Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Question on Showing points of discontinuities of a function are removable (or not) The question is as follows:
Given function: $F(x,y)=\frac{x + 2y}{sin(x+y) - cos(x-y)}$
Tasks:
a/ Find points of discontinuities
b/ Decide if the points (of
discontinuities) from part a are removable
Here is my work so far:
(1) For ... | Observe that F(x,y) =(x+2y) /(2*Cos(y+pi/4)*Sin(x - pi/4)).just use formula for sin(a) - sin(b) , and cos(b) = sin (pi/2 -b)
Thus the set of discontinuities are (x, pi/4 +n*pi) and (pi/4 + n*pi, y) for any x,y real. ie. they are lines parellel to x and y axis ie. a grid.
So we have to look at point of intersection of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/385256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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When are the binomial coefficients equal to a generalization involving the Gamma function? Let $\Gamma$ be the Gamma function and abbreviate $x!:=\Gamma(x+1)$, $x>-1$.
For $\alpha>0$ let us generalize the binomial coefficients in the following way:
$$\binom{n+m}{n}_\alpha:=\frac{(\alpha n+\alpha m)!}{(\alpha n)!(\alpha... | Set, for example, $m=1$ and consider the limit $n\rightarrow\infty$. Then
$$ {\alpha n+\alpha\choose \alpha n}\sim \frac{(\alpha n)^{\alpha}}{\alpha!},\qquad
{n+1 \choose n}\sim n.$$
It is clear that the only possibility for both asymptotics to agree is $\alpha=1$. In the more general situation, the same argument shows... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/385320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Existence of whole number between two real numbers $x$ and $x +1$? How to prove that there is a whole number, integer, between two real numbers $x$ and $x+1$ (in case $x$ is not whole)?
I need this for an exercise solution in my Topology class, so I can, probably, use more than just axioms from set theory.
Any ideas? ... | This can be proved using the decimal expansion of $x$. If $x$ has the decimal expansion $n_0.n_1n_2n_3\cdots$, where $n_0$ is some whole number, then $x+1$ has the decimal expanion $(n_0+1).n_1n_2n_3\cdots$.
Then it is clear that $x < n_0+1 < x+1$ if $x$ is not a whole number itself.
| {
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"url": "https://math.stackexchange.com/questions/385429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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What does "Solve $ax \equiv b \pmod{337}$ for $x$" mean? I have a general question about modular equations:
Let's say I have this simple equation:
$$ax\equiv b \pmod{337}$$
I need to solve the equation.
What does "solve the equation" mean? There are an infinite number of $x$'s that will be correct.
Does $x$ need to be ... | Write $p = 337$. This is a prime number.
First of all, if $a \equiv 0 \pmod{p}$ then there is a solution if and only if $b \equiv 0 \pmod{p}$, and then all integers $x$ are a solution.
If $a \not\equiv 0 \pmod{p}$, then the (infinite number of integer) solutions will form a congruence class modulo $p$. These can be fo... | {
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What is the precise statement of the theorem that allows us to "localize" our knowledge of derivatives? Most introductory calculus courses feature a proof that
Proposition 1. For the function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $x \in \mathbb{R} \Rightarrow f(x)=x^2$ it holds that $x \in \mathbb{R} \Right... | I’ll give a first try in answering this:
How about: Let $f : D_f → ℝ$ and $g : D_g → ℝ$ be differentiable in an open set $D ⊂ D_f ∩ D_g$. If $f|_D = g|_D$, then $f'|_D = g'|_D$.
I feel this is not what you want. Did I misunderstand you?
| {
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"url": "https://math.stackexchange.com/questions/385565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Evaluating this integral : $ \int \frac {1-7\cos^2x} {\sin^7x \cos^2x} dx $ The question :
$$ \int \frac {1-7\cos^2x} {\sin^7x \cos^2x} dx $$
I tried dividing by $\cos^2 x$ and splitting the fraction.
That turned out to be complicated(Atleast for me!)
How do I proceed now?
| The integration is $$\int \frac{dx}{\sin^7x\cos^2x}-\int\csc^7xdx$$
Using this repeatedly, $$\frac{m-1}{n+1}\int\sin^{m-2}\cos^n dx=\frac{\sin^{m-1}x\cos^{n+1}x}{m+n}+\int \sin^mx\cos^n dx,$$
$$\text{we can reach from }\int \frac{dx}{\sin^7x\cos^2x}dx\text{ to } \int \frac{\sin xdx}{\cos^2x}dx$$
Now use the Reduction... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Symmetric Groups and Commutativity I just finished my homework which involved, among many things, the following question:
Let $S_{3}$ be the symmetric group $\{1,2,3\}$. Determine the number of elements that commute with (23).
Now, solving this was unproblematic - for those interested; the answer is 2. However, it got... | *
*Let $\pi, \phi\in S_{\Omega}$. Then $\pi, \phi$ are disjoint if $\pi$ moves $\omega\in \Omega$ then $\phi$ doesn't move $\omega$.
For example, $(2,3)$ and $(4,5)$ in $S_6$ are disjoint. indeed, $\{2,3\}\cap\{4,5\}=\emptyset$.
Theorem: If $\pi, \phi\in S_{\Omega}$ are disjoint then $\pi\phi=\phi\pi$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Find the following integral: $\int {{{1 + \sin x} \over {\cos x}}dx} $ My attempt:
$\int {{{1 + \sin x} \over {\cos x}}dx} $,
given : $u = \sin x$
I use the general rule:
$\eqalign{
& \int {f(x)dx = \int {f\left[ {g(u)} \right]{{dx} \over {du}}du} } \cr
& {{du} \over {dx}} = \cos x \cr
& {{dx} \over {du}} = ... | You replaced $\cos^2 x$ by $\sqrt{1-u^2}$, it should be $1-u^2$.
Remark: It is easier to multiply top and bottom by $1-\sin x$. Then we are integrating $\frac{\cos x}{1-\sin x}$, easy.
| {
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clarity on a question The original question is :
Let $0 < a_{1}<a_{2}<\dots<a_{mn+1}\;\;$ be $\;mn+1\;$ integers. Prove that you can either $\;m+1\;$ of them no one of which divides any other or $\;n+1\;$ of them each dividing the following.
(1966 Putnam Mathematical Competition)
The question has words missing , so cou... | I think it is this question
Given a set of $(mn + 1)$ unequal positive integers, prove that we can either $(1)$ find $m + 1$ integers $b_i$ in the set such that $b_i$ does not divide $b_j$ for any unequal $i, j,$ or $(2)$ find $n+1$ integers $a_i$ in the set such that $a_i$ divides $a_{i+1}$ for $i = 1, 2, \dots , n$.
... | {
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reflection groups and hyperplane arrangement We know that for the braid arrangement $A_\ell$ in $\mathbb{C}^\ell$: $$\Pi_{1 \leq i < j \leq \ell} (x_i - x_j)=0,$$
$\pi_1(\mathbb{C}^\ell - A_\ell) \cong PB_\ell$, where $PB_\ell$ is the pure braid group.
Moreover, the reflection group that is associated to $A_\ell$ is th... | I have found out that Brieskorn proved the following (using the above notations):
$$
\pi_1(\mathbb{C}^\ell - A_\ell) \cong \text{ker}(A_L \rightarrow G_L)
$$
where $A_L$ is the corresponding Artin group, $G_L$ the reflection group.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Truth of Fundamental Theorem of Arithmetic beyond some large number Let $n$ be a ridiculously large number, e.g., $$\displaystyle23^{23^{23^{23^{23^{23^{23^{23^{23^{23^{23^{23^{23}}}}}}}}}}}}+5$$ which cannot be explicitly written down provided the size of the universe. Can a Prime factorization of $n$ still be possibl... | Yes, there exists a unique prime factorization.
No, we probably won't ever know what it is.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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When the ordinal sum equals the Hessenberg ("natural") sum Let $\alpha_1 \geq \ldots \geq \alpha_n$ be ordinal numbers. I am interested in necessary and sufficient conditions for the ordinal sum $\alpha_1 + \ldots + \alpha_n$ to be equal to the Hessenberg sum $\alpha_1 \oplus \ldots \alpha_n$, most quickly defined by ... | You are exactly right about the "asymmetrically absorptive" nature of standard ordinal addition (specifically with regard to Cantor normal form). Your condition is necessary and sufficient (sufficiency is easy, and you've shown necessity). I don't know of any standard name for such sequences, though. As for your $\omeg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/386274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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How to find length of a rectangular tile when viewing at some angle I have a question on angles.
