Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Proof of Plancherel formula I was looking at this question posted here some time ago.
How to Prove Plancherel's Formula?
I get it until in the third line he practically says that $\int _{- \infty}^{+\infty} e^{i(\omega - \omega')t} dt= 2 \pi \delta(\omega - \omega')$.
I mean, I would understand if we were integrating ... | I think he used that $$ 1 = \hat{\delta(w)} $$ so, $$\int _{-\infty}^{+\infty} e^{i(\omega-\omega ')t} dt $$ is the antitransform of $\delta$ values in $(\omega - \omega') $ plus $2\pi$ for definition of antitransform.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2303106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Is locally completeness a topological property? I know that completeness itself is not a topological property because a complete and a not complete metric space can be homeomorphic, e.g. $\Bbb R$ and $(0,1)$.
However, both $\Bbb R$ and $(0,1)$ are locally complete (each point has a neighborhood that is complete under t... | Another way of proving that the irrationals can be made complete with respect to a metric $d$ which is equivalent to the usual one consists in providing such a metric. This can be donne as follows: let $(q_n)_{n\in\mathbb N}$ be an enumeration of the rationals. Then, if $x,y\in\mathbb{R}\setminus\mathbb Q$, define$$d(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2303185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Infinite Series for Signal Energy and Power A doubt came up to me on an Oppenheim's Signals and Systems 2ed exercise:
1.3) (f) Determine $P_{\infty}$ and $E_{\infty}$ for the following signal: $x\left[n\right]=\cos\left(\frac{\pi}{4}n\right)$
On theory, $P_{\infty}$ and $E_{\infty}$ are defined as:
$$
\begin{align}
E_{... | I got to a solution!
All the previous assumtions about the sums were correct... Now:
$$
\begin{align}
E_{\infty}&=\lim\limits_{N\to \infty}{\frac{1}{4}\left[2+2\sum_{n=-N}^{+N}{1}+2\sum_{n=1}^{+N}{j^{n}}+2\sum_{n=1}^{+N}{\left(-j\right)^{n}}\right]}\\
&=\lim\limits_{N\to \infty}{\frac{1}{2}\left[\left(2N+2\right)+\unde... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2303326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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distribution in an inner product (inner product spaces) Sorry for the dumb question.
Suppose I'm in a general inner product space. How would I compute something like the following?
$$\langle x - \alpha y, x - \alpha y\rangle$$
where $\alpha$ is a complex scalar. Is the following right?
$$\begin{align}
\langle x - ... | The "real part" should appear after summing the two central terms because such a sum has the form $z+\overline{z}$ which equals to $2\Re(z)$ (in your case you have this for $z=\overline{-\alpha}\langle x,y\rangle$).
Here is the step-by-step:
$$\begin{align}
\langle x - \alpha y, x - \alpha y\rangle &= \langle x, x\rang... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2303431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Closure of closure of set equals closure of set Let $A\subseteq \mathbb{R}^n$. I want to prove that $cl(cl(A))\subseteq cl(A)$. Let $x\in cl(cl(A))$, then $x$ is adherent to $cl(A)$, which implies that $B(x;r)\cap cl(A)\neq\emptyset, \forall r>0$. How do I proceed from here? And do I really need to?
I'm guessing that i... | Closure of a closed set is equal to itself. As $cl(A)$ is closed, therefore
$$cl(cl(A))=cl(A)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2303492",
"timestamp": "2023-03-29T00:00:00",
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How to calculate $\lim_{n \to \infty} \frac{(\ln(n))^n}{(\ln(n+1))^{n+1}}$ How would I solve the following limit
$\lim_{n \to \infty} \frac{(\ln(n))^n}{(\ln(n+1))^{n+1}}$
Is there an intuitive way of solving it?
I have no clue on how to start. Any help would be appreciated.
| $\lim_{n \to \infty} \frac{(\ln(n))^n}{(\ln(n+1))^{n+1}}\\
\lim_{n \to \infty} \left(\frac{\ln(n)}{\ln(n+1)}\right)^n\left(\lim_{n \to \infty}\frac 1{\ln(n+1)}\right)
$
$\left(\frac{\ln(n)}{\ln(n+1)}\right)<1$ for all $n>0$
$\lim_{n \to \infty}\frac 1{\ln(n+1)} = 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2303594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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3d triangle in one equation without min or max How can I graph a triangle in 3d with a single equation that does not use min, max, floor, ceil, or absolute value?
If you believe not possible, then the closest approximation (equation that roughly appears to be a 3d triangle) will do.
| Consider the example of triangle with vertices $(0,0,0)$,$(3,3,0)$ and $(-3,3,0)$
$$((x+y) (x-y)(y-3)(\sqrt{9-x^2- (y-3)^2}+1))^2+z^2=0$$
$a^2+b^2=0$ ensures that $a=0$ and $b=0$. Here $z=0$ denotes equation of plane of triangle. $x+y=0$,$x-y=0$ and $y=3$ are 3 planes that contain 3 lines of triangle. The square root i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2303712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that a metric space $V$ is sequentially compact iff it is limit point compact Given is a metric space (V,d). Show that the following properties are equivalent:
a) V is sequential compact ( so each sequence in V has a convergent subsequence witha limit in V)
b) for each subset $A \subset V$ with an infinite amount ... | choose $a_n \in B(x;\frac{1}{n})∩A$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2303847",
"timestamp": "2023-03-29T00:00:00",
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Solve the Cauchy problem: $u_t + u^2 u_x = 0$, $ u(0,x) = 2+x$ Solve the Cauchy problem: For $t>0$, $$u_y + u^2 u_x = 0$$ $$u(x,0) = 2+x$$
So by the characteristic method:$\frac{dx}{dt} = z^2$, $\frac{dy}{dt}=1$, $\frac{dz}{dt}=0$, parametrized by $\Gamma: (s,0,2+s)$. Then we have $z=2+s$, $x=t(2+s)^2 +s $ and $y=t$, w... | $$u(x,y) \quad \begin{cases} u_y + u^2 u_x = 0 \\u(x,0) = 2+x \end{cases}$$
Your calculus is correct. The solution expressed on the form of implicit equation is :
$$u=x-yu^2 + 2$$
This agrees with the condition $u(x,0)=2+x$.
Probably you made a mistake in checking the agreement to the PDE.
$$u_x=1-2yuu_x \quad\to\quad ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Automorphisms of $\mathbb{Z}[x]$ (verification) Consider the ring $\mathbb{Z}[x]$. If $f$ is an automorphism of this ring, then since $f(1)=1$ so $f$ fixes $\mathbb{Z}$. To determine $f(x)$, since $0$ is a root of this polynomial with multiplicity $1$ we have $f(x)=ax$ for some $a\in\mathbb{Z}$. Applying evaluation map... | Hint: the map
$$
a_0+a_1x+\dots+a_nx^n\mapsto
a_0+a_1(x+1)+\dots+a_n(x+1)^n
$$
is clearly an automorphism of $\mathbb{Z}[x]$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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How large a hypercap do you need to intersect every basis? Consider the unit sphere $\mathscr{S}^d$. We can define a hyperspherical cap by the angle $\theta$ that its associated hypercone subtends at the centre of the sphere.
How large does $\theta$ have to be in order that for every set of orthogonal axes, the lines t... | This is equivalent to being given a set of axes, and asking for the largest spherical distance away from all of them a point can be. Without loss we can take the standard basis of $\mathbb{R}^d$.
Symmetry considerations imply such a point will be the same distance from all of the axes. Restricting to the section of the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2304227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Integrate $\int{ e^{{x}^{2}-x} \cdot x \cdot e^x} dx$
Integrate $\int{ e^{{x}^{2}-x} \cdot x \cdot e^x} dx$
I'd like to know how to do it because I need it for another task. Here is what I tried:
$$\int{ e^{{x}^{2}-x} \cdot x \cdot e^x} dx = \int{e^{x^{2}-x+x} \cdot x} \text{ }dx = \int{e^{x^{2}} \cdot x} \text{ }dx$... | You can always differentiate your answer and see what you get. In this case
$$
\frac{d}{dx}\left(\frac{1}{2}e^{x^{2}}+c\right)=\frac{1}{2}2x e^{x^{2}}=xe^{x^{2}}
$$
by the chain rule. So you are correct.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Function from $[0,1]^2$ to $[0,1]$ Consider $(x,y)\in [0,1]^2$. Is it true, that there exists only one $t\in [0,1]$ such that $(x,y)$ belongs to the line passing through $(0,t)\in [0,1]^2$ and $(1,t^2)\in [0,1]^2$? Could you help me to prove it? Is there a way to write in an "explicit" way this function $f$ from $[0,1]... | The line between $(0, t)$ and $(1, t^2)$ has the equation
$$y = t + (t^2-t)x$$
Given $(x,y)$ we can solve for $t$ and get
$$t = \frac{x-1}{2x} \pm \sqrt{\left(\frac{x-1}{2x}\right)^2 + \frac{y}{x}}
= \frac{(x-1) \pm \sqrt{(x-1)^2+4xy}}{2x}$$
Since $x,y \geq 0$ we have $(x-1)^2 + 4xy \geq (x-1)^2$ so taking the negative... | {
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Bounding the derivative of a function Let $f(x) = \frac{\exp(ax)}{1+\exp(ax)}$ for some $a > 0$ be a logistic function.
