Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Proof of $\sin^2 x+\cos^2 x=1$ using Euler's Formula How would you prove $\sin^2x + \cos^2x = 1$ using Euler's formula?
$$e^{ix} = \cos(x) + i\sin(x)$$
This is what I have so far:
$$\sin(x) = \frac{1}{2i}(e^{ix}-e^{-ix})$$
$$\cos(x) = \frac{1}{2} (e^{ix}+e^{-ix})$$
| Multiply $\mathrm e^{\mathrm ix}=\cos(x)+\mathrm i\sin(x)$ by the conjugate identity $\overline{\mathrm e^{\mathrm ix}}=\cos(x)-\mathrm i\sin(x)$ and use that $\overline{\mathrm e^{\mathrm ix}}=\mathrm e^{-\mathrm ix}$ hence $\mathrm e^{\mathrm ix}\cdot\overline{\mathrm e^{\mathrm ix}}=\mathrm e^{\mathrm ix-\mathrm ix}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/316936",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
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Prove: $a\equiv b\pmod{n} \implies \gcd(a,n)=\gcd(b,n)$ Proof: If $a\equiv b\pmod{n}$, then $n$ divides $a-b$. So $a-b=ni$ for some integer $i$. Then, $b=ni-a$. Since $\gcd(a,n)$ divides both $a$ and $n$, it also divides $b$. Similarly, $a=ni+b$, and since $\gcd(b,n)$ divides both $b$ and $n$, it also divides $a$.
Sinc... | HINT : Let $d = \gcd(b,n)$, then $b = dx$ and $n = dy$ for some $x$ and $y$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/317013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 2
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If $g$ is in $N_G(P)$ then $g\in P$ where $P$ is a $p$-Sylow subgroup Please help me to solve this problem. Let $P$ is a $p$-Sylow subgroup of the finite group $G$ and $g$ is an element such that $\lvert g \rvert=p^k$ then if $g$ is in $N_G(P)$ then $g\in P$. Where to start?
| If $H \leq G$ is a $p$-group, then $H \leq N_G(P)$ if and only if $H \leq P$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/317082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Graph with the smallest diameter. Consider a graph with $N$ vertexes where each vertex has at most $k$ edges.
I assume that $k < N$. What is the graph which have above property and has the smallest diameter?
Also, could you suggest good books in graph theory. Thanks.
| This question is quite difficult. The upper bound on the number of vertices given above by @Boris Novikov coincides with Moore's bound in the case of the Petersen graph. Moore's bound is not only achieved by the Petersen graph, but also by the Hoffman-Singleton graph, and in general, by the so-called Moore graphs. Unfo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/317134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Does a 3-Dimensional coordinate transformation exist such that its scale factors are equal? Let $\vec r=(x,y,z) $ be the position vector expressed in Cartesian coordinates. Let us define the coordinate transformation as
$\vec r(u,v,w)=(x(u,v,w),y(u,v,w),z(u,v,w)) $
The scale factors are defined by
$h_u=\vert \partial ... | Suppose we add another condition: not only $$\left\lVert\frac{\partial\mathbf r}{\partial u}\right\rVert=\left\lVert\frac{\partial\mathbf r}{\partial v}\right\rVert=\left\lVert\frac{\partial\mathbf r}{\partial w}\right\rVert,$$ but also $$\frac{\partial\mathbf r}{\partial u}\cdot\frac{\partial\mathbf r}{\partial v}=\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/317225",
"timestamp": "2023-03-29T00:00:00",
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Linear algebra - Dimension theorem. Suppose we have a vector space $V$, and $U$, $W$ subspaces of $V$.
Dimension theorem states:
$$ \dim(U+W)=\dim U+ \dim W - \dim (U\cap W).$$
My question is:
Why is $U \cap W$ necessary in this theorem?
| $U \cap W$ is the intersection of the vector spaces $U$ and $W$, that is, the set of all vectors of the space $V$ which are in both subspaces $U$ and $W$.
As $U$ and $W$ are both subspaces of $V$, their intersection $U \cap W$ is also a subspace of $V$ (this assertion can be easily proved). Because $U \cap W$ is a subs... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/317294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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There exists only two groups of order $p^2$ up to isomorphism. I just proved that any finite group of order $p^2$ for $p$ a prime is abelian. The author now asks to show that there are only two such groups up to isomorphism. The first group I can think of is $G=\Bbb Z/p\Bbb Z\oplus \Bbb Z/p\Bbb Z$. This is abelian and ... | A proof of that could be:
The center of a group is a subgroup, so it's order must divide $p^2$, but it's a known fact that if a group has order $p^m$, with $p$ prime, then the center of the group is different from $p^{m-1}$ and different from $1$, so in our case, the center has order $p^2$, so it's abelian.
That's wha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/317356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there a simple algorithm for factoring polynomials over the reals? Any real polynomial can be expressed as a product of quadratic and binomial factors like $(x+a)$ and $(x^2 + bx + c)$. Given a polynomial, is there an algorithm which will find such factors?
For example, how can I express $x^4 +1$ in the form $(x^2... | If you can find roots, you can find factors. As others have pointed out, this needs to be done using numerical methods for polynomials with degree greater than 4. In fact it's often a good idea to use numerical methods (rather than closed-form formulae) even in the degree 3 and degree 4 cases, too. There's lots of good... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/317400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How many ways are there to consider $\Bbb Q$ as an $\Bbb R$-module? How many ways are there to consider $\Bbb Q$ as an $\Bbb R$-module?
I guess there is only one way, and that is the trivial case. i.e.
$$\forall r\in \Bbb R,\, \forall a,b\in \Bbb Q \qquad r\cdot \frac{a}{b}=0$$
With an idea I can proof this:
$$\exist... | There are no ways to consider $\mathbb{Q}$ as an $\mathbb{R}$-module (i.e. $\mathbb{R}$-vector space), if you follow the usual convention of requiring that modules be unital, i.e. $1\cdot q=q$ for any $q\in\mathbb{Q}$. This is because a vector space over $\mathbb{R}$ (or indeed, any field) is free; because $\mathbb{R}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/317545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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"answer_id": 1
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If the sum of the digits of $n$ are divisible by 9, then $n$ is divisible by 9; help understanding part of a proof
Let $n$ be a positive integer such that $n<1000$. If the sum of the digits of $n$ is divisible by 9, then $n$ is divisible by 9.
I got up to here:
$$100a + 10b + c = n$$
$$a+b+c = 9k,\quad k \in\mathbb... | Simple way to see it: Take the number $N = \sum_{0 \le k \le n} d_k 10^k$ where the $d_k$ are the digits, modulo 9 you have:
$$
N = \sum_{0 \le k \le n} d_k 10^k
\equiv \sum_{0 \le k \le n} d_k \pmod{9}
$$
since $10 \equiv 1 \pmod{9}$, and so $10^k \equiv 1 \pmod{9}$ for all $k$.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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"answer_id": 2
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Is the set of integers with respect to the p-adic metric compact? Given the integers and a prime $p$. I thought I had successfully shown that $\mathbb{Z}$ was compact with respect to the metric $|\cdot |_p$, by showing that the open ball centered at zero contained all integers with more than a certain number of factor... | You don't prove compactness "by showing that the open ball centered at zero contained all integers with ..., and then showing that the remaining integers ... fell into a finite number of balls", i.e. by showing that there is a finite number of open balls covering the space.
Actually you can show more : $\Bbb Z$ with th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/317678",
"timestamp": "2023-03-29T00:00:00",
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Book Recommendations and Proofs for a First Course in Real Analysis I am taking real analysis in university. I find that it is difficult to prove some certain questions. What I want to ask is:
*
*How do we come out with a proof? Do we use some intuitive idea first and then write it down formally?