I have a rectangular tile. when looking straight I can find the width of the tile, but how do I find the apparent width when I see the same rectangular tile at some angle. Below I have attached an image for more clarity. S... | It depends on your projection. If you assume orthogonal projection, so that the apparent length of line segments is independent of their distance the way your images suggest, then you cannot solve this, since a rectangle of any aspect ratio might appear as a rectangle of any other aspect ratio by simply aligning it wit... | {
"language": "en",
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prove or disprove invertible matrix with given equations Given a non-scalar matrix $A$ in size $n\times n$ over $\mathbb{R}$ that maintains the following equation
$$A^2 + 2A = 3I$$
given matrix $B$ in size $n\times n$ too $$B = A^2 + A- 6I$$
Is $B$ an invertible matrix?
| Hint: The first equation implies that $A^2+2A-3I=(A-I)(A+3I)=0$. Hence the minimal polynomial $m_A(x)$ of $A$ divides $(x-1)(x+3)$. What happens if $x+3$ is not a factor of $m_A(x)$?
| {
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"timestamp": "2023-03-29T00:00:00",
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Estimate derivatives in terms of derivatives of the Fourier transform. Let us suppose that $f: \mathbb{R}^n \to \mathbb{R}$ is a smooth function. Furthermore, for every $\alpha$ multi-index, there exists $C_\alpha > 0$ such that
$$
|D^\alpha f(\xi)| \leq \frac{C_\alpha}{(1+|\xi|)^{|\alpha|}}.
$$
Does it follow that, f... | This settles it. See Theorem 9, it also settles regularity issues.
| {
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Unable to solve expression for $x$ I'm trying to solve this expression for $x$:
$$\frac{x^n(n(1-x)+1)}{(1-x)^2}=0$$
I'm not sure where to begin (especially getting rid of the $x^n$ part), any hints or tips are appreciated.
| Hint: Try multiplying both sides by the denominator, to get rid of it. Note that this may introduce extraneous solutions if what we multiplied by is $0$, so you have to consider the case of $(x - 1)^2 = 0$ separately.
Finally, to solve an equation of the form $a \cdot b = 0$, you can divide by $a \neq 0$ on both sides ... | {
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Summation of independent discrete random variables? We have a summation of independent discrete random variables (rvs) $Y = X_1 + X_2 + \ldots + X_n$. Assume the rvs can take non-negative real values. How can we find the probability mass function of $Y$?
Is there any efficient method like the convolution for integer ... | Since the random variables are continuous, you would speak of their probability density function (instead of the probability mass function). The probability density function (PDF) of $Y$ is simply the (continuous) convolution of the PDFs of the random variables $X_i$. Convolution of two continuous random variables is d... | {
"language": "en",
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Could someone please explain the theory behind finding if a given point is inside a circle on a grid? Let us say I have a grid of 1000 x 1000, and on that grid is drawn a circle, the circle could be anywhere.
If I then pick a random point from the grid with an x and y co-ordinate I can work out if the point is inside t... | Let me try to answer in words only. you have a circle. To fill it in, as with a paint program, it's every point whose distance from the center of the circle is less than the radius of the circle. Simple enough.
To test if a point is inside the circle, calculate the distance from the center point to your point. If less ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Mean number of particle present in the system: birth-death process, $E(X_t|X_0=i)$, $b_i=\frac{b}{i+1}$, $d_i=d$ Let $\{X_t\}$ be a birth–and–death process with birth rate
$$
b_i = \frac{b}{i+1},
$$
when $i$ particle are in the system, and a constant death rate
$$
d_i=d.
$$
Find the expected number of particle in the s... | Not sure one can get explicit formulas for $E[X_t]$ but anyway, your function $f$ is not rich enough to capture the dynamics of the process.
The canonical way to go is to consider $u(t,s)=E[s^{X_t}]$ for every $t\geqslant0$ and, say, every $s$ in $(0,1)$. Then, pending some errors in computations done too quickly, the... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Mnemonic for centroid of a bounded region The centroid of a region bounded by two curves is given by:
$ \bar{x} = \frac{1}{A}\int_a^b{x\left[f(x)-g(x)\right]dx} $
$ \bar{y} = \frac{1}{A}\int_a^b{\left[\frac{(f(x)+g(x)}{2}(f(x)-g(x))\right]dx} = \frac{1}{2A}\int_a^b{\left(f^2(x) - g^2(x)\right)dx}$
where A is just the a... | In order to remember those formulas, you have to use them repeatedly on many problems involving finding centroid of areas bounded by two curves. Have faith in the learning process and you will remember it after using it many times, just like playing online games.
| {
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calculate the number of possible number of words If one word can be at most 63 characters long. It can be combination of :
*
*letters from a to z
*numbers from 0 to 9
*hyphen - but only if not in the first or the last character of the word
I'm trying to calculate possible number of combinations for a given domai... | Close - but 26 letters plus 10 numbers plus the hyphen is 37 characters total, so it would be
(36^2)(37^61)
Now granted, that's just the number of alphanumeric combinations; whether those combinations are actually words would require quite a bit of proofreading.
| {
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Prove that connected graph G, with 11 vertices and and 52 edges, is Hamiltonian Is this graph always, sometimes, or never Eulerian? Give a proof or a pair of examples to justify your answer
Could G contain an Euler trail? Must G contain an Euler trail? Fully justify your answer
| $G$ is obtained from $K_{11}$ by removing three edges $e_1,e_2,e_3$. We label now the vertices of $G$ the following way:
$$e_1=(1,3)$$
Label the unlabeled vertices of $e_2$ by the smallest unused odd numbers, and label the unlabeled vertices of $e_3$ by the smallest unused odd numbers. Note that by our choices $e_3 \ne... | {
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Dense set in $L^2$ Let $ \Omega\subset \mathbb{R}^n$ with $m(\Omega^c)=0 $. Then how can we show that $ \mathcal{F}(C_{0}^{\infty}(\Omega))$ (here $ \mathcal{F}$ denotes the fourier transform) is dense in $L^2$(or $L^p$)?
Besides, I'm also interested to know if the condition that $m(\Omega^c)=0$ can be weakened to som... | To deal with the case $\Omega=\Bbb R^n$, take $f\in L^2$. Then by Plancherel's theorem, theorem 12 in these lecture notes, we can find $g\in L^2$ such that $f=\mathcal F g$. Now approximate $g$ by smooth functions with compact support and use isometry.
| {
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Triple Integral over a disk How do I integrate $$z = \frac{1}{x^2+y^2+1}$$ over the region above the disk $x^2+y^2 \leq R^2$?
| Use polar coordinates: $x^2+y^2 = r^2$, etc. An area element is $dx\, dy = r \, dr \, d\theta$. The integral over the disk is
$$\int_0^R dr \, r \: \int_0^{2 \pi} d\theta \frac{1}{1+r^2} = 2 \pi \int_0^R dr \frac{r}{1+r^2}$$
You can substitute $u=r^2$ to get for theintegral
$$\pi \int_0^{R^2} \frac{du}{1+u}$$
I trust... | {
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probable squares in a square cake There is a probability density function defined on the square [0,1]x[0,1].
The pdf is finite, i.e., the cumulative density is positive only for pieces with positive area.
Now Alice and Bob play a game: Alice marks two disjoint squares, Bob chooses the square that contains the maximum p... | I think Alice can always assure herself at least $1 \over 4$ cdf, in the following way.
First, in each of the 4 corners, mark a square that contains $1 \over 4$ cdf. Since the pdf is finite, it is always possible to construct such a square, by starting from the corner and increasing the square gradually, until it conta... | {
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How to simplify $\frac{(\sec\theta -\tan\theta)^2+1}{\sec\theta \csc\theta -\tan\theta \csc \theta} $ How to simplify the following expression :
$$\frac{(\sec\theta -\tan\theta)^2+1}{\sec\theta \csc\theta -\tan\theta \csc \theta} $$
| The numerator becomes
$(\sec\theta -\tan\theta)^2+1=\sec^2\theta+\tan^2\theta-2\sec\theta\tan\theta+1=2\sec\theta(\sec\theta -\tan\theta)$
So, $$\frac{(\sec\theta -\tan\theta)^2+1}{\sec\theta \csc\theta -\tan\theta \csc \theta}$$
$$=\frac{2\sec\theta(\sec\theta -\tan\theta)}{\csc\theta(\sec\theta -\tan\theta)}=2\frac{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Metric spaces and distance functions. I need to provide an example of a space of points X and a distance function d, such that the following properties hold:
*
*X has a countable dense subset
*X is uncountably infinite and has only one limit point
*X is uncountably infinite and every point of X is isolated
I'm rea... | For the first question, hint: $\mathbb{Q}$ is a countable set.
