Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Prove $(\vec A \times \vec B) \cdot (\vec C \times \vec D) = (\vec A \cdot \vec C)(\vec B \cdot \vec D) - (\vec A \cdot \vec D)(\vec C \cdot \vec B)$ Prove that $(\vec A \times \vec B) \cdot (\vec C \times \vec D) = (\vec A \cdot \vec C)(\vec B \cdot \vec D) - (\vec A \cdot \vec D)(\vec C \cdot \vec B)$.
The problem as... | Let us omit these horrible vector arrows. Then we have
$$
\begin{align}
( A \times B) \cdot ( C \times D) &\overset{(iii)}{=} D\cdot ((A\times B)\times C) \overset{(i)}{=}
D\cdot( (A\cdot C)B-(B\cdot C)A)\\
&=( A \cdot C)( B \cdot D) - ( A \cdot D)( C \cdot B),
\end{align}
$$
where in the last equality we have u... | {
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Absolute convergence of the series $sin(a_n)$ implies...? Can someone please give me a counterexample for the following statement?
If $\sum _1 ^\infty sin(a_n) $ absolutely converges , then $\sum_1^\infty a_n$ also absolutely converges.
I tried working with $a_n = \frac{ (-1)^n } {n} $ and $a_n = \frac{(-1)^n lnn}{n}$... | If $a_n=\pi$, then $\sin a_n=0$, and thus $\sum_{n=1}^\infty\sin a_n$ converges absolutely, while $\sum_{n=1}^\infty a_n$ does not.
Hence, you need to make an assumption: $a_n\in(-\pi/2,\pi/2)$.
Then use the fact that
$$
\frac{2|x|}{\pi}\le|\sin x|\le |x|,
$$
for all $x\in(-\pi/2,\pi/2)$, which implies that:
$$
\sum_{n... | {
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Prove $\{x \mid x^2 \equiv 1 \pmod p\}=\{1, -1\}$ for all primes $p$ One way to prove that $\{x \mid x^2 \equiv 1 \pmod p\}=\{1, -1\}$ is to use the fact that $\{1, -1\}$ is the only subgroup of the cyclic group of primitive residue classes modulo $p$ that has the order 2. Is there a more basic way to prove this?
| $$x^2\equiv1\mod p\iff p\text{ divides }(x-1)(x+1)$$
But a prime divides a product if and only if it divides one of the two factors.
| {
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Is the Series : $\sum_{n=1}^{\infty} \left( \frac 1 {(n+1)^2} + ..........+\frac 1 {(n+n)^2} \right) \sin^2 n\theta $ convergent?
Is the Series :
$$\sum_{n=1}^{\infty} \left( \frac 1 {(n+1)^2} + \ldots+\frac 1 {(n+n)^2} \right) \sin^2 n\theta $$ convergent?
Attempt:
$$\sum_{n=0}^{\infty} \left( \frac 1 {(n+1)^2} + ... | Notice that
$$\sum_{k=1}^n\frac 1{(n+k)^2}=\frac 1{n}\frac 1n\sum_{k=1}^nf\left(\frac kn\right),$$
with $f(x):=\frac 1{1+x^2}$, a continuous positive function. Hence the convergence of the initial series reduces to the convergence of
$$\sum_{n\geqslant 1}\frac{\sin^2(n\theta)}n.$$
Write $\sin^2(A)=\frac{1-\cos(2A)}2$... | {
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Identity of a Mathematician Mentioned in Euler I and several others are in the process of translating one of Euler's papers from Latin to English, in particular the one that the Euler Archive lists as E36. In it Euler proves the Chinese Remainder Theorem, but my question is this:
In Section 1, Euler refers to two (pres... | There is also a reference to Sauveur in Sandifer's book, "The Early Mathematics of Leonhard Euler", in the chapter about E-36.
For a preview, see:
http://books.google.com/books?id=CvBxLr_0uBQC&pg=PA351
| {
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A Probability Problem about the $Trypanosoma$ parasite Considering the distribution of Trypanosoma (parasite) lengths shown below, suppose we take a sample of two Tryoanosomes. What is the probability that:
a) both Trypanosomes will be shorter than 20um?
b) the first Trypanosome will be shorter than 20um and the second... | All three questions can be solved in a pretty similar fashion. The key to answering them is using the rule that, if $A$ and $B$ are independent events, then $P(A\cap B) = P(A)P(B)$ in other words, multiply the probabilities of each event. So for example, the probability of selecting a parasite shorter than 20 micro... | {
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PMF function: joint pmf question For the PMF of $X$
$$P(X = x) = (0.5)^x \;\text{ for }\;x\in\mathbb{Z}^+,$$
would the PMF of $Y= X-1$ just be
$$P(Y = x) = (0.5)^{x-1}.$$
| $$
\Pr(Y=x) = \Pr(X-1=x) = \Pr(X=x+1) = 0.5^{x+1}.
$$
| {
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Showing a set is open but not closed Let $X = \{ (x,y) \in \mathbb{R}^2 : x > 0, y \geq 0 \} $. I am claiming $X$ is open, but it is not closed.