*What books do you... | While this doesn't speak, directly, to Real Analysis, it is a recommendation that will help you there, and in other courses you're encounter, or will encounter soon:
In terms of both reading and writing proofs, in general, an excellent book to work through and/or have as a reference is Velleman's great text How to Prov... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/317753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Norm of the operator $Tf=\int_{-1}^0f(t)\ dt-\int_{0}^1f(t)\ dt$ Consider the operator $T:(C[-1, 1], \|\cdot\|_\infty)\rightarrow \mathbb R$ given by, $$Tf=\int_{-1}^0f(t)\ dt-\int_{0}^1f(t)\ dt,$$ is $\|T\|=2$. How to show $\|T\|=2$? On the one hand it is easy,
$$\begin{align}
|Tf|&=\left|\int_{-1}^0f(t)\ dt-\int_{0}^... | Try $f$ piecewise linear with $f(x)=-1$ if $x\leqslant -a$, $f(x)=x/a$ if $-a\leqslant x\leqslant a$ and $f(x)=+1$ if $x\geqslant a$, when $a\to0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Total Orders and Minimum/Maximum elements How can I prove that for any given Poset $(A,\preceq)$, $\preceq$ is a total order implies that $\forall a\in\preceq$, if a is a maximal, then a is maximum? Same goes for minimal/minimum.
| I assume that you are asking the following:
In a totally ordered set, why is every maximal element a greatest element?
The answer is simple: Assume $a$ is maximal. Let $b$ be an arbitrary element of our set. Since the set is totally ordered set we have either $a\leq b$ or $b\leq a$. Since $a$ is maximal, we must have $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Range and Kernel of a $3\times 3$ identity matrix? geometrically describe both.
the kernell would be just the zero vector, correct? and would also merely live in the 1st dimension, but geometrically speaking be non existent?
and for the range of the matrix, it would just be $\{(1,0,0),(0,1,0),(0,0,1)\}$. geometrically ... | The kernel is a subspace of the domain, regarding the matrix as a transformation. So the kernel is written as $\{ (0, 0, 0) \}$. Now, you can certainly regard this kernel as a space as well, but it is properly called 0-dimension, not non-existant.
Your range is correct. All of $\mathbb{R}^3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/317985",
"timestamp": "2023-03-29T00:00:00",
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How is GLB and LUB different than the maximum and minimum of a poset? As the subject asks, how is the greatest lower bound different than the minimum element in a poset, and subsequently, how is the least upper bound different than the minimum? How does a set having no maximum but multiple maximal elements affect the e... | The supremum coincides with maximum in the finite case. That said, a great deal of basic example are not finite, so supremum is a different operation in general. The most notorious thing here is that supremum is an element not necesarilly in the original set.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/318049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Surface area of a cylinders and prisms A cylinder has a diameter of 9cm and a height of 25cm. What is the surface area of the cylinder if it has a top and a base?
| The top and the bottom each have area $\pi(9/2)^2$.
For the rest, use a can opener to remove the top and bottom of the can. Then use metal shears to cut straight down, and flatten out the metal. We get a rectangle of height $25$, and width the circumference of the top. So the width is $9\pi$, and therefore the area o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/318123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Change of Variables in a 3 dimensional integral Let $\int_0^{\infty}\int_0^{\infty}\int_{-\infty}^{\infty}f(x_1,x_2,x_3)dx_1dx_2dx_3$ be a 3 dimensional variable ( i.e. $0\leq x_1,x_2\leq \infty,-\infty\leq x_3\leq \infty.)$ I am defining the following change of variables : $$x_1'=x_1,x_2'=x_2, x_3'=c_1x_1+c_2x_2+c_3x_... | New limits coincide with the old ones.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/318187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Derivatives of univalent functions must converge to derivative of univalent function? This is probably something basic that I am missing. I am reading the article Normal Families: New Perspectives by Lawrence Zalcman, and in one of his examples he makes the following assertion (I am paraphrasing, not quoting, for brevi... | Yes, this is a twist on Hurwitz's theorem: the limit of non-vanishing functions $f_n'$ is either identically zero, or nowhere zero. The first case is clear. In the second case, fix a point $z_0\in D$ and consider the functions $\tilde f_n=f_n-f_n(z_0)$. Since $\tilde f_n$ is pinned down at $z_0$, and the derivatives $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/318255",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Any open subset of $\Bbb R$ is a countable union of disjoint open intervals
Let $U$ be an open set in $\mathbb R$. Then $U$ is a countable union of disjoint intervals.
This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many d... | In a locally connected space $X$, all connected components of open sets are open. This is in fact equivalent to being locally connected.
Proof: (one direction) let $O$ be an open subset of a locally connected space $X$. Let $C$ be a component of $O$ (as a (sub)space in its own right). Let $x \in C$. Then let $U_x$ be a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/318299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "285",
"answer_count": 17,
"answer_id": 11
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$ \int_{0}^{1}x^m.(1-x)^{15-m}dx$ where $m\in \mathbb{N}$ $=\displaystyle \int_{0}^{1}x^m.(1-x)^{15-m}dx$ where $m\in \mathbb{N}$
My Try:: Put $x=\sin^2 \theta$ and $dx = 2\sin \theta.\cos \theta.d\theta$ and changing limit, We Get
$ = \displaystyle \int_{0}^{\frac{\pi}{2}}\sin^{2m}\theta.\cos^{30-2m}\theta.2\sin \thet... | $$f(m,n)=\int_0^1 x^m(1-x)^ndx$$
Repeated partial integration on the right hand side reveals that:
$$(m+1)f(m,n)=n\,f(m+1,n-1)$$
$$(m+2)(m+1)f(m,n)=n(n-1)\,f(m+2,n-2)$$
$$\cdots$$
$$(m+n)\cdots (m+1)\,f(m,n)=n!\,f(m+n,0)$$
$$\text{i.e}$$
$$(m+n)!f(m,n)=n!\,m!\,f(m+n,0)$$
Since
$$f(m+n,0)=\int_0^1 t^{m+n}dt=\frac{1}{m+n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to solve $ 13x \equiv 1 ~ (\text{mod} ~ 17) $?
How to solve $ 13x \equiv 1 ~ (\text{mod} ~ 17) $?
Please give me some ideas. Thank you.
| $$\frac{17}{13}=1+\frac4{13}=1+\frac1{\frac{13}4}=1+\frac1{3+\frac14}$$
The last but one convergent of $\frac{17}{13}$ is $1+\frac13=\frac43$
Using the relationship of the successive convergents of a continued fraction, $17\cdot3-13\cdot4=-1\implies 13\cdot4\equiv1\pmod{17}\implies x\equiv4\pmod{17}$
| {
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"timestamp": "2023-03-29T00:00:00",
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Is it necessary to know a lot of advance math to become a good junior high/high school teacher? By "advance math" I refer to Real Analysis, Abstract Algebra and Linear Algebra (to the level of Axler). I received mainly Bs in these courses with the exception of the intro-level Linear Algebra. Since I intend to be a teac... |
"Is it necessary to know a lot of advanced math to become a good junior high/high school teacher?"
It is neither necessary nor sufficient.
One of the best math teachers I've ever had probably doesn't remember much (if any) advanced math beyond linear algebra. I don't know how he performed as a math student himself, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/318481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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composition of continuous functions I was wondering if a function $f:[a,b]\rightarrow[c,d]$ is continuous, $g:[c,d]\rightarrow\mathbb{R}$ is continuous, does it necessarily imply that $g\circ f$ is continuous? Are there counterexamples? What is the necessary and sufficient condition for $g\circ f$ to be continuous?
Thi... | With the sequence definition of continuity it is obvious that $g\circ f$ is continous, because
$$\lim_{n\rightarrow \infty} g(f(x_n))=g(\lim_{n\rightarrow \infty} f(x_n)) = g(f(\lim_{n\rightarrow \infty} x_n))$$
because $f$ and $g$ are continuous.
It is hard to say what is necessary that the composition of function i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/318614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Lambda Calculus Equivalence I'm a bit new to lambda calculus and was wondering about the equivalence of two expressions
$$(\lambda x.\lambda y.xy)\lambda z.z\overset{?}=(\lambda x.\lambda y.xy)(\lambda z.z)$$
Can anyone help out?
| By convention the outer most parenthesis are dropped for minimal clutter.
$$\color{red}{(\lambda x.\lambda y.xy)}\color{blue}{\lambda z.z}\iff\color{red}{(\lambda x.\lambda y.xy)}\color{blue}{(\lambda z.z)}$$
The same thing is done in algebra:
$$\color{red}{(z)}\color{blue}{(x+y)}\iff \color{red}z\color{blue}{(x+y)}$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/318768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Ordinal exponentiation - $2^{\omega}=\omega$ This is my understanding of ordinal arithmetic - two ordinals are the same as one another if there is an order-preserving bijection between them. So for instance
$$1+\omega = \omega$$
because if
$$f(\langle x,y\rangle)=\begin{cases}y+1 & x=1\\ 1 &\text{otherwise}\end{cases... | Ordinal exponentiation is not cardinal exponentiation.