I am looking for an upper bound on the $k$-th derivative of $f$, i.e. $|f^{(k)}(x)|$ on $\mathbb{R}$.
Using Cauchy's integral formula I only obtain a bound of order $k! / (\pi-\epsilon)^{k+1}$ for $a=... | Given that $y=\dfrac{e^{ax}}{1+e^{ax}}$ we have
\begin{eqnarray}y^\prime&=&\frac{ae^{ax}}{1+e^{ax}}\\
&=&a(1-y)y
\end{eqnarray}
from which it follows that
$$ y^{\prime\prime}=a^2(1-2y)(1-y)y $$
etc, for $0< y<1$
So for each $n$ there is a polynomial $P_n(y)$ of degree $n$ such that
$$ y^{(n)}= a^nyP_n(y)\text{ for }y\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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On the arithmetic differential equation $n''=n'$
If $n'$ denotes the arithmetic derivative of non-negative integer $n$, and $n''=(n')'$, then solve the following equation $$n''=n'.$$
What I have found, you can read in one minute! I have tried to explain it very detailed so anyone, even with a little knowledge of elem... | We have that
$$(3\cdot 29\cdot 25733)'=3\cdot 29 + 3\cdot 25733 + 29\cdot 25733=7^7$$
So you are going to get non-trivial solutions. It's probably a difficult problem to come up with all solutions.
I was looking for "3 prime" solutions. So, if $n=abc$ with $n'=ab+ac+bc=(a+c)(b+c)-c^2$. Trying to solve with $q=5$ gives:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2304625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
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Area of curve in parametric equation Given the curve defined by the parametric equations:
$$
x=7\cos{3t}\\
y=7\sin{3t}\\
0\le t\le2\pi
$$
What is the area of the region bounded by this curve?
Clearly,
$$
x^2+y^2=(7\cos{3t})^2+(7\sin{3t})^2=7^2
$$
which is a circle centered at the origin and radius 7. Therefore, the ar... | In your second way of computing it, you have gone around the circle three times instead of one; thus you have (erroneously) tripled the answer. Also, there is a difference between clockwise and counterclockwise and you clearly got unlucky there, to get a negative area.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2304721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Need help in evaluating $\int_{-1}^1 (1-x^2)^k, k \in \mathbb{N}$ Can someone tell me how to evaluate this integral please?
$$\int_{-1}^1 (1-x^2)^k, k \in \mathbb{N}$$
I tried using the substitution x = sin(t), which would allow me to express this as:
$$\int_{-1}^1 cos^{2k+1}(t) dt$$
but this doesn't really help. Any o... | This is simply a Beta function. Using the substitution $t=x^2$, this reduces to
$$2\int_0^1 \frac{1}{2\sqrt{t}} (1-t)^k\ dt$$
$$\int_0^1 t^{-\frac{1}{2}} (1-t)^k$$
$$B\left(\frac{1}{2},k+1\right)$$
which can be expressed in a lot of different ways (see the linked Wikipedia page).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 3
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Least squares with variants Find $\mathbf X$ and $\mathbf Y$ such that the two equations hold
$$
(\mathbf X^t\mathbf X)^{-1}\mathbf X^t \mathbf Y=A
$$
$$
(\mathbf X^t\mathbf X)^{-1}=B
$$
$\mathbf{X}$ is an $n\times p$ matrix and $\mathbf{Y}$ is vector of length $n$
| I assume the first equation is $(X^tX)^{-1}X^tY = A$, otherwise the matrix multiplication is not defined if $p < n$.
I assume I have been given a $p \times p$ matrix B and a $p \times 1$ vector $A$ and a $n > p$ and I have to construct a $n \times p$ matrix $X$ and a $n \times 1$ vector $Y$ satisfying the two given equ... | {
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Discontinuity of the following function. I want to find out the discontinuities of the function $f$ defined as
$$f(x)=\begin{cases} \dfrac{1}{1+e^{1/(x-2)}+e^{1/(x-3)^2}}, & x\neq2,x\neq3, \\[6pt] 1, & x=2, \\[6pt] \dfrac{1}{1+e}, &x=3. \end{cases}$$
My attempt :
Here we only have to check the continuity of $f$ at $x=2... | hint
$$\lim_{x\to 2^-}e^{\frac {1}{x-2}}=e^{-\infty}=0$$
and at $2^+$, it is $+\infty $.
$$\lim_{x\to 3}e^{\frac {1}{(x-3)^2}}=+\infty $$
from this
$$\lim_{2^+}f (x)=\lim_3f (x)=\frac {1}{+\infty}=0$$
it is not continuous neither at $2$ nor at $3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2305020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Need help proving that $f(n) = 5n^2 - 2n + 16$ is not O(n) So I have tried time and time again but do not understand how to conclude this proof. The answer to the example states:
Prove that $5n^2 - 2n + 16$ is not O(n). Assume $5n^2 - 2n + 16$ is O(n). Then there exist constants C and k so that $5n^2 - 2n + 16 \le Cn$... | "How does n>C+2 disprove big O? What is the condition testing against, is it infinite in some way? and where do we ever assume that C "held for all large values of n"? and what does that even mean?"
You started by saying "$\underline{5n^2−2n+16≤Cn}$ for all $n > k$." So in particular there is some number $k$ so that t... | {
"language": "en",
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Convergence in probability implies convergence of $\lim_n P(X_n\in B)$ for all Borel $B$ The usual definition of convergence in probability is that $\lim_n P(|X_n-X|>\epsilon)\to 0$ for all $\epsilon>0$.
Is an equivalent definition that $\lim_n P(X_n\in B) = P(X\in B)$ for all measurable $B$?
Here's a sketch of why... | Let $X_{n}\stackrel{P}{\to}0$ and take $B=\mathbb{R}-\left\{ 0\right\} $.
It is quite well possible that$P\left(X_{n}\in B\right)=1$ for each
$n$, but next to that we have $P\left(0\in B\right)=0$.
So $\lim_{n\to\infty} P(X_n\in B)=P(X\in B)$ extremely fails to be true in that situation.
To get a concrete example:
if ... | {
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How can I compute eigenvalues or characteristic polynomial of this matrix? Please help. \begin{pmatrix}
2na & -a & -a & -a & -a & -a& -a\\
-a& a+b & 0 & 0 & -b & 0 & 0\\
-a& 0 & a+b & 0 & 0 & -b &0 \\
-a& 0 & 0 & a+b & 0 & 0&-b \\
-a& -b & 0 & 0 & a+b & 0 & 0\\
-a& 0&-b & 0 & 0 & a+b &0 \\
-a& 0& 0&-b... | Add to row 1 all other rows. Then subtract column 1 from every other column. Your matrix is then similar to
$$
B=\left[\begin{array}{c|c}0&0\\ \hline\ast&C\end{array}\right]
=\left[\begin{array}{c|cc}
0&0&0\\
\hline
\ast&(a+b)I_n+aE&-bI_n+aE\\
\ast&-bI_n+aE&(a+b)I_n+aE
\end{array}\right]
$$
where $E$ denotes the $n\tim... | {
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Existence of square root of a matrix Testing a method with the use of C.-H. theorem for finding square roots of real $ 2 \times 2 $ matrices I have noticed that some matrices probably don't have their square roots with real and complex entries.
An example the matrix $A= \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatri... | Two partial answers to your question make a full answer!
Let $A=\begin{bmatrix}0&1\\0&0\end{bmatrix}$.
*
*Answer 1: The set of matrices form a ring (rings are sets where you have algebraic operations of addition and multiplication). In abstract algebra, one learns about ring extensions, in other words, you can cons... | {
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Closed form for $\int_{0}^{2\pi} e^{\sin(x)+\cos(x)} dx$ I'm solving an integral in three coordinates. One of the coordinates is the integral: $\int_{0}^{2\pi} e^{\sin(x)+\cos(x)} dx$.
Is possible to get a closed form for that?
My efforts.
Let $u=-x$.