For the third question, hint: think about the discrete metric on a space.
For the second question: Let $X=\{x\in\mathbb{R}\:|\: x>1 \mbox{ or }x=\frac{1}{n}, n\in\mathbb{N}_{\geq 1}\}\cup\{0\}$. Let
*
*$d(x,y)=1$ if $x> 1$ or $y>1$,
*$d(x,y)=|x-y|$ if $... | {
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"url": "https://math.stackexchange.com/questions/387507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Group $\mathbb Q^*$ as direct product/sum Is the group $\mathbb Q^*$ (rationals without $0$ under multiplication) a direct product or a direct sum of nontrivial subgroups?
My thoughts:
Consider subgroups $\langle p\rangle=\{p^k\mid k\in \mathbb Z\}$ generated by a positive prime $p$ and $\langle -1\rangle=\{-1,1\}$.
T... | Yes, you're right.
Your statement can be generalized to the multiplicative group $K^*$ of the fraction field $K$ of a unique factorization domain $R$. Can you see how?
In fact, if I'm not mistaken it follows from this that for any number field $K$, the group $K^*$ is the product of a finite cyclic group (the group o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/387580",
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Compute the Centroid of a Semicircle without Calculus Can the centroid of a semicircle be computed without deferring to calculus or a limiting procedure?
| The following may be acceptable to you as an answer. You can use the centroid theorem of Pappus.
I do not know whether you really mean half-circle (a semi-circular piece of wire), or a half-disk. Either problem can be solved using the theorem of Pappus.
When a region is rotated about an axis that does not go through t... | {
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Determining probability of certain combinations Say I have a set of numbers 1,2,3,4,5,6,7,8,9,10 and I say 10 C 4 I know that equals 210. But lets say I want to know how often 3 appears in those combinations how do I determine that?
I now know the answer to this is $\binom{1}{1}$ $\binom{9}{3}$
I am trying to apply thi... | Corrected:
First off, your fraction is upside-down: $\binom{S-1}{N-1}$ is the total number of groups of $N$ students that include you, so it should be the denominator of your probability, not the numerator. Your figure of $\binom{M-1}K\binom{N-K-1}{S-K-1}$ also has an inversion: it should be $\binom{M-1}K\binom{S-K-1}{... | {
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"url": "https://math.stackexchange.com/questions/387713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proper way to define this multiset operator that does a pseudo-intersection? it's been a while since I've done anything with set theory and I'm trying to find a way to describe a certain operator.
Let's say I have two multisets:
$A = \{1,1,2,3,4\}$
$B = \{1,5,6,7\}$
How can I define the operator $\mathbf{O}$ such that... | Let us represent multisets by ordered pairs, $\newcommand{\tup}[1]{\langle #1\rangle}\tup{x,i}$ where $x$ is the element and $i>0$ is the number of times that $x$ is in the set.
Let me write the two two multisets in this notation now: $$A=\{\tup{1,2},\tup{2,1},\tup{3,1},\tup{4,1}\},\quad B=\{\tup{1,1},\tup{5,1},\tup{6,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Determining Fourier series for $\lvert \sin{x}\rvert$ for building sums My math problem is a bit more tricky than it sounds in the caption.
I have the following Task (which i in fact do not understand):
"Determine the Fourier series for $f(x)=\lvert \sin{x}\rvert$ in order to build the Sum for the series: $\frac{1}{1*... | You were on the right track. First, calculate the Fourier series of $f(x)$ (you can leave out the magnitude signs because $\sin x \ge 0$ for $0\le x \le \pi$ ):
$$a_n=\frac{2}{\pi}\int_{0}^{\pi}\sin x\cos nx\;dx=
\left\{ \begin{array}{l}-\frac{4}{\pi}\frac{1}{(n+1)(n-1)},\quad n \text{ even}\\
0,\quad n \text{ odd} \en... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/387820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove $n$ is prime? Let $n \gt 1$ and
$$\left\lfloor\frac n 1\right\rfloor + \left\lfloor\frac n2\right\rfloor + \ldots + \left\lfloor\frac n n\right\rfloor = \left\lfloor\frac{n-1}{1}\right\rfloor + \left\lfloor\frac{n-1}{2}\right\rfloor + \ldots + \left\lfloor\frac{n-1}{n-1}\right\rfloor + 2$$
and $\lfloor \... | You know that
$$\left( \left\lfloor\frac n 1\right\rfloor - \left\lfloor\frac{n-1}{1}\right\rfloor \right)+\left( \left\lfloor\frac n2\right\rfloor - \left\lfloor\frac{n-1}{2}\right\rfloor\right) + \ldots + \left( \left\lfloor\frac{n}{n-1}\right\rfloor - \left\lfloor\frac{n-1}{n-1}\right\rfloor\right) + \left\lfloor\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/387885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Examples of Diophantine equations with a large finite number of solutions I wonder, if there are examples of Diophantine equations (or systems of such equations) with integer coefficients fitting on a few lines that have been proven to have a finite, but really huge number of solutions?
Are there ones with so large nu... | Let $b$ is a non-zero integer, and let $n$ is a positive integer.
The equation $y(x-b)=x^n$ has only finitely many integer solutions.
The first solution is:
$x=b+b^n$ and $y=(1+b^{n-1})^n$.
The second solution is:
$x=b-b^n$ and $y=-(1-b^{n-1})^n$,
cf. [1, page 7, Theorem 9] and [2, page 709, Theorem 2].
The number of i... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Probability related finance question: Need a more formal solution You are offered a contract on a piece of land which is worth $1,000,000$ USD $70\%$ of the time, $500,000$ USD $20\%$ percent of the time, and $150,000$ USD $10\%$ of the time. We're trying to max profit.
The contract says you can pay $x$ dollars for som... | There is no unique arbitrage-free solution to the pricing problem with $3$ outcomes, so you will need to impose more assumptions to get a numerical value for the land.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/388014",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the centroid of a hollow spherical cap? I have a unit hollow sphere which I cut along a diameter to generate two equivalent hollow hemispheres. I place one of these hemispheres on an (x,y) plane, letting it rest on the circular planar face where the cut occurred.
If the hemisphere was solid, we could write th... | Use $z_0=\displaystyle {{\int z ds}\over {\int ds}}$ where $z=\cos\phi$ and $ds=\sin\phi d\theta d\phi$. That gives your 1/2.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/388083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Simple dice questions. There are two dice.
Assume dice are fair.
What does the following probability represent:
It's $$\frac{1}{6} + \frac{1}{6} - \left(\frac{1}{36} \right)$$
What does this represent:
$$\frac{1}{6} \cdot \frac{5}{6} + \frac{1}{6} \cdot \frac{5}{6}$$
This represents the probability of rolling just a si... | 1> Same as 3. [but used the inclusion exclusion principle]
2> P[You get different outcomes on rolling the pair of dice twice]
3> P[you get a particular outcome atleast once]
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/388132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A formal proof required using real analysis If $\int_0^1f^{2n}(x)dx=0$, prove that $f(x)=0$, where $f$ is a real valued continuous function on [0,1]?
It is obvious, since $f^{2n}(x) \geq 0$, the only way this is possible is when $f(x)=0$.
I am looking for any other formal way of writing this proof i.e. using concepts f... | Assume there exists $c\in[0,1]$ such that $f^{2n}(x)>0$, then by definition of continuity there exists $0<\delta< \min(c,1-c)$ such that $$|x-c|<\delta\implies |f^{2n}(x)-f^{2n}(c)|<\frac{1}{2} f^{2n}(c)$$
Especially, if $|x-c|<\delta$ then $f^{2n}(x)>\frac{1}{2} f^{2n}(c)$. Therefore
$$\int_0^1 f^{2n}(x) dx =\int_0^{c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/388200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Partitions of an interval and convergence of nets Let $\mathscr{T}$ be the set of partitions $\tau = (\tau_0 = 0 < \tau_1 < \dots < \tau_N = 1)$ of the interval $[0,1]$ (where $N$ is not fixed). This becomes a directed set by setting $\tau < \tau^\prime$ iff $\tau^\prime$ is a subdivision.
Now we can look at the nets
$... | A general hint and a partial illustration.