The cardinal exponentiation $2^\omega$ is indeed uncountable and has the cardinality of the continuum.
The ordinal exponentiation $2^\omega$ is the supremum of $\{2^n\mid n\in\omega\}$ which in turn is exactly $\omega$ again.
Also related:
*
*How is $\epsilon_0$... | {
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"url": "https://math.stackexchange.com/questions/318827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Derivative of $e^{\ln(1/x)}$ This question looks so simple, yet it confused me.
If $f(x) = e^{\ln(1/x)}$, then $f'(x) =$ ?
I got $e^{\ln(1/x)} \cdot \ln(1/x) \cdot (-1/x^2)$.
And the correct answer is just the plain $-1/x^2$. But I don't know how I can cancel out the other two function.
| Hint: the exponential and logarithm are inverse functions:
$$e^{\ln u}=u$$
for any $u>0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 1
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Are all integers fractions? In a college class I was asked this question on a quiz in regards to sets:
All integers are fractions. T/F.
I answered False because if an integer is written in fraction notation it is then classified as a rational number. The teacher said the answer was True and gave me the link http://ww... | Every integer $x \in \mathbb Z$ can be expressed as the fraction $x \over 1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/318948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 10,
"answer_id": 0
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Is $f(x)= \cos(e^x)$ uniformly continuous? As in the topic, my quest is to check (and prove) whether the given function $$f(x)= \cos(e^x)$$is uniformly continuous on $\left\{\begin{matrix}x \in (-\infty;0]
\\
x \in [0; +\infty)
\end{matrix}\right.$ .
My problem is that I have absolutely no idea how to do it. Any hin... | Note that
$f$ is differentiable on $\mathbb{R}$ and $f'(x)=-e^x\sin{(e^x)}.$
Let $x_n=\ln{\left(\dfrac{\pi}{6}+2\pi n\right)}, \;\; y_n= \ln{\left(\dfrac{\pi}{3}+2\pi n\right)}.$
Then $$|x_n-y_n| = {\ln{\left(\dfrac{\pi}{3}+2\pi n\right)} -\ln{\left(\dfrac{\pi}{6}+2\pi n\right)}}=\ln{\dfrac{\dfrac{\pi}{3}+2\pi n}{{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/319001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Value of limsup i? This is a part of my question.
$\lim \sup \cos(n\pi/12)$ as n goes to infinity
What is the value of this limit?
| When a sequence is bounded, the limsup is the largest limit of all convergent subsequences.
For all $n$, $\cos(n\pi/2)\leq 1$ so $\limsup\cos(n\pi/2) \leq 1$.
And for the extraction $n=4k$, $\cos(n\pi/2)=\cos(2k\pi)=1$. So $\limsup\cos(n\pi/2)\geq 1$.
Hence
$$
\limsup_{n\rightarrow +\infty}\cos(n\pi/2)=1.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/319072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A ‘strong’ form of the Fundamental Theorem of Algebra Let $ n \in \mathbb{N} $ and $ a_{0},\ldots,a_{n-1} \in \mathbb{C} $ be constants. By the Fundamental Theorem of Algebra, the polynomial
$$
p(z) := z^{n} + \sum_{k=0}^{n-1} a_{k} z^{k} \in \mathbb{C}[z]
$$
has $ n $ roots, including multiplicity. If we vary the valu... | To give a little expansion to @Andreas’s answer, let’s examine a little more closely the way the coefficients depend on the roots. Let’s take an $n$-tuple of roots, say $\rho=(\rho_1,\cdots,\rho_n)$ and form the corresponding $n$-tuple whose entries are the coefficients $a=(a_0,\cdots,a_{n-1})$ of the monic polynomial ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Number of ways to seat students at a round table subject to certain conditions. In an Olympic contest, there are $n$ teams. Each team is composed of $k$ students attending different subjects. How many ways are there to seat all the students at a round table such that $k$ students in a team sit together and there are ... | I think that your recurrence isn’t quite right. If you start with an acceptable arrangement of $n$ teams, you can insert an $(n+1)$-st team in any of the $n$ slots between adjacent teams. The members of the new team can be permuted in $k!$ ways; $(k-1)!$ of these have an unacceptable person at one end, $(k-1)!$ have an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/319200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Elementary topology problem. Let $ ((Y_{\alpha},\tau_{\alpha}) \mid \alpha \in J) $ be a $ J $-indexed family of topological spaces and $ X $ any non-empty set. Let $ (f_{\alpha} \mid \alpha \in J) $ be a $ J $-indexed family of functions, where $ f_{\alpha}: Y_{\alpha} \to X $. What topology $ \tau $ can you put on $ ... | One can define a topology $ \tau $ on $ X $ as follows:
Declare a subset $ U $ of $ X $ to be $ \tau $-open if and only if $ {f_{\alpha}^{\leftarrow}}[U] \in \tau_{\alpha} $ for each $ \alpha \in J $.
Then $ \tau $ is the finest topology on $ X $ that makes $ f_{\alpha}: (Y_{\alpha},\tau_{\alpha}) \to (X,\tau) $ cont... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/319252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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diagonalizability of matrix If $A$ is invertible, $F$ is algebraically closed, and $A^n$ is diagonalizable for some $n$ that is not an integer multiple of the characteristic of $F$, then $A$ is diagonalizable.
My question is:
(1) Why the condition "$A$ is invertible" essential?
(2) In wikipedia, diagonalizable matrix ... | If $A$ is invertible then $A^n$ is also invertible, so $0$ is not an eigenvalue of $A^n$.
$A^n$ is diagonalizable then its minimal polynomial $P$ is a product of distinct linear factors over $F$:
$$P(X)=(X-\lambda_1)\cdots(X-\lambda_k),\quad \lambda_i\neq\lambda_j$$
We know that $A^n$ is annihilated by $P$: $P(A^n)=... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\sum \limits_{k=1}^{\infty} \frac{6^k}{\left(3^{k+1}-2^{k+1}\right)\left(3^k-2^k\right)} $ as a rational number. $$\sum \limits_{k=1}^{\infty} \frac{6^k}{\left(3^{k+1}-2^{k+1}\right)\left(3^k-2^k\right)} $$
I know from the ratio test it convergest, and I graph it on wolfram alpha and I suspect the sum is 2; however, I... | That denominator should suggest the possibility of splitting the general term into partial fractions and getting a telescoping series of the form
$$\sum_{k\ge 1}\left(\frac{A_k}{3^k-2^k}-\frac{A_{k+1}}{3^{k+1}-2^{k+1}}\right)\;,$$
where $A_k$ very likely depends on $k$. Note that if this works, the sum of the series wi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
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} |
Eigenvalues - Linear algebra If $c$ is eigenvalue for matrix B.
How can I prove than $c^k$ is eigenvalue for matrix $B^k$?
am not sure what I should try here...will appreciate your help
| Induction...?
$$B(v)=cv\Longrightarrow B^k(v)=B(B^{k-1}v)\stackrel{\text{Ind. hypothesis}}=B(c^{k-1}v)=c^{k-1}Bv=c^{k-1}cv=c^kv$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Joint density with continuous and binary random variable Assume $X\in\mathbb{R}$, $Y\in\{0,1\}$ are two random variables. What allows us to claim that $$f_{X}(x) = f_{XY}(x,1) + f_{XY}(x,0)$$
where $f_X(x)$ and $f_{XY}(x,y)$ are densities.
| $$P(X\le x)=P(X\le x\mid Y=1)P(Y=1)+P(X\le x\mid Y=0)P(Y=0)\\
=P(X\le x, Y=1)+P(X\le x,Y=0)\\
f_X(x)=\frac{dP(X\le x)}{dx}=f_{XY}(x,1)+f_{XY}(x,0)$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Normal $T\in B(H)$ has a nontrivial invariant subspace I am wondering if the following is true:
Every normal $T\in B(H)$ has a nontrivial invariant subspace if $\dim(H)>1$?
| Let $ T \in B(\mathcal{H}) $ be a normal operator. Let $ \sigma(T) $ denote the spectrum of $ T $. We then have two cases to consider: (i) $ \sigma(T) $ is a singleton set, and (ii) $ \sigma(T) $ contains at least two points.