$\int_{0}^{2\pi} e^{\sin(x)+\cos(x)} dx =-\int_{0}^{2\pi} e^{\sqrt{... | Let
$$
I=\int_0^{2\pi}e^{\sin(x)+\cos(x)}\,dx.
$$
From the trigonometric addition formula $\cos(\alpha-\beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta$, with $\alpha=\pi/4, \beta=x$, using the special value $\cos(\pi/4) = \sin(\pi/4) = 1/\sqrt2$, we have
$$
\sin(x) + \cos(x) = \sqrt2\cos\left(\pi/4-x\right).
$$
From... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does the difference between a sequence of random variables and their conditional expectations converge in $L^1$? Let $(X_n)_n$ be a sequence of random variables on the probability space $(\Omega, \mathcal{F}, P)$, and let $(\mathcal{F}_n)_n$ be a filtration that increases to $\mathcal{F}$. We can assume $(X_n)_n$ is un... | It's not true, and a counterexample would be one similar to my comment. Take $\{\xi_i\}$ i.i.d. with $P(\xi_i = 1) = P(\xi_i = - 1) = 1/2$. Set $X_k = \prod\limits_{i = 1}^k \xi_k$ and $\mathcal{F}_n = \sigma(\{\xi_i\}_{i = 1}^{n-1})$. Then $$|X_n - E(X_n | \mathcal{F_{n}})| = |X_n - 0| = 1.$$
This means that $E|X_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Characterizing subsequences of the Thue-Morse sequence Consider the Thue-Morse sequence on the alphabet $\{0,1\}$ given by $T_0 = 0$ and $T_{n+1} = T_n \bar{T_n}$ where $\bar{T_n}$ is the bitwise negation of $T_n$. Then the Thue-Morse sequence is defined as
$$TM:=\lim\limits_{n\to\infty}T_n$$
(this is just one of man... | Every word of length at least 5 has a unique decomposition into 1-words, either beginning with the last letter of a 1-word or ending with the first letter of a 1-word. Moreover, if we put bars to indicate the separations between the 1-words occurring at positions 2k in the sequence, the sum of each 1-word is 1 exactly.... | {
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Improper integral $\int \limits_{2}^{4}\frac{\sqrt{(16 - x^2)^5}}{(x^2 - 9x + 20)^3}dx$ I can't figure out how to solve (say whether it converges or diverges) the following improper integral:
$$
\int \limits_{2}^{4}\frac{\sqrt{(16 - x^2)^5}}{(x^2 - 9x + 20)^3}dx
$$
I've tried to simplify this and got:
$$
\int \limits_{... | An alternate substitution is $x = 4 \, \sin(t)$ which leads to, with some difficulty,
\begin{align}
\int \frac{(16 - x^2)^{5/2}}{(x^2 - 9 x + 20)^3} \, dx = \frac{81}{2} \, \frac{x-6}{(x-5)^2} \, \sqrt{16 - x^2} - 63 \, \tan^{-1}\left(\frac{16 - 5x}{3 \, \sqrt{16 - x^2}}\right) - \sin^{-1}\left(\frac{x}{4}\right) + c_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2305911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
A theory of numbers problem I could really use some help here:
Prove that $17|2x+3y\iff 17|9x+5y$.
I don't even know how to start. Just pointing me at a similar problem that has a full solution would be helpful enough, but of course I will be extremely thankful if someone could explain how to approach solving this! :)
| Let's see the problem in terms of linear algebra.
Let $u=(2,3)$, $v=(9,5)$, and $w=(x,y)$. These are vectors in $\mathbb F_{17}^2$.
Then, $17\mid 2x+3y\iff 17\mid 9x+5y$ iff $\langle u,w \rangle = 0 \iff \langle v,w \rangle = 0 $, and this happens iff $u^{\perp} = v^{\perp}$, which happens iff $u$ and $v$ generate the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2306015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Convergence in compact open topology implies convergence pointwise I'm a beginner in topology, so pardon me if this problem is too easy.
Let $X$ and $Y$ be topological spaces. Let $C(X,Y)$ be the set of all continuous functions from $X$ to $Y$. Show that if a net $(f_i)$ converges to $f$ in the compact open topology o... | Suppose $f_n \to f$ in the compact-open topology.
Then for any $x \in X$, take an open set $U \subseteq Y$ that contains $f(x)$.
Then $f \in S\{x\}, U)$, (this is open in $C(X,Y)$ as finite sets are always compact) which means that there exists $N$ such that for all $n \ge N$ we have $f_n \in S(\{x\}, U)$, which means... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Existence of square root matrix $B \in \mathbb{C}^{2\times 2}$ for any $A \in \mathbb{C}^{2\times 2}$, where $A^2\neq 0$ I am trying to prove that for any $A \in \mathbb{C}^{2\times 2}$ with $A^2\neq 0$, there exists $B \in \mathbb{C}^{2\times 2}$ with $BB=A$.
I have tried the approach of a general matrix A andB with ... | We have the equations
\begin{eqnarray*}
a^2+bc= A \\
b(a+d)=B \\
c(a+d)=C \\
bc+d^2=D
\end{eqnarray*}
Multiply the first equation by $(a+d)^2$ and use the second & third we have
\begin{eqnarray*}
a^2(a+d)^2 +BC=A(a+d)^2 \\
d= -a +\sqrt{\frac{BC}{A-a^2}}.
\end{eqnarray*}
Now subtract the first & the fourth
\begin{eqna... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2306227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Formula using fibonacci numbers Let $a_n$ be the $n^{th}$ term of the sequence defined recursively by
$a_{n+1} = \frac {1}{1+a_n}$
and let $a_1 = 1.$ Find a formula for $a_n$ in terms of the Fibonacci numbers $F_n$. Prove that the formula you found is valid for all natural numbers $n.$
Wow can I solve this type of pr... | By induction $a_n=\frac{F_n}{F_{n+1}}$ because $a_1=\frac{F_1}{F_2}=1$ and
$$a_{n+1}=\frac{1}{1+a_n}=\frac{1}{1+\frac{F_n}{F_{n+1}}}=\frac{F_{n+1}}{F_{n+2}}.$$
$\{F_n\}:1,1,2,3,5,...$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2306311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Showing that a topological space is ${\rm T}_1$ Let $X$ be a topological space and let $\Delta = \{(x,x) : x\in X \}$ be the diagonal of $X\times X$ (with the product topology).
I was asked to prove that $X$ is ${\rm T}_1$ if and only if $\Delta$ can be written as intersection of open subsets of $X\times X$.
I think ... | Just an idea.
Suppose $\Delta=\bigcap O_\lambda$, where $(O_\lambda)_{\lambda\in\Lambda}$ is a family of open sets in $X\times X$. Let $(x,y)\in X\times X$ with $x\neq y$. Then $(x,y)\notin \Delta$ and therefore $(x,y)\notin\bigcap O_{\lambda}$. So choose $\mu\in\Lambda$ such that $(x,y)\notin\ O_\mu$; say $O_\mu=\pi_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2306420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Bounded and Closed but not Compact I had been proposed to construct not compact set that is however bounded and closed.
I could easily imagine from the different metric - such as discrete metric where
$d(x,y) =0$ if $x=y$ and $d(x,y) =1$ if $x \neq y$ then M itself is closed since it is metric itself and bounded by $D... | A subset of a banach space is compact if and only if it is closed and totally bounded. The closed unit ball in $l^1(\mathbb R)$ centered around $0$ is not totally bounded. For example because it is not a finite union of open balls of radius $\frac{1}{2}$, simply because each such ball can contain at most one of the seq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2306527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Probability that the withdrawn balls are the same color Question
An urn contains $n$ white and $m$ black balls, where
$n$ and $m$ are positive numbers.
*
*If two balls are randomly withdrawn, what is the probability that they are the same color?
*If a ball is randomly withdrawn and then
replace... | For the first one:
$$\large{\frac{\binom{m}{2}+\binom{n}{2}}{\binom{m+n}{2}}}$$
Explanation:
*
*$\large{\binom{m+n}{2}}$ ways of selecting two balls.
*$\large{\binom{m}{2}}$ ways of selecting two black balls.
*$\large{\binom{n}{2}}$ ways of selecting two white balls.
For the second one:
$$\frac{m^2+n^2}{(m+n)^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2306652",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Curvature of Regular Tilings of the Hyperbolic Plane Given a regular tiling of the hyperbolic plane, with p-sided polygons and q polygons meeting at each vertex, is the Gaussian curvature of the plane determined by (p,q)?
Intuitively it seems like having five pentagons meeting at each point would need greater (negative... | This depends on the edge length of the polygons.
Usually the convention is that you use a hyperbolic plane with curvature $-1$ unless indicated otherwise. In such a plane, you can observe that small figures look almost Euclidean, but the larger a figure is, the more pronounced the difference in geometry becomes. Actual... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2306739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Group homomorphisms from $\mathbb{C}^\ast$ to $ \mathbb{Z}$ I want to find $Hom_{\mathtt{Grp}}(\mathbb{C}^\ast,\mathbb{Z})$, where $\mathbb{C}^\ast$ is the multiplicative group, and $\mathbb{Z}$ is additive.
$\mathbb{C}$ is the additive group of complex numbers. We have the following map:
$\large{\mathbb{C} \xrightarr... | This isn't a full answer, but I suspect that this Hom group may be the trivial group. Suppose $\phi:\mathbb{C}^* \to \mathbb{Z}$ is a group homomorphism. We know that $\phi(1) = 0$ since $1$ is the identity. Then
$$
\phi(-1)^2 = \phi(1) = 0 \implies 2 \phi(-1) = 0 \implies \phi(-1) = 0
$$
By a similar argument, any co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2306895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to factorize :$f(x)=x^n+x+1 \ \ \ \ \ \ : n=3k+2 ,k\in \mathbb{N}$ How to factorize :
$$f(x)=x^n+x+1 \ \ \ \ \ \ : n=3k+2 ,k\in \mathbb{N}$$
And :
$$g(x)=x^n+x-1 \ \ \ \ \ \ : n=3k+2 ,k\in2m-1 \ \ \ , \ \ m\in\mathbb{N}$$
My try :
$$f(x)=x^n+x+1=x^{3k+2}+x+1$$
$$=(x^{3k+2}+x^{3k+1}+x^{3k})-(x^{3k+2}+x^{3k+1}+x... | Hint: evaluate $f(x)$ at $\omega$, where $\omega=e^{2\pi i/3}$, so $\omega^3=1$, and $\omega^2 + \omega + 1 = 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2306997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Lie derivative and representations of a Lie algebra I'm reading a book on integrable systems and am trying to understand Lie groups.