As I understood, you must show that for each $\varepsilon>0$ there are nets $\tau_1$ and $\tau_2$ such that $s_{\tau_1’}<\varepsilon$ and $r_{\tau_2’}<\varepsilon$ for each $\tau_1’>\tau_1$ and $\tau_2’>\tau_2$.
Since $(a+b)^2\ge a^2+b^2$, provided $a$ and $b$ are non-negat... | {
"language": "en",
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Unexpected approximations which have led to important mathematical discoveries On a regular basis, one sees at MSE approximate numerology questions like
*
*Prove $\log_{{1}/{4}} \frac{8}{7}> \log_{{1}/{5}} \frac{5}{4}$,
*Prove $\left(\dfrac{2}{5}\right)^{{2}/{5}}<\ln{2}$,
*Comparing $2013!$ and $1007^{2013}$
or... | The most famous, most misguided, and most useful case of approximation fanaticism comes from Kepler's attempt to match the orbits of the planets to a nested arrangement of platonic solids. Fortunately, he decided to go with his data instead of his desires and abandoned the approximations in favor of Kepler's Laws.
Kepl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/388345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "62",
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Determining power series for $\frac{3x^{2}-4x+9}{(x-1)^2(x+3)}$ I'm looking for the power series for $f(x)=\frac{3x^{2}-4x+9}{(x-1)^2(x+3)}$
My approach: the given function is a combination of two problems. first i made some transformations, so the function looks easier.
$$\frac{3x^{2}-4x+9}{(x-1)^2(x+3)})=\frac{3x^{2... | You can use the partial fraction decomposition:
$$
\frac{3x^{2}-4x+9}{(x-1)^2(x+3)}=
\frac{A}{1-x}+\frac{B}{(1-x)^2}+\frac{C}{1+\frac{1}{3}x}
$$
and sum up the series you get, which are known.
If you do the computation, you find $A=0$, $B=2$ and $C=1$, so
$$
\frac{3x^{2}-4x+9}{(x-1)^2(x+3)}=
\frac{2}{(1-x)^2}+\frac{1}{... | {
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"timestamp": "2023-03-29T00:00:00",
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Subgroup transitive on the subset with same cardinality Maybe there is some very obvious insight that i miss here, but i've asked this question also to other people and nothing meaningful came out:
If you have a subgroup G of $S_n$(the symmetric group on n elements), you can consider the natural action of G on the subs... | By the Livingstone Wagner Theorem, (Livingstone, D., Wagner, A., Transitivity of finite permutation groups on unordered sets. Math. Z. 90 (1965) 393–403), if $n \ge 2k$ and $G$ is transitive on $k$-subsets with $k \ge 5$, then $G$ is $k$-transitive. Using the classification of finite simple groups, $A_n$ and $S_n$ are... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "4",
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Prove A is symmetric matrix iff $A$ is square and $x^T Ay = (Ax)^T y$ Prove A is a symmetric matrix iff $A$ is square and $x^T Ay = (Ax)^T y$. (for all $x,y \in \mathbb{R}^n$)
Going from the assumption that it is symmetric to the two conditions is fairly straightforward.
However, going the other way, I am stuck at prov... | First, lets remember the rules of transposing the product: $(AB)^T = B^T A^T$
Using this, we start with the giver equation $ x^TAy=(Ax)^Ty$ and apply the rule above which yields
$x^TAy = x^TA^Ty$.
Note that the equation is true for all $x, y$, so the only way that is possible is if $A = A^T$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/388555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Congruence of invertible skew symmetric matrices I am asking for some hints to solve this excercise. Given an invertible skew symmetric matrix $A$, then show that there are invertible matrices $ R, R^T$ such that $R^T A R = \begin{pmatrix} 0 & Id \\ -Id & 0 \end{pmatrix}$, meaning that this is a block matrix that has t... | Hint: a skew-symmetric matrix commutes with its transpose, and so it is diagonalizable. Your block matrix on your right hand side is also skew-symmetric, and so it is also diagonalizable.
| {
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"url": "https://math.stackexchange.com/questions/388750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Probability of retirement event This is elementary, but not clear to me.
Suppose I know that the mean age of retirement is $\mu$ and the standard deviation $\sigma$.
What is the probability that someone of age $x$, who has not yet retired,
will retire sometime in the next year,
i.e., between $x$ and $x+1$?
Clearly for... | Integrate the Gaussian probability density function from x to x+1
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Do the premises logically imply the conclusion? $$b\rightarrow a,\lnot c\rightarrow\lnot a\models\lnot(b\land \lnot c)$$
I have generated an 8 row truth table, separating it into $b\rightarrow a$, $\lnot c\rightarrow\lnot a$ and $\lnot (b\land\lnot c)$. I know that if it was
$$\lnot c\rightarrow\lnot a\models\lnot(a \... | The second premise $\neg c\to\neg a$ implies that $a\to c$.
The first premise $b\to a$ leads to $b\to a\to c$, which implies $\neg c\to\neg b$.
The two last statements clearly prevent $b\land\neg c$ from being true.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/388888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does there exist a matrix $P$ such that $P^n=M$ for a special matrix $M$? Consider the matrix
$$
M=\left(\begin{matrix}
0&0&0&1\\
0&0&0&0\\
0&0&0&0\\
0&0&0&0\\
\end{matrix}\right).
$$
Is there a matrix $P\in{\Bbb C}^{4\times 4}$ such that $P^n=M$ for some $n>1$?
One obvious fact is that if such $P$ exists, then $P$ mu... | Since it has been proposed to treat this Question as a duplicate of the present one, it should be noted that there is a negative Answer to the "new" issue raised in Yeyeye's Answer here.
As earlier observed, since $M$ is nilpotent, for $P^n = M$ for $n\gt 1$ will require that $P$ is nilpotent. It follows that the mini... | {
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"source": "stackexchange",
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Let $W$ be a Wiener process and $X(t):=W^{2}(t)$ for $t\geq 0.$ Calculate $\operatorname{Cov}(X(s), X(t))$. Let $W$ be a Wiener process. If $X(t):=W^2(t)$ for $t\geq 0$, calculate $\operatorname{Cov}(X(s),X(t))$
| Assume WLOG that $t\geqslant s$. The main task is to compute $E(X_sX_t)$.
Write $X_t=(W_t-W_s+W_s)^2=(W_t-W_s)^2+2W_t(W_t-W_s)+W_s^2$, and use the fact that if $U$ and $V$ are independent random variables, so are $U^2$ and $V^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/389067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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stability and asymptotic stability: unstable but asymptotically convergent solution of nonlinear system Consider nonlinear systems of the form $X(t)'=F(X(t))$, where $F$ is smooth (assume $C^\infty$). Is it possible to construct such a system (preferably planar system) so that $X_0$ is an unstable equilibrium, but all ... |
Definition (Verhulst 1996)
The equilibrium solution $X_c$ is called asymptotically stable if there exists a $\delta(t_0)$ such that
\begin{equation}
||X_0 - X_c|| \le \delta(t_0) \implies \lim_{t\rightarrow\infty} ||X(t;t_0,X_0) - X_c|| =0
\end{equation}
So no, you're essentially asking can a system be both asymptoti... | {
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"timestamp": "2023-03-29T00:00:00",
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Where is wrong in this proof Suppose $a=b$.
Multiplying by $a$ on both sides gives $a^2 = ab$. Then we subtract $b^2$ on both sides, and get
$a^2-b^2 = ab-b^2$.
Obviously, $(a-b)(a+b) = b(a-b)$, so dividing by $a - b$, we find
$a+b = b$.
Now, suppose $a=b=1$. Then $1=2$ :)
| By a simple example:
if( A = 0 ) A * 5 = A * 7 and by dividing by A we have 5 = 7.
Then we can dividing by A if A not equal to zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/389180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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what is the value of this trigonometric expression I want to find out value of this expression
$$\cos^2 48°-\sin^2 12°$$
Just hint the starting step.Is there any any formula regarding $\cos^2 A-\sin^2 B$?
| I've got a formula :
$$\cos(A+B).\cos(A-B)=\cos^2A-\sin^2B$$
so from this formula this question is now easy
$$\cos^248-\sin^212$$
$$\cos60.\cos36$$
$$\frac{\sqrt{5}+1}{8}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/389276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is the difference of two recursively enumerable sets, reducible to $K$? Is the difference of two recursively enumerable sets, reducible to $K$?
$W_x/W_y=\{z|z \in W_x \& z \notin W_y\}$
$K=\{x|\Phi_x(x) \downarrow\}$
$W_x= \text{dom}(\Phi_x)$
| No.