Case (i): Suppose that $ \sigma(T) = \{ \lambda \} $ for some $ \lambda \in \mathbb{C} $. Let... | {
"language": "en",
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How do you solve linear congruences with two variables. For example: Solving for $x$ and $y$ given the following linear congruences.
$x + 2y \equiv 3 \pmod9\,$, $3x + y \equiv 2 \pmod9$
So far, I've tried taking the difference of the two congruences.
Since $x + 2y \equiv 3 \pmod9 \Rightarrow x + 2y = 3 + 9k\,$, and $3x... | The CRT is used solve systems of congruences of the form $\rm x\equiv a_i\bmod m_{\,i}$ for distinct moduli $\rm m_{\,i}$; in our situation, there is only one variable and only one moduli, but different linear congruences, so this is not the sort of problem where CRT applies. Rather, this is linear algebra.
Instead, yo... | {
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If $g$ is an element in an abelian group $G$ and $H\leqslant G$, must there exist an $n$ such that $g^n\in H$? Let $G$ be an abelian group and $H$ a subgroup of $G$. For each $g \in G$, does there always exist an integer $n$ such that $g^{n} \in H$?
| This is evident that if $[G:H]=n<\infty$ then $\forall g\in G, g^n\in H$.
| {
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solving a basic complex equation but using de Moivres theorem I have a question which should be super super easy!
If I was to solve $z^2 = 1+i$ how would I do this using de-moivres theorem?
I have the answer here in front of me so I don't want the answer, I just dont understand the method very well!
Any help would be a... | $$\forall w=x+iy\in\Bbb C:$$
$$w=|w|e^{i\phi}=|w|(\cos\phi+i\sin\phi)\;,\;\;\phi=\begin{cases}\arctan\frac{y}{x}+2k\pi &,\;\;y\neq 0\\{}\\2k\pi\end{cases}\;\;,\;\;\;k\in\Bbb Z$$
In our case:
$$w=1+i\Longrightarrow |w|=\sqrt 2\,\,,\,\,\arctan\frac{1}{1}=\frac{\pi}{4}+2k\pi\Longrightarrow$$
$$z^2=1+i=\sqrt 2\,e^{\frac{\p... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Differentiating piecewise functions. Say we have the piecewise function $f(x) = x^2$ on the interval $0 \le x < 4$; and it equals $x+1$ on the interval $ x \ge 4$. Why is it that, when I take the derivative, the intervals loose their equality and become strictly greater or strictly less than?
| Note that $\displaystyle \lim_{x\to 4^-}(f(x))=\lim_{x\to 4^-}(x^2)=16$ while $\displaystyle\lim_{x\to 4^+}(f(x))=f(4)=5.$ Therefore $f$ isn't continuous on $x=4$ and so it can't be differentiable there. Hence the domain of the derivative doesn't include $4$.
Around $0$ the left lateral derivative isn't even defined, t... | {
"language": "en",
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Show that $x$, $y$, and $z$ are not distinct if $x^2(z-y) + y^2(x-z) + z^2(y-x) = 0$. Suppose that $x^2(z-y) + y^2(x-z) + z^2(y-x) = 0$.
How can I show that $x$, $y$, and $z$ are not all distinct, that is, either $x=y$, $y=z$, or $x=z$?
| $x^2(z-y)+y^2(x-z)+z^2(y-x)=x^2(z-y)+x(y^2-z^2)+z^2y-y^2z=(z-y)(x^2-x(y+z)+zy)=(z-y)(x-y)(x-z)$.
| {
"language": "en",
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Logarithm proof problem: $a^{\log_b c} = c^{\log_b a}$ I have been hit with a homework problem that I just have no idea how to approach. Any help from you all is very much appreciated. Here is the problem
Prove the equation: $a^{\log_b c} = c^{\log_b a}$
Any ideas?
| If you apply the logarithm with base $a$ to both sides you obtain,
$$\log_a a^{\log_b c} = \log_a c^{\log_b a}$$
$$\log_b c = \log_b a \log_a c$$
$$\frac{\log_b c}{\log_b a} = \log_a c$$
however this last equality is the change of base formula and hence is true. Reversing the steps leads to the desired equality.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Flow of D.E what is the idea behind conjugacy? I got some kinda flow issue, ya know?
well enough with the bad jokes let A be a 2x2 matrix, T a change of Coordinate matrix, and $B=T^{-1}AT$ the canonical matrix ascoiated with A. Show that the function $h=T^{-1}: \mathbb R^{2} \to \mathbb R^{2}$, $h(x)=T^{-1}X$ is a co... | Suppose $X' = AX$. Let $Y = T^{-1} X T$. Then $$ Y' = (T^{-1} X T)' = T^{-1} X' T = T^{-1} (AX) T = (T^{-1} A T) (T^{-1} X T) = B (T^{-1} X T) = BY$$
So the flows of $B$ are simply conjugates of the flows of $A$. Is this what you wanted to know?
| {
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Show that $2 xy < x^2 + y^2$ for $x$ is not equal to $y$ Show that $2 xy < x^2 + y^2$ for $x$ is not equal to $y$.
| If $x\neq y$ then without loss of generality we can assume that $x>y$ $x-y>0\Rightarrow (x-y)^2>0\Rightarrow x^2-2xy+y^2>0\Rightarrow x^2+y^2>2xy$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/320244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Homogeneous equation I am trying to solve the following homogeneous equation:
Thanks for your tips
$xy^3y′=2(y^4+x^4)$
I think this isHomogeneous of order4
=> $xy^3dy/dx=2(y^4+x^4/1)$
=> $xy^3dy=2(x^4+y^4)dx$
=> $xy^3dy-2(x^4+y^4)dx=0$
I do not know how to continued
| Make the substitution $v=y^4$. Then by the chain rule we have $v'=4y^3 y'$. Now your DE turns into:
$$x \frac{v'}{4}=2(v+x^4)$$
Then can be simplified to:
$$v'-8\frac{v}{x}=8x^3$$
We first solve the homogeneous part:
$$v_h' -8\frac{v_h}{x}=0$$
This leads to $v_h=c\cdot x^8$ so that $v_h'=c\cdot 8x^7$. This is our homog... | {
"language": "en",
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Time complexity of a modulo operation I am trying to prove that if $p$ is a decimal number having $m$ digits, then $p \bmod q$ can be performed in time $O(m)$ (at least theoretically), if $q$ is a prime number. How do I go about this?
A related question is asked here, but it is w.r.t to MATLAB, but mine is a general o... | I will assume that $q$ is not part of your input, but rather a constant. Then, Algorithm D in 4.3.1 of Knuth's book "The Art of Computer Programming" (Volume 2) performs any long division in $O(m)$ steps.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/320420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
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Variance of the number of balls between two specified balls Question:
Assume we have 100 different balls numbered from 1 to 100, distributed in 100 different bins, each bin has 1 ball in it.
What is the variance of the number of balls in between ball #1 and ball #2?
What I did:
I defined $X_i$ as an indicator for ball ... | Denote the number of balls and the the number of bins by $b$. Suppose the first ball lands in bin $X_1$ and second ball lands in the bin $X_2$. The number of balls that will land in between them equals $Z = |X_2 -X_1| - 1$. Clearly
$$
\Pr\left( X_1 = m_1, X_2 = m_2 \right) = \frac{1}{b\cdot (b-1)} [ m_1 \not= m_2 ]
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/320528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Entire function dominated by another entire function is a constant multiple These two questions I didn't even find the way to solve
So please if you can help
*
*Suppose $f (z)$ is entire with $|f(z)| \le |\exp(z)|$ for every $z$
I want to prove that $f(z) = k\exp(z)$ for some $|k| \le 1$
*Can a non constant entire... | Hint:
1) Liouville + $\,\displaystyle{\frac{f(z)}{e^z}}\,$ is analytic and bounded...
