Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Simplify fraction - Where did the rest go? While studying maths I encountered the following fraction :
$\frac{5ab}{10b}$
Which I then had to simplify. The answer I came up with is:
$\frac{5ab}{10b} = \frac{ab}{2b}$
But the correct answer seemed to be:
$\frac{5ab}{10b} = \frac{a}{2} = \frac{1}{2}$a
Why is the above answ... | To get from $\dfrac{5ab}{10b}$ to $\dfrac{ab}{2b}$ you probably divided the numerator and denominator each by $5$.
Now divide them each by $b$ (if $b \not = 0$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/74275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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There exists a real number $c$ such that $A+cI$ is positive when $A$ is symmetric Without using the fact that symmetric matrices can be diagonalized: Let $A$ be a real symmetric matrix. Show that there exists a real number $c$ such that $A+cI$ is positive.
That is, if $A=(a_{ij})$, one has to show that there exists rea... | Whether $x^TAx$ is positive doesn't depend on the normalization of $x$, so you only have to consider unit vectors. The unit sphere is compact, so the sum of the first two sums is bounded. The third sum is $1$, so you just have to choose $c$ greater than minus the lower bound of the first two sums.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/74351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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Is there a reason why curvature is defined as the change in $\mathbf{T}$ with respect to arc length $s$ And not with respect to time $t$? (or whatever parameter one is using)
$\displaystyle |\frac{d\mathbf{T}(t)}{\mathit{dt}}|$ seems more intuitive to me.
I can also see that $\displaystyle |\frac{d\mathbf{T}(t)}{\math... | The motivation is that we want curvature to be a purely geometric quantity, depending on the set of points making up the line alone and not the parametric formula that happened to generate those points.
$\left|\frac{dT}{dt}\right|$ does not satisfy this property: if I reparameterize by $t\to 2t$ for instance I get a cu... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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A question about hyperbolic functions Suppose $(x,y,z),(a,b,c)$ satisfy $$x^2+y^2-z^2=-1, z\ge 1,$$ $$ax+by-cz=0,$$ $$a^2+b^2-c^2=1.$$ Does it follow that $$z\cosh(t)+c\sinh(t)\ge 1$$ for all real number $t$?
| The curve $(X_1,X_2,X_3)=\cosh(t)(x,y,z)+\sinh(t)(a,b,c), -\infty<t<\infty$ is continuous and satisfies $X_1^2+X_2^2-X_3^2=-\cosh^2(t)+\sinh^2(t)=-1$. One of its point $(x,y,z)$ (when $t=0$) lies on the upper sheet $X_1^2+X_2^2-X_3^2=-1, X_3\ge 1$. By connectness of the curve, the whole curve must lie in this connecte... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/74468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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On a finite nilpotent group with a cyclic normal subgroup I'm reading Dummit & Foote, Sec. 6.1.
My question is the following.
If $G$ is a finite nilpotent group with a cyclic normal subgroup $N$ such that $G/N$ is also cyclic, when is $G$ abelian?
I know that dihedral groups are not abelian, and I think the question is... | Rod, you are right when you say this can be brought back to every Sylow group being abelian. Since $G$ is nilpotent you can reduce to $G$ being a $p$-group. However, a counterexample is easily found, take the quaternion group $G$ of order 8, generated by $i$ and $j$ as usual. Let $N$ be the subgroup $<i>$ of index 2. $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/74544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the math notation for this type of function? A function that turns a real number into another real number can be represented like $f : \mathbb{R}\to \mathbb{R}$
What is the analogous way to represent a function that turns an unordered pair of elements of positive integers each in $\{1,...,n\}$ into a real numbe... | I would say that it might be best to preface your notation with a sentence explaining it, which will allow the notation itself to be more compact, and generally increase the understanding of the reader. For example, we could write:
Let $X=\{x\in\mathbb{N}\mid x\leq N\}$, and let $\sim$ be an equivalence relation on $X... | {
"language": "en",
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Proof of inequality $\prod_{k=1}^n(1+a_k) \geq 1 + \sum_{k=1}^n a_k$ with induction I have to show that $\prod_{k=1}^n(1+a_k) \geq 1 + \sum_{k=1}^n a_k$ is valid for all $1 \leq k \leq n$ using the fact that $a_k \geq 0$.
Showing that it works for $n=0$ was easy enough. Then I tried $n+1$ and get to:
$$\begin{align*}
... | We want to show :
$$\left(\frac{1}{a_{n+1}}+1\right)\prod_{i=1}^{n}\left(1+a_{i}\right)>1+\frac{1}{a_{n+1}}+\sum_{i=1}^{n}\frac{a_{i}}{a_{n+1}}$$
We introduce the function :
$$f(a_{n+1})=\left(\frac{1}{a_{n+1}}+1\right)\prod_{i=1}^{n}\left(1+a_{i}\right)-1+\frac{1}{a_{n+1}}+\sum_{i=1}^{n}\frac{a_{i}}{a_{n+1}}$$
If we d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/74636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 2
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Injective functions also surjective? Is it true that for each set $M$ a given injective function $f: M \rightarrow M$ is surjective, too?
Can someone explain why it is true or not and give an example?
| This statement is true if $M$ is a finite set, and false if $M$ is infinite.
In fact, one definition of an infinite set is that a set $M$ is infinite iff there exists a bijection $g : M \to N$ where $N$ is a proper subset of $M$. Given such a function $g$, the function $f : M \to M$ defined by $f(x) = g(x)$ for all $x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is $[0,1]^\omega$ a continuous image of $[0,1]$? Is $[0,1]^\omega$, i.e. $\prod_{n=0}^\infty [0,1]$ with the product topology, a continuous image of $[0,1]$? What if $[0,1]$ is replaced by $\mathbb{R}$?
Edit: It appears that the answer is yes, and follows from the Hahn-Mazurkiewicz Theorem ( http://en.wikipedia.org/wik... | So if I'm reading correctly you want to find out if there is a continuous (with respect product topology) surjective map $f: \mathbb{R} \rightarrow \mathbb{R}^{\omega}$?
No, there is not. Note that $\mathbb{R}$ is $\sigma$-compact, so write:
$$\mathbb{R} = \bigcup_{n \in \mathbb{N}} [-n,n]$$
Then using the fact that $f... | {
"language": "en",
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Extending to a holomorphic function Let $Z\subseteq \mathbb{C}\setminus \overline{\mathbb{D}}$ be countable and discrete (here $\mathbb{D}$ stands for the unit disc).
Consider a function $f\colon \mathbb{D}\cup Z\to \mathbb{C}$ such that
1) $f\upharpoonright \overline{\mathbb{D}}$ is continuous
2) $f\upharpoonright \ma... | No.
The function $g(z) = 1+ 2z + \sum_{n=1}^{\infty} 2^{-n^2} z^{2^n}$ is holomorphic on the open disk $\mathbb{D}$ and infinitely often real differentiable in any point of the closed disk $\overline{\mathbb{D}}$ but cannot be analytically extended beyond $\overline{\mathbb{D}}$:
The radius of convergence is $1$. For ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/74901",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How do you parameterize a sphere so that there are "6 faces"? I'm trying to parameterize a sphere so it has 6 faces of equal area, like this:
But this is the closest I can get (simply jumping $\frac{\pi}{2}$ in $\phi$ azimuth angle for each "slice").
I can't seem to get the $\theta$ elevation parameter correct. Help... | The following doesn't have much to do with spherical coordinates, but it might be worth noting that these 6 regions can be seen as the projections of the 6 faces of an enclosing cube.
In other words, each of the 6 regions can be parametrized as the region of the sphere
$$S=\{\{x,y,z\}\in\mathbb R^3\mid x^2+y^2+z^2=1\}$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do we check Randomness? Let's imagine a guy who claims to possess a machine that can each time produce a completely random series of 0/1 digits (e.g. $1,0,0,1,1,0,1,1,1,...$). And each time after he generates one, you can keep asking him for the $n$-th digit and he will tell you accordingly.
Then how do you check i... | All the sequences you mentioned have a really low Kolmogorov complexity, because you can easily describe them in really short space. A random sequence (as per the usual definition) has a high Kolmogorov complexity, which means there is no instructions shorter then the string itself that can describe or reproduce the st... | {
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How many ways can 8 people be seated in a row? I am stuck with the following question,
How many ways can 8 people be seated in a row?
if there are 4 men and 4 women and no 2 men or women may sit next to each other.
