Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Using the Casorati-Weierstrass theorem. Show that there is a complex number $z$ such that:$$\left|\cos{\left(\frac{1}{2z^4+3z^2+1}\right)}+100\tan^2{z}+e^{-z^2}\right|<1$$
It's easy to see that $z=i$ is a simple pole of $\frac{1}{2z^4+3z^2+1}$, but I want to know how to conclude that $z=i$ is an essential singularity o... | Denote $$f(z) = \cos\left(\frac{1}{2z^4+3z^2+1}\right)$$
To prove $z=i$ is an essential singularity of $f(z)$, just find two complex sequences of $\{z_n\}$ and $\{w_n\}$ so that $z_n, w_n\rightarrow i$ but $f(z_n) = 1$, $f(w_n)=-1$.
This means both $\lim_{z->i} f(z)$ and $\lim_{z->i} 1/f(z)$ do not exist. By definition... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Diameter of finite set of points is equal to diameter of its convex hull Let $M\subset \mathbb{R}^2$ be a finite set of points, $\operatorname{C}(M)$ the convex hull of M and
$$\operatorname{diam}(M) = \sup_{x,y\in M}\|x-y\|_2$$
be the diameter of $M$
What I want to show now is, that it holds
$$\operatorname{diam}(M) ... | Hint:
Prove this for a triangle and then use the fact that for every point of $C(M)$ there is a triangle that contains it, there are many ways to go from there.
I hope this helps ;-)
| {
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"url": "https://math.stackexchange.com/questions/439571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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are elementary symmetric polynomials concave on probability distributions? Let $S_{n,k}=\sum_{S\subset[n],|S|=k}\prod_{i\in S} x_i$ be the elementary symmetric polynomial of degree $k$ on $n$ variables. Consider this polynomial as a function, in particular a function on probability distributions on $n$ items. It is no... | (I know this question is ancient, but I happened to run into it while looking for something else.)
While I am not sure if $S_{n,k}$ is concave on the probability simplex, you can prove the result you want and many other similar useful things using Schur concavity. A sketch follows.
A vector $y\in \mathbb{R}_+^n$ majori... | {
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For $f,g~(f0$ let $\{h\in\mathcal C[0,1]:t-c
For $f,g~(f<g),t\in\mathcal C[0,1],c>0$ let $\{h\in\mathcal C[0,1]:t-c<h<t+c\}$$=\{h\in\mathcal C[0,1]:f<h<g\}.$ I want to show that $t-c=f,~t+c=g.$
$$t-c<t<t+c\text{ and } \\f<\dfrac{f+g}{2}<g.\\\text{Then }t-c<\dfrac{f+g}{2}<t+c\text{ and } f<t<g.$$
I don't know how to con... | Let $A=\{h\in\mathcal C[0,1]|\;t-c<h<t+c\}$ and $B=\{h\in\mathcal C[0,1]|\;f<h<g\}$.
For every $\epsilon\in(0,c)$ we have $t-\epsilon,t+\epsilon\in A$ by definition of $A$. Since $A=B$, this means that $t-\epsilon,t+\epsilon\in B$ for all $\epsilon\in(0,c)$, which by definition of $B$ means that $$f<t-\epsilon<t+\epsil... | {
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Techniques for determining how "random" a sample is? What techniques exist to determine the "randomness" of a sample?
For instance, say I have data from a series of $1200$ six-sided dice rolls. If the results were
1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, ...
Or:
1, 1, 1, ..., 2, 2, 2, ..., 3, 3, 3, ...
The confidence of ... | What I would do is to first take the samples one by one, and check to see whether it is uniform (You assign some value depending on how far the distribution is from uniform and the way you calculate this value depends on your application). I would then take the samples two by two and do the same thing above, and then t... | {
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Evaluate the integral $\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\,\mathrm dx$. Evaluate the integral $$\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\, \mathrm dx.$$
How can i evaluate this one? Didn't find any clever substitute and integration by parts doesn't lead anywhere (I think).
Any guidelin... | Symmetry! This is the same as the integral with $\cos^3 x$ on top.
If that is not obvious from the geometry, make the change of variable $u=\pi/2-x$.
Add them, you get the integral of $1$. So our integral is $\pi/4$.
| {
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Can someone explain the intuition behind this moment generating function identity? If $X_i \sim N(\mu, \sigma^2) $, we know that: $\bar{X} \sim N(\mu, \sigma^2 /n)$.
But why does:
$$\exp\left({\sigma^{2}\over 2}\sum_{i=1}^{n}(t_{i}-\bar{t})^{2}\right)= M_{X_{1}-\bar{X},X_{2}-\bar{X},...,X_{n}-\bar{X}}(t_1,t_2,...,t_n)$... | This identity relies on the fact that
$$\sum_{i=1}^nt_iX_i-\sum_{i=1}^nt_i\bar X=\sum_{j=1}^n(t_j-\bar t)X_j.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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covering space of $2$-genus surface I'm trying to build $2:1$ covering space for $2$- genus surface by $3$-genus surface.
I can see that if I take a cut of $3$-genus surface in the middle (along the mid hole) I get $2$ surfaces each one looks like $2$-genus surface which are open in one side so it's clear how to make t... | Take the dodecagon at the origin with one pair of edges intersecting the $y$-axis (call them the top and bottom faces) and one pair intersecting the $x$-axis. Cut the polygon along the $x$ axis, and un-identify the left and right faces. This gives two octagons, each with an opposing pair of unmatched edges. Identify th... | {
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"timestamp": "2023-03-29T00:00:00",
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Linear dependence of multivariable functions It is well known that the Wronskian is a great tool for checking the linear dependence between a set of functions of one variable.
Is there a similar way of checking linear dependance between two functions of two variables (e.g. $P(x,y),Q(x,y)$)?
Thanks.
| For checking linear dependency between two functions of two variables we can follow the follwing theorem given by "Green, G. M., Trans. Amer. Math. Soc., New York, 17, 1916,(483-516)".
Theorem: Let $y_{1}$ and $y_{2}$ be functions of two independence variables $x_{1}$ and $x_{2}$ i.e., $y_{1} = y_{1}(x_{1} ,x_{2}) $ a... | {
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Analysis of Differentiable Functions Suppose that $f : \Bbb{R} \to \Bbb{R}$ is a function such that $|f(x)− f(y)| ≤ |x−y|^2$ for all $x$ and $y$. Show that $f (x) = C$ for some constant $C$.
Hint: Show that $f$ is differentiable at all points and compute the derivative
I confused as to what I use as the function in or... | Hint: Let $y=x+h$, then you have
$$abs\left( \frac{f(x+h)-f(x)}{h}\right)\le |h|,$$ so that as $h \to 0$ you get that $f$ is differentiable. Maybe now you can use differentiability of $f$ to finish.
Actually once you know it's differentiable, the same inequality above shows the derivative is $0$, so not really more wor... | {
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"IFF" (if and only if) vs. "TFAE" (the following are equivalent) If $P$ and $Q$ are statements,
$P \iff Q$
and
The following are equivalent:
$(\text{i}) \ P$
$(\text{ii}) \ Q$
Is there a difference between the two? I ask because formulations of certain theorems (such as Heine-Borel) use the latter, while others use... | "TFAE" is appropriate when one is listing optional replacements for some theory. For example, you could list dozen replacements for the statements, such as replacements for the fifth postulate in euclidean geometry.
"IFF" is one of the implications of "TFAE", although it as $P \rightarrow Q \rightarrow R \rightarrow P... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/440211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What does $a\equiv b\pmod n$ mean? What does the $\equiv$ and $b\pmod n$ mean?
for example, what does the following equation mean?
$5x \equiv 7\pmod {24}$? Tomorrow I have a final exam so I really have to know what is it.
| Let $a=qn+r_{1}$ and $b=pn+r_{2}$, where $0\leq r_{1},r_{2}<n$. Then $$r_{1}=r_{2}.$$
$r_{1}$ and $r_{2}$ are remainders when $a$ and $b$ are divided by $n$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Constant growth rate? Say the population of a city is increasing at a constant rate of 11.5% per year. If the population is currently 2000, estimate how long it will take for the population to reach 3000.
Using the formula given, so far I've figured out how many years it will take (see working below) but how can I narr... | Let $a=1.115^{1/12}=\sqrt[12]{1.115}$, the twelfth root of $1.115$. Then
$$1.115^x=(a^{12})^x=a^{12x}\;,$$
and $12x$ is the number of months that have gone by. Thus, if you can solve $a^y=1.5$, $y$ will be the desired number of months. Without logarithms the best that you’ll be able to do is find the smallest integer ... | {
"language": "en",
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How to determine whether an isomorphism $\varphi: {U_{12}} \to U_5$ exists? I have 2 groups $U_5$ and $U_{12}$ , ..