The author states a property I cannot understand:
Let me define the protagonists: L is the Lie derivative, m is an element of a Lie group and X and Y are in its Lie algebra. The dot, unl... | Well, this comes from probably an older definition of a Lie Algebra, where someone defines for a given Lie Group $G$ its associated Lie Algebra, to be given by
$$
\mathfrak{g}= \{ A \in M_{n \times n}(\mathbb{C}) \thinspace | \thinspace \forall t \in \mathbb{R}: \exp(tA) \in G\}.
$$
So, if you have an action of the Li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2307085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Uniform digit generator The question goes as follows:
Let there be a random digit generator, generating the numbers $0,1,2,...,9$ uniformly and independently. The generator writes the numbers from left to right.
a) What is the expected value for the number of different digits appearing between two zeroes?
b) What is t... | One rigorous solution for b) can be this:
Let $d$ be the number of (not necessarily distinct) digits betweens two appearances of number 9, and $x$ the number of zeroes that appear in these digits, then $$P(x=k|d) = \frac{{d \choose k} 8^{d-k}}{9^d}$$ because that's the number of ways you can accomodate $k$ zeroes and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2307161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Area between $r=4\sin(\theta)$ and $r=2$ I'm trying to find the area between $r=4\sin(\theta)$ and $r=2$.
I found the points of intersections to be $\pi/6,5\pi/6$. Which implies the area is $$A=\frac{1}{2}\int_{\pi/6}^{5\pi/6}(4\sin(\theta))^2-2^2d\theta.$$
Is this correct? Or did I find the area for the following reg... | The desired red region is just the area of a circle with radius $2$ minus the area of the blue region:
$$
\pi(2)^2 - A
= 4\pi - \frac{1}{2}\int_{\pi/6}^{5\pi/6}(4\sin(\theta))^2-2^2d\theta
= 4\pi - \left( 2\sqrt 3 + \frac{4\pi}{3} \right)
= \frac{8\pi}{3} - 2\sqrt 3
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2307272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Functor $F$ is equivalence of categories implies $F$ is full. I am trying to understand the answer given to this question.
From what I understand they are saying that if there exists an $f$ such that $Ff = g$ then by naturality it must have the property that $GFf = Gg$ from which faithfulness of $G$ implies $Ff = g$.
T... | You are misreading the answer. They aren't assuming such an $f$ exists: they are explaining that you can find the formula for such an $f$ by assuming it exists and figuring out what it must be. Namely if $f:X\to Y$ exists, it must be given by the formula $f=\eta_Y^{-1} \circ Gg \circ \eta_X$.
Now stop assuming that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2307372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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How to create a 3 parameter function that gets its maximum in the center of the cube? I have an problem where I have a 3 dimensional board, and I want to assign to the center of that cube (aka the board) maximum values and to the edges of the board lower values (preferably in a way so that the upper left side will get ... | Let's assume each cell is identified by (i,j,d) and the maximum values are (I,J,D). Assume that i,j, and d start at 1, and that I, J, and D are odd so that there is a unique middle value on each axis, $M_I = (I+1)/2, M_J = (J+1)/2$, and $M_D = (D+1)/2$
Then the distance from the center, for example in i, is
$$|i-M_I|$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How To Calculate Length Of Screw Thread? I'm having a tough time searching for the answer to this as most of the results are in the realm of mechanical engineering and appear to be unrelated, I apologize if this is a really obvious question.
Say there is a circular arc in $2$ dimensions covering $90$ degrees at a radiu... | As an intuitively convincing method, you could consider straightening out the arc into a line of length $\frac{21}2\pi$ and then stretching it along another axis by length x and using Pythagorean theorem to calculate its new length: $\sqrt{(\frac{21}2\pi)^2+x^2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2307561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
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Find a basis for orthogonal complement in R⁴
How do I approach part 2? I found the projection of 1. to be (6,-2,2,-2) but what do I do now?
| For vector $\mathbf v = (x_1, x_2, x_3,x_4)$, the dot products of $\mathbf v$ with the two given vectors respectively are zero.
$$\begin{align*}
\begin{bmatrix}1&2&3&4\\2&5&0&1\end{bmatrix}
\begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix} &= \begin{bmatrix}0\\0\end{bmatrix}\\
\begin{bmatrix}1&2&3&4\\0&1&-6&-7\end{bmatrix... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that if $(C_n)$ is a sequence of connected subsets of $X$ such that $C_n\cap C_{n+1}\neq\emptyset$ then $\bigcup C_n$ is connected. Suppose that $(C_n)$ is a sequence of connected subsets of $X$ such that $C_n\cap C_{n+1}\neq\emptyset$ for each $n\in\mathbb{N}.$ It is required to prove that $\bigcup C_n$ is conne... | In your proof, since you have already assumed that $f:\cup_{1}^{n+1}C_k\rightarrow\{0,1\}$ is a continuous function (we know that there exists atleast one such continuous function, hence the assumption) , its clear that $\cup_1^nC_k$ and $C_{n+1}$ being connected sets will map to an individual element each in $\{0,1\}$... | {
"language": "en",
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Solution to $u'=ru$ in distributional sense? How do we show that $u(x)=ce^{rx}$ is the only solution to $u'=ru$ in $\mathcal D'(X)$?
I tried to decompose a $\phi\in\mathcal D(X)$ into parts and let $u$ acts on each of them, but I couldn't show that the sum converges in $\mathcal D(X)$. I am lost here, can anyone please... | The calculations are the same as in classical real analysis:
In $\mathcal D'(X)$ we can multiply with a $C^\infty$ functions, and since $e^{-rx}$ is never $0$ the equation $u'=ru$ will not change the set of solution when multiplied with $e^{-rx}$. Thus, $u'=ru$ is equivalent with $e^{-rx} u' - r e^{-rx} u = 0$. This ca... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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BMO2 2017 Question 4 - Bobby's Safe Bobby’s booby-trapped safe requires a $3$-digit code to unlock it. Alex
has a probe which can test combinations without typing them on
the safe. The probe responds Fail if no individual digit is correct.
Otherwise it responds Close, including when all digits are correct. For
example,... | We shall first prove that the task requires at least $13$ attempts. In the worst case scenario, Alex's first six attempts all fail, which means he has at least $(10-6)^3=64$ remaining combinations to guess. If he gets a "Close" outcome in the seventh round, he will then have at least
$$9+9+9+3+3+3+1=37$$ possibilitie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2308064",
"timestamp": "2023-03-29T00:00:00",
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Ellipse in a Rectangle What is the equation for an ellipse (or rather, family of ellipses) which has as its tangents the lines forming the rectangle $$x=\pm a, y=\pm b\;\; (a,b>0)$$?
This question is a modification/extension of this other question here posted recently.
| By exploiting the affine map $(x,y)\mapsto\left(\frac{x}{a},\frac{y}{b}\right)$ the question boils down to finding the family of ellipses inscribed in a square with vertices at $(\pm 1,\pm 1)$. In a ellipse the line joining the midpoints of parallel chords always go through the center. Additionally, the orthoptic curve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2308198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof using Induction? Any thoughts on how to prove the following proposition
If $x_1>0$ and $a>0$ and $x_{ n+1 }=\frac { 1 }{ 2 } \left( x_{ n }+\frac { a^{ 2 } }{ x_{ n } } \right) $ then $a\leq x_{n+1}\leq x_n$.
| hint
$$x_{n+1}-a=\frac {(x_n-a)^2 }{2x_n}$$
$$x_{n+1}-x_n=\frac {a^2-x_n^2}{2x_n} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2308304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What's the probability of getting 1000 heads in a row? I'm reading The Master Algorithm by Pedro Domingos and I'm having a hard time understanding something he wrote on page 74:
"If you do a million runs of a thousand coin flips, it's practically certain that at least one run will come up all heads."
My intuition tells... | The chances that flipping 1000 coins gives all heads is given by $\frac{1}{2^{1000}}$ as you predicted. However, the odds that this will happen, given that you try it $1000000$ times is:
$$1 - \left(1 - \frac{1}{2^{1000}}\right)^{1000000}$$
The certainty of which, I'm not too sure about. My calculator can't seem to han... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2308416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 6,
"answer_id": 0
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Prove the series $\sum_{n=1}^{\infty}\frac{(-1)^n}{\ln{n}+\sin{n}}$ converge How to prove this serie
$$\sum_{n=1}^{\infty}\frac{(-1)^n}{\ln{n}+\sin{n}}$$
converge? I can't do a comparison test with the Leibniz formula for $\pi$ because the series are not $>0$ for all $n$. I can't do a ratio test because I can't compute... | Let $$a_n = \frac{(-1)^n}{\sin n + \ln n }$$
We wish to show that $\sum a_n$ converges.
In order to do this, first note that $a_n$ is negative when $n$ is odd, and positive when $n$ is even.