I did it as follows,
As 4 men and 4 women must sit next to each other so we consider each of them as ... | If there is a man on the first seat, there has to be a woman on the second, a man on the third so forth. Alternatively, we could start with a woman, then put a man, then a woman and so forth. In any case, if we decide which gender to put on the first seat, the genders for the others seats are forced upon us. So there a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Probability that no two consecutive throws of some (A,B,C,D,E,F)-die show up consonants I have a question on probability. I am looking people presenting different approaches on solving this. I already have one solution but I was not satisfied like a true mathematician ;).....so go ahead and take a dig.....if no one ans... | Here is a solution different from the one given on the page @joriki links to. Call $c_n$ the probability that no two consecutive consonants appeared during the $n$ first throws and that the last throw produces a consonant. Call $b_n$ the probability that no two consecutive consonants appeared during the $n$ first throw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/75098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? How can one prove the statement
$$\lim_{x\to 0}\frac{\sin x}x=1$$
without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.
This is homework. In my math class, we are about to prove that $\sin$ is continuous. We found out... | Usual proofs can be circular, but there is a simple way for proving such inequality.
Let $\theta$ be an acute angle and let $O,A,B,C,D,C'$ as in the following diagram:
We may show that:
$$ CD \stackrel{(1)}{ \geq }\;\stackrel{\large\frown}{CB}\; \stackrel{(2)}{\geq } CB\,\stackrel{(3)}{\geq} AB $$
$(1)$: The quadrilat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/75130",
"timestamp": "2023-03-29T00:00:00",
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"answer_count": 28,
"answer_id": 18
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Euclidean distance vs Squared So I understand that Euclidean distance is valid for all of properties for a metric. But why doesn't the square hold the same way?
| You lose the triangle inequality if you don’t take the square root: the ‘distance’ from the origin to $(2,0)$ would be $4$, which is greater than $2$, the sum of the ‘distances’ from the origin to $(1,0)$ and from $(1,0)$ to $(2,0)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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mean and std deviation of a population equal? Hypothetically, if we have a population of size $n$ whose mean and std deviation are equal, I think with some work we have a constraint that the ratio, (Sum of squared points)/(Sum of points$)^2$ $= \frac{(2n-1)}{n^2}$, which gets small quickly as $n$ gets large. Are there ... | The distributions of exponential type whose variance and mean are related, so that $\operatorname{Var}(X) \sim (\mathbb{E}(X))^p$ for a fixed $p$, are called an index parameter, are known as Tweedie family.
The case you are interested in corresponds to index $p =2$. $\Gamma$-distribution possesses this property (the ex... | {
"language": "en",
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How many everywhere defined functions are not $ 1$ to $1$ I am stuck with the following question,
How many everywhere defined functions from S to T are not one to one
S={a,b,c,d,e}
T={1,2,3,4,5,6}
Now the teacher showed that there could be $6^5$ ways to make and everywhere defined function and $6!$ ways of it to be $1... | You have produced a complete and correct list of all ordered pairs $(x,y)$, where $x$ ranges over $S$ and $y$ ranges over $T$. However, this is not the set of all functions from $S$ to $T$.
Your list, however, gives a nice way of visualizing all the functions. We can produce all the functions from $S$ to $T$ by pickin... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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The Pigeon Hole Principle and the Finite Subgroup Test I am currently reading this document and am stuck on Theorem 3.3 on page 11:
Let $H$ be a nonempty finite subset of a group
$G$. Then $H$ is a subgroup of $G$ if $H$ is closed
under the operation of $G$.
I have the following questions:
1.
It suffices to... | To show $H$ is a subgroup you must show it's closed, contains the identity, and contains inverses. But if it's closed, non-empty, and contains inverses, then it's guaranteed to contain the identity, because it's guaranteed to contain something, say, $x$, then $x^{-1}$, then $xx^{-1}$, which is the identity.
$H$ is ass... | {
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"url": "https://math.stackexchange.com/questions/75371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $\lim \limits_{n\to\infty}\frac{n}{n^2+1} = 0$ from the definition This is a homework question:
Prove, using the definition of a limit, that
$$\lim_{n\to\infty}\frac{n}{n^2+1} = 0.$$
Now this is what I have so far but I'm not sure if it is correct:
Let $\epsilon$ be any number, so we need to find an $M... | First, $\epsilon$ should not be "any number", it should be "any positive number."
Now, you are on the right track. What do you need in order for $\frac{n}{n^2+1}$ to be smaller than $\epsilon$? You need $n\lt \epsilon n^2 + \epsilon$. This is equivalent to requiring
$$\epsilon n^2 - n + \epsilon \gt 0.$$
You want to fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/75429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Number of point subsets that can be covered by a disk Given $n$ distinct points in the (real) plane, how many distinct non-empty subsets of these points can be covered by some (closed) disk?
I conjecture that if no three points are collinear and no four points are concyclic then there are $\frac{n}{6}(n^2+5)$ distinct ... | When $n=6$, consider four points at the corner of a square, and two more points very close together near the center of the square. To be precise, let's take points at $(\pm1,0)$ and $(0,\pm1)$ and at $(\epsilon,\epsilon)$ and $(-2\epsilon,\epsilon)$ for some small $\epsilon>0$. Then if I'm not mistaken, the number of n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/75487",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Filter to obtain MMSE of data from Gaussian vector Data sampled at two time instances giving bivariate Gaussian vector $X=(X_1,X_2)^T$ with
$f(x_1,x_2)=\exp(-(x_1^2+1.8x_1x_2+x_2^2)/0.38)/2\pi \sqrt{0.19}$
Data measured in noisy environment with vector:
$(Y_1,Y_2)^T=(X_1,X_2)^T+(W_1,W_2)^T$
where $W_1,W_2$ are both $i... | What you need is $\mathbb{E}(X_1 \mid Y_1, Y_2)$. We have
$$
\operatorname{var}\begin{bmatrix} X_1 \\ Y_1 \\ Y_2 \end{bmatrix} = \left[\begin{array}{r|rr} 1 & 1 & -0.9 \\ \hline1 & 1.02 & -0.9 \\ -0.9 & -0.9 & 1.02 \end{array}\right]= \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{12}^\top & \Sigma_{22} \end... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to prove that $\sum\limits_{n=1}^\infty\frac{(n-1)!}{n\prod\limits_{i=1}^n(a+i)}=\sum\limits_{k=1}^\infty \frac{1}{(a+k)^2}$ for $a>-1$? A problem on my (last week's) real analysis homework boiled down to proving that, for $a>-1$,
$$\sum_{n=1}^\infty\frac{(n-1)!}{n\prod\limits_{i=1}^n(a+i)}=\sum_{k=1}^\infty \frac{... | This uses a reliable trick with the Beta function. I say reliable because you can use the beta function and switching of the integral and sum to solve many series very quickly.
First notice that $$\prod_{i=1}^{n}(a+i)=\frac{\Gamma(n+a+1)}{\Gamma(a+1)}.$$ Then
$$\frac{(n-1)!}{\prod_{i=1}^{n}(a+i)}=\frac{\Gamma(n)\Gamma... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Conditional expectation of $\max(X,Y)$ and $\min(X,Y)$ when $X,Y$ are iid and exponentially distributed I am trying to compute the conditional expectation $$E[\max(X,Y) | \min(X,Y)]$$ where $X$ and $Y$ are two iid random variables with $X,Y \sim \exp(1)$.
I already calculated the densities of $\min(X,Y)$ and $\max(X,Y... | For two independent exponential distributed variables $(X,Y)$, the joint distribution is
$$
\mathbb{P}(x,y) = \mathrm{e}^{-x-y} \mathbf{1}_{x >0 } \mathbf{1}_{y >0 } \, \mathrm{d} x \mathrm{d} y
$$
Since $x+y = \min(x,y) + \max(x,y)$, and $\min(x,y) \le \max(x,y)$ the joint distribution of
$(U,V) = (\min(X,Y), \m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/75732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Showing $f^{-1}$ exists where $f(x) = \frac{x+2}{x-3}$ Let $f(x) = \dfrac{x + 2 }{x - 3}$.
There's three parts to this question:
*
*Find the domain and range of the function $f$.
*Show $f^{-1}$ exists and find its domain and range.
*Find $f^{-1}(x)$.