$U_5 = \{1,2,3,4\}, U_{12} = \{1,5,7,11\}$.
I have to determine whether an isomorphism $\varphi: {U_{12}} \to U_5$ exists.
I started with the "$yes$" case: there is an isomorphism.
So I searched an isom... | Note that $x^2\equiv 1\pmod {12}$ for all elements of $U_{12}$ whereas the corresponding property does not hold in $U_5$.
| {
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For what integers $n$ does $\phi(2n) = \phi(n)$? For what integers $n$ does $\phi(2n) = \phi(n)$?
Could anyone help me start this problem off? I'm new to elementary number theory and such, and I can't really get a grasp of the totient function.
I know that $$\phi(n) = n\left(1-\frac1{p_1}\right)\left(1-\frac1{p_2}\righ... | Hint: You may also prove in general that
$$\varphi(mn)=\frac{d\varphi(m)\varphi(n)}{\varphi(d)}$$
where $d=\gcd(m,n).$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/440557",
"timestamp": "2023-03-29T00:00:00",
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Higher Moments of Sums of Independent Random Variables Let $X_1 \dots X_n$ be independent random variables taking values $\{-1,1\}$ with equal probability 1/2. Let $S_n = \sum X_i$. Is there a closed form expression for $E[(S_n)^{2j}]$. If not a closed form expression then can we hope to get a nice tight upper bound. I... | The random variable $S_n^{2j}$ takes the value $(2k-n)^{2j}$ with probability $\binom nk\frac 1{2^n}$, hence
$$\mathbb E\left[S_n^{2j}\right]=\sum_{k=0}^n\binom nk(2k-n)^{2j}.$$
It involves computations of terms of the form $\sum_{k=0}^n\binom nk k^p$, $p\in\Bbb N$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Asking for a good starting tutorial on differential geometry for engineering background student. I just jumped into a project related to an estimation algorithm. It needs to build measures between two distributions. I found a lot of papers in this field required a general idea from differential geometry, which is like ... | I suggest you read Lee's introduction to topological manifolds followed by his introduction to smooth manifolds.
| {
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"url": "https://math.stackexchange.com/questions/440824",
"timestamp": "2023-03-29T00:00:00",
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What justifies assuming that a level surface contains a differentiable curve? My textbook's proof that the Lagrange multiplier method is valid begins:
Let $X(t)$ be a differentiable curve on the surface $S$ passing through $P$
Where $S$ is the level surface defining the constraint, and $P$ is an extremum of the funct... | By the Implicit Function Theorem, near $P$ you can represent your level surface as a graph, say $z=\phi(x,y)$, where $\phi$ is continuously differentiable. If $P=\phi(a,b)$, take any line through $(a,b)$ and you get a nice curve.
| {
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What does it exactly mean for a subspace to be dense? My understanding of rationals being dense in real numbers:
I know when we say the rationals are dense in real is because between any two rationals we can find a irrational number. In other words we can approximate irrational numbers using rationals. I think a more p... | In general topological spaces, a dense set is one whose intersection with any nonempty open set is nonempty. For metric spaces, since we have a topological base of open balls, this is equivalent to every point in space space being arbitrarily close, with regards to the metric, to point in the dense set.
Note that $L^2$... | {
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Must certain rings be isomorphic to $\mathbb{Z}[\sqrt{a}]$ for some $a$ Consider the group $(\mathbb{Z}\times\mathbb{Z},+)$, where $(a,b)+(c,d)=(a+c,b+d)$. Let $\times$ be any binary operation on $\mathbb{Z}\times\mathbb{Z}$ such that $(\mathbb{Z}\times\mathbb{Z},+,\times)$ is a ring. Must there exist a non-square inte... | Probably the most natural counterexample is the following:
If the operation $\times$ is defined such that the resulting ring is simply product of two copies of the usual ring $(\mathbb{Z},+,\times)$ (that is, if we set $(a,b)\times(c,d)=(ac,bd)$), then, again, no isomorphism exists, since the resulting ring $\mathbb{Z}... | {
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How to check if three coordinates form a line Assume I have three coordinates from a world map in Longitude + Latitude. Is there a way to determine these three coordinates form a straight line?
What if I was using a system with bounds that defines the 2 corners (northeast - southwest) in Long/Lat?
The long & lat are ex... | I'll assume that by "line" you mean "great circle" -- that is, if you want to go from A to C via the shortest possible route, then keep going straight until you circle the globe and get back to A, you'll pass B on the way.
The best coordinates for the question to be in are cartesian -- the 3D vector from the center of ... | {
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When factoring polynomials does not result in repeated factors I found the following statement in the book introduction to finite fields and their applications:
Let $x^n-1 = f_1(x)f_2(x)\dots f_m(x)$ be the decomposition of $x^n-1$ into monic irreducible factors over $\mathbb{F}_q$. If $\text{GCD}(n,q)=1$, then there ... | If $f(x)=g(x)^2h(x)$ then by the product rule of polynomial derivatives: $$f'(x)=2g(x)g'(x)h(x)+g(x)^2h'(x) =g(x)\left(2g'(x)h(x)+g(x)h'(x)\right)$$
So when $f(x)$ has a repeated factor, $f'(x)$ has a common factor with $f(x)$.
| {
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"timestamp": "2023-03-29T00:00:00",
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How to prove uniform distribution of $m\oplus k$ if $k$ is uniformly distributed? All values $m, k, c$ are $n$-bit strings. $\oplus$ stands for the bitwise modulo-2 addition.
How to prove uniform distribution of $c=m\oplus k$ if $k$ is uniformly distributed? $m$ may be of any distribution and statistically independant ... | This is not true. For example, if $m = k$, $c$ is not uniformly distributed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/441329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why does $\lim_{n \to \infty} \sqrt[n]{(-1)^n \cdot n^2 + 1} = 1$? As the title suggests, I want to know as to why the following function converges to 1 for $n \to \infty$:
$$
\lim_{n \to \infty} \sqrt[n]{(-1)^n \cdot n^2 + 1} = 1
$$
For even $n$'s only $n^2+1$ has to be shown, which I did in the following way:
$$\sqrt... | It's common for CAS's like Wolfram Alpha to take $n$th roots that are complex numbers with the smallest angle measured counterclockwise from the positive real axis. So the $n$th root of negative real numbers winds up being in the first quadrant of the complex plane. As $n\to\infty$, this $n$th root would get closer to ... | {
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Explaining the physical meaning of an eigenvalue in a real world problem Contextual Problem
A PhD student in Applied Mathematics is defending his dissertation and needs to make 10 gallon keg consisting of vodka and beer to placate his thesis committee. Suppose that all committee members, being stubborn people, refuse ... | An interpretation of eigenvalues and eigenvectors of this matrix makes little sense because it is not in a natural fashion an endomorphism of a vector space: On the "input" side you have (liters of vodka, liters of beer) and on the putput (liters of liquid, liters of alcohol). For example, nothing speaks against switch... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Expression for the Maurer-Cartan form of a matrix group I understand the definition of the Maurer-Cartan form on a general Lie group $G$, defined as
$\theta_g = (L_{g^{-1}})_*:T_gG \rightarrow T_eG=\mathfrak{g}$.
What I don't understand is the expression
$\theta_g=g^{-1}dg$
when $G$ is a matrix group. In particular, I... | This notation is akin to writing $d\vec x$ on $\mathbb R^n$. Think of $\vec x\colon\mathbb R^n\to\mathbb R^n$ as the identity map and so $d\vec x = \sum\limits_{j=1}^n \theta^j e_j$ is an expression for the identity map as a tensor of type $(1,1)$ [here $\theta^j$ are the dual basis to the basis $e_j$]. In the Lie grou... | {
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Mean value theorem application for multivariable functions
Define the function $f\colon \Bbb R^3\to \Bbb R$ by $$f(x,y,z)=xyz+x^2+y^2$$
The Mean Value Theorem implies that there is a number $\theta$ with $0<\theta <1$ for which
$$f(1,1,1)-f(0,0,0)=\frac{\partial f}{\partial x}(\theta, \theta, \theta)+\frac{\partia... | Hint: Consider $g(t):=f(t,t,t)$. What is $g'(t)$?
| {
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Orthogonal Subspaces I am reading orthogonality in subspaces and ran into confusion by reading this part:
Suppose S is a six-dimensional subspace of nine-dimensional space $\mathbb R^9$.
a) What are the possible dimensions of subspace orthogonal to $S$? Answer: Sub spaces orthogonal to S can have dimensions $0,1,2,3.... | Take all the vectors linearly independent vectors in $S$ and put them in a matrix (as rows). Since $S$ has dim=6, so there are 6 linearly independent vectors in S. Thus the matrix will have size 6x9. Now, rank of this matrix is $6$, and the orthogonal complement to $S$ is the rank of its null space. So, by rank nullity... | {
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Recursion Question - Trying to understand the concept Just trying to grasp this concept and was hoping someone could help me a bit. I am taking a discrete math class. Can someone please explain this equation to me a bit?