My Try:
To show it is not closed, I found a sequence that converges to a point outside $X$. For instance, $(x_n, y_n) = (\frac{1}{n}, \frac{1}{n} ) \to (0,0) \notin X$. Hence... | The set is not open. Take for example the point $P=(5,0)$. There is no ball with centre $P$ which is fully contained in our set.
| {
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Pumping Lemma Squares Proof Explanation I'm looking for some help understand this perfect squares proof using the pumping lemma. Here is the proof:
I don't understand how n^2 + k < n^2 + n towards the end of the proof. Would anyone be able to explain this to me? I also think that the last line n^2 < n^2+k < (n+1)^2... | First from the statement of the pumping lemma we know that $|xy|\le n$. Also we have set $k$ to be the length of $y$, i.e. $|y|=k$. Moreover the length of $xy$ is certainly greater or equal than the length of $y$: $|xy|\ge |y|$. Combining these we get
$$n\ge |xy|\ge |y|=k$$
therefore $n^2+k\le n^2+n<n^2+2n+1=(n+1)(n+1)... | {
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An Euler-Mascheroni-like sequence How does one compute the limit of the sequence:
$$\sum_{k = 0}^{n}\frac{1}{3k+1} - \frac{\ln(n)}{3}$$
I would apreciate a hint.
| Hint: use a comparison series/integral, by writing $$\frac{\ln n}{3} = \int_{1}^n \frac{dx}{3x}=\sum_{k=1}^{n-1} \int_{k}^{k+1}\frac{dx}{3x}$$ and
$$\sum_{k=0}^n \frac{1}{3k+1} = \sum_{k=0}^n \int_{k}^{k+1}\frac{dx}{3k+1}$$
Edit: is not a straightforward approach -- I was thinking about proving convergence.
| {
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Proof that if $A\subset B$ then $A^* = B^*$ I prove here that if $B$ is a Banach space, and $A$ is a closed subspace of $B$, $A\subset B$, then
$$A^* = B^*.$$
($A^*$ stands for the dual of $A$.)
There is obviously something wrong here but where?
We already have that $A^* \subset B^*$. The other inclusion comes from th... | This question has been answered in comments:
The mistake is that we already have $A^∗\subset B^∗$. Taking the dual acts contravariantly, i.e. it reverses inclusions and arrows of morphisms. – Your Ad Here Jan 28 at 16:09
| {
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Check if a graph is Eulerian
Let $G=((2,3,4,5,6,7),E)$ be a graph such that {$x$,$y$} $\in E$ if and only if the product of $x$ and $y$ is even, decide if G is an Eulerian graph.
My attempt
I tried to plot the graph, this is the result:
So, if my deductions are true, this is not an Eulerian graph because it's connec... | You don't need to draw the graph --- you can easily do this analytically.
You have 3 odd-numbered vertices and 3 even-numbered vertices. A product $xy$ is even iff at least one of $x,y$ is even.
A graph has an eulerian cycle iff every vertex is of even degree. So take an odd-numbered vertex, e.g. 3. It will have an ev... | {
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Proving that $\frac{e^x + e^{-x}}2 \le e^{x^2/2}$
Prove the following inequality:
$$\dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$$
This should be solved using Taylor series.
I tried expanding the left to the 5th degree and the right site to the 3rd degree, but it didn't help.
Any tips?
| Obscene overkill: since
$$ \cosh(z)=\prod_{n\geq 0}\left(1+\frac{4z^2}{(2n+1)^2\pi^2}\right) $$
we have
$$ \log\cosh(z)\leq \sum_{n\geq 0}\frac{4z^2}{(2n+1)^2\pi^2} =\frac{z^2}{2}$$
and the claim follows by exponentiating both sides. In other terms, it is sufficient to integrate both sides of $\tanh(z)\leq z$ over $[0,... | {
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Is the Hausdorff condition redundant here? This is a question in Algebraic Topology by Hatcher, Chapter 0:
21) If $X$ is a connected Hausdorff space that is a union of a finite number of 2 spheres,
any two of which intersect in at most one point, show that $X$ is homotopy equivalent
to a wedge sum of $S^1$’s and $... | Note that
a union of a finite number of 2 spheres, any two of which intersect in at most one point
is not the same as
a space $X$ as a finite disjoint union of spaces homeomorphic to $S^2$ where we identify at most one point of each pair,
the union Hatcher speaks of need not arise as a quotient of a disjoint union ... | {
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Why does f' = 0 gives the min or max? I understand how to calculate it, but I am just curious, why actually it works? Do we have a proof that it always works?
| Consider the case where a differentiable function $f(x)$ has a minimum at the point $x_{0}$. By definition, there exists a $\epsilon-$neighborhood
$$
x = \left\{ x\in\mathbb{R} \colon \left| x - x_{0} \right| < \epsilon \right\}
$$
with $\epsilon>0$ where $f(x) > f(x_{0})$.
If we constrain $\delta < \epsilon$, then ... | {
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Lagrange's Theorem exercise Be $u : \mathbb{R} \rightarrow \mathbb{R}$ a $C^2$ function. Proof that exists a $x \in \mathbb{R}$ with $0<x<2$ for which $u(2)-2u(1)+u(0)=u''(x)$
Applying Lagrange's Theorem I showed:
$u(2)-u(1)=u'(x_1)$ with $0<x_1<2$
$u'(x_1) - u'(0) = u''(x_2) x_1$ with $0<x_2<x_1$
$u(2)-u(1)-u'(0) = u'... | If $f \in C^2(a,b)$ then for every $x\in (a,b)$ and suitable $h>0$
$$
f(x+h)=f(x)+hf^{\prime}(x) + \frac{h^2}{2}f^{\prime\prime}(\xi_+), \qquad \xi_+ \in (x,x+h)
$$
and
$$
f(x-h)=f(x)-hf^{\prime}(x) + \frac{h^2}{2}f^{\prime\prime}(\xi_-), \qquad \xi_- \in (x-h,x).
$$
Summing we get
$$
f(x+h)+f(x-h) - 2f(x) = h^2\frac... | {
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What is a good book for multivariate stochastic processes? I'm looking for a good book that introduces (preferred without measure-theoretic proofs though that may have to do) multivariate stochastic processes. So suppose you have $\{\mathbf{X}_n : n \in \mathbb{Z}_0^\infty\}$ where $\mathbf{X}_n \in \Re^k$ or $\in \mat... | Spectral Analysis and Its Applications 1968 by G. M. Jenkins and D. G. Watts has all the information I requested. Looks like works from the 60s can often trump tons others from 2000s.
| {
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Kalman Filter Process Noise Covariance I want to model the movement of a car on a straight 300m road in order to apply Kalman filter on some noisy discrete data and get an estimate of the position of the car.