I'm at a loss for #2, showing that the inverse function exists... | It is a valid way to find the inverse by solving for x first and then verify that $f^{-1}(f(x))=x$ for all $x$ in your domain. It is quite preferable to do it here because you need it for 3. anyways.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/75839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Second countability and products of Borel $\sigma$-algebras We know that the Borel $\sigma$-algebra of the Cartesian product space (with the product topology) of two topological spaces is equal to the product
of the Borel $\sigma$-algebras of the factor spaces. (The product
$\sigma$-algebra can be defined via pullback... | This is not even true for a product of two spaces: see this Math Overflow question. To rephrase, second countability can be important even for products of two topological spaces.
| {
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"url": "https://math.stackexchange.com/questions/75890",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
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Expected value of the stochastic integral $\int_0^t e^{as} dW_s$ I am trying to calculate a stochastic integral
$\mathbb{E}[\int_0^t e^{as} dW_s]$. I tried breaking it up into a Riemann sum
$\mathbb{E}[\sum e^{as_{t_i}}(W_{t_i}-W_{t_{i-1}})]$, but I get expected value of $0$, since $\mathbb{E}(W_{t_i}-W_{t_{i-1}}) =0... | The expectation of the Ito integral $\mathbb{E}( \int_0^t \mathrm{e}^{a s} \mathrm{d} W_s )$ is zero as George already said.
To compute $\mathbb{E}( W_t \int_0^t \mathrm{e}^{a s} \mathrm{d} W_s )$, write $W_t = \int_0^t \mathrm{d} W_s$. Then use Ito isometry:
$$
\mathbb{E}( W_t \int_0^t \mathrm{e}^{a s} \mathrm{d} W_s... | {
"language": "en",
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The tricky time complexity of the permutation generator I ran into tricky issues in computing time complexity of the permutation generator algorithm, and had great difficulty convincing a friend (experienced in Theoretical CS) of the validity of my reasoning. I'd like to clarify this here.
Tricky complexity question ... | Your friend's bound is rather weak. Let the input number be $x$. Then the output is $x!$ permutations, but the length of each permutation isn't $x$ bits, as you claim, but $\Theta(\lg(x!)) = \Theta(x \lg x)$ bits. Therefore the total output length is $\Theta(x! \;x \lg x)$ bits.
But, as @KeithIrwin has already pointed ... | {
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"answer_id": 2
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Homeomorphism between two spaces I am asked to show that $(X_{1}\times X_{2}\times \cdots\times X_{n-1})\times X_{n}$ is homeomorphic to $X_{1}\times X_{2}\times \cdots \times X_{n}$. My guess is that the Identity map would work but I am not quite sure. I am also wondering if I could treat the the set $(X_{1}\times X_{... | Let us denote $A = X_1\times \cdots \times X_{n-1}$ and $X = X_{1}\times \cdots\times X_{n-1}\times X_n$. The box topology $\tau_A$ on $A$ is defined by the basis of open product sets:
$$
\mathcal B(A) = \{B_1\times\cdots \times B_{n-1}:B_i \text{ is open in } X_i,1\leq i\leq n-1\}.
$$
The box topology $\tau_X$ on $X... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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If G is a group of order n=35, then it is cyclic I've been asked to prove this.
In class we proved this when $n=15$, but our approached seemed unnecessarily complicated to me. We invoked Sylow's theorems, normalizers, etc. I've looked online and found other examples of this approach.
I wonder if it is actually unnec... | Another explicit example:
Consider
$$
A = \left( \begin{array}{cc}
1 & -1
\\
0 & -1
\end{array} \right), \quad \text{and} \quad B = \left(\begin{array}{cc}
1 & 0
\\
0 & -1
\end{array}
\right).
$$
Then, $A^2 = B^2 = I$, but
$$
AB = \left( \begin{array}{cc}
1 & 1
\\
0 & 1
\end{array}
\right)
$$
has... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/76112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 7,
"answer_id": 4
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What would be the radius of convergence of $\sum\limits_{n=0}^{\infty} z^{3^n}$? I know how to find the radius of convergence of a power series $\sum\limits_{n=0}^{\infty} a_nz^n$, but how does this apply to the power series $\sum\limits_{n=0}^{\infty} z^{3^n}$? Would the coefficients $a_n=1$, so that one may apply D'A... | What do you do for the cosine and sine series? There, you cannot use the Ratio Test directly because every other coefficient is equal to $0$. Instead, we do the Ratio Test on the subsequence of even (resp. odd) terms. You can do the same here. We have $a_{3^n}=1$ for all $n$, and $a_j=0$ for $j$ not a power of $3$. Def... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/76173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Infinitely many $n$ such that $p(n)$ is odd/even?
We denote by $p(n)$ the number of partitions of $n$. There are infinitely many integers $m$ such that $p(m)$ is even, and infinitely many integers $n$ such that $p(n)$ is odd.
It might be proved by the Euler's Pentagonal Number Theorem. Could you give me some hints?
| Hint: Look at the pentagonal number theorem and the recurrence relation it yields for $p(n)$, and consider what would happen if $p(n)$ had the same parity for all $n\gt n_0$.
| {
"language": "en",
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"source": "stackexchange",
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Lie bracket and covariant derivatives I came across the following equality
$[\text{grad} f, X] = \nabla_{\text{grad} f} X + \nabla_X \text{grad} f$
Is this true, and how can I prove this (without coordinates)?
| No. Replace all three occurrences of the gradient by any vector field, call it $W,$ but then replace the plus sign on the right hand side by a minus sign, and you have the definition of a torsion-free connection, $$ \nabla_W X - \nabla_X W = [W,X].$$ If, in addition, there is a positive definite metric, the Levi-Civit... | {
"language": "en",
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Prove sequence $a_n=n^{1/n}$ is convergent How to prove that the sequence $a_n=n^{1/n}$ is convergent using definition of convergence?
| Well, the easiest proof is that the sequence is decreasing and bounded below (by 1); thus it converges by the Monotone Convergence Theorem...
The proof from definition of convergence goes like this:
A sequence $a_{n}$ converges to a limit L in $\mathbb{R}$ if and only if $\forall \epsilon > 0 $, $\exists N\in\mathbb{N}... | {
"language": "en",
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Simplify an expression to show equivalence I am trying to simplify the following expression I have encountered in a book
$\sum_{k=0}^{K-1}\left(\begin{array}{c}
K\\
k+1
\end{array}\right)x^{k+1}(1-x)^{K-1-k}$
and according to the book, it can be simplified to this:
$1-(1-x)^{K}$
I wonder how is it done? I've tried t... | Simplify[PowerExpand[Simplify[Sum[Binomial[K, k + 1]*x^(k + 1)*(1 - x)^(K - k - 1), {k, 0, K - 1}], K > 0]]] works nicely. The key is in the use of the second argument of Simplify[] to add assumptions about a variable. and using PowerExpand[] to distribute powers.
| {
"language": "en",
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"source": "stackexchange",
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What row am I on in a matrix if I only have the column index of an element, and the total number of columns and rows? Assume a 3 x 3 grid, but the elements are actually contained in a zero-based index array. So I know if I'm on element 5, I'm in row 2, and if I'm on element 7 then I'm in row 3. How would I actually cal... | The logic used is fairly simple. If you have a 3 by 3 grid, starting at 0, the elements would look as:
0 1 2
3 4 5
6 7 8
You count from left to right, until the number of columns matches the number of columns the grid has, then start again with the next row.
Mathematically, the row is floor(elementNumber / numberOfR... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Cauchy's Theorem (Groups) Question? I'm afraid at this ungodly hour I've found myself a bit stumped. I'm attempting to answer the following homework question:
If $p_1,\dots,p_s$ are distinct primes, show that an abelian group of order $p_1p_2\cdots p_s$ must be cyclic.
Cauchy's theorem is the relevant theorem to th... | Start by proving that, in an abelian group, if $g$ has order $a$, and $h$ has order $b$, and $\gcd(a,b)=1$, then $gh$ has order $ab$. Clearly, $(gh)^{ab}=1$, so $gh$ has order dividing $ab$. Now show that if $(gh)^s=1$ for some $s\lt ab$ then you can find some $r$ such that $(gh)^{rs}$ is either a power of $g$ or of $h... | {
"language": "en",
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Prove that: set $\{1, 2, 3, ..., n - 1\}$ is group under multiplication modulo $n$? Prove that:
The set $\{1, 2, 3, ..., n - 1\}$ is a group under multiplication modulo $n$ if and only if $n$ is a prime number without using Euler's phi function.
| Assume that $H=\{1,2,3,...n-1\}$ is a group. Suppose that $n$ is not a prime.
Then $n$ is composite, i.e $n=pq$ for $1<p,q<n-1$ . This implies that $pq \equiv0(mod n)$ but $0$ is not in H. Contradiction, hence $n$ must be prime.
Conversely, Suppose $n$ is a prime then $gcd(a,n)=1$ for every a in H. Therefore, $ax=1-ny... | {
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How many correct answers does it take to fill the Trivial Pursuit receptacle? My friends and I likes to play Trivial Pursuit without using the board.