$f(0) = 3$
$f(n+1) = 2f(n) + 3$
$f(1) = 2f(0) + 3 = 2 \cdot 3 + 3 = 9$
$f(2) = 2f(1) + 3 = 2 \cdo... | Perhaps by considering a different sequence this may become clearer:
$$f(0)=0$$
$$f(n+1)=n+1$$
therefore
$$\begin{align}
f(n=1)&=f(n=0)+1=0+1=1\\
f(n=2)&=f(n=1)+1=(f(n=0)+1)+1=(0+1)+1=2\\
f(n=3)&=f(n=2)+1=(f(n=1)+1)+1=((f(n=0)+1)+1)+1=((0+1)+1)+1=3\\
\end{align}$$
So this will generate all the natural numbers.
| {
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"answer_id": 2
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Prove that every irreducible cubic monic polynomial over $\mathbb F_{5}$ has the form $P_{t}(x)=(x-t_{1})(x-t_{2})(x-t_{3})+t_{0}(x-t_{4})(x-t_{5})$? For a parameter $t=(t_{0},t_{1},t_{2},t_{3},t_{4},t_{5},)\in\mathbb F_{5}^{6}$ with $t_{0}\ne 0$ and {$t_{i},i>0$} are ordering of elements in $\mathbb F_{5}$ (t1~t5 is a... | Hints:
*
*A cubic is reducible, only if it has a linear factor. But then it should have a zero in $\mathbb{F}_5$, so it suffices to check that none of $t_1,t_2,t_3,t_4,t_5$ is a zero of $P_t(x)$.
*This part is tricky. I would go about it as follows. Let $t$ and $t'$ be
two vectors of parameters. Consider the differ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Homeomorphism from the interior of a unit disk to the punctured unit sphere I need help constructing a homeomorphism from the interior of the unit disk, $\{(x,y)\mid x^2+y^2<1\}$, to the punctured unit sphere, $\{(x,y,z)\mid x^2+y^2+z^2 = 1\} - \{(0,0,1)\}$. I was thinking you could take a line passing through $(0,0,1... | Stereographic projection, as mentioned in the other answers, usually comes up in this context. Indeed, it is a beautiful way of identifying the punctured sphere with $\mathbb{R}^2$, in a conformal manner (preserving angles).
However, if you only care about finding a homeomorphism to the disk (and there is no way to mak... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/441965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Evaluating the integral of $f(x, y)=yx$ Evaluate the integral $I = \int_C f(x,y) ds$ where $f(x,y)=yx$. and the curve $C$ is given by $x=\sin(t)$ and $y=\cos(t)$ for $0\leq t\leq \frac{pi}{2}$. I got the answer for this as $\frac{\sqrt{2}}{2}$ is that right?
| Evaluate the parameterized integral
$$ \int_0^{\pi/2} \cos t \sin t \sqrt{\cos^2 t + \sin^2 t} \, dt = \int_0^{\pi/2} \cos t \sin t \, dt. $$
I don't see a square root of 2 appearing anywhere.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Weakly compact implies bounded in norm The weak topology on a normed vector space $X$ is the weakest topology making every bounded linear functionals $x^*\in X^*$ continuous.
If a subset $C$ of $X$ is compact for the weak topology, then $C$ is bounded in norm.
How does one prove this fact?
| The first key point is that an element of $x$ can be identified with a linear functional of norm $\|x\|$ on the dual $X^*$. Indeed, it follows from Hahn-Banach that there exists $x^*\in X^*$ such that $x^*(x)=\|x\|$ with $\|x^*\|=1$. Therefore, denoting by $e_x$ the linear functional (called point evaluation) $e_x:x^*... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof that $ \lim_{x \to \infty} x \cdot \log(\frac{x+1}{x+10})$ is $-9$ Given this limit:
$$ \lim_{x \to \infty} x \cdot \log\left(\frac{x+1}{x+10}\right) $$
I may use this trick:
$$ \frac{x+1}{x+1} = \frac{x+1}{x} \cdot \frac{x}{x+10} $$
So I will have:
$$ \lim_{x \to \infty} x \cdot \left(\log\left(\frac{x+1}{... | I'll use the famous limit
$$\left(1+\frac{a}{x+1}\right)^x\approx\left(1+\frac{a}{x}\right)^x\to e^a$$
We have
$$x \ln \frac{x+1}{x+10}=x \ln \frac{x+1}{x+1+9}=-x\ln\left( 1+\frac{9}{x+1} \right)=-\ln\left( 1+\frac{9}{x+1} \right)^x\to-9$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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Poisson Estimators Consider a simple random sample of size $n$ from a Poisson distribution with mean $\mu$. Let $\theta=P(X=0)$.
Let $T=\sum X_{i}$. Show that $\tilde{\theta}=[(n-1)/n]^{T}$ is an unbiased estimator of $\theta$.
| We have $\Pr(X_1=0)=e^{-\mu}=\theta$.
Therefore
$$
\theta=\mathbb E(\Pr(X_1=0\mid X_1+\cdots+X_n)).
$$
So what is
$$
\Pr(X_1=0\mid X_1+\cdots+X_n=x)\text{ ?}
$$
It is
$$
\begin{align}
& {}\qquad \frac{\Pr(X_1=0\text{ and } X_1+\cdots+X_n=x)}{\Pr(X_1+\cdots+X_n=x)} = \frac{\Pr(X_1=0)\cdot\Pr(X_2+\cdots+X_n=x)}{e^{-n\mu}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculation for absolute value pattern I have a weird pattern I have to calculate and I don't quite know how to describe it, so my apologies if this is a duplicate somewhere..
I want to solve this pattern mathematically. When I have an array of numbers, I need to calculate a secondary sequence (position) based on the i... | For an array with $n$ elements, the function is:
$$f(n,k)=1+2(k-[n/2])$$
when $k\geq [n/2]$ and
$$f(n,k)=2+2([n/2]-k-1)$$
when $k<[n/2]$, where $[\cdot]$ is the floor function. Example:
$$f(5,4)=1+2(4-2)=5$$.
$$f(5,1)=2+2(1-2+1)=2$$
| {
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"source": "stackexchange",
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For the Fibonacci numbers, show for all $n$: $F_1^2+F_2^2+\dots+F_n^2=F_nF_{n+1}$ The definition of a Fibonacci number is as follows:
$$F_0=0\\F_1=1\\F_n=F_{n-1}+F_{n-2}\text{ for } n\geq 2$$
Prove the given property of the Fibonacci numbers for all n greater than or equal to 1.
$$F_1^2+F_2^2+\dots+F_n^2=F_nF_{n+1}$$
... | This identity is clear from the following diagram:
(imagine here a generalized picture with $F_i$ notation)
The area of the rectangle is obviously
$$F_n(F_{n}+F_{n-1})=F_nF_{n+1}$$
On the other hand, since the area of a square is x^2, it is obviously:
$$F_1^2+F_2^2+\dots+F_n^2$$
Therefore:
$$F_1^2+F_2^2+\dots+F_n^2=F_... | {
"language": "en",
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Is [0,1] closed? I thought it was closed, under the usual topology $\mathbb{R}$, since its compliment $(-\infty, 0) \cup (1,\infty)$ is open.
However, then then intersection number would not agree mod 2, since it can arbitrarily intersect a compact manifold even or odd times.
P.S. The corollary.
$X$ and $Z$ are closed... | A closed manifold is a compact boundaryless manifold. So the last line "Let [0,1] be the closed manifold Z" is wrong, for $\partial[0,1]\ne\phi$.
| {
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Is't a correct observation that No norm on $B[0,1]$ can be found to make $C[0,1]$ open in it? There's a problem in my text which reads as:
Show that $C[0,1]$ is not an open subset of $(B[0,1],\|.\|_\infty).$
I've already shown in a previous example that for any open subspace $Y$ of a normed linear space $(X,\|.\|),~Y... | This statement is actually true under more general settings. It seems convenient to talk about topological vector spaces, of which normed spaces are a very special kind.