Let $\omega$ denote the set of natural numbers. $K$ is c.e. but incomputable. If a set $A$ and its complement $\bar{A} = \omega - A$ is c.e., then $A$ must be computable. Hence $\bar{K} = \omega - K$, the complement of $K$, is not a c.e. set.
It is clear that $\omega$, the set of natural numbers, is c.e. (it is c... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove $L=\{ 1^n| n\hspace{2mm}\text{is a prime number} \}$ is not regular. Prove $L=\{ 1^n| n\hspace{2mm}\text{is a prime number} \}$ is not regular.
It seems to use one Lemma: Pumping Lemma.
| In addition:
Instead of Pumping lemma one can use the following fact: $L$ is regular iff it is an union of $\lambda$-classes for some left congruence $\lambda$ on the free monoid $A^*$ such that $|A^*/\lambda|<\infty$. Here $A=\{a\}$ (in your notation $a=1$) is commutative, so $\lambda$ is two-sided. The structure o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/389421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Weaker/Stronger Topologies and Compact/Hausdorff Spaces In my topology lecture notes, I have written:
"By considering the identity map between different spaces with the same underlying set, it follows that for a compact, Hausdorff space:
$\bullet$ any weaker topology is compact, but not Hausdorff
$\bullet$ any stronger... | Hint. $X$ being a set, a topology $\tau$ is weaker than a topology $\sigma$ on $X$ if and only if the application
$$ (X, \sigma) \to (X,\tau), x \mapsto x $$
is continuous.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/389485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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$\int_0^1 \frac{{f}(x)}{x^p} $ exists and finite $\implies f(0) = 0 $ Need some help with this question please.
Let $f$ be a continuous function and
let the improper inegral
$$\int_0^1 \frac{{f}(x)}{x^p} $$
exist and be finite for any $ p \geq 1 $.
I need to prove that
$$f(0) = 0 $$
In this question, I really wante... | Hint: Let
$$
g(t) = \int_t^1 \frac{f(x)}{x^p}dx
$$
Now investigate properties of $g(t)$ around $t=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/389527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Even weighted codewords and puncturing My question is below:
Prove that if a binary $(n,M,d)$-code exists for which $d$ is even, then a binary $(n,M,d)$-code exists for which each codeword has even weight.
(Hint: Do some puncturing and extending.)
| Breaking this down to individual steps. Assume that $d=2t$ is an even integer.
Assume that an $(n,M,d)$ code $C$ exists.
*
*Show that puncturing the last bit from the words of $C$, you get a code $C'$ with parameters $(n-1,M,d')$, where $d'\ge d-1$. Actually we could puncture any bit, but let's be specific. Also we ... | {
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Introduction to Abstract Harmonic Analysis for undergraduate background I'm looking for a good starting book on the subject which only assumes standard undergraduate background.
In particular, I need to gain some confidence working with properties of Haar measures, so I can better understand the spaces $L^{p}(G)$ for a... | I suggest the short book by Robert, "Introduction to the Representation Theory of Compact and Locally Compact Groups" which is leisurely and has plenty of exercises. The only prerequisite of this book is some familiarity of finite dimensional representations.
A second book you should look at is Folland's "A Course in A... | {
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$d\mid p\!-\!1\Rightarrow x^d-1\pmod{\!p^2}$ has exactly $d$ roots for odd prime $p$ I'm trying to figure out the number of solutions to the congruence equation $x^d \equiv1 \pmod{p^2}$ where $p$ is prime and $d\mid p-1$.
For the congruence equation ${x^d}\equiv1 \pmod p$ where $p$ is prime and $d\mid p-1$ I've shown t... | Let $p$ be an odd prime, and work modulo $p^k$ (so $k$ does not have to be $2$). Then if $d$ divides $p-1$, the congruence $x^d\equiv 1\pmod{p^k}$ has exactly $d$ solutions.
To prove this, we use the fact that there is a primitive root $g$ of $p^k$, that is, a generator of the group of invertibles modulo $p^k$. Then f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/389710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Arc length of logarithm function I need to find the length of $y = \ln(x)$ (natural logarithm) from $x=\sqrt3$ to $x=\sqrt8$.
So, if I am not mistake, the length should be $$\int^\sqrt8_\sqrt3\sqrt{1+\frac{1}{x^2}}dx$$
I am having trouble calculating the integral. I tried to do substitution, but I still fail to think o... | $\int^{\sqrt{8}}_{\sqrt{3}}\sqrt{1+\frac{1}{x^{2}}}dx$=$\int^{\sqrt{8}}_{\sqrt{3}}\frac{\sqrt{1+x^{2}}}{x}dx$=$\int^{\sqrt{8}}_{\sqrt{3}}\frac{1+x^{2}}{x\sqrt{1+x^{2}}}$=$\int^{\sqrt{8}}_{\sqrt{3}}\frac{x}{\sqrt{1+x^{2}}}dx$+
+$\int^{\sqrt{8}}_{\sqrt{3}}\frac{1}{x\sqrt{1+x^{2}}}dx$=
$=\frac{1}{2}\int^{\sqrt{8}}_{\sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/389781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Order of this group Given $k\in{\mathbb{N}}$, we denote $\Gamma _2(p^k)$ the multiplicative group of all matrix $\begin{bmatrix}{a}&{b}\\{c}&{d}\end{bmatrix}$ with $a,b,c,d\in\mathbb{Z}$, $ad-bc = 1$, $a$ and $d$ are equal to $1$ module $p^k$ and $b$ and $c$ are multiples of $p^k$.
How can I show that $|\Gamma _2(p)/\G... | By exhibiting a set of at most $N=p^{4k}$ matrices $A_1,\ldots,A_N\in\Gamma_2(p)$ such that for each $B\in\Gamma_2(p)$ we have $A_iB\in\Gamma_2(p^k)$ for some $i$.
Matrices of the form $A_i=I+pM_i$ suggest themselves.
Or much simpler: by counting how many matrices in $\Gamma_2(p)$ and $\Gamma_2(p^k)$ map to the same el... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/389845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How do I see that $F$ is a vector field defined on all of $\mathbb{R}^3$? $$\vec{F}(x,y,z)= y^3z^3\mathbf{i} + 2xyz^3\mathbf{j} + 3xy^2z^2\mathbf{k}$$
How do I see that $F$ is a vector field defined on all of $\mathbb{R}^3$? And then is there an easy way to check if it has continuous partial derivatives?
I am looking a... | The vector field $F$ is defined on all of $\mathbb{R}^3$ because all of its component functions are (there are no pints where the functions are undefined, i.e., they make sense when plugging in any point of $\mathbb{R}^3$ into them). If say one of the component functions was $\frac{1}{x-y}$ then the vector field wouldn... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/389896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Artinian rings are perfect Definition. A ring is called perfect if every flat module is projective.
Is there a simple way to prove that an Artinian ring is perfect (in the commutative case)?
| The local case is proved here, Lemma 10.97.2, and then extend the result to the non-local case by using that an artinian ring is isomorphic to a finite product of artinian local rings and a module $M$ over a finite product of rings $R_1\times\cdots\times R_n$ has the form $M_1\times\cdots\times M_n$ with $M_i$ an $R_i$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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For any $11$-vertex graph $G$, show that $G$ and $\overline{G}$ cannot both be planar Let $G$ be a graph with 11 vertices. Prove that $G$ or $\overline{G}$ must be nonplanar.