2) Develop the function in power series and extend analytically into the "other" halpf plane.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is the set of discontinuity of $f$ countable? Suppose $f:[0,1]\rightarrow\mathbb{R}$ is a bounded function satisfying: for each $c\in [0,1]$ there exist the limits $\lim_{x\rightarrow c^+}f(x)$ and $\lim_{x\rightarrow c^-}f(x)$. Is true that the set of discontinuity of $f$ is countable?
| Hopefully this more tedious proof is more illustrative...
In light of the assumptions on $f$, there are two ways that $f$ can fail to be continuous: (1) the left and right hand limits differ, or (2) the limits are equal, but the function value differs from the limit (thanks to @GEdgar for pointing this out).
Let $\epsi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to prove that if group $G$ is abelian, $H = \{x \in G : x = x^{-1}\}$ is a subgroup? $G$ is abelian, $H = \{x \in G : x = x^{-1}\}$ is a subgroup?
I know to prove that a subset $H$ of group $G$ be a subgroup, one needs to (i) prove $\forall x,y \in H:x \circ y \in H$ and (ii) $\forall x \in H:x^{-1} \in H.$
| Unless you know $H$ is a nonempty subset of $G$, we need also to show that the identity $e \in G$ is also in $H$:
(o) Clearly, the identity $e \in G$ is its own inverse, hence $e = e^{-1} \in H$.
$(ii)\;$ $\forall x \in H$, $x\in H \implies x = x^{-1} \in H$. Hence we have closure under inverses.
$(i)$ $x\circ y \in H... | {
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Cardinals of set operations without AC Given info: $|A|=\mathfrak{c}$ , $|B|=\aleph_0$ in ZF (no axiom of choice).
Prove: $|A\cup B|=\mathfrak{c}$
If $B \subset A\implies|A \backslash B|=\mathfrak{c}$?
I have found several places proving that for $|\mathbb{R} \backslash \mathbb{Q}|,$ but none of the solutions appears ... | Prove it first for disjoint $A$ and $B$, relaxing the condition on $B$ to $|B|\le \aleph_0$. You then recover the full statement by considering $A\cup B = A\cup(B\setminus A)$.
You can restrict your attention even further to, say $A=(0,1)$ and $B$ being a subset of the integers. Once you have proved it for that case, t... | {
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Show that that $\lim_{n\to\infty}\sqrt[n]{\binom{2n}{n}} = 4$ I know that
$$
\lim_{n\to\infty}{{2n}\choose{n}}^\frac{1}{n} = 4
$$
but I have no Idea how to show that; I think it has something to do with reducing ${n}!$ to $n^n$ in the limit, but don't know how to get there. How might I prove that the limit is four?
| Hint: By induction, show that for $n\geq 2$ $$\frac{4^n}{n+1} < \binom{2n}{n} < 4^n.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/320846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Convergence of $\int_3^{\infty} \frac{1}{(\ln(x))^2(x-\ln(x))}$ Does this integral converge?
$$\int_3^{\infty} \frac{1}{(\ln(x))^2(x-\ln(x))}$$
I've been trying to solve this for the past 2 hours...literally. I know the answer is fairly simple, but I just can't think of it
| We have $$\dfrac{x}e > \ln(x) \,\,\, \forall x > 0$$
Hence,
$$x - \ln(x) > x - \dfrac{x}e = x \left(1 - \dfrac1e\right)$$
Hence, we get that
$$I = \int_3^{\infty} \dfrac{dx}{\ln^2(x)(x-\ln(x))} < \left(\dfrac{e}{e-1} \right) \times \int_3^{\infty} \dfrac{dx}{x \ln^2(x)} = \dfrac{e}{(e-1)\ln(3)}$$
EDIT
A way to approxim... | {
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Can three distinct points in the plane always be separated into bounded regions by four lines? How can I show that for any three points in the plane, four lines can be drawn that separate the three points into distinct enclosed regions?
Can any six points be enclosed in distinct regions formed by five lines?
Clarificat... | Okay, I think this works. By scaling and rotation, we can assume that two of the points are $(0,0)$ and $(0,1)$. Then the other point is $(x,y)$. Now the problem can be solved if the third point is $(1,0)$, with something like
Now if $x\ne 0$, the linear transformation $A=\pmatrix{x&0\\y&1}$ maps the point $(0,1)$ to ... | {
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Give the combinatorial proof of the identity $\sum\limits_{i=0}^{n} \binom{k-1+i}{k-1} = \binom{n+k}{k}$ Given the identity
$$\sum_{i=0}^{n} \binom{k-1+i}{k-1} = \binom{n+k}{k}$$
Need to give a combinatorial proof
a) in terms of subsets
b) by interpreting the parts in terms of compositions of integers
I should not use... | $\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
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\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 \ov... | {
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Expected value uniform decreasing function We are given a function $f(n,k)$ as
for(i=0;i < k;i++)
n = rand(n);
return n;
rand is defined as a random number generator that uniformly generates values in the range $[0,n)$. It returns a value strictly less than $n$; also $\operatorname{rand}(0)=0$.
What is the expected ... | If the random numbers are not restricted to integers but are uniformly generated on the entire interval $[0,n)$, the expected value is halved in each iteration, and thus by linearity of expectation the expected value of $f(n,k)$ is $2^{-k}n$.
If $n$ and the random numbers are restricted to integers, the expected value ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/321068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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What are some open research problems in Stochastic Processes? I was wondering, what are some of the open problems in the domain of Stochastic Processes. By Stochastic Processes.
Any examples or recent papers or similar would be appreciated.
The motivation for this question is that I was studying stochastics from a hig... | Various academics have lists on their website
*
*Richard Weber, University of Cambridge (operations research)
*David Aldous, University of California, Berkeley with updates from Thomas M. Liggett
*Krzysztof Burdzy, University of Washington
*Hermann Thorisson, University of Iceland
Other resources that might be ... | {
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"timestamp": "2023-03-29T00:00:00",
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Solving equation $\log_y(\log_y(x))= \log_n(x)$ for $n$ I'm just wondering, if I log a constant twice with the same base $y$,
$$\log_y(\log_y(x))= \log_n(x)$$
Can it be equivalent to logging the same constant with base $n$? If yes, what is variable $n$ equivalent to?
| Yes you can do that with the initial conditions for a logarithm satisfying.
The conditions for log(x) to the base n are: x > 0, n > 0 and n != 1.
so you should be careful with the domain that you are choosing.
| {
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"timestamp": "2023-03-29T00:00:00",
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Universal algorithm to estimate probability of drawing certain combination of coloured balls I am writing AI for a board game and would be happy for some guidance in creating the function to calculate probabilities.
So, there's a pool of N coloured balls. Up to 7 colours, the quantity of balls in each colour is given (... | We use your notation. Imagine that we have put identity numbers on all the balls.
There are $\binom{N}{M}$ ways to choose $M$ balls from $N$. Note that the binomial coefficient counts the number of possible "hands" of $M$ balls. Order is irrelevant.
In case you are unfamiliar with binomial coefficients, $\binom{n}{k}... | {
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Expectation of a stochastic exponential In class a while ago we used the following simplification:
$$ \mathbb E \left[ \exp\left(\langle \boldsymbol a,\mathbf W_t\rangle \right) \right] \quad =\quad \exp\left(\frac12 |\boldsymbol a|^2 t\right) $$
with $\boldsymbol a$ a constant $n$-dim vector, $\mathbf W_t$ an $n$-di... | Let $W_t = (W_t^1,\ldots,W_t^n)$ a $n$-dimensional Brownian motion. Then the processes $(W_t^j)_{t \geq 0}$ are independent 1-dimensional Brownian motions ($j=1,\ldots,n$). Thus
$$\mathbb{E}\big(e^{\langle a,W_t \rangle} \big) = \prod_{j=1}^n \mathbb{E}\big(e^{a_j \cdot W_t^j} \big) \stackrel{W_t^j \sim N(0,t)}{=} \pro... | {
"language": "en",
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How can a continuous function map closed sets to open sets (and vice versa)? Definition of continuity: A function $f: X \to Y$ (where $X$ and $Y$ are topological spaces) is continuous if and only if for any open subset $V$ of $Y$, the preimage $f^{-1}(V)$ is open in $X$.