We will write $\sum a_n < \sum b_i + \sum c_j$, where $b_i$ are negative terms (with only odd indices $i$) that are smaller in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2308532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
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Show that the derivative of this function is positive
Suppose that $n>1$, $g\in(0,1)$ and $f(g)\in(0,1)$. Suppose that
$\frac{df(g)}{dg}\geq0$.
Define $B(g,n)$ as:
$B(g,n)=\sum _{k=1}^n \frac{n!}{k!(n-k)!}g^{n-k}(1-g)^{k-1}(1-(1-f(g))^k)$
Show that:
$\frac{dB(g,n)}{dg}>0$.
Some of these assumptions can be relaxed ... | This is not a full solution, but some possibly helpful thoughts too long to fit a comment.
Notice that
$$B(g,n) = \sum _{k=1} ^n \frac {n!} {k!(n-k)!} g^{n-k} (1-g)^{k-1} - \sum _{k=1} ^n \frac {n!} {k!(n-k)!} g^{n-k} (1-g)^{k-1} (1-f(g))^k = \\
\frac 1 {1-g} \sum _{k=1} ^n \frac {n!} {k!(n-k)!} g^{n-k} (1-g)^k - \frac... | {
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Why doesn't $[a]^{[b]} = [a^b]$? When we construct the modular numbers $\Bbb Z/n\Bbb Z$ so that addition works as $[a]+[b]=[a+b]$ and multiplication works as $[a][b]=[ab]$, we get for free that $a[b]=[ab]$ also works. Why can't we also define a sense of exponentiation $[a]^{[b]}$ as $[a^b]$? We do have that $[a]^b = ... | The key notion here is a congruence — "congruent modulo $n$" was constructed to be an equivalence relation that respects addition and multiplication, but need not preserve anything not constructed from them.
So, it's not really about addition/multiplication being special at all — it's about the relation being construct... | {
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} |
Calculus Spivak Chapter 2 problem 16(c) The question asks to prove that if $\frac mn \lt \sqrt{2}$, then there is another rational number $\frac {m'}{n'}$ with $\frac mn \lt \frac {m'}{n'} \lt \sqrt{2}$.
Intuitively, it's clear that such a number exists, but I don't understand the solution to this problem. It states: l... | I actually have a slightly different answer to the above, which I think is closer to the book as it relies directly on parts (a) and (b).
If anyone spots anything wrong I'd appreciate if you comment below and point out any mistakes:
We have proven in part (a) that:
$\frac{m^2}{n^2} < 2 \implies \frac{(m + 2n)^2}{(m + n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2308909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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$(T^* T)^{-1}$ is self adjoint Suppose $U$, $V$ are finite-dimensional inner product spaces, and $T \in \mathcal{L}(U, V )$ is invertible. I need to show that $T^* T \in \mathcal{L}(V)$ is positive definite and invertible. Also show that the inverse of $T^* T$ is self-adjoint and positive definite.
For the first part I th... | Let $S = (T^{\ast}T)^{-1}$, then for any $v\in V$, let $u = S(v)$, so
$$
0\leq \langle T^{\ast}Tu,u\rangle = \langle v,S(v)\rangle
$$
so $S$ is positive semidefinite. Since it is invertible, it must be positive definite. Also, it follows that all the eigen-values of $S$ are real (in fact, non-negative reals). Hence, $S... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2309024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving a limit using the epsilon delta definition in C I am trying to prove that the
$$\lim_{z \rightarrow z_0} z^2+c = z_0^2+c,$$
where $c$ is some complex constant.
I was taught that typically it is a good idea to begin from the end of the proof and work backwards. Therefore, I will start with
$$|z^2-z^2_0| < \epsi... | You want to show that you can find a $\delta > 0$ such that for all $z \in U_{\delta}(z_0) := \{z \in \mathbb{C}: |z-z_0| < \delta \}$, you have that $z \in U_{\epsilon}(z_0)$. Now for any $\delta > 0$, there exists a constant $C_\delta$ such that $|z+z_0| < C_{\delta}$ for all $z \in U_{\delta}(z_0)$:
$$|z+z_0| \leq |... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2309139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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} |
Class $K^{\infty}$ is elementary provided that $K$ is.
Let $K$ be an elementary class of $\Sigma$-structures. Show that the class $K^{\infty}$ formed by the structures of $K$ with infinite domain is also elementary.
I'm really clueless about this problem. I know that $K$ is elementary if there is som $\Phi \subseteq ... | Suppose $\Phi$ is a set of sentences defining $K$ - that is, $K=\{\mathcal{A}: \mathcal{A}\models\Phi\}$. Now $K^\infty\subseteq K$ as you observe, so you want to find a bigger set of sentenecs $\Psi\supseteq \Phi$ such that $K^\infty=\{\mathcal{A}: \mathcal{A}\models\Psi\}$.
(Why bigger? Well, satisfying more sentenc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2309284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Ideal in the ring of symetric polynomials I need to find generators of the ideal $J$ in the ring of symmetric polynomials of 3 variables $(x_1, x_2, x_3)$ such that $\forall j \in J: x_1 = x_2 \Rightarrow
j=0$.
It is clear that polynomials $(x_1-x_2)(x_2-x_1)$ and $(x_1 - x_2)^2$ are generators. So are this polynomia... | Take a symmetric function $f(x_1, x_2, x_3)$. View it as an element of $\mathbf{Q}(x_2,x_3)[x_1]$. For polynomials over a field, we know that $g(t) \in k[t]$ has a root at $a$ if and only if $(t - a) \mid g$.
Thus since $f(x_2,x_2,x_3) = 0$ it must be that $(x_2 - x_1) \mid f$ in $\mathbf{Q}(x_2,x_3)[x_1]$. By Gauss's ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2309438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $f'(x)\le g'(x)$, prove $f(x)\le g(x)$ I have to do the following exercise:
Let $f$ and $g$ two differentiable functions such that $f(0)=g(0)$ and $f'(x)\leq g'(x)$ for all $x$ in $\mathbb{R}$. Prove that $f(x)\leq g(x)$ for any $x\geq0$.
Now, I know this is true because the first derivative of a function is the a... | For $x \ge 0$, we have
$f(x) = f(0) + \displaystyle \int_0^x f'(s) ds = g(0) + \displaystyle \int_0^x f'(s) ds, \tag{1}$
since $f(0) = g(0)$; and since $f'(x) \le g'(x)$, we have
$\displaystyle \int_0^x f'(s) ds \le \displaystyle \int_0^x g'(s) ds; \tag{2}$
thus,
$f(x) = g(0) + \displaystyle \int_0^x f'(s) ds \le g(0) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2309532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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What does it mean, that "The coordinates of $a_n$ in the ... canonical base are ...? I would like to understand an article, which explains vectorial products in $n$ dimension.
As far as I know, the bases are vectors. If I multiply them with scalars, and add them together, then I can produce all the vectors in that vect... | The determinant defining vector $a_n$ is simply an abuse of notation. It means that $a_n$ may be expressed as the cofactor expansion of the determinant of the "matrix"
$$
A = \left( \begin{matrix}
e_1 & e_2 & \cdots & e_n \\
a_{11} & a_{12} & \cdots & a_{1n} \\
\cdots & \cdots & \cdots & \cdots \\
a_{n-1,1} & a_{n-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2309633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
How to find a 3x3 matrix with determinant =0 from which I can delete random column and random row to make it nonzero? I need to find a $3 \times 3$ matrix and the determinant of this matrix has to be $0$.
I also need to be able to delete randomly chosen column and row to make the determinant nonzero? Is it even possib... | The matrix
$$\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{pmatrix}$$works just fine :
*
*the columns are in arithmetic progression, which means that the middle column is the arithmetic mean of the extremal columns, and thus they are not independent.
*for any $2\times 2$ submatrix, the difference of the c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2309785",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 3
} |
How many $4$-letter strings either start with c or end with two vowels? If the alphabet consists of $\{a,b,c,d,e,f\}$, how many four letter strings either start with c or end with two vowels?
My reasoning was as follows:
Starts with c but does not have vowel in third position + starts with c but does not have a vowel i... | Easier to do inclusion-exclusion.
Starts with C + ends with two vowels - (starts with C and ends with two vowels)
$1*6*6*6 + 6*6*2*2 - 1*6*2*2 = 336$
You were okay but a double adding snuck in. You counted those that start with C and do not have vowels in either the 3rd or 4th position twice.
Because those things have... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2309888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Why an open upper half plane is not homeomorphic to infinite band and its boundary? Why an open upper half plane is not homeomorphic to infinite band and its boundary?
| It is due to the fact that any homeomorphism maps the boundary of one to the other. The boundary of the range which is the infinite band is not connected, but the domain i.e the upper hand plane has a connected boundary. A homeomorphism is a continuous map, and images of connected sets under continuous maps are connect... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2310084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is T one-to-one iff T is onto? In Friedberg's Linear Algebra, the following proof that T is one-to-one if and only if T is onto is presented. Here, $T:V\rightarrow F$ is linear.
"We have that T is one-to-one if and only if N(T)=0, if and only if nullity(T) = 0, if and only if rank(T) = dim(V), if and only if rank(T... | This holds only if V and W have equal (finite) dimensions.