In a Kalman filter the matrix $A$ and process noise covariance $Q$ is what describes the system. I have chosen ... | Process noise can be viewed as any disturbance that affect the dynamics, for example steep on the roads, wind affects, friction on the roads etc. In general you can just say it as environmental disturbances.
| {
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Equivalence and Order Relations I have the following problem: Provide an example of a set $S$ and a relation $\sim$ on $S$ such that $\sim$ is both an equivalence relation and an order relation. Conjecture for which sets and this is possible, and prove your conjecture.
Below is the example I came up with but I don't ... | Assuming your order relations are required to be strict (that is, irreflexive), then you're almost done.
You have already shown that $\varnothing$ has an appropriate relation. What remains is to show that no nonempty set can have a relation that is both an equivalence and an order.
Thus assume that $A$ is a set, $x\in ... | {
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Help with inequality estimate, in $H^1$, Given a bilinear form on $H^1 \times H^1$, where $H^1 = W^{1,2}$
\begin{align*}
B[u,v] = \int_U \sum_{i,j}a^{i,j}(x)u_{x_i}v_{x_j} + \sum_ib^i(x)u_{x_i}v + c(x)uv \, \mathrm{d}x
\end{align*}
the book gives the inequality bound $\vert B[u,v] \vert \leq \alpha \Vert u \Vert_{H^1} ... | The claim follows since both $\|u\|_{L^2}$ and $\|u_{x_i}\|_{L^2}$ are bounded by $\|u\|_{H^1}$.
| {
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logarithm and exponent computation performance Using glibc on a x86 processor, which takes more CPU time? $a\ log\ b$ or $b^a$? For which values of $a$ is one faster than the other? Optional: Does the base used matter?
See also: What algorithm is used by computers to calculate logarithms?
Because I know someone will ... | Look through the code used by the FreeBSD operating system:
http://svnweb.freebsd.org/base/head/lib/msun/src/
http://svnweb.freebsd.org/base/head/lib/msun/src/e_pow.c?view=markup
http://svnweb.freebsd.org/base/head/lib/msun/src/e_log.c?view=markup
It is claimed that these are rather high quality algorithms, better t... | {
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Polar Integral Confusion Yet again, a friend of mine asked for help with a polar integral, we both got the same answer, the book again gave a different answer.
Question
Use a polar integral to find the area inside the circle $r=4 \sin \theta$ and outside $r=2$.
Proposed Solution
We see that the two circles intersect ... | The two circles intersect when:
$$\sin \theta = 1/2 \ \to \ \theta = \pi/6, \pi-\pi/6$$
So the integral should be from $\pi/6$ until $5\pi/6$, and not as you wrote.
Note that you could have eliminated both answers by noting that the area should be larger than half of the circle, or $A>2\pi$, whereas your book's answer... | {
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Intuition behind smooth functions. Smooth functions $f(t)$ are those such that $\frac{d^nf(t)}{dt^n}$ exists for all $n\in\Bbb{N}$.
I understand the intuition behind smoothness for functions like $f(t)=| t|$ and $f(t)=\sqrt{t}$. $f(t)$ has a "sharp" (and hence non-smooth) turn at $t=0$. Similarly, $f(t)=\sqrt{t}$ ends... | The problem with the function $f(t) = t^{1/3}$ is not that it is something else. The problem is that in the point $0$, the slope of the curve is infinite as the function turns completely vertical. The curve drawn is actually smooth, as you can find a smooth parametrisation of it.
The parametrisation $$t\rightarrow (t, ... | {
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Example of quotient mapping that is not open I have the following definition:
Let ($X$,$\mathcal{T}$) and ($X'$, $\mathcal{T'}$) be topological spaces. A surjection $q: X \longrightarrow X'$ is a quotient mapping if $$U'\in \mathcal{T'} \Longleftrightarrow q^{-1}\left( U'\right) \in \mathcal{T} \quad \text{i.e. if } \... | An open map is:
for any open set in the domain, its image is open
A quotient map only requires:
for any open preimage in the domain, its image is open
A quotient map may not be an open map because there may exist a set such that it is open but is not a preimage, i.e., the preimage of the image of an open set may not be... | {
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Evans PDE chapter 2 problem 4 Problem is
Give a direct proof that if $u \in C^{2}(U) \cap C(\overline{U})$ is harmonic within a bounded open set $U$, then
$\max_{\overline{U}} u =\max_{\partial U} u$.
What I think is that;
Let $u^{\epsilon} = u+ \epsilon |x|^{2}$, where $\epsilon >0 $. Then $\bigtriangleup u^{\ep... | If the maximum is attained inside the domain, at point $x_0$, then the Hessian $H$ of $u$ at $x_0$ is symmetric definite negative. You first have that
$$Tr(H) = \Delta u$$
and because H is definite negative,
$$Tr(H) < 0$$
This implies a contradiction with $u$ is harmonic.
| {
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When are complex conjugates of solutions not also solutions? I've heard that for "normal" equations (e.g. $3x^2-2x=0$), if $(a+bi)$ is a solution then $(a-bi)$ will be a solution as well.
This is because, if we define $i$ in terms of $i^2=-1$ then we might as well define $i^\prime=-i$. Since ${i^\prime}^2=-1$ we find $... | If the polynomial has real coefficients, and there is a nonreal root, then its conjugate is also a root. Otherwise, there would be at least one nonreal coefficient.
| {
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sequence defined by $u_0=1/2$ and the recurrence relation $u_{n+1}=1-u_n^2$ I want to study the sequence defined by $u_0=1/2$ and the recurrence relation $$u_{n+1}=1-u_n^2\qquad n\ge0.$$ I calculated sufficient terms to understand that this sequence does not converge because its odd and even subsequences converge to di... | The function $f:[0,1]\longrightarrow[0,1]$ has a unique fixed point but is repellent. And we can't reach it in a finite number of steps because is irrational.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/656147",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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supremum equals to infumum The question is:
What can you say about the set M if sup M = inf M.