So let $X$ be a topological vector space and $Y$ be an open subspace. So we know $Y$ contains some open set.
Since the topology on topological vector... | {
"language": "en",
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Finding perpendicular bisector of the line segement joining $ (-1,4)\;\text{and}\;(3,-2)$
Find the perpendicular bisector of the line joining the points $(-1,4)\;\text{and}\;(3,-2).\;$
I know this is a very easy question, and the answer is an equation. So any hints would be very nice. thanks
| Hint: The line must be orthogonal to the difference vector $(3-(-1),-2-4)$ and pass through the midpoint $(\frac{-1+3}2,\frac{4-2}2)$.
| {
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Can we use the Second Mean Value Theorem over infinite intervals? Let $[a,b]$ be any closed interval and let $f,g$ be continuous on $[a,b]$ with $g(x)\geq 0$ for all $x\in[a,b]$. Then the Second Mean Value Theorem says that
$$\int_a^bf(t)g(t)\text{d}t = f(c)\int_a^b g(t)\text{d}t,$$
for some $c\in(a,b)$.
Does this theo... | No. Consider $g(t)=\frac{1}{t}$, $f=g$ on $[1,\infty)$.
| {
"language": "en",
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Floor Inequalities Proving the integrality of an fractions of factorials can be done through De Polignac formula for the exponent of factorials, reducing the question to an floored inequality.
Some of those inequalities turn out to be very hard to proof if true at all.
The first is, given $x_i \in \mathbb{R}$ and $\{x_... | Let $\theta_i=\{x_i\}$, so that the second inequality reads
$$\sum_{i=1}^{n}\left \lfloor q_i \theta_i \right \rfloor \geq \left \lfloor \sum_{i=1}^{n}\theta_i \right \rfloor. \tag{1}$$
Also let $L$ denote the right side, as in the proof of the OP.
Now if for each $i$ we had $\theta_i<L/q_i$ then we would have
$$\sum_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Indefinite integral $\int{\frac{dx}{x^2+2}}$ I cannot manage to solve this integral:
$$\int{\frac{dx}{x^2+2}}$$
The problem is the $2$ at denominator, I am trying to decompose it in something like $\int{\frac{dt}{t^2+1}}$:
$$t^2+1 = x^2 +2$$
$$\int{\frac{dt}{2 \cdot \sqrt{t^2-1} \cdot (t^2+1)}}$$
But it's even hard... | Hint:
$$x^2+2 = 2\left(\frac{x^2}{\sqrt{2}^2}+1\right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/442991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Is a Whole Number A Rational Number Is a Whole Number part of A Rational Number or a whole number??
| The real answer, as usual, is "it depends". As the other answers have indicated, it is possible to identify whole numbers with certain rational numbers. On the other hand, it's also possible to identify rational numbers with certain ordered pairs of integers. So it really depends on your perspective/purpose. If you're ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/443152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Steady State Solution Non-Linear ODE I'm working through some problems studying for a numerical methods course, but I'm stuck on how to answer the following question analytically. It says to find the steady state solution of the following equation:
$\frac{dy}{dx} = -ay + e^{-y}$ where $y(0)=0$. It says that the steady ... | Firstly, I will expand on my comment.
If $a\neq 0$, then $0=-ay+e^{-y}$, which can be rearranged to $ye^y=\dfrac{1}{a}$. This has the "analytic" solution using the LambertW function $y=LambertW(1/a)$.
If $a=0$, then the original ODE is $y'=e^{-y}$, which has solution $y(x)=\log(x+c)$. This does not have a steady state.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Intuition for orthogonal vectors in $\Bbb R^n$ Two vectors in $\Bbb R^n$ are orthogonal iff their dot product is $0$. I'm aware that the dot product can be defined in other spaces, but to keep things simple let's restrict ourselves to $\Bbb R^n$.
Given that the idea of orthogonality is roughly to identify when two vect... | Firstly, let'd consider the geometry, to expand on Raskolnikov's comments:
Consider two independent vectors in $\mathbb R^n$. Then there's a unique plane through them (and the origin). Pick a linear transformation from $\mathbb R^n\to\mathbb R^2$ that sends that plane onto all of $\mathbb R^2$ and preserves lengths. Th... | {
"language": "en",
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A strange trigonometric identity in a proof of Niven's theorem I can't understand the inductive step on Lemma A in this proof of Niven's theorem. It asserts, where $n$ is an integer:
$$2\cos ((n-1)t)\cos (t) = \cos (nt) + \cos ((n-2)t)$$
I tried applying the angle subtraction formula to both sides, but all that does is... | As
$\cos(A-B)+\cos(A+B)=\cos A\cos B+\sin A\sin B+\cos A\cos B-\sin A\sin B=2\cos A\cos B$
Put $A+B=nt,A-B=(n-2)t$
Alternatively use $\cos C+\cos D=2\cos\frac{C+D}2\cos\frac{C-D}2$
| {
"language": "en",
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Visual proof of the addition formula for $\sin^2(a+b)$? Is there a visual proof of the addition formula for $\sin^2(a+b)$ ?
The visual proof of the addition formula for $\sin(a+b)$ is here :
Also it is easy to generalize (in any way: algebra , picture etc) to an addition formula for $\sin^n(a+b)$ where $n$ is a given... | It's not entirely clear what you mean by "the addition formula for $\sin^2(\alpha+\beta)$", but if it's this ...
$$\sin^2(\alpha+\beta) = \cos^2 \alpha + \cos^2\beta - 2 \cos\alpha\cos\beta \cos\left(\alpha+\beta\right)$$
... then here's a picture-proof that relies on the Law of Cosines (which itself has a nice picture... | {
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Confused by proof of Abelian group whose order divisible by prime has element divisible by prime. I have a problem understanding the assumptions for the proof of this theorem
Theorem: If $A$ is abelian with order $a$ divisible by prime $p$, then $A$ has an element of order $p$.
The proof goes as follows: Obviousl... | Your first assumption is correct. We are indeed using (strong) induction on the number $\frac{|A|}{p}$ for fixed, but arbitrary $p$.
Any non-identity element $a$ in any group $A$ of order a prime $p$ is itself of order $p$. That is to say, for each prime $p$ there is exactly one group (even without assuming abelian) of... | {
"language": "en",
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What is the derivative of $x\sin x$? Ok so I know the answer of $\frac{d}{dx}x\sin(x) = \sin(x)+ x\cos(x)$...but how exactly do you get there? I know $\frac{d}{dx} \sin{x} = \cos{x}$. But where does the additional $\sin(x)$ (in the answer) come in?
| A bit of intuition about the product rule:
Suppose that you have a rectangle whose height at time $t$ is $h(t)$ and whose width at time $t$ is $w(t)$. Then the area at time $t$ is $A(t)=h(t)w(t)$. Now, as the time changes, how does the area change?
(Please, forgive my use of paint here.)
Say the white rectangle was f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/443509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Weighted uniform convergence of Taylor series of exponential function
Is the limit
$$
e^{-x}\sum_{n=0}^N \frac{(-1)^n}{n!}x^n\to e^{-2x} \quad \text{as } \ N\to\infty \tag1
$$
uniform on $[0,+\infty)$?
Numerically this appears to be true: see the difference of two sides in (1) for $N=10$ and $N=100$ plotted ... | Thanks, this was a fun problem.
From the integral representation
$$
\sum_{k=0}^{n} \frac{x^k}{k!} = \frac{1}{n!} \int_0^\infty (x+t)^n e^{-t} \,dt \tag1
$$
we can derive the expression
$$
e^{-x} \sum_{k=0}^{n} \frac{(-x)^k}{k!} = e^{-2x} - \frac{e^{-2x} (-x)^{n+1}}{n!} \int_0^1 t^n e^{xt}\,dt. \tag2
$$
Now
$$
\int_0^1... | {
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"timestamp": "2023-03-29T00:00:00",
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Find the coefficient of $x^{20}$ in $(x^{1}+⋯+x^{6} )^{10}$ I'm trying to find the coefficient of $x^{20}$ in
$$(x^{1}+⋯+x^{6} )^{10}$$
So I did this :
$$\frac {1-x^{m+1}} {1-x} = 1+x+x^2+⋯+x^{m}$$
$$(x^1+⋯+x^6 )=x(1+x+⋯+x^5 ) = \frac {x(1-x^6 )} {1-x} = \frac {x-x^7} {1-x}$$
$$(x^1+⋯+x^6 )^{10} =\left(\dfrac {x-x^7}... | Since $(x+x^2+\cdots+x^6)^{10}=x^{10}(1+x+\cdots+x^5)^{10}$ and $1+x+\cdots+x^5=\frac{1-x^6}{1-x},$ we need to find the coefficient of $x^{10}$ in $(\frac{1-x^6}{1-x})^{10}=(1-x^6)^{10}(1-x)^{-10}.$
Since $(1-x^6)^{10}(1-x)^{-10} = (1-10x^6+45x^{12}+\cdots) \sum_{m=0}^{\infty}\binom{m+9}{9}x^{m},$ the coefficient of
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 0
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Define the linear transformation T: P2 -> R2 by T(p) = [p(0) p(0)] Find a basis for the kernel of T. Pretty lost on how to answer this question.