From the rank-nullity theorem one has:
$nullity(T)+rank(T) = dim(V)$
Since $T$ is injective then $nullity (T)=0$. Then this implies $rank (T)=dim (V)=dim (W) $ which suggests surjectivity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2310189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How can we prove that there does not exist any metric space $4,5,6$ dense subsets? Let $(X,d)$ be a metric space. Which of the following is possible?
(A) $X$ has exactly $3$ dense subsets.
(B) $X$ has exactly $4$ dense subsets.
(C) $X$ has exactly $5$ dense subsets.
(D) $X$ has exactly $6$ dense subsets.
I think it's a... | Hint: Consider the set $X = A\cup D$ with $A=\{0\}$ and $D=\left\{\frac1n: n\geq 1\right\}$ equipped with the metric from ${\Bbb R}$. Show that the only dense subsets are $A\cup D$ and $D$ (so there are precisely two dense subsets).
Now, this example is constructed using a discrete set $D$ with precisely one accumulati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2310446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to solve this absolute value equation I have the following integral $$\int_0^{\pi/2} |\sin x-\cos x|dx. $$It's a simple integral but when I try to solve the module I get stuck. I took $\sin x-\cos x>0$ and squaring this I found that $\sin 2x<1$. When I apply $\arcsin$ it would mean $$2x<\frac\pi2 \implies x<\frac\p... | \begin{align}
\int_0^{\pi/2} |\sin(x)-\cos(x)|dx &= \int_0^{\pi/4} |\sin(x)-\cos(x)|dx+\int^0_{\pi/4} |\sin(x)-\cos(x)|dx\\
&=-\int_0^{\pi/4} \sin(x)-\cos(x)dx+\int_{\pi/4}^{\pi/2} \sin(x)-\cos(x)dx\\
&=..
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2310519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 3
} |
Help with finding side of a triangle I have a right angled triangle question and have through my working figure out that the angles are $90^\circ$, $67^\circ$, and $23^\circ$. I have one side opposite $23^\circ$ which is $13$cm. I need to find the side length opposite the right angle.
I've tried sine rule and I get a n... | This is a more complete solution to the ones mentioned above:
I think that a more straightforward way would be to use the sine here. As @zubzub mentioned, the side opposite a right angle is a hypotenuse (by definition). If we call the hypotenuse $h$, since sine = opposite/hypotenuse, $\sin 23º = 13/h$, so by rearrangem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2310617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Set with sum of elements equal to sum of squares The question I'm attempting to solve is
For a set $S$ such that $\sum S_i = \sum S_i^2$, which is greater: $\sum S_i^3$ or $\sum S_i^4$?
My first thought was that $$\sum S_i - \sum S_i^2 = 0 \implies \sum S_i(1-S_i) = 0$$ hints at some kind of symmetry which may be lev... | If the $S_i$ can have any sign nothing can be said (see e.g. the example of Paul, where $\sum S_i^3 < \sum S_i^4$, whereas for $S = \{.5, .5, (1-\sqrt{3})/2\}$ the opposite inequality holds).
Hence I am assuming that $S_i \geq 0$ for every $i$.
We have that
$$
S_i^2 = (S_i)^{1/2} (S_i)^{3/2} \leq \frac{1}{2} S_i + \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2310705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Show that if $AB=BA=0$ then $rank(A+B)=rank(A)+rank(B)$ We are supposed to show that if $AB=BA=0$ then $rank(A+B)=rank(A)+rank(B)$
Maybe this fact help to answer the question:
$rank(A+B)\le rank(A)+rank(B)$
and
$rank(A)+rank(B)=dim(Im(B))$.
But I got stuck when trying to prove that $rank(A+B)\ge rank(A)+rank(B)$
| Take
$$
A=B=\begin{pmatrix} 0 & 1 \cr 0 & 0 \end{pmatrix}.
$$
Then $AB=BA=0$, but
$$
1=rank(A+B)\neq rank(A)+rank(B)=2.
$$
So Daniel is right.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2310819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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recurrence relation for loans You borrow $4000$ dollars, at $12$ percentage compounded monthly, to
buy a car. If the loan is to be paid back over two years, what is the
monthly payment? Note you pay back same amount every month. Use a recurrence relation (for loans) to solve the problem.
What I have tried so far is
$P(... | Not gonna do your problem for you. However, here is a full explanation for the derivation of the formula for the monthly payment, and a more in-depth description can be found at my blog post on this page.
Given the initial balance, the number of payments, and the monthly interest amount, finding the monthly payment can... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2310906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Why doesn't this linear transformation exist? In Friedberg's Linear Algebra, the following theorem is presented:
"Let V and W be vector spaces over F, and suppose that V is finite-dimensional with a basis $\left\{v_1,...,v_n\right\}$. For any vectors $w_1,...,w_n$ in W there exists exactly one linear transformation $T:... | Let $V=W=\mathbb{R}^2$. Let $x_1 = (1,0), x_2 = (2,0), y_1=(0,1), y_2=(1,3)$. Suppose there exits a linear transformation $T$ as required. Then
$$T(2,0)=(1,3).$$ Also $$T(2,0)=2T(1,0)=2(0,1)=(0,2).$$
This gives a contradiction. The reason this happens, as mentioned in the comments, is that $x_1$ and $x_2$ are linearly ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2311024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Studying the derivative of the integral function $\frac{1}{x}\int_0^x\arctan(e^t)\mathrm{d}t$ I was trying to calculate the derivative of the function
$$
F(x) =\frac{1}{x}\int_0^x\arctan(e^t)\mathrm{d}t
$$
I thought the fastest way was to use the Leibniz's rule for the derivative of a product,
$$
(f\cdot g)' = f'g + g'... | With your notation
$$
g(x)=\int_0^x\arctan(e^t)\,dt
$$
we have
$$
F'(x)=\frac{xg'(x) - g(x)}{x^2}=\frac{x\arctan(e^x)-g(x)}{x^2}
$$
for $x\ne0$. On the other hand, the function $F$ can be extended by continuity at $0$ as
$$
\lim_{x\to0}F(x)=\frac{\pi}{4}
$$
and
$$
\lim_{x\to0}F'(x)=
\lim_{x\to0}\frac{1}{2x}\frac{xe^x}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2311168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Find $\lim_{x\rightarrow \frac{\pi}{4}}\frac{\cos(2x)}{\sin(x-\frac{\pi}{4})}$ without L'Hopital I tried:
$$\lim_{x\rightarrow \frac{\pi}{4}}\frac{\cos(2x)}{\sin(x-\frac{\pi}{4})} =
\frac{\frac{\cos(2x)}{x-\frac{\pi}{4}}}{\frac{\sin(x-\frac{\pi}{4})}{x-\frac{\pi}{4}}}$$ and $$\begin{align}\frac{\cos(2x)}{x-\frac{\pi}... | Just as an alternate approach:
I said: $\frac{cos(2\theta)}{sin\theta*cos(\frac{\pi}{4})-cos\theta*(\frac{\pi}{4})}$ =
$\frac{cos(2\theta)}{1/\sqrt(2)*sin\theta-cos\theta}$ Noting that $cos(2\theta)=cos^2\theta-sin^2\theta$ and $sin\theta-cos\theta=-(cos\theta-sin\theta)$
So, we have
$\frac{cos^2\theta-sin^2\theta}{-1/... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2311265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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Help with this integral: $\int_{0}^{1}\frac{1}{1+x^{2}}dx$ (Riemann) I'm stuck when I try to solve this integral:
$$\int_{0}^{1}\frac{1}{1+x^{2}}dx$$
I try this:
$$\int_{0}^{1}\frac{1}{1+x^{2}}dx=\lim_{n\rightarrow\infty}\sum_{i=1}^{n}\frac{1}{1+(\frac{i}{n})^{2}}\frac{1}{n}=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=... | Not sure why you can't use the well-known antiderivative $arctanx$, to get $$\int_0^1\frac1 {1+x^2}dx=[arctanx]_0^1=\frac \pi4-0=\frac \pi4$$...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2311351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
What does the group of the geometric images of z, such that $z+\overline{z}+|z^2|=0$ define?
What does the group of the geometric images of z, such that
$z+\overline{z}+|z^2|=0$ define?
A) A circumference of center (0,-1) and radius 1.
B) Two lines, of equations $y=x+2$ and $y=x$.
C)A circumference of center (1,0) a... | Put $z=\rho (\cos (t)+i \sin (t))$.
We have
$$z+\overline {z}=2\rho \cos (t) $$
$$|z^2|=|z|^2=\rho^2$$
thus $$\rho=-2\cos (t) $$
and
$$z=-2\cos^2 (t)-2i\cos (t)\sin (t) $$
or
$$x_z=-1-\cos (2t) $$
$$y_z=-\sin (2t) $$
thus
$$(x_z+1)^2+y_z^2=1$$
it is a circle with radius $1$ and center $(-1,0) $.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2311464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Volume of revolution setup I've been given a question as so
Consider the region $R = \{(x,y)\; |\; 0 ≤ y \;and \;y^2≤ x ≤ y+2\}$
Draw R and calculate the exact volume by rotating the region R about the line x=0
So this is what $R$ looks like (shaded in black).. I think.