We play it like this:
*
*Throw a die to determine what color you get to answer.
*Ask a question, if the answer is correct you get a point.
*If enough points are awa... | This is the coupon collector's problem. For six, on average you will need $6/6+6/5+6/4+6/3+6/2+6/1=14.7$ correct answers, but the variability is high. This is the expectation, not the number to have 50% chance of success.
| {
"language": "en",
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"source": "stackexchange",
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Measure-theoretical isomorphism between interval and square What is an explicit isomorphism between the unit interval $I = [0,1]$ with Lebesgue measure, and its square $I \times I$ with the product measure? Here isomorphism means a measure-theoretic isomorphism, which is one-one outside some set of zero measure.
| For $ x \in [0,1]$, let $x = .b_1 b_2 b_3 \ldots$ be its base-2 expansion (the choice in the ambiguous cases doesn't matter, because that's a set of measure 0). Map this to
$(.b_1 b_3 b_5 \ldots,\ .b_2 b_4 b_6 \ldots) \in [0,1]^2$
| {
"language": "en",
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Solution to an ODE, can't follow a step of a Stability Example In my course notes, we are working on the stability of solutions, and in one example we start out with:
Consider the IVP on $(-1,\infty)$:
$x' = \frac{-x}{1 + t}$ with $x(t_{0}) = x_{0}$.
Integrating, we get $x(t) = x(t_{0})\frac{1 + t_{0}}{1 + t}$.
I can'... | Separate variables and get $\int 1/x \,dx = \int -1/(1+t)\,dt$. Then $\ln|x|=-\ln|1+t|+C$
Exponentiate both sides and get $|x| = e^{-\ln|1+t|+C}$ and so $|x|=e^{\ln|(1+t)^{-1}|}e^C$
Relabel the constant drop absolute values and recover lost zero solution (due to division by $x$) and get $x=Ce^{\ln|(1+t)^{-1}|}=C(1+t)^{... | {
"language": "en",
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Adjunction of a root to a UFD Let $R$ be a unique factorization domain which is a finitely generated $\Bbbk$-algebra for an algebraically closed field $\Bbbk$. For $x\in R\setminus\{0\}$, let $y$ be an $n$-th root of $x$. My question is, is the ring
$$ A := R[y] := R[T]/(T^n - x) $$
a unique factorization domain as wel... | I think you can get a counterexample to the unit question, even in characteristic zero, and even in an integral domain (in contrast to Georges' example), although there are a few things that need checking.
Let $R={\bf C}[x,1/(x^2-1)]$, so $1-x^2$ is a unit in $R$. Then $$(1+\sqrt{1-x^2})(1-\sqrt{1-x^2})=xx$$
It rem... | {
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Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be entire and $\exists M \in\mathbb{R}: $Re$(f(z))\geq M$ $\forall z\in\mathbb{C}$. Prove $f(z)=$constant
Possible Duplicate:
Liouville's theorem problem
Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be entire and suppose $\exists M \in\mathbb{R}: $Re$(f(z))\geq M$ $\forall z\in... | Consider the function $\displaystyle g(z)=e^{-f(z)}$. Note then that $\displaystyle |g(z)|=e^{-\text{Re}(f(z))}\leqslant \frac{1}{e^M}$. Since $g(z)$ is entire we may conclude that it is constant (by Liouville's theorem). Thus, $f$ must be constant.
| {
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Calculate the expansion of $(x+y+z)^n$ The question that I have to solve is an answer on the question "How many terms are in the expansion?".
Depending on how you define "term" you can become two different formulas to calculate the terms in the expansion of $(x+y+z)^n$.
Working with binomial coefficients I found that ... | For the non-trivial interpretation, you're looking for non-negative solutions of $a + b + c = n$ (each of these corresponds to a term $x^a y^b z^c$). Code each of these solutions as $1^a 0 1^b 0 1^c$, for example $(2,3,5)$ would be coded as $$110111011111.$$ Now it should be easy to see why the answer is $\binom{n+2}{n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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what is the tensor product $\mathbb{H\otimes_{R}H}$ I'm looking for a simpler way of thinking about the tensor product: $\mathbb{H\otimes_{R}H}$, i.e a more known algbera which is isomorphic to it.
I have built the algebra and played with it for a bit, but still can't seem to see any resemblence to anything i already k... | Hint :
(1) Show that the map
$$H \otimes_R H \rightarrow End_R(H), x \otimes y \mapsto (a \mapsto xay).$$
is an isomorphism of $R$-vector spaces (I don't know the simplest way to do this, but try for example to look at a basis (dimension is 16...)).
(2) Denote by $H^{op}$ the $R$-algebra $H$ where the multiplication is... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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The norm of $x\in X$, where $X$ is a normed linear space Question:
Let $x\in X$, $X$ is a normed linear space and let $X^{*}$ denote the dual space of $X$.
Prove that$$\|x\|=\sup_{\|f\|=1}|f(x)|$$ where $f\in X^{*}$.
My proof:
Let $0\ne x\in X$, using HBT take $f\in X^{*}$ such that $\|f\|=1$ and $f(x)=\|x\|$.
N... | Thanks for the comments. Let see....
Let $0\ne x\in X$, using the consequence of HBT (analytic form) take $g\in X^{*}$ such that $\|g\|=1$ and $
g(x)=\|x\|$.
Now, $\|x\|=g(x)\le|g(x)|\le\sup_{\|f\|=1}|f(x)|$, this implies $$\|x\|\le\sup_{\|f\|=1}|f(x)|\quad (1)$$
Since $f$ is a bounded linear functional (given): $|f... | {
"language": "en",
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Right angles in the clock during a day Can someone provide a solution for this question ...
Given the hours , minutes and seconds hands calculate the number of right angles the three hands make pairwise with respect to each other during a day... So it asks for the second and hour angle , minute and hour and second and ... | Take two hands: a fast hand that completes $x$ revolutions per day, and a slow hand that completes $y$ revolutions per day. Now rotate the clock backwards, at a rate of $y$ revolutions per day: the slow hand comes to a standstill, and the fast hand slows down to $x-y$ revolutions per day. So the number of times that th... | {
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How to prove that $\lim\limits_{h \to 0} \frac{a^h - 1}{h} = \ln a$ In order to find the derivative of a exponential function, on its general form $a^x$ by the definition, I used limits.
$\begin{align*}
\frac{d}{dx} a^x & = \lim_{h \to 0} \left [ \frac{a^{x+h}-a^x}{h} \right ]\\ \\
& =\lim_{h \to 0} \left [ \frac{a^... | It depends a bit on what you're prepared to accept as "basic algebra and exponential and logarithms properties". Look first at the case where $a$ is $e$. You need to know that $\lim_{h\to0}(e^h-1)/h)=1$. Are you willing to accept that as a "basic property"? If so, then $a^h=e^{h\log a}$ so $$(a^h-1)/h=(e^{h\log a}-1)/h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/77348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Question about two simple problems on covering spaces Here are two problems that look trivial, but I could not prove.
i) If $p:E \to B$ and $j:B \to Z$ are covering maps, and $j$ is such that the preimages of points are finite sets, then the composite is a covering map.
I suppose that for this, the neighborhood $U$ tha... | Lets call an open neighborhood $U$ of a point $y$ principal (wrt. a covering projection $p: X \to Y$), if it's pre image $p^{-1}(U)$ is a disjoint union of open sets, which are mapped homeomorphically onto $U$ by $p$.
By definition a covering projection is a surjection $p: X \to Y$, such that every point has a principa... | {
"language": "en",
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A property of Hilbert sphere Let $X$ be (Edit: a closed convex subset of ) the unit sphere $Y=\{x\in \ell^2: \|x\|=1\}$ in $\ell^2$ with the great circle (geodesic) metric. (Edit: Suppose the diameter of $X$ is less than $\pi/2$.) Is it true that every decreasing sequence of nonempty closed convex sets in $X$ has a non... | No. For example, let $A_n$ be the subset of $X$ consisting of vectors that are zero in the first $n$ co-ordinates.
EDIT: this assumes that when $x$ and $y$ are antipodal, convexity of $S$ containing $x$, $y$ only requires that at least one of the great-circle paths is contained in $S$. If it requires all of them, then ... | {
"language": "en",
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Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$. Why is
$$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$
Can we generalize it to any exponent $x \in \Bbb R$? This is to say, is
$$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$
This is being repurposed in an effort to cut down on duplic... | The Stirling's formula says that:
$$ n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n, $$
inasmuch as
$$ \lim_{n \to \infty} \frac{n!}{\sqrt{2 \pi n} \left(\displaystyle\frac{n}{e}\right)^n} = 1, $$
thearebfore
$$
\begin{aligned}
\lim_{n \to \infty} \frac{2^n}{n!} & = \lim_{n \to \infty} \frac{2^n}{\sqrt{2 \pi n} \lef... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove that $ f(x) = \sum_{k=1}^\infty \frac{\sin((k + 1)!\;x )}{k!}$ is nowhere differentiable This function is continuous, it follows by M-Weierstrass Test. But proving non-differentiability, I think it's too hard. Does someone know how can I prove this? Or at least have a paper with the proof?