I know that supremum is the lowest upper bound and infumum is the biggest lower bound. But I cant figure out what you can say about the set M. And we only had one lession about supremum and infumum.
| You know that for any $x \in M$, we have $\inf M \leq x \leq \sup M$, by definition of upper and lower bounds (not necessarily least upper or greatest lower bound). Now, if $\inf M = \sup M$, what does that say about the possible values of $x$?
| {
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Intuitively, why is the Euler-Mascheroni constant near $\sqrt{1/3}$? Questions that ask for "intuitive" reasons are admittedly subjective, but I suspect some people will find this interesting.
Some time ago, I was struck by the coincidence that the Euler-Mascheroni constant $\gamma$ is close to the square root of $1/3$... | I have an interesting approach. The Shafer-Fink inequality and its generalization allow to devise algebraic approximations of the arctangent function with an arbitrary uniform accuracy. By a change of variable, the same holds for the hyperbolic arctangent function over the interval $(0,1)$ and for the logarithm functio... | {
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How do i prove that $\mathfrak{M}\oplus\Sigma$ is the sigma algebra generated by products of elements of generating sets? Let $(X,\mathfrak{M}),(Y,\Sigma)$ be measurable spaces.
Let $\mathscr{A},\mathscr{B}$ be sets such that $\sigma(\mathscr{A})=\mathfrak{M}$ and $\sigma(\mathscr{B})=\Sigma$.
How do i prove that $\mat... | Define $H := \{ A \times B : A \in \mathfrak M, B \in \Sigma\}$. The goal is to show that $\sigma(H) = \sigma(G)$. The inclusion $\sigma(G) \subseteq \sigma(H)$ follows from $G \subseteq H$. For the other inclusion,
it suffices to prove that $H \subseteq \sigma(G)$. Note that for all $B \in \mathscr B$ the family $R_B ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to find the shortest distance from a line to a solid? The equation $x^2 + y^2 + z^2 - 2x + 6y - 4z + 5 = 0$ describes a sphere. Exactly
how close does the line given by $x = -1+t; y = -3-2t; z = 10+7t$ get to this sphere?
So the sphere is centered at $(1,-3,-2)$ and the radius is $3$.
I want to find the point wher... | Let's just find a vector in the direction of the line, find a vector connecting a point on the line t0 the center, and then make sure they're perpendicular.
Any two points on the line will allow us to find a vector in the direction of the line. With,say, $t=0$ and $t=1$, we get $(-1,-3,10)$ and $(0,-5,17)$, yielding a ... | {
"language": "en",
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"source": "stackexchange",
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An identity involving the Pochhammer symbol I need help proving the following identity:
$$\frac{(6n)!}{(3n)!} = 1728^n \left(\frac{1}{6}\right)_n \left(\frac{1}{2}\right)_n \left(\frac{5}{6}\right)_n.$$
Here,
$$(a)_n = a(a + 1)(a + 2) \cdots (a + n - 1), \quad n > 1, \quad (a)_0 = 1,$$
is the Pochhammer symbol. I do no... | Pochhammer symbols (sometimes) indicate rising factorials, i.e., $n!=(1)_n$ . This is obviously the case here, since the left hand side is never negative, assuming natural n.
$$\bigg(\frac16\bigg)_n=\prod_{k=0}^{n-1}\bigg(\frac16+k\bigg)=\prod_{k=0}^{n-1}\bigg(\frac{6k+1}6\bigg)=6^{-n}\cdot\prod_{k=0}^{n-1}(6k+1)$$
$$\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/656505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Help me with this limit $$ \lim_{x\to0} {{xe^x \over e^x-1}-1 \over x}$$
I know it should equal ${1 \over 2}$ because when i calculate with number like $0.0001$ the limit $\approx {1 \over 2}$ but i can't prove it.
| Divide the top and bottom by $x$ to clean stuff up:
$$\dots={ {{e^x \over e^x-1}-\frac{1}{x} }}\normalsize=\frac{x\cdot e^x-e^x+1}{x\cdot(e^x-1)}$$
Can you do it now?
| {
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If $H \leq G$ and $H \subset Z(G)$, the center of $G$, is $H \trianglelefteq G$? This is probably a very dumb question. Is it true that, in general, if $H$ is a subgroup of a group $G$, and $H \subset Z(G)$, the center of $G$, does it follow that $H$ is normal in $G$?
What I know so far that could potentially be useful... | This can be understood intuitively with group actions. Say $G$ acts on a set $X$:
*
*A subset $Y\subseteq X$ is pointwise fixed if $gy=y$ for all $g\in G$.
*A subset $Y\subseteq X$ is setwise fixed if $gY:=\{gy:y\in Y\}=Y$ for all $g\in G$.
The group $G$ acts on itself by conjugation. Then:
*
*A subset $H\subs... | {
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Find $\exp(D)$ where $D = \begin{bmatrix}5& -6 \\ 3 & -4\end{bmatrix}. $ The question is
Find $\exp(D)$ where $D = \begin{bmatrix}5& -6 \\ 3 & -4\end{bmatrix}. $
I am wondering does finding the $\exp(D)$ requires looking for the canonical form... Could someone please help?
| Hint:
*
*Write the Jordan Normal Form (it is diagonalizable) with unique eigenvalues.
*$e^{D} = P \cdot e^J \cdot P^{-1}$
The Jordan Normal Form is:
$$A = P J P^{-1} = \begin{bmatrix}1&2 \\ 1 & 1\end{bmatrix}~\begin{bmatrix}-1&0 \\0 & 2\end{bmatrix}~\begin{bmatrix}-1&2 \\ 1 & -1\end{bmatrix}$$
Now use the above f... | {
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Are all metabelian groups linear Are all metabelian groups linear? (i.e. isomorphic to a subgroup of
invertible matrices over a field)
| Every finitely generated metabelian group $\Gamma$ has a faithful representation over a finite product of fields (I think it's due to Remeslennikov). If $\Gamma$ is (virtually) torsion-free, one field of characteristic zero is enough. But for instance, denoting $C_n$ the cyclic group of order $n$, the wreath product $C... | {
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Prove ${\rm tr}\ (AA^T)={\rm tr}\ (A^TA)$ for any Matrix $A$ Prove ${\rm tr}\ (AA^T)={\rm tr}\ (A^TA)$ for any Matrix $A$
I know that each are always well defined and I have proved that, but I am struggling to write up a solid proof to equate them. I know they're equal.