Now, if $U$ is a closed subset of $X$ (meaning ... | Let $X$ be your favorite topological space, perhaps $\mathbb{R}$, and $Y$ be the space with just a single point.Then the only function $f:X\to Y$ is continuous, and, just like with Brian M. Scott's example (indeed, $Y$ has the discrete topology here), every set maps to a set which is both open and closed.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is the quotient map $SL_n(\mathbb{Z})$ to $SL_n(\mathbb{Z}/p\mathbb Z)$ is surjective? Recall that $SL_n(\mathbb{Z})$ is the special linear group, $n\geq 2$, and let $q\geq 2$ be any integer. We have a natural quotient map $$\pi: SL_n(\mathbb{Z})\to SL_n(\mathbb{Z}/q).$$ I remember that this map is surjective (is i... | The result is true for $n\geq 1$ and any integer $q\geq 1$.
The group $SL_n(\mathbb{Z}/q\mathbb{Z})$ is generated by the elementary (transvection) matrices.
It is easily seen that every elementary matrix is in the image of $\pi$, as the image of an elementary matrix in $SL_n(\mathbb{Z})$.
So $\pi$ is surjective.
| {
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Proving a set is a subset of a union with itself and another set Let A and B be sets. Show that
$$\space A \subseteq (A \cup B)$$
My proof is as follows, but I don't feel confident that what I've done is correct. Any input is much appreciated.
$$x \in A \cup B \equiv x \in A \wedge x \in B \equiv x \in A$$
Therefore, A... | You're going in the wrong direction. Don't assume $x\in A\cup B$, assume $x\in A$: you want to show that each element in $A$ is also an element of $A\cup B$ (not vice versa). Also, $x\in A\cup B\iff x\in A\textrm{ OR (not and) }x\in B$.
Let $x\in A$, and note that $x\in A\cup B\implies x\in A$ or $x\in B$. But, as $x\i... | {
"language": "en",
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Existence of an improper integral without the existence of a limit Can a continuous non-decreasing function exist for all $x \in [0, \infty)$ with $\int_0^\infty f \, dx$ existing, but the $\lim_{x \to \infty} f$ not existing? If it does, what does it look like? I feel that if the limit does not exist, how can the imp... | The limit must exist, and it must be $0$.
If $f$ is not bounded above, the integral clearly doesn't exist. If $f(x)$ is bounded above, then since $f$ is non-decreasing, $\lim_{x\to\infty} f(x)$ exists. One can show that if that limit is non-zero, then the integral doesn't converge.
The assumption of continuity is not... | {
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Alternating Series Using Other Roots of Unity $\sum (-1)^n b_n$ is representative of an alternating series.
We look at whether $\sum b_n$ converges and if $b_{n+1}<b_n$ $\forall n\in \mathbb{Z}$.
What if our alternating series is of the form $\sum z^n b_n$ where $z$ is any primitive root of unity. Do the same tests st... | In Dirichlet's test you can replace $(-1)^n$ in the alternating series test by any sequence with bounded partial sums, and thus as a special case also by $z^n$ where $z$ is a root of unity.
Thus the answers to your questions are "yes" and "no", respectively.
| {
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Is there an exponential bound for $(1+p)^{-n}$ when $p$ is small? Is there an exponential upper bound for $(1+p)^{-n}$ when $0<p<1$? Similarly a exponential lower bound for $(1+p)^n$ will also be good. Do you know of any resources where one can pick up bounds as such quickly? Thanks.
| When $np \ll 1, (1\pm p)^n \approx 1 \pm np$ with the accuracy improving as $np$ gets smaller. The error is of order $\frac {(np)^2}2$. When you say exponential bound, do you want less than exponential growth? It certainly doesn't fall exponentially
| {
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"timestamp": "2023-03-29T00:00:00",
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Adjacency graph of cutting plane is a bipartite graph Each time you draw a line on a plane, you are cutting it in half. Suppose you keep doing this without drawing a line parallel to a previous one. An adjacency graph can be constructed to represent this where each node represents an undivided portion of the plane, and... | Theorem: A graph with atleast one edge is 2-chromatic if and only if it has no circuit of odd length.
Reference: Page 168 Theorem 8-2 in Graph Theory by narsingn Deo.
Your adjacency graph doesnt have any circuit with odd degree.
| {
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If every subsequence is convergent, prove that the sequence is convergent
If every subsequence of a given sequence of real numbers is convergent, prove that the sequence is convergent.
Help me please. I could not understand how to solve this question.
| Your question is trivial unless you change the "If" to "If and only if". I'll prove the "if and any if" version.
(Trivial direction) Any sequence is a subsequence of itself, so if all subsequences of a given sequence converge, so does the original sequence.
(Nontrivial direction) Suppose $(x_n)$ converges to $L$. Let... | {
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How to evaluate $\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$ This problem appears at the end of Trig substitution section of Calculus by Larson. I tried using trig substitution but it was a bootless attempt
$$\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$$
| First make the substitution $x=\tan t$ to find
$$I=\int_0^1 dx\,\frac{\ln(x+1)}{x^2+1}=\int_0^{\pi/4} dt\,\ln(1+\tan t).$$
Now a substitution $u=\frac{\pi}{4}-t$ gives that
$$I=\int_0^{\pi/4} du\,\ln\left(\frac{2\cos u}{\cos u+\sin u}\right).$$
If you add these, you get
$$2I=\int_0^{\pi/4} dt\,\ln\left(\frac{\sin t+\co... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If we have $f$ is one-to-one, why can we conclude that $n\mathbb{Z}_m=\mathbb{Z}_m? $ Suppose $m,n \in \mathbb{Z},m,n\geq1.$ Define a map $$f:\mathbb{Z}_m \rightarrow \mathbb{Z}_m$$ where $[x] \rightarrow [nx]$ If we have $f$ is one-to-one, why can we conclude that $n\mathbb{Z}_m=\mathbb{Z}_m? $
| Any one to one function $f$ from a set $A$ with $m$ elements to a set $B$ with $m$ elements is onto. And any onto function is one to one.
This does not hold for infinite sets of the same cardinality.
The proof for finite sets is a matter of counting. Since $f$ is one to one, the values of $f$ at the $m$ elements of $... | {
"language": "en",
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Finding $G$ with normal $H_1,\ldots,H_n$ such that $G=H_1\cdots H_n$ and $H_i\cap H_j=\{e\}$ for $i\neq j$, but $G\not\cong H_1\times\cdots\times H_n$ How can I find a group $G$ with normal subgroups $H_1,\ldots,H_n$ such that $G=H_1H_2\cdots H_n$ and $H_i\cap H_j=\{e\}$ for all $i\neq j$, but $G\not\cong H_1\times H_2... | $Q_{8}$ is not ok, as in it two non-trivial normal subgroups intersect nontrivially.
Take instead the Klein $4$-group $V = \{e, a_1, a_2, a_3\}$, and $H_i = \langle a_i \rangle = \{ 1, a_i \}$ for $i = 1, 2, 3$.
You have indeed $G = H_1 H_2 H_3$ (actually, two factors already suffice) and $H_i \cap H_j = \{ e \}$ for $... | {
"language": "en",
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For which $\alpha$ does the series $\sum_{n=1}^\infty n^\alpha x^{2n} (1-x)^2$ converges uniformly at $[0,1]$. $$\sum_{n=1}^\infty n^\alpha x^{2n} (1-x)^2$$
Find for which $\alpha$ this series converges uniformly at $[0,1]$.