The function is ... | (Edited: handwaving replaced by rigor)
For conciseness, define the helper functions $\gamma_k(x)=\sin((k+1)!x)$. Then $f(x)=\sum_k \frac{\gamma_k(x)}{k!}$.
Fix an arbitrary $x\in\mathbb R$. We will construct a sequence $(x_n)_n$ such that
$$\lim_{n\to\infty} x_n = x \quad\land\quad \lim_{n\to\infty} \left|\frac{f(x_n)-... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is the power set of the natural numbers countable? Some explanations:
A set S is countable if there exists an injective function $f$ from $S$ to the natural numbers ($f:S \rightarrow \mathbb{N}$).
$\{1,2,3,4\}, \mathbb{N},\mathbb{Z}, \mathbb{Q}$ are all countable.
$\mathbb{R}$ is not countable.
The power set $\mathcal ... | Power set of natural numbers has the same cardinality with the real numbers. So, it is uncountable.
In order to be rigorous, here's a proof of this.
| {
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An inequality for graphs In the middle of a proof in a graph theory book I am looking at appears the inequality
$$\sum_i {d_i \choose r} \ge n { m /n \choose r},$$
and I'm not sure how to justify it. Here $d_i$ is the degree of vertex $i$ and the sum is over all $n$ vertices. There are $m$ edges. If it is helpful I... | Fix an integer $r \geq 1$. Then the function $f: \mathbb R^{\geq 0} \to \mathbb R^{\geq 0}$ given by
$$
f(x) := \binom{\max \{ x, r-1 \}}{r}
$$
is both monotonically increasing and convex.* So applying Jensen's inequality to $f$ for the $n$ numbers $d_1, d_2, \ldots, d_n$, we get
$$
\sum_{i=1}^n f(d_i) \geq n \ f\le... | {
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Equal simple field extensions? I have a question about simple field extensions.
For a field $F$, if $[F(a):F]$ is odd, then why is $F(a)=F(a^2)$?
| Since $[F(a):F]$ is odd the minimal polynomial of $a$ is an odd degree polynomial, say $p(x)=b_{0}x^{2k+1}+b_1x^{2k}+...b_{2k+1}$, now since $a$ satisfies $p(x)$ we have: $b_{0}a^{2k+1}+b_1a^{2k}+...b_{2k+1}=0$ $\implies$ $a(b_0a^{2k}+b_2a^{2k-2}+...b_{2k})+b_1a^{2k}+b_3a^{2k-2}+...b_{2k+1}=0$ $\implies$ $a= -(b_1a^{2k... | {
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Is it possible that $\mathbb{Q}(\alpha)=\mathbb{Q}(\alpha^{n})$ for all $n>1$? Is it possible that $\mathbb{Q}(\alpha)=\mathbb{Q}(\alpha^{n})$ for all $n>1$ when $\mathbb{Q}(\alpha)$ is a $p$th degree Galois extension of $\mathbb{Q}$?
($p$ is prime)
I got stuck with this problem while trying to construct polynomials w... | If you mean for some $\alpha$ and $p$, then yes: if $\alpha=1+\sqrt{2}$, then
$\mathbb{Q}(\alpha)$ is of degree 2, which is prime, and $\alpha^n$ is never a rational number, so $\mathbb{Q}(\alpha)=\mathbb{Q}(\alpha^n)$ for all $n>1$.
If you mean for all $\alpha$ such that the degree of $\mathbb{Q}(\alpha)$ is a prime ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Prove the identity $ \sum\limits_{s=0}^{\infty}{p+s \choose s}{2p+m \choose 2p+2s} = 2^{m-1} \frac{2p+m}{m}{m+p-1 \choose p}$ $$ \sum\limits_{s=0}^{\infty}{p+s \choose s}{2p+m \choose 2p+2s} = 2^{m-1} \frac{2p+m}{m}{m+p-1 \choose p}$$
Class themes are: Generating functions and formal power series.
| I will try to give an answer using basic complex variables here.
This calculation is very simple in spite of some more complicated intermediate expressions that appear.
Suppose we are trying to show that
$$\sum_{q=0}^\infty
{p+q\choose q} {2p+m\choose m-2q}
= 2^{m-1} \frac{2p+m}{m} {m+p-1\choose p}.$$
Introduce the i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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"answer_id": 2
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Countable or uncountable set 8 signs Let S be a set of pairwise disjoint 8-like symbols on the plane. (The 8s may be inside each other as well) Prove that S is at most countable.
Now I know you can "map" a set of disjoint intervals in R to a countable set (e.g. Q :rational numbers) and solve similar problems like this,... | Let $\mathcal{E}$ denote the set of all your figure eights. Then, define a map $f:\mathcal{E}\to\mathbb{Q}^2\times\mathbb{Q}^2$ by taking $E\in\mathcal{E}$ to a chosen pair of rational ordered pairs, one sitting inside each loop. Show that if two such figure eights were to have the same chosen ordered pair, they must i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/78018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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"Best practice" innovative teaching in mathematics Our department is currently revamping our first-year courses in mathematics, which are huge classes (about 500+ students) that are mostly students who will continue on to Engineering.
The existing teaching methods (largely, "lemma, proof, corollary, application, lemma... | “...no matter how way out they may seem.”
In that case you might want to consider the public-domain student exercises for mathematics that I have created. The address is: http://www.public-domain-materials.com/folder-student-exercise-tasks-for-mathematics-language-arts-etc---autocorrected.html
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 4,
"answer_id": 3
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The constant distribution If $u$ is a distribution in open set $\Omega\subset \mathbb R^n$ such that ${\partial ^i}u = 0$ for all $i=1,2,\ldots,n$. Then is it necessarily that $u$ is a constant function?
| It's true if we assume that $\Omega$ is connected. We will show that $u$ is locally constant, and the connectedness will allow us to conclude that $u$ is indeed constant. Let $a\in\Omega$ and $\delta>0$ such that $\overline{B(a,2\delta)}\subset \Omega$. Consider a test function $\varphi\in\mathcal D(\Omega)$ such that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/78142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Optimal number of answers for a test with wrong-answer penalty Suppose you have to take a test with ten questions, each with four different options (no multiple answers), and a wrong-answer penalty of half a correct answer. Blank questions do not score neither positively nor negatively.
Supposing you have not studied s... | Let's work this all the way through. Suppose you answer $n$ questions. Let $X$ be the number you get correct. Assuming $\frac{1}{4}$ chance of getting an answer correct, $X$ is binomial$(n,1/4)$. Let $Y$ be the actual score on the exam, including the penalty. Then $Y = X - \frac{1}{2}(n-X) = \frac{3}{2}X-\frac{1}{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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What kinds of non-zero characteristic fields exist? There are these finite fields of characteristic $p$ , namely $\mathbb{F}_{p^n}$ for any $n>1$ and there is the algebraic closure $\bar{\mathbb{F}_p}$. The only other fields of non-zero characteristic I can think of are transcendental extensions namely $\mathbb{F}_{q}(... | The basic structure theory of fields tells us that a field extension $L/K$ can be split into the following steps:
*
*an algebraic extension $K^\prime /K$,
*a purely transcendental extension $K^\prime (T)/K^\prime$,
*an algebraic extension $L/K^\prime (T)$.
The field $K^\prime$ is the algebraic closure of $K$ in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/78266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Are polynomials dense in Gaussian Sobolev space? Let $\mu$ be standard Gaussian measure on $\mathbb{R}^n$, i.e. $d\mu = (2\pi)^{-n/2} e^{-|x|^2/2} dx$, and define the Gaussian Sobolev space $H^1(\mu)$ to be the completion of $C_c^\infty(\mathbb{R}^n)$ under the inner product
$$\langle f,g \rangle_{H^1(\mu)} := \int f g... | Nate, I once needed this result, so I proved it in Dirichlet forms with polynomial domain (Math. Japonica 37 (1992) 1015-1024). There may be better proofs out there, but you could start with this paper.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Confused about modular notations I am little confused about the notations used in two articles at wikipedia.