I tried to show that the main diagonal elements w... | To give you an idea of how to properly write these sort of proofs down, here's the proof.
For a matrix $X$, let $[X]_{ij}$ denote the $(i,j)$ entry of $X$. Let $A$ be $m\times n$ and $B$ be $n\times m$. Then
\begin{align*}
\mathrm{tr}\,(AB)
&= \sum_{i=1}^n[AB]_{ii} \\
&= \sum_{i=1}^n\sum_{k=1}^m[A]_{ik}\cdot[B]_{ki} \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/656918",
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"source": "stackexchange",
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Exercice on a differential form Let $\omega$ be a $q$-form on $\mathbb{R}^2$ and let
*
*$Z_{\mathbb{R}^2}(dx_1)=\{p \in \mathbb{R}^2 \colon (dx_1)_{|p}=0\}$
*$Z_{S^1}(dx_1)=\{p \in S^1 \colon (dx1_{|S^1})_{|p}=0\}$
where $(dx1_{|S^1})$ is the pull-back of $q$. I have to show that $Z_{\mathbb{R}^2}(dx_1) \cap S^1... | The $1$-form $dx_1$ is the differential of the coordinate function $$x_1:\>{\mathbb R}^2\to{\mathbb R}, \quad (p_1,p_2)\mapsto p_1\ .$$
For a vector $X=(X_1,X_2)\in T_p$ one has
$$x_1(p+X)-x_1(p)= X_1\ .$$
Since the right side is obviously linear in $X$ we can already conclude that $$dx_1(p).X=X_1\ .$$
This means that ... | {
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Real and distinct roots of a cubic equation The real values of $a$ for which the equation $x^3-3x+a=0$ has three real and distinct roots is
| Here is a graphical approach. Note that '+ a' shifts the graph up or down.
1) Disregard '+ a'.
2) Graph $ f(x) = x^3 - 3x $
http://www.wolframalpha.com/input/?i=plot%28x%5E3+-+3x%29#
3) from the graph we see that we must find the y values at the local extrema. You did that already getting y = 2 and -2
4) It should be o... | {
"language": "en",
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Zeros of $e^{z}-z$, Stein-Shakarchi Complex Analysis Chapter 5, Exercise 13 This is an exercise form Stein-Shakarchi's Complex Analysis (page 155) Chapter 5, Exercise 13:
Prove that $f(z) = e^{z}-z$ has infinite many zeros in $\mathbb{C}$.
Attempt:
If not, by Hadamard's theorem we obtain $$e^{z}-z = e^{az+b}\prod_{1... | It is easy to see that $e^z-z$ has order of growth 1. So we can use Hadamard's theorem.
Assume that $f(z)=e^z-z$ has only finitely many zeroes, $a_1,a_2,\ldots, a_n$. Then by Hadamard's factorization theorem, for some $a\in \mathbb{C}$ we have $$e^z-z=e^{az}\prod_{i=1}^{n}\Big(1-\frac{z}{a_i}\Big).$$ Then by using th... | {
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Continuosly differentation on composite functions Let $f: \mathbb{R}\rightarrow\mathbb{R}$ a $C^1$ function and defined $g(x) = f(\|x\|)$.
Prove $g$ is $C^1$ on $\mathbb{R}^n\setminus\{0\}$. Give an example of $f$ such that $g$ is $C^1$ at the origin and an exemple of $f$ such that $g$ is not. Find a necessary conditio... | It is clear that $g$ is $C^1$ on $\mathbb{R}^n\backslash\{0\}$ as $x\mapsto \|x\|$ is also $C^1$ there. The derivative is given by
$$\nabla g(x)=\frac{x}{\|x\|} f^\prime(\|x\|)$$
for $x\not=0$.
Now let us investigate differentiability at $g$. For a real number $h\not=0$ we have
$$\frac{1}{h}(g(he_i)-g(0))=\frac{1}{h}(f... | {
"language": "en",
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Prove a square can't be written $5x+ 3$, for all integers $x$. Homework question, should I use induction?..
Help please
| Hint $\ $ Let $\,n\,$ by any integer. $\ {\rm mod}\ 5\!:\ n \equiv 0,\pm1,\pm2\,\Rightarrow\, n^2\equiv 0,1,4,\,$ so $\ n^2\not\equiv 3\!\pmod{\!5}$
Remark $ $ It's easier if you know $\mu$Fermat: $\ 3 \equiv n^2\overset{\rm square}\Rightarrow\color{#0a0}{-1}\equiv \,n^4\ [\,\equiv \color{#c00}1$ by $\mu$Fermat]
This ... | {
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Show that f has a unique zero and prove that f′(λ)>0 with f being continuous and differentiable. Let $f: [a,b] \longrightarrow \mathbb{R}( a<b)$, $f$ is continuous and differentiable.
We assume that $f$ and $f'$ are increasing and $f(a)<0, 0<f(b)$.
Show that $f$ has a unique zero which we denote $\lambda$ and prove th... | You don't need the fact that $f$ is increasing, only $f'$.
First, as you said, use IVT for existence.
Second, consider your integral $f(a)-f(0)$ where $f(a)=0$ and conclude that $f'(a)$ must be positive. (Edit: as per your comment, if you don't like integral, use MVT to show the same fact.)
Finally, given this, conclud... | {
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Generalization of principle of inclusion and exclusion (PIE) The PIE can be stated as $$|\cup_{i=1}^n Y_i| = \sum_{J\subset[n], J\neq \emptyset} (-1)^{|J|-1} |Y_J|$$ where $[n]=\{1,2,...,n\}$ and $Y_J=\cap_{i \in J} Y_i$.