Define the linear transformation $T:P_2 \rightarrow \Bbb{R}^2$ by
$$
T(p) =\left[\begin{array}{c}p(0)\\p(0)\end{array}\right]
$$
Find a basis for the kernel of $T$.
So a $P_... | Recall the definition of kernel. Let $V,W$ be vector spaces over the same field of scalars and let $T:V\to W$ be a linear map. The kernel of $T$ denoted by $\ker T$ is the set of all $v \in V$ such that $T(v) = 0$.
So, let $T:P_2(\Bbb R)\to \Bbb R^2$ be the map you defined, i.e.: $T(p) = (p(0),p(0))$. One arbitrary ele... | {
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Suggestions on how to prove the following equality. $a^{m+n}=a^m a^n$ Let $a$ be a nonzero number and $m$ and $n$ be integers. Prove the following equality: $a^{m+n}=a^{m}a^{n}$
I'm not really sure what direction to go in. I'm not sure if I need to show for $n$ positive and negative separately or is there an easier way... | As I mention in the comment, you need some definition of $a^n$ in order to get started. It turns out that $a^n: \mathbb{R}\setminus\{0\} \times \mathbb{Z}\to \mathbb{R}$ is uniquely determined by specifying the relationships
*
*$a^1 = a$;
*$a^n = a\cdot a^{n-1}$.
(Can you see why both of these are necessary?)
As ... | {
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combination related question suppose that dinner cooker has 500 mint,500 orange and 500 strawberry,and he wished to do packets containing 10 mint,5 orange and 5 strawberry,question is what is a maximum number of packets he can make by this way?
so as i think,it is a combination related problem,which means tha... | After making 50 packets, you've used up $500=50\times 10$ mint, $250=50\times 5$ orange, and $250=50\times 5$ strawberry. There is no mint left, so you can't make any more.
| {
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Infinite Coins Tossed Infinitely Often If an infinite number of coins are tossed infinitely often, is it true that there will be infinite subsets of those coins that repeat any finite sequence of heads/tails infinitely often? I.e., infinitely many coins will always produce heads, infinitely many always produce tails, i... | The answer to the question in your first sentence is yes, with probability $1$. However, the assertion of your second sentence, beginning with "I.e.", is false, also with probability $1$. The same applies to the question and assertion of your second paragraph.
For any coin, the probability that a given finite sequence ... | {
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Interaction of two values I am a mathematically-challenged guy struggling (or to say better - having no clue at all) about a problem. I have two values (let's call them value ONE and TWO). The first can go from 55 to 190. The second can be anything i want. By making these two values interact i get different results as ... | I have to warn you that this is not a generally valid procedure. It just worked because the data seemed to exhibit a linear relationship. And in that case, it is not too difficult to find out what that relationship is. First, look at VALUE TWO and the RESULT. You'll notice that in the second group of data, when VALUE T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/444109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $A$ is a symmetric matrix in $\mathbb{R}$, why is $PAP^t$ diagonal? In Linear Algebra Why is following correct:
Given a symmetric matrix $A$ on the field of the real numbers, why is that true that there exists an unitary matrix $P$ such that $PAP^t$ is a diagonal matrix?
I know that from the Spectral Theorem there e... | This is a straightforward consequence of the spectral theorem. Let $u_1=p_1+iq_1,\ \ldots,\ u_n=p_n+iq_n$ be an orthonormal eigenbasis of $A$, where $u_1,\ldots,u_{k_1}$ correspond to the eigenvalues $\lambda_1$. Since $\lambda_1$ is real, we have $Ap_\ell=\lambda_1p_\ell$ and $Aq_\ell=\lambda_1q_\ell$ for $\ell=1,2,\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/444179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Proving statement $(A \cup C)\setminus B=(A\setminus B)\cup C \iff B\cap C= \varnothing$ I want to prove the following statment:
$$(A \cup C)\setminus B=(A\setminus B)\cup C \iff B\cap C= \varnothing$$
Do I need to prove each side? Or is one side enough? I mean, if I get from the left side to the right is it enough?
Ho... | Let me try to explain how I'd think about this problem, since I'd work directly from definitions and thus use fewer formulas than other people seem to like. For the left-to-right implication, notice that $(A\setminus B)\cup C$ contains (by definition) all the elements of $C$. But it equals $(A\cup C)\setminus B$ and th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/444248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving the length of a circle's arc is proportional to the size of the angle How can I prove that:
The length of the arc is proportional to the size of the angle.
Every book use this fact in explaining radians and the fundamental arc length equation $s = r\theta$. However no book proofs this fact.
Is this fact some ... | Let o = theta(angle subtended by the arc)
Ur equation can be written as s/r=o
We surely have two radii of circle subtending that angle o and forming arc s.
In the space between them we can fit more such radii.
The number of that radii will be decided by theta and they will form arc s.
Now divide arc s with the radius t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/444326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 7,
"answer_id": 5
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Show $\cos(x+y)\cos(x-y) - \sin(x+y)\sin(x-y) = \cos^2x - \sin^2x$
Show $\cos(x+y)\cos(x-y) - \sin(x+y)\sin(x-y) = \cos^2x - \sin^2x$
I have got as far as showing that:
$\cos(x+y)\cos(x-y) = \cos^2x\cos^2y -\sin^2x\sin^2y$
and
$\sin(x+y)\sin(x-y) = \sin^2x\cos^2y - \cos^2x\sin^2y$
I get stuck at showing:
$\cos^2x\co... | Hint: $$\cos(a+b)=\cos a \cos b-\sin a \sin b$$$$\cos(2a)=\cos^2 a -\sin ^2a $$
$$\begin{align}\cos(x+y)\cos(x-y) - \sin(x+y)\sin(x-y) &= \cos((x+y)+(x-y))\\&=\cos2x\\&=\cos^2x - \sin^2x\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/444407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How many positive integers $n$ satisfy $n = P(n) + S(n)$ Let $P(n)$ denote the product of digits of $n$ and let $S(n)$ denote the sum of digits of $n$. Then how many positive integers $n$ satisfy
$$
n = P(n) + S(n)
$$
I think I solved it, but I need your input.
I first assumed that $n$ is a two digit number. Then $n... | hint : consider the inequality $10^x\leq 9^x+9x$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/444463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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About the addition formula $f(x+y) = f(x)g(y)+f(y)g(x)$ Consider the functional equation
$$f(x+y) = f(x)g(y)+f(y)g(x)$$
valid for all complex $x,y$. The only solutions I know for this equation are $f(x)=0$, $f(x)=Cx$, $f(x)=C\sin(x)$ and $f(x)=C\sinh(x)$.
Question $1)$ Are there any other solutions ?
If we set $x=y$ w... | As leshik pointed out, this equation has plenty of discontinuous solutions (e.g. for $g=1$ it becomes Cauchy's functional equation), so let's just consider continuous solutions. $f=0$ is the trivial solution; from now on we will assume that $f$ is not identically zero. We consider two cases:
*
*If $f(x)$ and $g(x)$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/444517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
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Construct a compact set of real numbers whose limit points form a countable set. I searched and found out that the below is a compact set of real numbers whose limit points form a countable set.
I know the set in real number is compact if and only if it is bounded and closed.
It's obvious it is bounded since $\,d(1/4, ... | The limit points are $\{\frac{1}{2^m}\mid m\in \mathbb{N}\}$. These are contained in the set (to get $\frac{1}{2^k}$ (for $k>1$), take $m=k-1$, $n=2$).