According to the page on Fermat Primality test
$
a^{p-1}\equiv 1 \pmod{m}$ means that when $a^{p-1}$ is divided by $m$, the remainder is 1.
And according to the page on Modular Exponential
$c \equiv b^e \pmod... | Congruence is similar to equations where they could be interpreted left to right or right to left and both are correct.
Another way to picture this is:
a ≡ b (mod m) implies there exists an integer k such that k*m+a=b
There are various ways to visualize a (mod m) number system as the integers mod 3 could be viewed as ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is there any intuition behind why the derivative of $\cos(x)$ is equal to $-\sin(x)$ or just something to memorize? why is $$\frac{d}{dx}\cos(x)=-\sin(x)$$ I am studying for a differential equation test and I seem to always forget \this, and i am just wondering if there is some intuition i'm missing, or is it just one ... | I am with Henning Makholm on this and produce a sketch
The slope of the blue line is the red line, and the slope of the red line is the green line. I know the blue line is sine and the red line is cosine; the green line can be seen to be the negative of sine. Similarly the partial area under the red line is the blue... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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No. of solutions of equation? Given an equation $a_1 X_1 + a_2 X_2 + \cdots + a_n X_n = N$ where $a_1,a_2,\ldots,a_n$ are positive constants and each $X_i$ can take only two values $\{0,1\}$. $N$ is a given constant. How can we calculate the possible no of solutions of given equation ?
| This is counting the number of solutions to the knapsack problem. You can adapt the Dynamic Programming algorithm to do this in pseudopolynomial time. (I'm assuming the input data are integers.) Let $s[i,w]$ = # of ways to achieve the sum $w$ using only the first $i$ variables.
Then $$s[0,w]=\begin{cases}1 & w=0 \\0 & ... | {
"language": "en",
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Conditional expectation for a sum of iid random variables: $E(\xi\mid\xi+\eta)=E(\eta\mid\xi+\eta)=\frac{\xi+\eta}{2}$ I don't really know how to start proving this question.
Let $\xi$ and $\eta$ be independent, identically distributed random variables with $E(|\xi|)$ finite.
Show that
$E(\xi\mid\xi+\eta)=E(\eta\mid\xi... | $E(\xi\mid \xi+\eta)=E(\eta\mid \xi+\eta)$ since $\xi$ and $\eta$ are exchangeable, i.e. $(\xi,\eta)$ and $(\eta,\xi)$ are identically distributed. (Independent does not matter here.)
So $2E(\xi\mid \xi+\eta)=2E(\eta\mid \xi+\eta) = E(\xi\mid \xi+\eta)+E(\eta\mid \xi+\eta) =E(\xi+\eta\mid \xi+\eta) = \xi+\eta$ since th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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How do I prove equality of x and y? If $0\leq x,y\leq\frac{\pi}{2}$ and $\cos x +\cos y -\cos(x+y)=\frac{3}{2}$, then how can I prove that $x=y=\frac{\pi}{3}$?
Your help is appreciated.I tried various formulas but nothing is working.
| You could also attempt an geometric proof. First, without loss of generality you can assume
$0 <x,y < \frac{\pi}{2}$.
Construct a triangle with angles $x,y, \pi-x-y$.
Let $a,b,c$ be the edges. Then by cos law, you know that
$$\frac{a^2+b^2-c^2}{2ab}+ \frac{a^2+c^2-b^2}{2ac}+ \frac{b^2+c^2-a^2}{2bc}=\frac{3}{2}$$
and y... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving that if $G/Z(G)$ is cyclic, then $G$ is abelian
Possible Duplicate:
Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative
If $G/Z(G)$ is cyclic, then $G$ is abelian
If $G$ is a group and $Z(G)$ the center of $G$, show that if $G/Z(G)$ is cyclic, then $G$ is abelian.
This is what I have so far:
We k... | Here's part of the proof that $G$ is abelian. Hopefully this will get you started...
Let $Z(G)=Z$. If $G/Z$ is cyclic, then it has a generator, say $G/Z = \langle gZ \rangle$. This means that for each coset $xZ$ there exists some $i \in \mathbb{Z}$ such that $xZ=(gZ)^i=g^iZ$.
Suppose that $x,y \in G$. Consider $x \in x... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Breaking a variable out of a trigonometry equation $A = 2 \pi r^2 - r^2 (2 \arccos(d/2r) - \sin(2 \arccos(d/2r)))$
Given $A$ and $r$ I would like to solve for $d$. However, I get stuck breaking the $d/2r$ out the trig functions.
For context this is the area of two overlapping circles minus the overlapping region. Give... | This is a transcendental equation for $d$ which can't be solved for $d$ in closed form. You can get rid of the trigonometric functions in the last term using
$$\sin2x=2\sin x\cos x$$
and
$$\sin x=\sqrt{1-\cos^2 x}$$
and thus
$$\sin\left(2\arccos\frac d{2r}\right)=\frac dr\sqrt{1-\left(\frac d{2r}\right)^2}\;,$$
but tha... | {
"language": "en",
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Example of a function that is not twice differentiable Give an example of a function f that is defined in a neighborhood of a s.t. $\lim_{h\to 0}(f(a+h)+f(a-h)-2f(a))/h^2$ exists, but is not twice differentiable.
Note: this follows a problem where I prove that the limit above $= f''(a)$ if $f$ is twice differentiable a... | Consider the function $f(x) = \sum_{i=0}^{n}|x-i|$. This function is continuous everywhere but not differentiable at exactly n points. Consider the function
$G(x) = \int_{0}^{x} f(t) dt$. This function is differentiable, since $f(t)$ is continuous due to FTC II. Now $G'(x) = f(x)$, which is not differentiable at $n$ p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/78825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 2
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Counting words with parity restrictions on the letters Let $a_n$ be the number of words of length $n$ from the alphabet $\{A,B,C,D,E,F\}$ in which $A$ appears an even number of times and $B$ appears an odd number of times.
Using generating functions I was able to prove that $$a_n=\frac{6^n-2^n}{4}\;.$$
I was wondering ... | I don’t know whether you’d call them combinatorial, but here are two completely elementary arguments of a kind that I’ve presented in a sophomore-level discrete math course. Added: Neither, of course, is as nice as Didier’s, which I’d not seen when I posted this.
Let $b_n$ be the number of words of length $n$ with an o... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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State-space to transfer function I’m looking into MATLAB’s state-space functionality, and I found a peculiar relation that I don’t believe I’ve seen before, and I’m curious how one might obtain it. According to this documentation page, when converting a state-space system representation to its transfer function, the fo... | They are using the Sherman-Morrison formula, which I remember best in the form
$$
\det(I+MN) = \det(I+NM);
$$
this holds provided that both products $MN$ and $NM$ are defined. Note that if $M$ is
a column vector and $N$ a row vector, then $I+NM$ is a scalar. Now
$$\begin{align}
\det(sI-A+BC) &= \det\left((sI-A)(I+... | {
"language": "en",
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Is this a valid function? I am stuck with the question below,
Say whether the given function is one to one. $A=\mathbb{Z}$, $B=\mathbb{Z}\times\mathbb{Z}$, $f(a)=(a,a+1)$
I am a bit confused about $f(a)=(a,a+1)$, there are two outputs $(a,a+1)$ for a single input $a$ which is against the definition of a function. Plea... | If $f\colon A\to B$, then the inputs of $f$ are elements of $A$, and the outputs of $f$ are elements of $B$, whatever the elements of $B$ may be.
If the elements of $B$ are sets with 17 elements each, then the outputs of the function will be sets with 17 elements each. If the elements of $B$ are books, then the output ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Distance between $N$ sets of reals of length $N$ Let's say, for the sake of the question, I have 3 sets of real numbers of variate length:
$$\{7,5,\tfrac{8}{5},\tfrac{1}{9}\},\qquad\{\tfrac{2}{7},4,\tfrac{1}{3}\},\qquad\{1,2,7,\tfrac{4}{10},\tfrac{5}{16},\tfrac{7}{8},\tfrac{9}{11}\}$$
Is there a way to calculate the ov... | Let $E(r)$ be the set of all points at distance at most $r$ from the set $E$. This is called the closed $r$-neighborhood of $E$. The Hausdorff distance $d_H(E_1,E_2)$ is defined as the infimum of numbers $r$ such that $E_1\subseteq E_2(r)$ and $E_2\subseteq E_1(r)$. There are at least two reasonable ways to generalize ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The field of fractions of a field $F$ is isomorphic to $F$ Let $F$ be a field and let $\newcommand{\Fract}{\operatorname{Fract}}$ $\Fract(F)$ be the field of fractions of $F$; that is, $\Fract(F)= \{ {a \over b } \mid a \in F , b \in F \setminus \{ 0 \} \}$. I want to show that these two fields are isomorphic. I sugges... | Let $F$ be a field and $Fract(F)=\{\frac{a}{b} \;|\; a\in F, b\in F, b\not = 0 \} $ modulo the equivalence relation $\frac{a}{b}\sim \frac{c}{d}\Longleftrightarrow ad=bc$. We exhibit a map that is a field isomorphism between $F$ and $Fract(F)$.