Problems using it are usually reduced to counting problems and require finding a good union decom... | The principle of inclusion and exclusion is intimately related to Möbius inversion, which can be generalized to posets. I'd start digging in this general area.
| {
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Shading in a simple closed curve I started thiking about this today and I have an answer I feel I can justify intuitively but not rigorously.
Let $\mathbb{S} = \{(x,y) \ | \ f(x,y)=c\}$ define the points on a simple closed curve (at least I think this is the right terminology; examples would be a circle, elipse or hear... | With a few additional assumptions your statement becomes true.
Suppose $f$ is continuous. You've assumed that $S$ is a simple closed curve, so by the Jordan curve theorem, the complement of $S$ in the plane consists of two components, one unbounded and one bounded. The bounded component is what you are calling the poin... | {
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Find m.g.f. given $E(X^r)$ function?
"Let $X$ be a random variable with $E(X^r) = 1 / (1 + r)$, where $r = 1, 2, 3,\ldots,n$. Find the series representation for the m.g.f. of $X$, $M(t)$. Sum this series. Identify (name) the probability distribution of $X$?
As a hint, use the Taylor Formula."
The expectation is what ... | The definition of mgf is
$$
M(t) = \mathbb{E}[e^{tX}]
$$
and so
$$
E[X^r] = M^{(r)}(0).
$$
Notice that if we represent $M(t)$ as a McLaurin series, say
$$
M(t) = \sum_{n=0}^\infty \frac{m_n}{n!} t^n
$$
then $M^{(n)}(0) = \frac{m_n}{n!}$. We can now equate them, getting
$m_n = \frac{n!}{1+n}$ and so
$$
M(t) = \sum_{n=0}... | {
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ZF Set Theory Axiom of Infinity Could someone please state and explain the axiom of infinity in ZF set theory? This isn't homework, it's just something that has interested me for awhile.
| The axiom of infinity says that there exists a set $A$ such that $\varnothing\in A$, that is the empty set is an element of $A$, and for every $x\in A$ the set $x\cup\{x\}$ is also an element of $A$.
The definable function $f(x)=x\cup\{x\}$ is an injection from $A$ into itself, and since $f(x)\neq\varnothing$ for every... | {
"language": "en",
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Need help with an integral. I am asked to show that
$\displaystyle \frac{1}{2\pi} \int_{0}^{2\pi} e^{2 \cos \theta} \cos \theta \, d\theta = 1 + \frac{1}{2!} + \frac{1}{2!3!} + \frac{1}{3!4!} + \cdots$
by considering $\displaystyle \int e^{z + \frac{1}{z}} \, dz$. I don't really know how to incorporate the hint, and an... | Hint: the usual way to attempt a real integral from $0$ to $2\pi$ by using complex methods is to substitute $z=e^{i\theta}$. At some stage you may need to take the real part or imaginary part of a complex integral.
| {
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'uniform approximation' of real in $[0,1]$ Good evening,
Prove that: For every $\varepsilon>0$, there exist an $n\in \mathbb{N}$, such that for every $x\in[0,1]$, there exist $(p,q)\in \mathbb{N^2}$, with $0\leq p\leq q\leq n$ and |$qx−p|≤\varepsilon$.
I have tried to prove the result like the proof of density but I di... | Hint: Look at Dirichlet's approximation theorem.
| {
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"timestamp": "2023-03-29T00:00:00",
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I'm trying to prove there is infinite rational numbers between any two rational number I understand how to explain but can't put it down on paper.
$\displaystyle \frac{a}{b}$ and $\displaystyle \frac{c}{d}$ are rational numbers. For there to be any rational between two numbers I assume $\displaystyle \frac{a}{b} < \fra... | One natural way to complete your proof is by contradiction.
You noticed that, between any two rational numbers, there is a third.
Now, pick any two rational numbers $x < y$, and assume that there are a finite number of (say, only $n$) rational numbers between them. Call these numbers $a_0 < a_1 < \cdots < a_{n-1}$. Now... | {
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Maple Input and string conversion In Maple how does one prompt a user to input an equation with variable x? Then convert that equation into a data type that will enable me to perform functions on said equation?
| One way to do this is to use the readstat command, which prompts the user to enter a Maple statement whose value is returned. For example:
Test := proc()
local p, x;
x := readstat( "Enter Variable: " );
p := readstat( "Enter Polynomial: " );
diff( p, x )
end proc:
Then you can do:
> Test();
Enter Variable: x;... | {
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(Geometry) Proof type questions
Can someone please explain to me the given question and proof? otherwise I might just have to end up dropping my maths course because unfortunately I'm not understanding anything from my teacher. I'm not really 'seeing' the concept, if that makes sense.
Regards,
| The thing you might be tripped up on is the 180. The 180 comes from the 'line' DE, which is a straight angle, you know, a 180. And since 180 is a constant, subtracting either $\angle CBD$ or $\angle ABD$ you'll end up with the same number.
| {
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The kernel of homomorphism of a local ring into a field is its maximal ideal? I have a question about the proof of Theorem 3.2. of Algebra by Serge Lang.
In the theorem $A$ is a subring of a field $K$ and $\phi:A \rightarrow L$ is a homomorphism of $A$ into an algebraically closed field $L$.
In the beginning of the pro... | I believe, Lang meant that one can consider only the case of a local ring and the kernel being $\mathfrak{m}$ (i.e. $A:=A_{\mathfrak{p}}$). For the kernel: let $R$ be a ring of germs of infintely-differentiable functions, and take a homomorphism of $R$ to formal power series. That is embeddeble into the field of Lauren... | {
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give an example of algebraic numbers $\alpha, \beta$ such that.... Question is to find algebraic numbers $\alpha, \beta$ such that :
$$[\mathbb{Q}(\alpha):\mathbb{Q}]>[\mathbb{Q}(\beta):\mathbb{Q}]>[\mathbb{Q}(\alpha\beta):\mathbb{Q}]$$
It is not so difficult to find algebraic numbers $\alpha, \beta$ such that :
$$[\ma... | One can argument this way. Suppose that we can find some $\alpha$ with the following properties:
1) $[\mathbb Q(\alpha)\colon \mathbb Q]=n$ and there is some prime $p$ such that $p(p+1)\mid n$.