This question was given as extra study material but a little stuck. Any intuitive explanation would be great!
| It seems the following. Euler formula implies that $E\le 3V-6$ for each planar graph. If both $G$ and $\bar G$ are planar, then $55=|E(K_{11})|\le 6|V(K_{11})|-12=54$, a contradiction.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Total no. of ordered pairs $(x,y)$ in $x^2-y^2=2013$ Total no. of ordered pairs $(x,y)$ which satisfy $x^2-y^2=2013$
My try:: $(x-y).(x+y) = 3 \times 11 \times 61$
If we Calculate for positive integers Then $(x-y).(x+y)=1.2013 = 3 .671=11.183=61.33$
my question is there is any better method for solving the given questi... | You can solve this pretty quickly, since you essentially need to solve a bunch of linear systems. One of them is e.g.
$$x-y = 3\times 11$$
$$x+y = 61$$
Just compute the inverse matrix of
$$\left[\begin{array}{cc}
1 & -1\\
1 & 1
\end{array}\right]$$
and multiply that with the vectors corresponding to the different combi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/390101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Bernoulli differential equation help? We have the equation $$3xy' -2y=\frac{x^3}{y^2}$$ It is a type of Bernouli differential equation. So, since B. diff equation type is
$$y'+P(x)y=Q(x)y^n$$
I modify it a little to:
$$y'- \frac{2y}{3x} = \frac{x^2}{3y^2}$$
$$y'-\frac{2y}{3x}=\frac{1}{3}x^2y^{-2}$$
Now I divide both s... | $$\text{We have $3xy^2 y'-2y^3 = x^3 \implies x (y^3)' - 2y^3 = x^3 \implies \dfrac{(y^3)'}{x^2} + y^3 \times \left(-\dfrac2{x^3}\right) = 1$}$$
$$\text{Now note that }\left(\dfrac1{x^2}\right)' = -\dfrac2{x^3}. \text{ Hence, we have }\dfrac{d}{dx}\left(\dfrac{y^3}{x^2}\right) = 1\implies \dfrac{y^3}{x^2} = x + c$$
$$\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/390162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
} |
Find quotient space on $\mathbb{N} $ On $\mathbb{N}$ is given equivalence relation R with $nRm \iff 4|n-m$. Topology on $\mathbb{N}$ is defined with $\tau=\{\emptyset\}\cup\{U\subseteq\mathbb{N}|n\in U \wedge m|n \implies m\in U\}$.
I need to find quotient space $(\mathbb{N}/R,\tau_{R})$.
I have solution: $\tau_{R}=\{\... | Note that $p^{-1}(\{[1],[2],[3]\}) = \mathbb{N}\backslash 4\mathbb{N}$.
We need to prove that $n\in\mathbb{N}\backslash 4\mathbb{N}$ and $m|n$ implies $m\in\mathbb{N}\backslash 4\mathbb{N}$ and we do this by contraposition.
Suppose $m\notin\mathbb{N}\backslash 4\mathbb{N}$, then $m = 4k$ and thus $m|n$ implies $n = pm ... | {
"language": "en",
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"source": "stackexchange",
"question_score": "1",
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Composition of $\mathrm H^p$ function with Möbius transform Let $f:\mathbb D\rightarrow \mathbb C$ be a function in $\mathrm{H}^p$, i.e. $$\exists M>0,\text{ such that }\int_0^{2\pi}|f(re^{it})|^pdt\leq M<\infty,\forall r\in[o,1)$$
Consider a Möbius transform of the disk $\varphi :\mathbb D\rightarrow\mathbb D$, which ... | Yes. If $\varphi$ is a holomorphic map of unit disk into itself, the composition operator $f\mapsto f\circ \varphi$ is bounded on $H^p$. In fact,
$$\|f\circ \varphi\|_{H^p}\le \left(\frac{1+|\varphi(0)|}{1-|\varphi(0)|}\right)^{1/p}\|f \|_{H^p} \tag1$$
Original source:
John V. Ryff, Subordinate $H^p$ functions, Duke... | {
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What is the meaning of "independent events " and how can we logically conclude independence of two events in probability?
What is the meaning of "independent events " in probability
For eg: Two events (say A and B)are independent , what I understand is the occurrence of A is not affected by occurrence of B .But I am... | I think you are asking why is the notion of independent events when there is no relation between them!
Notion of independent events help to calculate the probability of occurring two independent events on like p(A)Up(B),P(A) intersection P(B) on contrary to calculating dependent events.
Independent events--> their prob... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
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on the commutator subgroup of a special group Let $G'$ be the commutator subgroup of a group $G$ and $G^*=\langle g^{-1}\alpha(g)\mid g\in G, \alpha\in Aut(G)\rangle$.
We know that always $G'\leq G^*$.
It is clear that if $Inn(G)=Aut(G)$, then $G'=G^*$.
Also if $G$ is a non abelian simple group or perfect group, then ... | Serkan's answer is part of a more general family and a more general idea.
The more general idea is to include non-inner automorphisms that create no new subgroup fusion. The easiest way to do this is with power automorphisms, and the simplest examples of those are automorphisms that raise a single generator to a power.... | {
"language": "en",
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Newbie vector spaces question So browsing the tasks our prof gave us to test our skills before the June finals, I've encountered something like this:
"Prove that the kernel and image are subspaces of the space V: $\ker(f) < V, \operatorname{im}(f) < V$, where $<$ means a subspace."
Is it just me or there's something wr... | Let $f$ is a linear transformation from $V$ to $V$. So what is the kernel of $f$? Indeed, it is $$\ker(f)=\{v\in V\mid f(v)=0_V\}$$ It is obvious that $\ker(f)\subseteq V$. Now take $a,b\in F$ the field associated to $V$, and let $v,w\in\ker(f)$. We have $$f(av+bw)=f(av)+f(bw)=af(v)+bf(w)= 0+0=0$$ So the subset $\ker(f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $\exp: \mathfrak{sl}(n,\mathbb R)\to \operatorname{SL}(n,\mathbb R)$ is not surjective It is well known that for $n=2$, this holds. The polar decomposition provides the topology of $\operatorname{SL}(n,\mathbb R)$ as the product of symmetric matrices and orthogonal matrices, which can be written as the produc... | Over $\mathbb{R}$, for a general $n$, a real matrix has a real logarithm if and only if it is nonsingular and in its (complex) Jordan normal form, every Jordan block corresponding to a negative eigenvalue occurs an even number of times. So, you may verify that $\pmatrix{-1&1\\ 0&-1}$ (as given by rschwieb's answer) and... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Determine the number of elements of order 2 in AR
So i have completed parts a and b. For b i reduced R to smith normal form and ended up with diagonals 1,2,6. From this i have said that the structure of the group is $Z_2 \oplus Z_6 \oplus Z$. But i have no idea what so ever about part c.
| As stated in the comments, let us focus on $\,\Bbb Z_2\times\Bbb Z_6\,$ , which for simplicity (for me, at least) we'll better write multiplicatively as $\,C_2\times C_6=\langle a\rangle\times\langle b\rangle\,\;,\;\;a^2=b^6=1$
Suppose the first coordinate is $\,1\,$ , then the second one has to have order $\,2\,$ , a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Closed form for $\sum_{n=1}^\infty\frac{1}{2^n\left(1+\sqrt[2^n]{2}\right)}$ Here is another infinite sum I need you help with:
$$\sum_{n=1}^\infty\frac{1}{2^n\left(1+\sqrt[2^n]{2}\right)}.$$
I was told it could be represented in terms of elementary functions and integers.
| Notice that
$$
\frac1{2^n(\sqrt[2^n]{2}-1)}
-\frac1{2^n(\sqrt[2^n]{2}+1)}
=\frac1{2^{n-1}(\sqrt[2^{n-1}]{2}-1)}
$$
We can rearrange this to
$$
\left(\frac1{2^n(\sqrt[2^n]{2}-1)}-1\right)
=\frac1{2^n(\sqrt[2^n]{2}+1)}
+\left(\frac1{2^{n-1}(\sqrt[2^{n-1}]{2}-1)}-1\right)
$$
and for $n=1$,
$$
\frac1{2^{n-1}(\sqrt[2^{n-1}]... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/390801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "34",
"answer_count": 2,
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Extending transvections/generating the symplectic group The context is showing that the symplectic group is generated by symplectic transvections.
At the very bottom of http://www-math.mit.edu/~dav/sympgen.pdf it is stated that any transvection on the orthogonal space to a hyperbolic plane (a plane generated by $u,v$ s... | Assume that $V$ is whole space with $2n$ dimensional. Then the orthogonal space you mentioned (let it be $W$) is $2n-2$ dimensional symplectic vector space (It is very easy to check the conditions). Then take another space spanned by $v$ and $w$ such that $v$, $w$ $\in W$ and $\omega(v,w)=1$. The new space orthogonal t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/390853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Calculus II, Curve length question. Find the length of the curve
$x= \int_0^y\sqrt{\sec ^4(3 t)-1}dt, \quad 0\le y\le 9$
A bit stumped, without the 'y' in the upper limit it'd make a lot more sense to me.
Advice or solutions with explanation would be very appreciated.
| $$\frac{dx}{dy} = \sqrt{\sec^4{3 y}-1}$$
Arc length is then
$$\begin{align}\int_0^9 dy \sqrt{1+\left ( \frac{dx}{dy} \right )^2} &= \int_0^9 dy\, \sec^2{3 y} \\ &= \frac13 \tan{27} \end{align}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What are some relationships between a matrix and its transpose? All I can think of are that
If symmetric, they're equivalent
If A is orthogonal, then its transpose is equivalent to its inverse.