Every fraction field of an integral domain $D$ comes with a canonical ring ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/79188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 1,
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A deceiving Taylor series When we try to expand
$$
\begin{align}
f:&\mathbb R \to \mathbb R\\
&x \mapsto
\begin{cases}
\mathrm e^{-\large\frac 1{x^2}} &\Leftarrow x\neq 0\\
0 &\Leftarrow x=0
\end{cases}
\end{align}$$
in the Taylor series about $x = 0$, we wrongly conclude that $f(x) \equiv 0$, because ... | To add to the other answers, in many cases one uses some simple sufficient (but not necessary) condition for analyticity: for example, any elementary function (polynomials, trigonometric functions, exponential,logarithms) is analytic in any open subset of its domain,; the same for compositions, sums, products, recipro... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
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Inner product space computation
If $x = (x_1,x_2)$ and $y = (y_1,y_2)$ show that $\langle x,y\rangle = \begin{bmatrix}x_1 & x_2\end{bmatrix}\begin{bmatrix}2 & -1 \\ 1 & 1\end{bmatrix}\begin{bmatrix}y_1 \\ y_2\end{bmatrix}$ defines an inner product on $\mathbb{R}^2$.
Is there any hints on this one? All I'm thinking is... | Use the definition of an inner product and check whether your function satisfies all the properties. Note that in general, as kahen pointed out in the comment, $\langle \mathbf{x}, \mathbf{y}\rangle = \mathbf{y}^*A\mathbf{x}$ defines an inner product on $\mathbb{C}^n$ iff $A$ is a positive-definite Hermitian $n \times ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prime reciprocals sum Let $a_i$ be a sequence of $1$'s and $2$'s and $p_i$ the prime numbers.
And let $r=\displaystyle\sum_{i=1}^\infty p_i^{-a_i}$
Can $r$ be rational, and can r be any rational $> 1/2$ or any real?
ver.2:
Let $k$ be a positive real number and let $a_i$ be $1 +$ (the $i$'th digit in the binary decimal... | The question with primes in the denominator:
The minimum that $r$ could possibly be is $C=\sum\limits_{i=1}^\infty\frac{1}{p_i^2}$. However, a sequence of $1$s and $2$s can be chosen so that $r$ can be any real number not less than $C$. Since $\sum\limits_{i=1}^\infty\left(\frac{1}{p_i}-\frac{1}{p_i^2}\right)$ diverges... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Difference between maximum and minimum? If I have a problem such as this:
We need to enclose a field with a fence. We have 500m of fencing material and a building is on one side of the field and so won’t need any fencing. Determine the dimensions of the field that will enclose the largest area.
This is obviously a m... | The Extreme Value Theorem guarantees that a continuous function on a finite closed interval has both a maximum and a minimum, and that the maximum and the minimum are each at either a critical point, or at one of the endpoints of the interval.
When trying to find the maximum or minimum of a continuous function on a fin... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof: If $f'=0$ then is $f$ is constant I'm trying to prove that if $f'=0$ then is $f$ is constant WITHOUT using the Mean Value Theorem.
My attempt [sketch of proof]: Assume that $f$ is not constant. Identify interval $I_1$ such that $f$ is not constant. Identify $I_2$ within $I_1$ such that $f$ is not constant. Repea... | So we have to prove that $f'(x)\equiv0$ $\ (a\leq x\leq b)$ implies $f(b)=f(a)$, without using the MVT or the fundamental theorem of calculus.
Assume that an $\epsilon>0$ is given once and for all. As $f'(x)\equiv0$, for each fixed $x\in I:=[a,b]$ there is a neighborhood $U_\delta(x)$ such that
$$\Biggl|{f(y)-f(x)\over... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What was the notation for functions before Euler? According to the Wikipedia article,
[Euler] introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.
— Leonhard Euler, Wikipedia
What was the notation for functions b... | Let's observe an example :
$a)$ formal description of function (two-part notation)
$f : \mathbf{N} \rightarrow \mathbf{R}$
$n \mapsto \sqrt{n}$
$b)$ Euler's notation :
$f(n)=\sqrt{n}$
I don't know who introduced two-part notation but I think that this notation must be older than Euler's notation since it gives more inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/79613",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 2
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Upper bound for $-t \log t$ While reading Csiszár & Körner's "Information Theory: Coding Theorems for Discrete Memoryless Systems", I came across the following argument:
Since $f(t) \triangleq -t\log t$ is concave and $f(0) = 0$ and $f(1) = 0$, we have for every $0 \leq t \leq 1-\tau$, $0 \leq \tau \leq 1/2$,
\begin... | The function $g$ defined on the interval $I=[0,1-\tau]$ by $g(t)=f(t)-f(t+\tau)$ has derivative $g'(t)=-\log(t)+\log(t+\tau)$. This derivative is positive hence $g$ is increasing on $I$ from $g(0)=-f(\tau)<0$ to $g(1-\tau)=f(1-\tau)>0$. For every $t$ in $I$, $g(t)$ belongs to the interval $[-f(\tau),f(1-\tau)]$, in par... | {
"language": "en",
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} |
limit of $f$ and $f''$ exists implies limit of $f'$ is 0
Prove that if $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}f''(x)$ exist, then $\lim\limits_{x\to\infty}f'(x)=0$.
I can prove that $\lim\limits_{x\to\infty}f''(x)=0$. Otherwise $f'(x)$ goes to infinity and $f(x)$ goes to infinity, contradicting t... | This is similar to a recent Putnam problem, actually. By Taylor's theorem with error term, we know that for any $x$,
$$
f(x+1) = f(x) + f'(x) + \tfrac12f''(t)
$$
for some $x\le t\le x+1$. Solve for $f'(x)$ and take limits....
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/79755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
$f'(x)-xf(x)=0$ has more roots than $f(x)=0$ Let $f(x)$ be a polynomial with real coefficients. Show that the equation $f'(x)-xf(x)=0$ has more roots than $f(x)=0$.
I saw the hint, nevertheless I can't prove it clearly. The hint is that $f(x)e^{-x^2/2}$ has a derivative $(f'(x)-xf(x))e^{-x^2/2}$, and use the Rolle's th... | $f(x)e^{-x^2/2}$ is zero at $\alpha_1$, and tends to zero at $-\infty$. So it must have a zero derivative somewhere in $(-\infty,\alpha_1)$.
Edited to reply to Gobi's comment
You can use Rolle's theorem after a little work. Let us write $g(x)$ for $f(x)e^{-x^2/2}$. Take any point $t \in (-\infty,\alpha_1)$. Since $g(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/79821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
ArcTan(2) a rational multiple of $\pi$? Consider a $2 \times 1$ rectangle split by a diagonal. Then the two angles
at a corner are ArcTan(2) and ArcTan(1/2), which are about $63.4^\circ$ and $26.6^\circ$.
Of course the sum of these angles is $90^\circ = \pi/2$.
I would like to know if these angles are rational multipl... | Lemma: If $x$ is a rational multiple of $\pi$ then $2 \cos(x)$ is an algebraic integer.
Proof
$$\cos(n+1)x+ \cos(n-1)x= 2\cos(nx)\cos(x) \,.$$
Thus
$$2\cos(n+1)x+ 2\cos(n-1)x= 2\cos(nx)2\cos(x) \,.$$
It follows from here that $2 \cos(nx)= P_n (2\cos(x))$, where $P_n$ is a monic polynomial of degree $n$ with integer coe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/79861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 3,
"answer_id": 1
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If a Fourier Transform is continuous in frequency, then what are the "harmonics"? The basic idea of a Fourier series is that you use integer multiples of some fundamental frequency to represent any time domain signal.
Ok, so if the Fourier Transform (Non periodic, continuous in time, or non periodic, discrete in time)... | In order to talk about a fundamental frequency, you need a fundamental period. But the Fourier transform deals with integrable functions ($L^1$, or $L^2$ if you go further in the theory) defined on the whole real line, and they are not periodic (except the zero function).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/79893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Show $X_n {\buildrel p \over \rightarrow} X$ and $X_n \le Z$ a.s., implies $X \le Z$ a.s. Suppose $X_n {\buildrel p \over \rightarrow} X$ and $X_n \le Z,\forall n \in \mathbb{N}$. Show $X \le Z$ almost surely.