2) The minimal polynomial $m(x)$ of $\alpha$ is of the form $f(x^{p(p+1)})$ for some $f(x)\in \mathbb Z[x]$.
Then $[\mathbb Q(... | {
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Very silly permutation question Okay let me briefly explain my doubt.
I'll explain some easy problems,so that you can study easily my mind and you can guess what confusion i might be going through right now.
This may be silly.But please help me out.
How do you permute the letters ABC with no character can be repeated?
... | This can be understood by renaming the second A to A'. This gives you obviously 3!=6 possible permutations:
AA'B
ABA'
A'AB
A'BA
BAA'
BA'A
Now you can group these into pairs whose elements are equal except of A and A' being swapped:
AA'B, A'AB
ABA', A'BA
BAA', BA'A
Since A and A' shall not be distinguished we only kee... | {
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Integrating $\int_0^\infty\frac{1}{1+x^6}dx$ $$I=\int_0^\infty\frac{1}{1+x^6}dx$$
How do I evaluate this?
| $\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... | {
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Functional analysis and physics Some branches of mathematics like functional analysis do not on first encounters seem to have any possible applications.Can someone please give me some examples of applications of functional analysis in phsics?I believe there can hardly be any absolutely unapplied branch of mathematics... | Historically functional analysis emerged from applications; it is therefore obvious that it has "applications". In quantum mechanics the obserables are operators on a hilbert space, and one wants in particular to realize the "canonical commutation relation" $[q,p]=1$ for self-adjoint operators $q$ and $p$ (position and... | {
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At a closed monoidal category, how can I derive a morphism $C^A\times C^B\to C^{A+B}$? Let $A$, $B$ and $C$ be objects of a closed monoidal category which is also bicartesian closed. How can I derive a morphism $C^A\times C^B\to C^{A+B}$?
$(-)\times (-)$ denotes the product, $(-)+(-)$ the coproduct and $(-)^{(-)}$ the ... | Cartesian closure applies to cartesian categories, i.e. categories which are (symmetric) monoidal with respect to the (binary) product bifunctor (basically any finitely complete category is cartesian). Cartesian closed categories are those categories where each functor $A\times-$ has a right adjoint $(-)^A$ realizing t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/658848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Solve the following differential equation: $ty' + 2y = \sin(t)$ An exercise from the book:
Solve the following differential equation: $ty' + 2y = \sin(t)$
This is the first time I approch a differential equation, and the book doesn't provide an example how to solve an differential equation,
So I need your help to sho... | Hint:
Consider the following ODE:
$$y'(t) + p(t) y(t) = F(t).$$
Suppose you are interested in rewriting this equation as follows:
$$ \frac{d}{dt}(I y) = I F,$$
for some function $I(t)\neq 0$ (called integrating factor). Expand the product of derivatives to find:
$$y' + \frac{I'}{I} y = F, $$
so it must hold:
$$\frac{I'... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/658961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is it possible to swap vectors into a basis to get a new basis? Let $V$ be a vector space in $\mathbb{R}^3$. Assume we have a basis, $B = (b_1, b_2, b_3)$, that spans $V$. Now choose some $v \in V$ such that $v \ne 0$. Is is always possible to swap $v$ with a column in $B$ to obtain a new basis for $V$?
My initial thou... | If $(v_1,\cdots, v_n)$ is a basis for any vector space $V$, and $w\in V$ is an arbitrary vector, then swapping $w$ for $v_i$ will result in a basis iff $w$ is not in the span of $\{v_j\mid j\ne i\}$. The proof is a good exercise.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/659113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How prove $\sum\limits_{cyc}(f(x^3)+f(xyz)-f(x^2y)-f(x^2z))\ge 0$ let $x,y,z\in (0,1)$, and the function
$$f(x)=\dfrac{1}{1-x}$$
show that
$$f(x^3)+f(y^3)+f(z^3)+3f(xyz)\ge f(x^2y)+f(xy^2)+f(y^2z)+f(yz^2)+f(z^2x)+f(zx^2)$$
For this problem simlar this Schur inequality:
http://www.artofproblemsolving.com/Wiki/index.php/... | In fact, this nice problem can indeed be solved using Schur's Inequality of 3rd degree, albeit indirectly. First we will prove a lemma:
Lemma: For $x, y, z>0$, we have:
$$e^{x^3}+e^{y^3}+e^{z^3}+3e^{xyz}\ge e^{x^2y}+e^{xy^2}+e^{y^2z}+e^{yz^2}+e^{z^2x}+e^{zx^2}$$
Proof: By Schur's inequality, for each $n\in\mathbb{N}$, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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A question about the action of $S_n$ on $K[x_1,...,x_n]$ Let ${K}$be the field ($\,Char\,K\not=0)$. Let $n\in \mathbb{Z}^{+}$. $S_n$ acts on $K[x_1,...,x_n]$in the following way:
If $p\in K[x_1,...,x_n]$ and $\sigma\in S_n$, then $\sigma p$ is the polynomial $p(x_{\sigma(1)},x_{\sigma(2)},...,x_{\sigma (n)})$.
Questi... | Yes.
Let $f(x_1,\ldots, x_n)=\prod_{k=1}^n x_k^k$. Then
$p=\sum_{h\in H} h(f)$ has the desired property. (This works for all fields, also those of nonzero characteristic)
| {
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"url": "https://math.stackexchange.com/questions/659471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How prove this $x_{2^n}\ge 2^n-n+1$ the sequence $ (x_n)_{n\ge 1}$, $ x_n$ being
the exponent of 2 in the decomposition of the numerator of
$$\dfrac{2}{1}+\dfrac{2^2}{2}+\cdots+\dfrac{2^n}{n}$$
goes to infinity as $ n\to\infty$.