We can tell there are no other limit points, since the closest points to $\frac{1}{2^k}(1-\frac{1}{l})$ (for $l>2$) are $\frac{1}{2^k}(1-\frac{1}{l+1})$ and $\frac{1}{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 2
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Impossible identity? $ \tan{\frac{x}{2}}$ $$\text{Let}\;\;t = \tan\left(\frac{x}{2}\right). \;\;\text{Show that}\;\dfrac{dx}{dt} = \dfrac{2}{1 + t^2}$$
I am saying that this is false because that identity is equal to $2\sec^2 x$ and that can't be equal. Also if I take the derivative of an integral I get the function so... | You're wrong, the identity is correct. Note that $t = \tan(x/2)$ implies
$x = 2 \arctan(t) + 2 n \pi$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/444642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
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Saturation of the Cauchy-Schwarz Inequality Consider a vector space ${\cal S}$ with inner product $(\cdot, \cdot)$. The Cauchy-Schwarz Inequality reads
$$
(y_1, y_1) (y_2, y_2) \geq \left| (y_1, y_2) \right|^2~~\forall y_1, y_2 \in {\cal S}
$$
This inequality is saturated when $y_1 = \lambda y_2$. In particular, this i... | You have $|y_1|^2|y_2|^2\geq|(y_1,y_2)|^2\geq|\text{Im}(y_1,y_2)|$. If the right hand side is equal to the left-most hand side then, in particular, you have 'saturation' of Cauchy's inequality. Then $y_2=\lambda y_1$ because the saturation of Cauchy's inequality happens if and only if the vectors are proportional. You ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A problem with the proof of a proposition I have a problem with the proof of Proposition 5.1. of the article of Ito.(Noboru Itˆo. On finite groups with given conjugate types. I.
Nagoya Math. J., 6:17–28, 1953.).
I don't know what is "e" and "e-1" in the proof.
I'd be really greatfull if someone help me.You can find the... | I would appear that $e$ is a group. Whenever the author writes $e-1$ they actually have "some group":$e-1$, so they just mean the index of $e$ in the group, minus one.
As to what $e$ is I am not sure, but I would guess it is related to $E$. I am not even sure whether $E$ is a group or an element, but most of the time i... | {
"language": "en",
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How to prove that càdlàg (RCLL) functions on $[0,1]$ are bounded? While studying the space $\mathbb{D}[0,1]$ of right continuous functions with left hand limits (i.e. càdlàg functions) on $[0,1]$, I came across the following theorem:
Theorem. If $f$ is càdlàg on $[0,1]$, it is bounded.
My proof attempt:
I am aware th... | Billigsley gives an excellent one-line proof. I am repeating it here. First note that this Lemma is true :
For every $\epsilon >0$, $ \exists$ partition $ 0=t_0<t_1<\ldots,t_k=1$ such that for the set $S_i = [t_i,t_{i+1})$, we have $\sup_{s,t \in S_i} |f(s)-f(t)| <\epsilon$ for all $i$.
This Lemma is easily proved.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/444853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Solving the trigonometric equation $A\cos x + B\sin x = C$ I have a simple equation which i cannot solve for $x$:
$$A\cos x + B\sin x = C$$
Could anyone show me how to solve this. Is this a quadratic equation?
| $A\cos x+B\sin x=C$ so if $A\neq 0, B\neq 0$ then $$\frac{A}{\sqrt{A^2+B^2}}\cos x+\frac{B}{\sqrt{A^2+B^2}}\sin x=\frac{C}{\sqrt{A^2+B^2}}$$ in which $$\frac{A}{\sqrt{A^2+B^2}}\le1,~~\frac{B}{\sqrt{A^2+B^2}}\le1,~~\frac{C}{\sqrt{A^2+B^2}}\le1$$ This means you can suppose there is a $\xi$ such that $\cos(\xi)=\frac{A}{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/444887",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving existence of $T$-invariant subspace Let $T:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ be a linear transformation. I'm trying to prove that there exists a T-invariant subspace $W\subset \mathbb{R}^3$ so that $\dim W=2$.
How can I prove it? Any advice?
| If you've not learned about minimal polynomials or Jordan normal form, here's a simple proof involving the Cayley-Hamilton Theorem.
Let $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a linear map and $A = [T]_{\text{std}}$ be the matrix representation of $T$ in the standard basis.
Take the set $\{v,Tv\}$. If this ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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Evaluating $\int\cos\theta~e^{−ia\cos\theta}~\mathrm{d}\theta$ Is anybody able to solve this indefinite integral :
$$
\int\cos\theta~e^{\large −ia\cos\theta}~\mathrm{d}\theta
$$
The letter $i$ denotes the Imaginary unit;
$a$ is a constant;
Mathematica doesn't give any result.
Thanks for any help you would like to pro... | $\int \cos\theta(−ia\cos\theta) d\theta$
=$-ia \int \cos^2\theta d\theta$
Hint: $\cos^2\theta=\frac {1+\cos(2\theta)} 2$
I do not think that you have to do anything special because of the imaginary part. That just means your result is imaginary
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Evaluate $\int_{1}^{\infty}e^{-x}\ln^{2}\left(x\right)dx$ Evaluate :$$\int_{1}^{\infty}e^{-x}\ln^{2}\left(x\right)\mathrm{d}x$$
I've tried to solve this with some elegant substitutions like $t=e^x$ or $t=\ln\left(x\right)$ . I've also tried to integrate by parts without any success. any help would be good.
| Let's define (for $\Re(a)>-1$) the function $$ f(a):=\int_{1}^{\infty}x^a\,e^{-x}\left(x\right)\,\mathrm{d}x$$ then this is the incomplete gamma function $\,f(a)=\Gamma(a+1,1)$.
But since $\,\displaystyle x^a=e^{a\ln(x)}\,$ we have :
$$f''(a)=\int_{1}^{\infty}x^a\;\ln^{2}(x)\;e^{-x}\,\mathrm{d}x$$
So that your function... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/445082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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"answer_id": 1
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$z_0$ non-removable singularity of $f\Rightarrow z_0$ essential singularity of $\exp(f)$
Let $z_0$ be a non-removable isolated singularity of $f$. Show that $z_0$ is then an essential singularity of $\exp(f)$.
Hello, unfortunately I do not know how to proof that. To my opinion one has to consider two cases:
*
*$... | We can also look at it from the other side.
If $z_0$ is a removable singularity of $e^f$, then $\lvert e^{f(z)}\rvert < K$ in some punctured neighbourhood of $z_0$. Since $\lvert e^w\rvert = e^{\operatorname{Re} w}$, that means $\operatorname{Re} f(z) < K'\; (= \log K)$ in a punctured neighbourhood $\dot{D}_\varepsilon... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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$20$ hats problem I've seen this tricky problem, where $20$ prisoners are told that the next day they will be lined up, and a red or black hat will be place on each persons head. The prisoners will have to guess the hat color they are wearing, if they get it right the go free. The person in the back can see every hat i... | To solve the problem with $n$ prisoners and $k$ colours, do as follows:
Wlog. the colors are the elements of $\mathbb Z/k\mathbb Z$.
If $c_i$ is the colour of th ehat of the $i$th prisoner, then
the $i$th prisoner can easiliy compute $s_i:=\sum_{j<i}c_j$.
Let the $n$th prisoner announce $s_n$.