And the equation I came up with was to integrat... | Close. Just remember that:
$$
\pi(R - r)^2 \neq \pi(R^2 - r^2)
$$
So the correct setup is:
$$
V = \pi\int_0^2 \left[(y + 2)^2 - (y^2)^2 \right] \, dy
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2311564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Evaluate $\sqrt{41}$ to $n$ decimal places without using a calculator. Evaluate $\sqrt{41}$ to $n$ decimal places without using a calculator.
$$x^2 -10x-16=0$$
I was asked to solve the above quadratic giving the solutions to one decimal place.
Using the quadratic formula I got: $x= 5\pm \sqrt{41}$
Using a calculator t... | Hint. Construct a recurrence sequence $x_n=f(x_{n-1})$ for $n\geq 1$, with
$$f(x)=\frac{1}{2}\left(x+\frac{41}{x}\right).$$
A reasonable starting point would be $x_0=7$ (note that $6^2<41<7^2$).
For details take a look here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2311678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
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In a nice protomodular category, is normality stable under joins? Call a subobject normal if it's an equivalence class of some internal equivalence relation on the codomain. In a finitely complete category, normality is closed under finite intersection and finite product and is also pullback stable. This can be proved ... | This paper of Tomas Everaert and Tim Van der Linden shows that it is true for semi-abelian categories (see Proposition 2.7). Exactness implies that every normal monomorphism is a kernel, and they need the existence of coproduct to define joins of subobjects; so I'm not sure that this would still hold in a category that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2311770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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CDF of max($X$, $Y$) - where is the mistake? $X$ and $Y$ are independent r.v.'s and we know $F_X(x)$ and $F_Y(y)$. Let $Z=max(X,Y)$. Find $F_Z(z)$.
Here's my reasoning:
$F_Z(z)=P(Z\leq z)=P(max(X,Y)\leq z)$.
I claim that we have 2 cases here:
1) $max(X,Y)=X$. If $X<z$, we are guaranteed that $Y<z$, so $F_Z(z)=P(Z\le... | I think you are fine with separating the cases, but then do not take care of them correctly. Since when you say in your case 1 that the maximum is $X$, you are "conditioning on" $X>Y$ and that changes the space over which you calculate the probabilities.
We have two cases that either of which happens:
case 1: $X<Y<z$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2311848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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Multiple integral related to zeta function I am trying to calculate the following integral
$$\int_V \frac{d^d\vec{r}}{e^{x_1+...+x_d}-1},$$
where $V=[0,\infty)^d$ and $\vec{r}=(x_1,...,x_d)$. I know that the result should be related to the Riemann zeta function, but I do not see how to do it quickly and elementary (i.e... | By Fubini's theorem
$$\int_{(0,+\infty)^d}\frac{x_1}{e^{x_1+\ldots+x_d}-1}\,d\mu_d = \int_{(0,+\infty)^{d-1}}\int_{0}^{+\infty}\frac{x_1}{e^{x_1+\ldots+x_d}-1}\,dx_1\,d\mu_{d-1} $$
and the last integral equals
$$ \int_{(0,+\infty)^{d-1}}\text{Li}_2\left(e^{-(x_2+\ldots+x_d)}\right)\,d\mu_{d-1}=\int_{(0,1)^{d-1}}\frac{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2311941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Can the following trigonometric equation be transformed into the other? Can $$16\sec^2(x)\tan^4(x)+88\sec^4(x)\tan^2(x)+16\sec^6(x)$$ be proven equal to
$$24\sec^6(x)-8\sec^4(x)+96\sec^4(x)\tan^2(x)-16\sec^2(x)\tan^2(x)$$
I have made about six attempts, but I keep getting stuck. I thought I'd ask, maybe someone el... | Sometimes the easiest thing to do is convert everything into sines and cosines.
\begin{array}{l}
16 \sec^2(x) \tan^4(x) + 88 \sec^4(x) \tan^2(x) + 16 \sec^6(x) \\
=\dfrac{16\sin^4(x)}{\cos^6(x)}+\dfrac{88\sin^2(x)}{\cos^6(x)}
+\dfrac{16}{\cos^6(x)} \\
=\dfrac{16\sin^4(x) + 88 \sin^2(x) + 16}{\cos^6... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2312031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Calculate the value of the series $\,\sum_{n=1}^\infty\frac{1}{2n(2n+1)(2n+2)}$ Calculate the infinite sum
$$\dfrac{1}{2\cdot 3\cdot 4}+ \dfrac{1}{4\cdot 5\cdot 6}+\dfrac{1}{6\cdot 7\cdot 8}+\cdots$$
I know this series is convergent by Comparison Test, but I can't understand how can I get the value of the sum.
Is th... | Hint: $$(2n)(2n+1)(2n+2) = (2n+1)^3 - (2n+1)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2312143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Whether a series is convergent or divergent Is it true that if $\sum u_n$ is convergent, where $u_n$'s are positive real numbers then $\sum \dfrac{u_1+u_2+...+u_n}{n}$ is divergent?
I know that if $\lim_{n\to\infty}u_n =0$ then $\lim_{n\to\infty}\dfrac{u_1+u_2+...+u_n}{n}=0$ and it is the necessary condition for a se... | The post has been edited to say the $u_i$'s are positive. Under that assumption, $\sum \dfrac{u_1+u_2+...+u_n}{n}$ diverges because $\sum_{n=1}^m \dfrac{u_1+u_2+...+u_n}{n}>\sum_{n=1}^m \dfrac{u_1}{n}$ and the latter goes to infinity as $m\to\infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2312256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Solve the recurrence relation. Need some help x(n)= x(n-1) + 2*n + 3 ; Given x(0)=4;
I'm using backwards substitution
x(n-1)= [x(n-2) + 2(n-1) + 3] + 2*n+ 3
x(n-2)= [x(n-3) + 2(n-2) + 3] + x(n-2) + 2(n-1) + 3 + 2*n + 3 so on and so forth......
so i write the general expression as :
x(n) = x(n-i) + [something here] + i*... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2312352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
On proof of showing uniqueness of adjoint equivalence functor
Suppose that $C,D$ are categories, and fix a functor $F:C\to D$. Then if there is some $G$ such that $(F,G,e,\epsilon)$ is an adjoint equivalence, and if $G'$ is another such functor, $G$ and $G'$ are isomorphic by a unique isomorphism.
(Here adjoint equiv... | The previous line is equivalent to
$$G_2(\epsilon_{2,y})\circ e_{2,G_2(y)}\circ f_y= G_2(\epsilon_{1,y})\circ e_{2,G_1(y)}.$$
The "therefore" follows from the "triangle identity" between the unit and counit for $F\dashv G_2$:
$$G_2(\epsilon_{2,y})\circ e_{2,G_2(y)}=id_{G_2(y)}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2312459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to handle integrals of the form $\int_0^1 \sin^2(\pi x) f(x) dx$? In another question here on the site I was looking for an integral bound and it turned out one has to know how to deal with oscillatory integrals of the form $$\int_0^{1} \sin^2(\pi x) f(x) dx,$$
where $f$ is a smooth function. There should be some t... | $$\int_{0}^{1}\sin^2(\pi x)\,f(x)\,dx = \frac{1}{2}\int_{0}^{1}f(x)\,dx-\frac{1}{2}\int_{0}^{1}\cos(2\pi x)f(x)\,dx $$
and if $f$ is a $C^1$ function we have
$$ \frac{1}{2}\int_{0}^{1}\cos(2\pi x)\,f(x)\,dx = -\frac{1}{4\pi}\int_{0}^{1}\sin(2\pi x)\,f'(x)\,dx $$
by integration by parts. If $f(x)$ behaves like $\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2312545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Show that if $\frac1a+\frac1b+\frac1c = a+b+c$, then $\frac1{3+a}+\frac1{3+c}+\frac1{3+c} \leq\frac34$
Show that if $a,b,c$ are positive reals, and
$\frac1a+\frac1b+\frac1c = a+b+c$, then
$$\frac1{3+a}+\frac1{3+b}+\frac1{3+c} \leq\frac34$$
The corresponding problem replacing the $3$s with $2$ is shown here:
How... | For the case $k≥3$, $(k\in\mathbb Z)$
Given equality implies $ab+bc+ca=abc(a+b+c)$
We know that $(x+y+z)^2≥3(xy+yz+zx)$
put $x=ab,y=bc,z=ca\Rightarrow (ab+bc+ca)^2\geq3abc(a+b+c)=3(ab+bc+ca)$
$\Rightarrow abc(a+b+c)=ab+bc+ca\geq3$ and $a+b+c\geq3$ (obvious)
$\dfrac {1}{k+a}+\dfrac {1}{k+b}+\dfrac {1}{k+c}≤\dfrac {3}{k+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2312672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Prove that a function $f : X \mapsto X$ is injective if and only if it has a left inverse. Prove that a function $f : X \mapsto X$ is injective if and only if it has a left
inverse.
Can someone help me with this? What I've done is form predicates of the form $LI(f) \Rightarrow INJ(f)$ to begin step-by-step.
I understan... | You can show injective implies a left inverse by constructing g.