They have the same rank and determinant.
Is there any relationship between their images/kernels or even eigenvalues?
| Fix a ring $R$, and let $A \in M_n(R)$. The characteristic polynomial for $A$ is $$\chi_A(x)=\det (xI-A),$$
so that $\chi_{A^T}(x) = \det (x I -A^T)= \det ((xI-A)^T)=\det(xI-A)$. Since the eigenvalues of $A$ and $A^T$ are the roots of their respective characteristic polynomials, $A$ and $A^T$ have the same eigenvalue... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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Infinitely many primes of the form $4n+3$ I've found at least 3 other posts$^*$ regarding this theorem, but the posts don't address the issues that I have.
Below is a proof that for infinitely many primes of the form $4n+3$, there's a few questions I have in the proof which I'll mark accordingly.
Proof: Suppose there w... |
Therefore, it is divisible by a prime (How did they get to this conclusion?).
All integers are divisible by some prime!
So every prime which divides N must be of the form 4n+1 (Why must it be of this form?).
Because we've assumed that $p_1, \dots, p_k$ are the only primes of the form 4n+3. If none of those divide N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/391096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "36",
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generalized eigenvector for 3x3 matrix with 1 eigenvalue, 2 eigenvectors
I am trying to find a generalized eigenvector in this problem. (I understand the general theory goes much deeper, but we are only responsible for a limited number of cases.)
I have found eigenvectors $\vec {u_1}$ and $\vec {u_2}.$
When I try $u_1... | We are given the matrix:
$$\begin{bmatrix}2 & 1 & 1\\1 & 2 & 1\\-2 & -2 & -1\\\end{bmatrix}$$
We want to find the characteristic polynomial and eigenvalues by solving
$$|A -\lambda I| = 0 \rightarrow -\lambda^3+3 \lambda^2-3 \lambda+1 = -(\lambda-1)^3 = 0$$
This yields a single eigenvalue, $\lambda = 1$, with an algebr... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Free online mathematical software What are the best free user-friendly alternatives to Mathematica and Maple available online?
I used Magma online calculator a few times for computational algebra issues, and was very much satisfied, even though the calculation time there was limited to $60$ seconds.
Very basic computa... | I use Pari/GP. SAGE includes this as a component too, but I really like GP alone, as it is. In fact, GP comes with integer relation finding functions (as you mentioned) and has enough rational/series symbolic power that I have been able to implement Sister Celine's method for finding recurrence relations among hypergeo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/391224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 9,
"answer_id": 5
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Integral of $\int^1_0 \frac{dx}{1+e^{2x}}$ I am trying to solve this integral and I need your suggestions.
I think about taking $1+e^{2x}$ and setting it as $t$, but I don't know how to continue now.
$$\int^1_0 \frac{dx}{1+e^{2x}}$$
Thanks!
| $\int^{1}_{0}\frac{dx}{1+e^{2x}}=$ $ \int^{1}_{0}\frac{e^{-2x}dx}{1+e^{-2x}}=$ $-\frac{1}{2}\int^{1}_{0}\frac{(1+e^{-2x})'dx}{1+e^{-2x}}=$
$=-\frac{1}{2}ln(1+e^{-2x})|^{1}_{0}= \frac{1}{2}ln\frac{2e^{2}}{1+e^{2}}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/391257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 2
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Zorn's lemma and maximal linearly ordered subsets Let $T$ be a partially ordered set. We say $T$ is a tree if $\forall t\in T$ $\{r\in T\mid r < t\}$ is linearly ordered (such orders can be considered on connected graphs without cycles, i.e. on trees). By a branch we mean a maximal linearly ordered subset in $T$.
It is... | Here is a nice way of proving the well-ordering principle from "Every tree has a branch":
Let $A$ be an infinite set, and let $\lambda$ be the least ordinal such that there is no injection from $\lambda$ into $A$. Consider the set $A^{<\lambda}$, that is all the functions from ordinals smaller than $\lambda$ into $A$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/391336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Number of ways to arrange $n$ people in a line I came across this confusing question in Combinatorics.
Given $n \in \mathbb N$. We have $n$ people that are sitting in a row. We mark $a_n$ as the number of ways to rearrange them such that a person can stay in his seat or move one seat to the right or one seat to the le... | HINT: It’s a little more convenient to think of this in terms of permutations: $a_n$ is the number of permutations $p_1 p_2\dots p_n$ of $[n]=\{1,\dots,n\}$ such that $|p_k-k|\le 1$ for each $k\in[n]$. Call such permutations good.
Suppose that $p_1 p_2\dots p_n$ is such a permutation. Clearly one of $p_{n-1}$ and $p_n$... | {
"language": "en",
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Why do we use open intervals in most proofs and definitions? In my class we usually use intervals and balls in many proofs and definitions, but we almost never use closed intervals (for example, in Stokes Theorem, etc). On the other hand, many books use closed intervals.
Why is this preference? What would happen if we... | My guess is that it's because of two related facts.
*
*The advantage of open intervals is that, since every point in the interval has an open neighbourhood within the interval, there are no special points 'at the edge' like in closed intervals, which require being treated differently.
*Lots of definitions rely on t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Computing $\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}$? How would you compute$$\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}\, \, ?$$
| HINT:
$$\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}= \int_0^{\pi\over2} \frac{\csc^2xdx}{\csc^2x+1}=\int_0^{\pi\over2} \frac{\csc^2xdx}{\cot^2x+2}$$
Put $\cot x=u$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/391537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
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Building a tower using colorful blocks How many possibilities are there to build a tower of n height, using colorful blocks, where:
*
*white block is of 1 height
*black, red, blue, green, yellow, pink blocks are equal to 2 heights
I need to find the generating function formula for this.
So, for n = 1 I get 1 poss... | Let the number of towers of height $n$ be $a_n$. To build a tower of height $n$, you start with a tower of height $n - 1$ and add a white block ($a_{n - 1}$ ways) or with a tower of height $n - 2$ and add one of 6 height-2 blocks. In all, you can write:
$$
a_{n + 2} = a_{n + 1} + 6 a_n
$$
Clearly $a_0 = a_1 = 1$.
Defin... | {
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"url": "https://math.stackexchange.com/questions/391581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Matrix Norm Inequality $\lVert A\rVert_\infty \leq \sqrt{n} \lVert A\rVert_2$ So I'm trying to prove that
$\lVert A\rVert_\infty \leq \sqrt{n} \lVert A\rVert_2$.
I've written the right hand side in terms of rows, but this method doesn't seem to be getting me anywhere.
Where else should I go?
| Writing $A=(A_1,\dots,A_n)^\mathrm{T}$ with $A_i$ being the $i$-th row of the matrix, let $A_j$ be the row for which
$$
\lVert A\rVert_\infty = \max_{1\leq i\leq n }\lVert A_i\rVert_1 = \lVert A_j\rVert_1 = \sum_{k=1}^n \left|A_{i,j}\right|
$$
Then
$$
n\lVert A\rVert_2^2 = n\sum_{i=1}^n \lVert A_i\rVert_2^2 \geq n\lV... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/391744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Radius of convergence of the Bernoulli polynomial generating function power series. The generating function of the Bernoulli Polynomials is: $$\frac{te^{xt}}{e^t-1}=\sum_{k=0}^\infty B_k(x)\frac{t^k}{k!}.$$
Would it be right to say that the radius of convergence of this power series is $2\pi$ ? I'm not sure since the p... | For every fixed $x=c$, the radius of convergence of the power series is $2\pi$. This is because $$\frac{ze^{cz}}{e^z-1}$$ is analytic everywhere except at $z=i2\pi n, n=\pm 1,\pm 2,\cdots$ (not at $0$ though.) The disk $B(0,2\pi)$ is the smallest one centered at $0$ that contains a singularity on its boundary, so the r... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Does this ODE have an exact or well-established approximate analytical solution? The equation looks like this:
$$\frac{\mathrm{d}y}{\mathrm{d}t} = A + B\sin\omega t - C y^n,$$
where $A$, $B$, $C$ are positive constants, and $n\ge1$ is an integer. Actually I am mainly concerned with the $n=4$ case.