As $(1-x)^2$ is not dependent of $n$, I thought about rewriting it as:
$$(1-x)^2 \sum_{n=1}^\in... | Let's denote by
$$f_n(x)=n^\alpha x^{2n} (1-x)^2.$$
First, we search for which values of $\alpha$ we have the normal convergence of the series that implies the uniform convergence. The supremum of $f_n$ is attained at $x_n=\frac{n}{n+1}$ so
$$||f_n||_{\infty}=f_n(x_n)=n^\alpha (1+\frac{1}{n})^{-2n}(\frac{1}{n+1})^2\si... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Principal $n$th root of a complex number This is really two questions.
*
*Is there a definition of the principal $n$th root of a complex number? I can't find it anywhere.
*Presumably, the usual definition is $[r\exp(i\theta)]^{1/n} = r^{1/n}\exp(i\theta/n)$ for $\theta \in [0,2\pi)$, but I have yet to see this anyw... | There really is not a coherent notion of "principal" nth root of a complex number, because of the inherent and inescapable ambiguities.
For example, we could declare that the principal nth root of a positive real is the positive real root (this part is fine), but then the hitch comes in extending this definition to inc... | {
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Sequences and Languages Let $U$ be the following language. A string $s$ is in $U$ if it can be written as:
$s = 1^{a_1}01^{a_2}0 ... 1^{a_n}01^b$,
where $a_1,..., a_n$ are positive integers such that there is a 0-1 sequence $x_1, ..., x_n$ with $x_1a_1 + ... + x_na_n = b$. Show that $U \in P$.
Not sure how to even appr... | Hint for constructing a pushdown automaton recognising your language $U$ by empty stack:
[I didn't re-read the question before writing this, so I forgot the $a_i$'s had to be positive. You have to make a slight modification, probably adding two extra states, because of that, but the basic idea is still the same.]
Us... | {
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Totient function and Euler's Theorem Given $\big(m, n\big) = 1$, Prove that
$$m^{\varphi(n)} + n^{\varphi(m)} \equiv 1 \pmod{mn}$$
I have tried saying
$$\text{let }(a, mn) = 1$$
$$a^{\varphi(mn)} \equiv 1 \pmod{mn}$$
$$a^{\varphi(m)\varphi(n)} \equiv 1 \pmod{mn}$$
$$(a^{\varphi(m)})^{\varphi(n)} \equiv 1 \pmod{mn}$$... | Hint What is $m^{\varphi(n)} + n^{\varphi(m)} \pmod{m}$? What about $\pmod n$?
| {
"language": "en",
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If a group $G$ has only finitely many subgroups, then show that $G$ is a finite group. If a group $G$ has only finitely many subgroups, then show that $G$ is a finite group. I have no idea on how to start this question. Can anyone guide me?
| Suppose $G$ were infinite. If $G$ contains an element of infinite order, then _.
Otherwise, every element of $G$ has finite order, and if $x_1, x_2, \ldots, x_n$ are any finite set of elements of $G$, then the subgroups $\langle x_i \rangle$ cover only finitely many elements of $G$. Therefore __.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Injectivity is a local property Let $R$ be a commutative noetherian ring, and let $M$ be an $R$-module. How can I show that if any localization $M_p$ at a prime ideal $p$ of the ring $R$ is injective over $R_p$, then $M$ is injective?
| Baer's criterion shows that it suffices to show that $\hom(B,M) \to \hom(A,M)$ is surjective for $B=R$ and $A=$ an ideal, in particular both are finitely presented. But then $\hom$ commutes with localization and we are done.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do I show the equivalence of the two forms of the Anderson-Darling test statistic? It's stated in many places regarding the Anderson-Darling test statistic, which is defined as
$$n\int_{-\infty}^\infty \frac{(F_n(x) - F(x))^2}{F(x)(1 - F(x))}dF(x)$$
that this is functionally equivalent to the statistic
$$A^2 = -n -... | I think you can prove it by dividing the integral into (n+1) integrals on the intervals $[Y_k; Y_{k+1})$.
| {
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Choose 38 different natural numbers less than 1000, Prove among these there exists at least two whose difference is at most 26. Choose any 38 different natural numbers less than 1000.
Prove that among the selected numbers there exists at least two whose difference is at most 26.
I think I need to use pigeon hole princi... | Pigeonhole-principle is a good idea.
Hint: Think about partitioning $\{1,2,\ldots,999\}$ into subsets $\{1,2,\ldots,27\}$, $\{28,29,\ldots, 54\}$, $\{55,56,\ldots,81\}$, ..., $\{\ldots,998,999\}$ of size $27$ each.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Counting problem - How many times an inequality holds? Let $k>2$ be a natural number and let $b$ be a non-negative real number.
Now, for each $n$, $m \in \{ 1, 2, ... k \}$, consider the following inequalities:
$$ mb < k - n $$
We have $k^2$ inequalities. How can I count the couples of $n$ and $m$ such that the inequa... | Presumably $b$ is given. We can note that the right side ranges from $0$ to $k-1$, so define $p=k-n \in [0,k-1]$ and ask about $mb \lt p$. For a given $p$, the number of allowable $m$ is $\lfloor \frac {p-1}b \rfloor$ So we are asking for $\sum_{p=1}^{k-1}\lfloor \frac {p-1}b \rfloor$ Let $q=\lfloor \frac {k-2}b\rf... | {
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example of a set that is closed and bounded but not compact
Find an example of a subset $S$ of a metric space such that $S$ is closed and bounded but not compact.
One such example that comes from analysis is probably a closed and bounded set in $C[0,1]$. I attempt to construct my own example to see if it works.
Is $\... | The "closed" ball $\lVert x \rVert \leq 1$ in any infinite dimensional Banach space is closed and bounded but not compact. It is closed because any point outside it is contained in a small open ball disjoint from the first one, by the triangle inequality. That is, if $\lVert y \rVert = 1 + 2 \delta,$ then the sets $\lV... | {
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"url": "https://math.stackexchange.com/questions/323033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
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Express $\sin 4\theta$ by formulae involving $\sin$ and $\cos$ and its powers. I have an assignment question that says "Express $\sin 4\theta$ by formulae involving $\sin$ and $\cos$ and its powers."
I'm told that $\sin 2\theta = 2 \sin\theta \cos\theta$ but I don't know how this was found.
I used Wolfram Alpha to get... | That's a trig identity.
So...
$\sin{4\theta} = 2\sin{2\theta}cos{2\theta}$
Can you take it from there?
| {
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"timestamp": "2023-03-29T00:00:00",
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Compact injections and equivalent seminorms Let $V$ and $H$ be two Banach spaces with norm $\lVert \cdot \rVert$ and $\lvert \cdot \rvert$ respectively such that $V$ embeds compactly into $H$. Let $p$ be a seminorm on $V$ such that $p(u) + \lvert u \rvert$ is a norm on $V$ that is equivalent to $\lVert \cdot \rVert$. ... | We will first show that, given $x\in X$, there exists $z\in N$ s.t. $\|x-z\|_X=d(x-z,N)$ and $p(x)=p(x-z)$. Let $\{z_n\}\subset N$ s.t. $\|x-z_n\|_X\to d(x,N)$. Then $\infty>\|x\|_X+\sup_{n}\|x-z_n\|_X>\sup_{m}\|z_m\|_X$. Since $N$ is finite dimensional, there exists a subsequence $z_{n_k}$ and $z\in N$ s.t. $z_{n_k}\t... | {
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If $H \triangleleft G$ and $|G/H|=m$, show that $x^m \in H$ for $\forall x \in G$
If $H \triangleleft G$ and $|G/H|=m$, show that $x^m \in H$ for $\forall x \in G$.
My attempt is: since $|G/H|=\frac{|G|}{|H|}=m$, we have $x^{|G|}=x^{m|H|}=e$, then I stuck here. Can anyone guide me ?
| Since $(xH)^m=H$, it follows that $x^mH=H$, and thus that $x^m\in H$.
| {
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What was the first bit of mathematics that made you realize that math is beautiful? (For children's book) I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament," and found t... | When I was a kid I realized that
$$0^2 + 1\ (\text{the first odd number}) = 1^2$$
$$1^2 + 3\ (\text{the second odd number}) = 2^2$$
$$2^2 + 5\ (\text{the third odd number}) = 3^2$$
and so on...