First create a helper relation $ g' = \{ ( f(x), x ) : x \in X \}$ . This relation is a (partial) function due to the injective property ensuring that for every element in the domain there is at most one image.
Then you can construct $ g(x) = \begin{cases... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2312764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How shall I identify all assumptions required by a Theorem in Evan's PDE textbook? My general question is given in the title. Let us consider Hopf-Lax formula - Theorem 4 of Section 3.3.2 of the PDE book by Evans, Click here.
[Theorem 4] If $x\in \mathbb R^n$ and $t>0$, then the solution $u= u(x, t)$ of the minimizati... | The hypotheses are listed directly above the theorem statement (e.g., "Recall we are assuming..."). It is common in textbooks or papers to list some of the hypotheses right before the theorem to make the theorem statements more concise. In general, you should aim to understand the proof in sufficient detail that you ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2312931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Problem with strong law of large numbers, but not identical Suppose that $X_n$ are independent random variables where $\sup_n E|X_n| < \infty$.
Then, $\frac{\sum_{i=1}^{n}X_i}{n^a}$ converges to 0 almost surely for all $a>1$.
I think that $\frac{1}{n^{a-1}}$ converges to 0 and $\frac{\sum_{i=1}^{n}X_i}{n}$ converges be... | Let $\Delta\equiv \sup_n\mathsf{E}|X_n|$. Then
$$
\sum_{k=1}^\infty\mathsf{P}\left(\left|\frac{X_k}{k^\alpha}\right|\ge 1\right)\le \sum_{k=1}^\infty \frac{\Delta}{k^\alpha}<\infty,
$$
and for $Y_n=X_n/n^{\alpha}1\{|X_n|\le n^{\alpha}\}$,
$$
\sum_{k=1}^\infty\mathsf{E}Y_k \quad\text{and}\quad \sum_{k=1}^\infty \operato... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2313051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find exponent of prime $p$ in prime factorization of a number $x$ Say we have a number $x$ such that
$$ x = a^{r}.b^{s}.c^{t}.p^{u} $$
Is there a formula or method which can directly give me the exponent of a particular prime in this prime factorization.
For small $x$ calculating it is not a problem but when $x$ is of ... | Expanding on FredAkalin's answer, we can write an algorithm in pseudocode as follows:
prime_exp(x,p):
count = 0
while x mod p is 0: (loop while x is exactly divisible by p)
count += 1 (increase exponent value by 1)
x = x / p (divide x by p)
return count
We can run this on the number $364=2^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2313162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Prove that a real polynomial $x^n+ a_1x^{n-1}+ \cdots +a_n$ cannot be completely resolved into linear factors if $a_1^2
Prove that a real polynomial $x^n+ a_1x^{n-1}+ \cdots +a_n$ cannot be completely resolved into linear factors if $a_1^2<a_2$.
Here's what I've got. Let $\alpha_1, \dots, \alpha_n$ be the roots of the ... | Assume that $\alpha_1,...,\alpha_n$ are $n$ real roots of the polynomial. Notice that $$\sum_{i<j} \alpha_i \alpha_j = a_2 > a_1^2 \geq 0\implies a_1^2-a_2 = \sum_i \alpha_i^2+\sum_{i<j}\alpha_i \alpha_j>0$$
which is a contradiction with our assumption that $a_1^2-a_2<0$. Hence, these real roots cannot exist at the sam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2313216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
In a metric space $(X,d)$ if two sequences {$x_n$} and {$y_n$} are Cauchy then d($x_n$,$y_n$) is convergent Let (X,d) be a metric space and{ ${x_n}$} , {$y_n$} be two arbitrary Cauchy sequences in X then {$d(x_n,y_n)$} is convergent.
I think it is sufficient to show that {$d(x_n,y_n)$} is Cauchy in $\mathbb R$. But I... | Let $\epsilon>0$ be given. As $\{x_n\}$ is Cauchy, there exists $N_1$ such that $d(x_n,x_m)<\frac\epsilon2$ for all $n,m>N_1$. Similarly, there exists $N_2$ such that $d(y_n,y_m)<\frac\epsilon2$ for all $n,m>N_1$.
Therefore, with $N:=\max\{N_1,N_2\}$, we have
$$d(x_n,y_n)\le d(x_n,x_m)+d(x_m,y_m)+d(y_m,y_n)< d(x_m,y_m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2313325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
What is the difference between orthogonal subspaces and orthogonal complements? I am learning linear algebra through professor Giblert Strang's lectures on MIT OCW.
The professor says that the row space and the null space of a matrix are orthogonal subspaces. This I can follow, since any vector in the nullspace takes a... | There are two conceptual issues here:
(1) The orthogonal complement $X^{\perp}$ of a vector subspace $X \leq \Bbb V$ consists of the vectors in $\Bbb V$ that are perpendicular to all of the vectors in $X$, not just at least one vector.
In practice, it is convenient to use the characterization that $v \in \Bbb V$ is ort... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2313448",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
For a polynomial with integer coefficients, is it true that if constant term is prime then it cannot be the root of the polynomial.
For a polynomial with integer coefficients, is it true that if constant term is prime then it cannot be the root of the polynomial.
Let $p$ be a polynomial with constant term $a_0$ and... | Counter example:
$$
p(x) = (x-3)(x-1) = x^{2}-4x+3
$$
where the root are, as noted by @hardmath, $\left( 3, 1 \right)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2313539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
} |
How would one find the number that is a given percentage between two numbers? Assume I have two numbers, say, 23 and 1150. Also assume I have a given percentage, say, 32.6%. What formula would I use to find the number that is 32.6% between 23 and 1150?
| The number that is $x\%$ of the way between $a$ and $b$ is given by $$a + (b-a)\cdot \frac x{100}.$$
For example, the number $30\%$ of the way between $10$ and $20$ is $$10 + (20-10) \cdot \frac{30}{100} = 13.$$
Why this formula? The factor $b-a$ gives you the length of the interval. The factor $\dfrac x{100}$ tells ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2313598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Splitting a quadrilateral into two triangles if it has a vertex with an inner angle bigger than 180º I'm trying to write an app in which I need to test if a 3D quadrilateral has some angle equal or bigger to 180 degrees, Ie., it is a degenerated or concave quadrilateral. If this is true, I have to split it into two tri... | Let's assume you have a quadrilateral in 3D, defined by four vertices
$$\begin{array}{l}
\vec{v}_1 = ( x_1 , y_1 , z_1 ) \\
\vec{v}_2 = ( x_2 , y_2 , z_2 ) \\
\vec{v}_3 = ( x_3 , y_3 , z_3 ) \\
\vec{v}_4 = ( x_4 , y_4 , z_4 ) \end{array}$$
If one of the vertices is not in the same plane as the other three, the quadrila... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2313715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
If $f(x_n)$ converges for all bounded continuous functions $f$, does then $x_n$ converge? Let $x_n\in \mathbb{R}$ be a sequence and $x\in \mathbb{R}$ such that $f(x_n) \longrightarrow f(x)$ (as $n \longrightarrow \infty$) for all bounded continuous functions $f \colon \mathbb{R} \rightarrow \mathbb{R}$. Do we then have... | Suppose otherwise. Then there is an an interval $(x-a,x+a)$ such that $x_n\notin(x-a,x-a)$ for infinitely many $n$'s. Define$$f(y)=\left\{\begin{array}{ll}x+a&\text{ if }y\geqslant x+a\\y&\text{ if }y\in(x-a,x+a)\\x-a&\text{ if }y\leqslant x-a.\end{array}\right.$$Then $\bigl|f(x_n)-x\bigr|\geqslant a$ for infinitely ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2313955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Validity of Ito's formula for "piecewise-defined" Ito processes Let $b_i: \mathbb{R}^{m} \to \mathbb{R}$ be smooth functions, for each $i \in \{ 1, \ldots,m\}$ and let $(\Omega, \mathbb{F}, \mathbb{P})$ be a probability space equipped with a one-dimensional Brownian motion $W$. We divide the time interval $[0,T]$ eve... | You can write
$$
X_t = \int_0^t b(X_s)ds + \int_0^t \sigma(s) dW_s,
$$
where
$$
\sigma(t) = \sum_{n=1}^{2N} (-1)^n \mathbf{1}_{[t_{n-1},t_n)}(t).
$$
By the Lévy martingale characterization theorem, the process
$$
B_t = \int_0^t \sigma(s) dW_s
$$
is a standard Wiener process. Therefore, the answers to your questions a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2314055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$f(z)=\int_C\frac {g(x)}{x-z}dx$ is holomorphic? Problem :
C is a smooth simple closed curve in $\mathbb C$
$g(x)$ is a continuous function on C
$f(z)=\int_C\frac {g(x)}{x-z}dx\quad (z\notin C)$
Show that $f(z)$ is holomorphic in $\mathbb C\setminus C$
My try :
$\lim_{h\to0}\frac{f(z+h)-f(z)}{h} =\lim_{h\to0}\int_C\fra... | The interchange of the limit and the integral is not justified, as you note,
so you need to try something else. The following is the usual wey to get round your problem, and consists of expanding the integral kernel $(z-\xi)^{-1}$ into
a uniformly convergent powerseries, so that you can interchange summation and
integ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2314183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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