The $n=1$ case is tr... | If $\frac{dy}{dt} = F(t,y)$ has $F$ and $\partial_2F$ continuous on a rectangle then there exists a unique local solution at any particular point within the rectangle. This is in the basic texts. For $F(t,y) = f(t)-g(y)$ continuity requires continuity of $f$ and $g$ whereas continuity of the partial derivative of $F$ w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/391862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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relationship of polar unit vectors to rectangular I'm looking at page 16 of Fleisch's Student's Guide to Vectors and Tensors. The author is talking about the relationship between the unit vector in 2D rectangular vs polar coordinate systems. They give these equations:
\begin{align}\hat{r} &= \cos(\theta)\hat{i} + \si... | The symbols on the left side of those equations don't make any sense. If you wanted to change to a new pair of coordinates $(\hat{u}, \hat{v})$ by rotating through an angle $\theta$, then you would have
$$
\left\{\begin{align}
\hat{u} &= (\cos \theta) \hat{\imath} + (\sin \theta)\hat{\jmath} \\
\hat{v} &= (-\sin \thet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/391947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 1
} |
Differentiate $\log_{10}x$ My attempt:
$\eqalign{
& \log_{10}x = {{\ln x} \over {\ln 10}} \cr
& u = \ln x \cr
& v = \ln 10 \cr
& {{du} \over {dx}} = {1 \over x} \cr
& {{dv} \over {dx}} = 0 \cr
& {v^2} = {(\ln10)^2} \cr
& {{dy} \over {dx}} = {{\left( {{{\ln 10} \over x}} \right)} \over {2\ln 1... | $${\rm{lo}}{{\rm{g}}_{10}}x = {{\ln x} \over {\ln 10}} = \dfrac{1}{\ln(10)}\ln x$$
No need for the chain rule, in fact, that would lead you to your mistakes, since $\dfrac 1 {\ln(10)}$ is a constant.
So we differentiate only the term that's a function of $x$: $$\dfrac{1}{\ln(10)}\frac d{dx}(\ln x)= \dfrac 1{x\ln(10)}$$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
} |
If $\lim_{t\to\infty}\gamma(t)=p$, then $p$ is a singularity of $\gamma$. I'm trying to solve this question:
Let $X$ be a vectorial field of class $C^1$ in an open set
$\Delta\subset \mathbb R^n$. Prove if $\gamma(t)$ is a trajectory of
$X$ defined in a maximal interval $(\omega_-,\omega_+)$ with
$\lim_{t\to\inf... | For $n \geq 0$, let $t_n \in (n,n+1)$ such that $\gamma'(t_n)=\gamma(n+1)-\gamma(n)$ (use mean value theorem).
Then $\gamma'(t_n)=X(\gamma(t_n)) \underset{n \to + \infty}{\longrightarrow} X(p)$ and $\gamma'(t_n) \underset{n \to + \infty}{\longrightarrow} p-p=0$. Thus $X(p)=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/392108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Sequence of Functions Converging to 0 I encountered this question in a textbook. While I understand the intuition behind it I am not sure how to formally prove it.
Define the sequence of functions $(g_n)$ on $[0,1]$ to be $$g_{k,n}(x) = \begin{cases}1 & x \in \left[\dfrac{k}n, \dfrac{k+1}n\right]\\ 0 & \text{ else}\end... | 1) What is $\lVert g_n\rVert_2$?
2) Show that for any $x$, $g_n(x)=0$ infinitely often and $g_n(x)=1$ infinitely often.
3) Note that $\frac1n\to 0 $ and $\frac2n\to 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/392169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Do there exist some non-constant holomorphic functions such that the sum of the modulus of them is a constant Do there exist some non-constant holomorphic functions $f_1,f_2,\ldots,f_n$such that $$\sum_{k=1}^{n}\left|\,f_k\right|$$ is a constant? Can you give an example? Thanks very much
| NO. Suppose $f, g$ are holomorphic functions on the unite disc.
$$
2\pi r M=2\pi r( |f(z_0)|+|g(z_0)|)=|\int_{|z-z_0|=r} fdz|+|\int_{|z-z_0|=r} gdz|\le \int_{|z-z_0|=r} (|f|+|g|)|dz|=2\pi r M
$$
so all equal sign hold, then $f, g$ are constants.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/392245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Solve $\sqrt{2x-5} - \sqrt{x-1} = 1$ Although this is a simple question I for the life of me can not figure out why one would get a 2 in front of the second square root when expanding. Can someone please explain that to me?
Example: solve $\sqrt{(2x-5)} - \sqrt{(x-1)} = 1$
Isolate one of the square roots: $\sqrt{(2x-5... | To get rid of the square root, denote: $\sqrt{x-1}=t\Rightarrow x=t^2+1$. Then:
$$\sqrt{2x-5} - \sqrt{x-1} = 1 \Rightarrow \\
\sqrt{2t^2-3}=t+1\Rightarrow \\
2t^2-3=t^2+2t+1\Rightarrow \\
t^2-2t-4=0 \Rightarrow \\
t_1=1-\sqrt{5} \text{ (ignored, because $t>0$)},t_2=1+\sqrt{5}.$$
Now we can return to $x$:
$$x=t^2+1=(1+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/392308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 5
} |
Guides/tutorials to learn abstract algebra? I recently read up a bit on symmetry groups and was interested by how they apply to even the Rubik's cube. I'm also intrigued by how group theory helps prove that "polynomials of degree $\gt4$ are not generally solvable".
I love set theory and stuff, but I'd like to learn som... | I really think that Isaacs book Algebra: A Graduate Course introduces the group theory in detail without omitting any proof. It may sound difficult because of the adjective "Graduate" but I do not think that the explanations are that difficult to follow for undergraduates as long as they know how to write proofs.
The b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/392374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 7,
"answer_id": 4
} |
properties of recursively enumerable sets $A \times B$ is an r.e.(recursively enumerable) set, I want to show that $A$ (or $B$) is r.e. ($A$ and $B$ are nonempty)
I need to find a formula. I've got an idea that I should use the symbolic definition of an r.e. set. That is, writing a formula for the function that speci... | The notion of computable or c.e. is usually defined on $\omega$. To understand what $A \times B$ being computable or c.e., you should identify order pairs $(x,y)$ with the natural number under a bijective pairing function. Any of the usual standard pairing function, the projection maps $\pi_1$ and $\pi_2$ are computabl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/392407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Cyclic shifts when multiplied by $2$. I was trying to solve the following problem:
Find a number in base $10$, which when multiplied by $2$, results in a number which is a cyclic shift of the original number, such that the last digit (least significant digit) becomes the first digit.
I believe one such number is $10526... | Suppose an $N$ digit number satisfies your condition, write it as $N= 10a + b$, where $b$ is the last digit. Then, your condition implies that
$$ 2 (10a + b) = b\times 10^{N-1} + a $$
Or that $b \times (10^{N-1} -1 ) = 19 a$.
Clearly, $b$ is not a multiple of 19, so we must have $10^{N-1} -1$ to be a multiple of 19. Yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/392475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Approximation of alternating series $\sum_{n=1}^\infty a_n = 0.55 - (0.55)^3/3! + (0.55)^5/5! - (0.55)^7/7! + ...$ $\sum_{n=1}^\infty a_n = 0.55 - (0.55)^3/3! + (0.55)^5/5! - (0.55)^7/7! + ...$
I am asked to find the no. of terms needed to approximate the partial sum to be within 0.0000001 from the convergent value of... | Hint:
$\sin x=x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dfrac{x^7}{7!}+ \dots$
Your expression is simply $\sin (0.55)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/392534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
how to apply hint to question involving the pigeonhole principle The following question is from cut-the-knot.org's page on the pigeonhole principle
Question
Prove that however one selects 55 integers $1 \le x_1 < x_2 < x_3 < ... < x_{55} \le 100$, there will be some two that differ by 9, some two that differ by 10, a ... | Here is $9$ done explicitly: Break the set into subsets with a difference of $9$: $\{1,10,19,28,\ldots,100\},\{2,11,20,\ldots 92\},\ldots \{9,18,27,\ldots 99\}$. Note that there is one subset with $12$ members and eight with $11$ members. If you want to avoid a pair with a difference of $9$ among your $55$ numbers, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/392605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How to integrate $\int \sqrt{x^2+a^2}dx$ $a$ is a parameter. I have no idea where to start
| I will give you a proof of how they can get the formula above. As a heads up, it is quite difficult and long, so most people use the formula usually written in the back of the text, but I was able to prove it so here goes.
The idea is to, of course, do trig-substitution.
Since $$\sqrt{a^2+x^2} $$ suggests that $x=a\tan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/392663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
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