I've try the following, but I didn't succeed.
By the triangle inequality, $X=X-X_n+X_n \le |X_n-X|+|X_n|$.... | $X_n {\buildrel p \over \rightarrow} X$ implies that there is a subsequence $X_{n(k)}$ with $X_{n(k)}\to X$ almost surely.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/79946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Sequential continuity for quotient spaces Sequential continuity is equivalent to continuity in a first countable space $X$. Look at the quotient projection $g:X\to Y$ to the space of equivalence classes of an equivalence relation with the quotient topology and a map $f:Y\to Z$. I want to test if $f$ is continuous.
Can ... | You certainly can’t do it in general if $X$ isn’t a sequential space, i.e., one whose structure is completely determined by its convergent sequences. $X$ is sequential iff it’s the quotient of a metric space, and the composition of two quotient maps is a quotient map, so if $X$ is sequential, $Y$ is also sequential, th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/80017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Taking the derivative of $\frac1{x} - \frac1{e^x-1}$ using the definition Given $f$:
$$
f(x) = \begin{cases}
\frac1{x} - \frac1{e^x-1} & \text{if } x \neq 0 \\
\frac1{2} & \text{if } x = 0
\end{cases}
$$
I have to find $f'(0)$ using the definition of derivative (i.e., limits). I alr... | Hmm, another approach, which seems simpler to me. However I'm not sure whether it is formally correct, so possibly someone else can also comment on this.
The key here is that the expression $\small {1 \over e^x-1 } $ is a very well known generation function for the bernoulli-numbers
$$\small {1 \over e^x-1 } =
x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/80078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
A question about composition of trigonometric functions A little something I'm trying to understand:
$\sin(\arcsin{x})$ is always $x$, but $\arcsin(\sin{x})$ is not always $x$
So my question is simple - why?
Since each cancels the other, it would make sense that $\arcsin(\sin{x})$ would always
result in $x$.
I'd app... | It is a result of deriving an inverse function for non-bijective one. Let $f:X\to Y$ be some function, i.e. for each $x\in X$ we have $f(x)\in Y$. If $f$ is not a bijection then we cannot find a function $g:Y\to X$ such that $g(f(x)) = x$ for all $x\in X$ and $f(g(y)) =y$ for all $y\in Y$.
Consider your example, $f = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/80146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_count": 1,
"answer_id": 0
} |
Variance over two periods with known variances? If period 1 has variance v1 and period 2 has variance v2, what is the variance over period 1 and period 2? (period 1 and period 2 are the same length)
I've done some manual calculations with random numbers, and I can't seem to figure out how to calculate the variance over... | If you only know the variances of your two sets, you can't compute the variance of the union of the two. However, if you know both the variances and the means of two sets, then there is a quick way to calculate the variance of their union.
Concretely, say you have two sets $A$ and $B$ for which you know the means $\mu_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/80195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
The Mathematics of Tetris I am a big fan of the old-school games and I once noticed that there is a sort of parity associated to one and only one Tetris piece, the $\color{purple}{\text{T}}$ piece. This parity is found with no other piece in the game.
Background: The Tetris playing field has width $10$. Rotation is a... | My colleague, Ido Segev, pointed out that there is a problem with most of the elegant proofs here - Tetris is not just a problem of tiling a rectangle.
Below is his proof that the conjecture is, in fact, false.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/80246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "277",
"answer_count": 4,
"answer_id": 3
} |
A book asks me to prove a statement but I think it is false The problem below is from Cupillari's Nuts and Bolts of Proofs.
Prove the following statement:
Let $a$ and $b$ be two relatively prime numbers. If there exists an
$m$ such that $(a/b)^m$ is an integer, then $b=1$.
My question is: Is the statement true?
I b... | Your counterexample is valid. But the statement is true if $m$ is required to be a positive natural number or positive integer.
Alternatively, note it's not if true $m$ is required to be negative.
In my opinion, it seems like you were supposed to assume $m>0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/80303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Crafty solutions to the following limit The following problem came up at dinner, I know some ways to solve it but they are quite ugly and as some wise man said: There is no place in the world for ugly mathematics.
These methods are using l'Hôpital, but that becomes quite hideous very quickly or by using series expansio... | By Taylor's series we obtain that
*
*$\sin (\tan x)= \sin\left(x + \frac13 x^3 + \frac2{15}x^5+ \frac{17}{315}x^7+O(x^9) \right)=x + \frac16 x^3 -\frac1{40}x^5 - \frac{55}{1008}x^7+O(x^9)$
and similarly
*
*$\tan (\sin x)= x + \frac16 x^3 -\frac1{40}x^5 - \frac{107}{5040}x^7+O(x^9)$
*$\arcsin (\arctan x)= x - \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/80364",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "36",
"answer_count": 2,
"answer_id": 1
} |
How strong does a matrix distort angles? How strong does it distort lengths anisotrolicly? let there be given a square matrix $M \in \mathbb R^{N\times N}$. I would like to have some kind of measure in how far it
*
*Distorts angles between vectors
*It stretches and squeezes discriminating directions.
While I am f... | Consider the singular value decomposition of the matrix.
Look at the singular values. These tell you, how the unit sphere is
stretched or squeezed by the matrix along the directions corresponding to the
singular vectors.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/80420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Decoding and correcting $(1,0,0,1,0,0,1)$ Hamming$ (7,4)$ code Correct any error and decode $(1,0,0,1,0,0,1)$ encoded using Hamming $(7,4)$ assuming at most one error. The message $(a,b,c,d)$ is encoded $(x,y,a,z,b,c,d)$
The solution states $H_m = (0,1,0)^T$ which corresponds to the second column in the Standard Hammin... | The short answer is that you get the syndrome $H_m$ by multiplying the received vector $r$ with the parity check matrix: $H_m=H r^T$.
There are several equivalent parity check matrices for this Hamming code, and you haven't shown us which is the one your source uses. The bits that you did give hint at the possibility ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/80472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How can one prove that $\sqrt[3]{\left ( \frac{a^4+b^4}{a+b} \right )^{a+b}} \geq a^ab^b$, $a,b\in\mathbb{N^{*}}$? How can one prove that $\sqrt[3]{\left ( \frac{a^4+b^4}{a+b} \right )^{a+b}} \geq a^ab^b$, $a,b\in\mathbb{N^{*}}$?
| Since $\log(x)$ is concave,
$$
\log\left(\frac{ax+by}{a+b}\right)\ge\frac{a\log(x)+b\log(y)}{a+b}\tag{1}
$$
Rearranging $(1)$ and exponentiating yields
$$
\left(\frac{ax+by}{a+b}\right)^{a+b}\ge x^ay^b\tag{2}
$$
Plugging $x=a^3$ and $y=b^3$ into $(2)$ gives
$$
\left(\frac{a^4+b^4}{a+b}\right)^{a+b}\ge a^{3a}b^{3b}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/80550",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Convergence of $b_n=|a_n| + 1 - \sqrt {a_n^2+1}$ and $b_n = \frac{|a_n|}{1+|a_{n+2}|}$ here's my daily problem:
1) $b_n=|a_n| + 1 - \sqrt {a_n^2+1}$. I have to prove that, if $b_n$ converges to 0, then $a_n$ converges to 0 too. Here's how I have done, could someone please check if this is correct? I'm always afraid to ... | You seem to have a fundamental misconception regarding the difference between the limit of a sequence and an element of a sequence.
When we say $b_n$ converges to $0$, it does not mean $b_n = 0$ for all $n$. For instance $b_n = \frac{1}{n}$ is convergent to $0$, but there is no natural number $n$ for which $b_n = 0$.
I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/80607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How to prove such a simple inequality? Let $f\in C[0,\infty)\cap C^1(0,\infty)$ be an increasing convex function with $f(0)=0$ $\lim_{t->\infty}\frac{f(t)}{t}=+\infty$ and $\frac{df}{dt} \ge 1$.
Then there exists constants $C$ and $T$ such that for any $t\in [T,\infty)$, $\frac{df}{dt}\le Ce^{f(t)}.$
Is it correct? If... | It is false as can be seen by the following proof by contradiction. Suppose it is true, and such a $C$ and $T$ exist. Then consider the following sequence of functions $f_n(t)=n(t-T) + 1+2T$ for $t\geq T$ and then extend $f_n$ smoothly for $t < T$ while keeping $f_n > 1$ and $f_n'\geq 1$. Then we have that $f_n'(T)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/80729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
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