from $a$ to $ab$ in Step 1 is equivalent to the integral having limits from $1$ to $b$. I'm a beginner here. Please explain in detail.
\begin{align*}
\ln(ab) = \int^{ab}_{1} \frac{1}{x} dx &= \int^{... | Given that the integral $\int_a^b f(x)\>dx$ is "the area under the curve $y=f(x)$ for $x$ between $a$ and $b$", the equality of the two integrals
$$\int_1^b{dx\over x},\quad \int_a^{ab} {dt\over t}$$
follows with an elementary geometric argument: The map $$(x,y)\mapsto\left(a x,\>{y\over a}\right)$$
which stretches by ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/771015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
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