Then the $(n-1)$st priso... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/445321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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What is the definition of "formal identity"? In Ahlfors' Complex Analysis he remarks that harmonic $u(x,y)$ can be expressed as
$$
u(x,y) = \frac{1}{2}[f(x + i y) + \overline{f}(x - i y)]
$$
when $x$ and $y$ are real. He then writes
"It is reasonable to expect that this is a formal identity,
and then it holds even w... | The word "formal" as it's being used here doesn't have an entirely rigorous meaning. The archetypal example of a formal argument is manipulating a power series without worrying about convergence, which gives rise to the notion of formal power series. In general, a formal argument is one based on the "form" of the mathe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/445394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
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Closed forms for $\lim_{x\rightarrow \infty} \ln(x) \prod_{x>(p-a)>0}(1-(p-a)^{-1})$ Im looking for closed forms for $\lim_{x \rightarrow \infty} \ln(x) \prod_{x>(p-a)>0}(1-(p-a)^{-1})$ where $x$ is a positive real, $a$ is a given real, $p$ is the set of primes such that the inequation is valid. Lets call this Limit $L... | I always take the log when I
run into a product,
so,
pressing on regardless,
let's look at
$\begin{align}
f(x, a)
&=\sum_{x>(p-a)>0}\ln(1-(p-a)^{-1})\\
&=\sum_{a < p < x+a}\ln(1-(p-a)^{-1})\\
&=\sum_{a < p < x+a}-(\frac1{p-a}+\frac1{2(p-a)^2}+...)\\
&=-\sum_{a < p < x+a}\frac1{p-a}+C\\
&=C-\sum_{a < p < x+a}\frac1{p(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/445452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Deriving the series representation of the digamma function from the functional equation By repeatedly using the functional equation $ \displaystyle\psi(z+1) = \frac{1}{z} + \psi(z)$, I get that $$ \psi(z) = \psi(z+n) - \frac{1}{z+n-1} - \ldots - \frac{1}{z+1} - \frac{1}{z}$$
or $$\psi(z+1) = \psi(z+n+1) - \frac{1}{z... | The functional equation tells us:
$$\frac{1}{z+n} -\frac{1}{n}=-\left(
\Psi \left( z+n+1 \right) -\Psi \left( z+n \right)\right) +\left(
\Psi \left( n+1 \right) -\Psi \left( n \right)\right)$$
and so we can form the partial sum:
$$-\sum _{n=1}^{N} \frac{1}{z+n} -\frac{1}{n}=-\sum _{n=1}^{N}\left(
\Psi \left( z+n+1 \rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/445531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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finitely generated & finitely related = finitely presented module? Let $R$ be a ring $M$ an $R$-module. How can I prove that if
*
*$M\cong R^n/N$ for some $n\!\in\!\mathbb{N}$ and some submodule $N\leq R^n$
and if
*
*$M\cong R^{(I)}/\langle u_1,\ldots,u_m\rangle$ for some set $I$ and some vectors $u_1,\ldots... | Every generating system $E$ of a finitely generated module $M$ contains a finite generating system (namely, look at those generators of $E$ which are needed to generate a finite generating system of $M$). Now assume that there are only finitely many relations between the generators of $E$. But these only use finitely m... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Double integral $\iint_D |x^3 y^3|\, \mathrm{d}x \mathrm{d}y$ Solve the following double integral
\begin{equation}
\iint_D |x^3 y^3|\, \mathrm{d}x \mathrm{d}y
\end{equation}
where $D: \{(x,y)\mid x^2+y^2\leq y \}$.
Some help please? Thank you very much.
| This may be done easily in polar coordinates; the equation of the circle is $r=\sin{\theta}$, $\theta \in [0,\pi]$. The integrand is $r^6 |\cos^3{\theta} \sin^3{\theta}|$, and is symmetric about $\theta = \pi/2$. The integral is then
$$2 \int_0^{\pi/2} d\theta \, \cos^3{\theta} \, \sin^3{\theta}\, \int_0^{\sin{\theta... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Homogeneous differential equations of second order I can not find other solutions of the equation
$$
y'' + 4y = 0
$$
addition of the solutions $y = 0$ and $y = \sin 2x$. There are positive solutions? Or solutions in terms of exponential function? Thanks.
| So for these equations we take a guess that the solution is exponential form($e^{rt}$).
$$y''+4y=0$$
$$\frac {d^2} {dt^2} e^{rt}+4e^{rt}$$
$$r^2e^{rt}+4e^{rt}=0$$
$$e^{rt}(r^2+4)=0$$
$$r^2=-4$$
$$r=\pm2i$$
So the solution to your equation is $Ae^{\pm2ix}$. If you covered this in class, you know that you can expand th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/445792",
"timestamp": "2023-03-29T00:00:00",
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If $f(t) = 1+ \frac{1}{2} +\frac{1}{3}+....+\frac{1}{t}$, find $\sum^n_{r=1} (2r+1)f(r)$ in terms of $f(n)$ If $f(t) = 1+ \frac{1}{2} +\frac{1}{3}+....+\frac{1}{t}$,
Find $x$ and $y$ such that $\sum^n_{r=1} (2r+1)f(r) =xf(r) -y$
| Since $(n+1)^2 - n^2 = 2n+1$, we'd expect $\sum_{r=1}^n (2r+1)f(r)$ to be something like $n^2 f(n)$. A little experimentation shows that it's actually $(n+1)^2 f(n) - n(n+1)/2$. $\textbf{Proof}$: At $n=1$, this is $4 - 1 = 3 = (2+1)f(1)$.
The forward difference of $(n+1)^2 f(n) - n(n+1)/2$ is $(n+2)^2 f(n+1) - (n+1)(n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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black and white balls in the box A box contains $731$ black balls and $2000$ white balls. The following process is to be repeated as long as possible.
(1) arbitrarily select two balls from the box. If they are of the same color, throw them out and put a black ball into the box. (We have sufficient black balls for this)... | First, the number of balls in the box decrease by one at each step. Suppose $B(t)$ and $W(t)$ are the number of black and white balls present after $t$ steps. Since we start with $B(0)+W(0)=2731$ and $B(t+1)+W(t+1)=B(t)+W(t)-1$, we have that $B(2730)+W(2730)=1$. At this point we can no longer continue the process. So a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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solve congruence $x^{59} \equiv 604 \pmod{2013}$ This is an exercise from my previous exam; how should I approach this?
Solve congruence $\;x^{59} \equiv 604 \pmod{2013}$
Thanks in advance :)
| Hint We have that $3 \cdot 11\cdot 61=2013$. Break up your congruence into three.
By $x^2\equiv 1\mod 3$, the first one turns into $x\equiv 1\mod 3$, for example, since we can deduce $3\not\mid x$. Glue back using CRT.
ADD Just in case you want the solution. First we may write $x^{59}\equiv 604\mod 3$ as $ x^{2\cdot 2... | {
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Getting equation from differential equations I have:
$\dfrac {dx} {dt}$=$-x+y$
$\dfrac {dy}{dt}$=$-x-y$
and I am trying to find $x(t)$ and $y(t)$ given that $x(0)=0$ and $y(0)=1$
I know to do this I need to decouple the equations so that I only have to deal with one variable but the decoupling is what I am having troub... | If I understand your question correctly, you are seeking for a possiblity to separate the equations as common with some of the systems, by shifting the system parallel to the existing $x$ and $y$ coordinates. In simple words you want to find a way by addition or subtraction or multiplication or division. A fair answer ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/446061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Do I have enough iMac boxes to make a full circle? My work has a bunch of iMac boxes and because of their slightly wedged shape we are curious how many it would take to make a complete circle.
We already did some calculations and also laid enough out to make 1/4 of a circle so we know how many it would take, but I'm cu... | Trying to avoid trigonometric functions: The outer circle must be longer than the inner circle by $2\pi$ times the height, so compute $$\frac{2\pi\text{height}}{(\text{bottom}-\text{top})} $$
as a good approximation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/446110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
How find that $\left(\frac{x}{1-x^2}+\frac{3x^3}{1-x^6}+\frac{5x^5}{1-x^{10}}+\frac{7x^7}{1-x^{14}}+\cdots\right)^2=\sum_{i=0}^{\infty}a_{i}x^i$ let
$$\left(\dfrac{x}{1-x^2}+\dfrac{3x^3}{1-x^6}+\dfrac{5x^5}{1-x^{10}}+\dfrac{7x^7}{1-x^{14}}+\cdots\right)^2=\sum_{i=0}^{\infty}a_{i}x^i$$
How find the $a_{2^n}=?$
my idea:l... | The square of the sum is $$\sum_{u\geq0}\left[\sum_{\substack{n,m,k,l\geq0\\(2n+1)(2k+1)+(2m+1)(2l+1)=u}}(2n+1)(2m+1)\right]x^u.$$
It is easy to use this formula to compute the first coefficients, and we get (starting from $a_1$)
$$0, 1, 0, 8, 0, 28, 0, 64, 0, 126, 0, 224, 0, 344, 0, 512, 0, 757, 0,
1008, 0, 1332, 0, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/446272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
} |
Equation of motion Pendulum using $w=e^{ix}$ I'm working with the equation of motion for a pendulum as follows:
$$x''+ \frac{g}{l} \sin (x)=0$$
Where $x$ is the angle between the pendulum and the vertical rest position.
I am required to use the complex variable $w=e^{ix}$ to rewrite the equation of motion in the form $... | Multiply the equation through by $x'$ and integrate once to get
$$x'^2-\frac{2 g}{\ell} \cos{x} = C$$
where $C$ is a constant of integration. Now, if $w=e^{i x}$, then $\cos{x}=(w+w^{-1})/2$ and
$$w' = i x' e^{i x} \implies x'=-i w'/w$$
Then the equation is equivalent to
$$-\frac{w'^2}{w^2} - \frac{g}{\ell} \left (w+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/446457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding the value to which a sequence converges The question is $f_1=\sqrt2 \ \ \ , \ \ f_{n+1}=\sqrt{2f_n}$, I have to show that it converges to 2. The book proceeds like this:
let $\lim f_n=l$.