I checked it for A LOT of numbers :D
Years passed before someone taught me the basics of multiplication of polynomial and hen... | {
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"question_score": "662",
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"answer_id": 4
} |
What was the first bit of mathematics that made you realize that math is beautiful? (For children's book) I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament," and found t... | If you've ever heard of $3,529,411,764,705,882$ being multiplied by $3/2$ to give $5,294,117,647,058,823$ (which is the same as the 3 being shifted to the back), you might consider including that in the book.
There are lots of other examples, like $285,714$ turning into $428,571$ (moving the 4 from back to front) when ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/323334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "662",
"answer_count": 162,
"answer_id": 36
} |
What was the first bit of mathematics that made you realize that math is beautiful? (For children's book) I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament," and found t... | For me, it was the beauty of the number 1, how it can be multiplied with anything , and it won't change the number it is being multiplied with, also how it can be represented as any number divided by itself such as 4/4=1
I would also love to share this beautiful poem by Dave Feinberg that is titled "the square root of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/323334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "662",
"answer_count": 162,
"answer_id": 68
} |
What was the first bit of mathematics that made you realize that math is beautiful? (For children's book) I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament," and found t... | Maybe the fact that the homotopy category of a model category is equivalent to the full subcategory of fibrant-cofibrant objects with homotopy classes of morphisms.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/323334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "662",
"answer_count": 162,
"answer_id": 100
} |
What was the first bit of mathematics that made you realize that math is beautiful? (For children's book) I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament," and found t... | I think one of my early favourite mathematical things was simply "proof by contradiction" -- any of them.
I think its appeal is that you nearly have proof by example, except that you're proving a negative.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/323334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "662",
"answer_count": 162,
"answer_id": 132
} |
How do we integrate, $\int \frac{1}{x+\frac{1}{x^2}}dx$? How do we integrate the following integral?
$$\int \frac{1}{x+\large\frac{1}{x^2}}\,dx\quad\text{where}\;\;x\ne-1$$
| The integral is equivalent to
$$\int dx \frac{x^2}{1+x^3} = \frac{1}{3} \int \frac{d(x^3)}{1+x^3} = \frac{1}{3} \log{(1+x^3)} + C$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/323404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Proving a matrix is positive definite using Cholesky decomposition If you have a Hermitian matrix $C$ that you can rewrite using Cholesky decomposition, how can you use this to show that $C$ is also positive definite?
$C$ is positive definite if $x^\top C x > 0$ and $x$ is a vector.
| From Wikipedia:
If A can be written as LL* for some invertible L, lower triangular or
otherwise, then A is Hermitian and positive definite.
$A=LL^*\implies x^*Ax=(L^*x)^*(L^*x)\ge 0$
Since $L$ is invertible, $L^*x\ne 0$ unless $x=0$, so $x^*Ax>0\ \forall\ x\ne 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/323474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Show $ \varlimsup_{n\rightarrow\infty}{\sqrt[n]{a_{1}+a_{2}+\cdots+a_{n}}}=1 $ if $\displaystyle \varlimsup_{n\rightarrow\infty}{\sqrt[n]{a_{n}}}=1 $ Let $\{a_{n}\}$ be a positive sequence with $\displaystyle \varlimsup_{n\rightarrow\infty}{\sqrt[n]{a_{n}}}=1 $. How can we show that
$$ \varlimsup_{n\rightarrow\infty}{\... | This is true.
1) Pick $\epsilon>0$. Then there exists $N$ such that
$$
\sqrt[n]{a_n}\leq 1+\epsilon\quad\Rightarrow\quad a_n\leq (1+\epsilon)^n\qquad\forall n\geq N.
$$
Now
$$
\sum_{k=1}^Ka_k =\sum_{k=1}^{N-1}a_k+\sum_{k=N}^Ka_k\leq C+\sum_{k=N}^K(1+\epsilon)^k=C+ (K-N+1)(1+\epsilon)^K\leq C+K(1+\epsilon)^K
$$
where $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/323558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 3
} |
The definition of continuous function in topology I am self-studying general topology, and I am curious about the definition of the continuous function. I know that the definition derives from calculus, but why do we define it like that?I mean what kind of property we want to preserve through continuous function?
| If you mean the definition that $f\colon X\to Y$ is continuous if $f^{-1}(U)$ is open for every open $U\subseteq Y$, then this is because this property is equivalent to continuity in metric spaces, but doesn't refer to the metric. This makes it a natural candidate for the general definition.
It is a little harder to mo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/323610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 2
} |
How to prove a generalized Gauss sum formula I read the wikipedia article on quadratic Gauss sum. link
First let me write a definition of a generalized Gauss sum.
Let $G(a, c)= \sum_{n=0}^{c-1}\exp (\frac{an^2}{c})$, where $a$ and $c$ are relatively prime integers.
(Here is another question. Is the function $e(x)$ def... | The quadratic Gauss sum is given by
\begin{eqnarray*}
G(s;k) = \sum_{x=0}^{k-1} e\left(\frac{sx^2}k\right),
\end{eqnarray*}
where $\displaystyle e(\alpha) = e^{2\pi \imath \alpha}$, $s$ is any integer coprime to $p$ and $k$ is a positive integer. The generalized Gauss sums is given by
\begin{eqnarray*}
G(a,b,c) = \sum_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/323774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Trigonometry functions How do I verify the following identity:
$$\frac{1-(\sin x - \cos x)^2}{\sin x} = 2\cos x$$
I have done simpler problems but got stuck with this one.
Please help.
Tony
| $$\frac{1-(\sin x -\cos x)^2}{\sin x}=\frac{1-(\sin^2x+\cos^2x)+2\sin x\cos x}{\sin x}=\frac{2\cos x\sin x}{\sin x}=2\cos x$$
Here I used the identity: $\sin^2x+\cos^2x=1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/323843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Intersection of a Collection of Sets I am trying to figure out how to write out the answer to this.
If I am given: $$A_i = \{i,i+1,i+2,...\}$$
And I am trying to find the intersection of a collection of those sets given by:
$$\bigcap_{i=1}^\infty A_i $$
First off, am I right in saying its the nothing since:
$$A_1 = \{ ... | Yes, that's correct, the answer is the empty set. To explain this properly you want to take a number $n \in \mathbb N$ and explain why $n \notin \bigcap A_i$. If you do this with no conditions on $n$ you will have shown that the intersection doesn't contain any positive integers, hence it's empty.
To show $n \notin \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/323886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Discuss the eigenvalues and eigenvectors of $A=I+2vv^T$.. I need some help with this question: Let $v\in\mathbb R^n$. Discuss the eigenvalues and eigenvectors of $$A=I+2vv^T$$. Can anyone help me?
| The matrix $vv^T$ is real symmetric, so it's diagonalizable, then there's an invertible matrix $P$ and a diagonal matrix $D$ such that
$$vv^T=PDP^{-1}.$$
Since the rank of $vv^T$ is $0$ or $1$ then $D=diag(||v||^2,0,\ldots,0).$
Now, we have:
$$A=I+2vv^T=P(I+2D)P^{-1}$$
so $A$ is diagonalizable in the same basis of eige... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/323931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Complex representation of sinusoids question A sum of sinusoids defined as:
$$\tag1f(t) = \sum_{n=1}^{N}A\sin(2\pi tn) + B\cos(2\pi tn)$$ is said to be represented as:
$$\tag2f(t) = \sum_{n=-N}^{N}C\cdot e^{i2\pi tn}$$
which is derived from Euler's identity $e^{ix} = \cos(x) + i\sin(x)$ from which follows that:
$$\cos(... | Using the identities:
$$\begin{aligned}
f(t)
&= \sum_{n=1}^N A \cos(2 \pi t n) + B \sin(2 \pi t n) \\
&= \sum_{n=1}^N \left[A\left(\frac{e^{i 2 \pi t n} + e^{-i 2 \pi t n}}{2} \right) +
B \left(\frac{e^{i 2 \pi t n} - e^{-i 2 \pi t n}}{2i} \right)\right] \\
&= \sum_{n=1}^N \left[\left(\frac{A}{2}+\frac{B}{2i}\right)e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/324057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
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