We have, $f_{n+1}=\sqrt{2f_n} \implies (f_{n+1})^2=2f_n$.
Also, $\lim f_n=l \implies \lim f_{n+1}=l$. [HOW ?]
Thus, $l^2=2... | See this. In short, you may express the limit as
$$2^{1/2 + 1/4 + 1/8 + \cdots} = 2^1 = 2$$
There are very interesting generalizations.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/446536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
If $A+B+C+D+E = 540^\circ$ what is $\min (\cos A+\cos B+\cos C+\cos D+\cos E)$? Let each of $A, B, C, D, E$ be an angle that is less than $180^\circ$ and is greater than $0^\circ$. Note that each angle can be neither $0^\circ$ nor $180^\circ$.
If $A+B+C+D+E = 540^\circ,$ what is the minimum of the following function? $... | What you're looking at is a constrained optimization problem.
Put another way, you are being asked to minimize $\cos A+\cos B+\cos C+\cos D+\cos E$ given the constraint $A+B+C+D+E=540$.
Using lagrange multipliers, we can rewrite this as the minimization of the function
$L(A,B,C,D,E,\lambda)=\cos A+\cos B+\cos C+\cos ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/446629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 5,
"answer_id": 1
} |
Strategies for Factoring Expressions with Four Terms I'm trying to come up with a general strategy for factoring expressions with four terms on the basis of the symmetries of the expressions. One thought I had was the following: count up the number of terms in which each variable appears, and compare.
For example, if ... |
Factorization in $\mathbb{C}[x_1,x_2, .. .x_n]$ is polynomial-time given a
suitable representation of the algebraic number coefficients.
Paper i'm quoting:
http://www.aimath.org/pastworkshops/polyfactorrep.pdf
Paper that is cited:
http://scholar.google.co.uk/scholar?cluster=697805526622901646&hl=en&as_sdt=0,5&sciod... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/446694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
NURBS, parametrized curves and manifolds Let's start with the definitions:
A parametrized curve is a map $γ : (α,β) → R^n$ , for some $α,β$ with
$−∞ ≤ α < β ≤ ∞$.
A NURBS curve is defined by $C(u)=\sum_{i=1}^n R_{i,p}(u)\mathbf{P_i}$ as a rational function from the domain $\Omega=[0,1]$ to $R^n$.
A parametrized manifol... | To answer your question in brief, All curves and surfaces are manifolds, parametrized or not. Manifolds are abstractions of surfaces or curves.Just look up the definition of a chart and atlas on Wikipedia. In a way , a parametrization is a kind of chart. A precise answer might be possible if you add a bit more to your ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/446747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Different meanings of math terms in different countries Does anyone know of a list of math terms that have (slightly) different meanings in different countries?
For example, "positive" could mean $\geq 0 $ in some places, and "strictly positive" means $>0$ - See Dutch wikipedia page on Positive numbers, which states "I... | There is a declining but still existent tendency in French to not assume fields are commutative,
English (field, division ring) = French (corps commutatif, corps)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/446811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
One Point Derivations on locally Lipschitz functions Let $A$ be the algebra of $\mathbb{R}\to\mathbb{R}$ locally Lipschitz functions. What is the vector space of derivations at $0$? The proof that for continuous functions there aren't really any doesn't seem to work in this case.
I was thinking about trying to define t... | As Etienne pointed out in a comment, the book Lipschitz Algebras by Weaver is very much relevant. Section 4.7 introduces and describes derivations on the algebra of Lipschitz functions on a compact metric space. Weaver's constructions can be viewed as a way to introduce differentiable structures on metric spaces. See t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/446882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
is this the right truth table? When I filled out the table I tried my best to figure it out. But If I made any mistakes please help me correct them. Thanks!
sorry 5th one should be false
| EDIT: UPDATE
Now your table is mostly correct.
*
*Check your truth value assignment columns; we need to cover all possible $2^3$ truth-value assignments, and you've missed, for example, $P = F, Q = T, R = T$, but double counted another.)
*The only truth value combination that is false is when we have $P = F, Q = T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/446953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How many times is the print statement executed?
Hello,
I've gotten far on this exercise, with the following insight:
Here is a matrix of examples (vertical-axis is n=1,2,3,4,5,6,7,8; horizontal-axis is k=1,2,3,4)
1: 1 1 1 1
2: 2 3 4 5
3: 3 6 10 15
4: 4 10 20 35
5: 5 15 35 70
... | This is a standard "stars and bars" problem.
For given $n\geq1$ and $k\geq1$ we have to count the number of $k$-tuples $(i_1,\ldots,i_k)$ such that
$$1\leq i_k\leq i_{k-1}\leq\ldots\leq i_2\leq i_1\leq n\ .$$
Each such $k$-tuple can be encoded as a $0$-$1$-sequence of length $n+k-1$ as follows: Begin by writing $n-1$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/447007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Inverse of $(A + B)$ and $(A + BCD)$? Consider $A$ as an arbitrary matrix and $B$ as a symmetric matrix. Since $B$ is symmetric, therefore, it can be written as a $\Gamma \Delta \Gamma'$, where $\Delta$ is a diagonal matrix with eigen-values on the the main diagonal and $\Gamma$ is a matrix of corresponding eigenvector... | Have you considered the woodbury matrix inversion AKA Matrix Inversion Lemma?
https://en.wikipedia.org/wiki/Woodbury_matrix_identity
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/447081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
decoupling and integrating differential equations I am having trouble with the process of decoupling. If I have
$$\frac{dx}{dt}=-x+y$$
$$\frac{dy}{dt}=-x-y$$
I am trying to figure out how to solve for $x(t)$ and $y(t)$ by decoupling the system so that I only have one variable but I can't seem to get anywhere
| Compute $d^2 x/dt^2$, giving
$$\frac{d^2 x}{dt^2} = - \frac{dx}{dt} + \frac{dy}{dt}$$
Use the second equation to replace $\frac{dy}{dt}$ with an expression in x and y, and use the first equation to replace $y$ with an expression in $dx/dt$ and $x$. The result is a second order equation in x.
Edit: For clarity, this giv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/447162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Implication of an inequality relation Suppose I have linear function $f$ on $\mathbb{R}^n$ and another function $p$, which is positive homogeneous, i.e. $p(\lambda x)=\lambda p(x)$ for all $\lambda\ge 0$. We have the following implication
$$f(x)>-1\Rightarrow p(x)>-1$$
Since we can multiply by positive scalars, we get
... | "Since we can multiply by positive scalars": what does it mean?
Is this something like: for $x = λy$ with $λ>0$ and y in $ℝ^n$:
$f(x)>−1 ⇒ p(x)>−1$
$f(λy)>−1 ⇒ p(λy)>−1$
$λf(y)>−1 ⇒ λp(y)>−1$
$f(y)>−1/λ ⇒ p(y)>−1/λ$
?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/447212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Say I have a bag of 100 unique marbles. With I replacement, I pick 10 marbles at a time, at random. Say I have a bag of 100 unique marbles. With replacement, I pick 10 marbles at a time, at random. How many times will I have to pick the marbles (10 marbles a pick) in order to have a 95% chance of having seen every uniq... | Simulations show that you need 73 or 72 picks (probably this, $10^7$ turns give $p = 0.950612$) and I don't know any efficient method to calculate that exactly (of course, that doesn't mean there isn't one).
irb(main):021:0> average(1000000) do g(100,10,73)? 1:0 end
=> 0.955118
irb(main):022:0> average(1000000) do g(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/447301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
norms and sparsity Could anyone please elaborate on why $L^2$ norm moves toward the outliers compared to $L^1$ norm. I mean, what property/quantity in the mathematical expression of the norms makes it perform such way.
One more thing is, how $L^1$ norm introduces more sparsity in the solution?
Thank you.
Praveen
| I think the overall context you're referring to is the problem of $L^0$ minimization, i.e. compressed sensing. So, the goal of your question is to find the relationship between the following problems:
$(P_0)\qquad \min \|x\|_0 \;s.t.\;Ax=y\\
(P_1)\qquad \min \|x\|_1 \;s.t.\;Ax=y\\
(P_2)\qquad \min \|x\|_2 \;s.t.\;Ax=y$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/447370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
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