Even more, prove that $$x_{2^n}\ge 2^n-n+1$$
My idea: maybe
$$\dfrac{2}{1}+\dfrac{2^2}{2}+... | Now,I have solution This nice problem,
we only note that
$$\dfrac{2}{1}+\dfrac{2^2}{2}+\cdots+\dfrac{2^n}{n}=\dfrac{2^n}{n}\sum_{k=0}^{n-1}
\dfrac{1}{\binom{n-1}{k}}$$
This indentity proof can see:http://www.artofproblemsolving.com/Forum/viewtopic.php?p=371496&sid=e0319f030d85bf1390a8fb335fd87c9d#p371496
also I reme... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
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Combinatorial proof that $\sum_{j=0}^k (-1)^j {\binom n j}=(-1)^k \binom{n-1}{k}$
Prove $\sum_{j=0}^k (-1)^j {\binom n j}=(-1)^k \binom{n-1}{k}$.
It can be proven easily using induction and Pascal's identity, but I want some insight. The alternating sum reminds of inclusion-exclusion, but the RHS can be negative.
I'v... | Reverse the terms in your LHS and multiply by $(-1)^k$ to see that your identity is equivalent to this: $\sum_{j=0}^k (-1)^{j} {\binom n {k-j}}= \binom{n-1}{k}$. The right hand side is the number of $k$-subsets of positive integers less than $n$. That equals the number of $k$-subsets of positive integers less than or e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/659653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How can you find the $x$-coordinate of the inflection point of the graphs of $f'(x)$ and $ f''(x)$? So I understand how to find the inflection points for the graph of $f(x)$.
But basically, I have been shown a graph of an example function $f(x)$ and asked the state the inflection points of the graph. (I am just shown a... | I've always thought that the points where the f(x) graph cuts the x-axis is where the points of inflection for f'(x) are.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/659729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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LCM of First N Natural Numbers Is there an efficient way to calculate the least common multiple of the first n natural numbers? For example, suppose n = 3. Then the lcm of 1, 2, and 3 is 6. Is there an efficient way to do this for arbitrary n that is more efficient than the naive approach?
| If you don't want to assume having or constructing list if primes, you can just recursively apply the two argument LCM, resulting in something that is likely a bit worse than $O(n \cdot log n)$ time. You can likely improve to close to $O(n \cdot log n)$ by replacing the linear application with divide and conquer:
Assum... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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"answer_id": 5
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Graph with vertices having certain degrees Let $G$ be a simple graph with $n$ vertices, such that $G$ has exactly $7$ vertices of degree $7$ and the remaining $n-7$ vertices of degree $5$. What is the minimum possible value for $n$?
I have gotten that $n$ could equal $14$ with $G$ as the following graph:
i) $G=G_1 \cup... | Assuming you are concerned with simple graphs, you can use the Erdos-Gallai theorem that characterizes when does a graph sequence admit a graphical representation.
Using the above theorem you can verify that you need at least three vertices of degree $5$ thus giving you the degree sequence $ds = (7,7,7,7,7,7,7,5,5,5).$... | {
"language": "en",
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Find the differential of an n-variable function The problem goes like this:
If $f:\mathbb{R}^n\to\mathbb{R}, f(x)=\arctan||x||^4$, prove that $Df(x)(x)=\displaystyle\frac{4||x||^4}{1+||x||^8}$
Now, I've calculated each of the partial differentials (if that's the right word) and applied that $1\times n$ matrix to a vect... | $\mathrm{Arctan}$ is differentiable on $\mathbb{R}$ with :
$$ \big( \mathrm{Arctan}' \big)(x) = \frac{1}{1+x^{2}} $$
And $\varphi \, : \, x \in \mathbb{R}^{n} \, \longmapsto \, \Vert x \Vert^{4}$ is differentiable on $\mathbb{R}^{n}$ with :
$$ \mathrm{D}_{x}\varphi \cdot h = 4 \Vert x \Vert^{2} \left\langle x,h \right\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Confusion about a way to compute a residue at a pole Suppose I have a function of the form $$f(x)=\frac{1}{(x-a)(x-b)^2(x-c)^3}$$
Clearly, I have a simple pole at $a$, and poles of order 2,3 at $b,c$, respectively.
By definition, the residue at $x=\alpha$ is the coeffecient of the term $(x-\alpha)^{-1}$ in the Laurent... | Yes, there is such a rule. If you have a partial fraction decomposition
$$f(z) = h_\alpha(z) + h_\beta(z) + h_\gamma(z) + g(z),$$
where $h_w$ is the principal part of $f$ in $w$, and $g$ is the remaining holomorphic part of $f$, then you can develop $h_\beta,\, h_\gamma$, and $g$ in Taylor series about $\alpha$ to obta... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Help solve simultaneous inequality that has $\leq$ and $\geq$ in it
I only have problems determining the values of $\alpha$ and $\beta$, so I will only show the solution used to derive their values:
So, I know how to get the inequalities at ★ and † but I don't know how use them to deduce $\alpha$ and $\beta$.
All I k... | The reasoning of the solution is the following.
Firstly, the inequalities $(\star)$ and $(\dagger)$ represent the two principle minors of the Hessian matrix of $f$. It is known that $f$ is convex iff it's Hessian matrix is positive semidefinite and the Hessian matrix is positive semidefinite if it's principle minors ar... | {
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Triple of powers is subvariety? Consider the set $B=\{(t^2,t^3,t^4)\mid t\in \mathbb{C}\}$. Is it a subvariety of $\mathbb{C}^3$? That is, is it the set of common zeros of some (finite number of) polynomials?
I'm thinking about $y^2-x^3$ and $z-x^2$. But suppose $x=t^2$, then we're allowing $y=\pm t^3$. How to get rid ... | That $x = t^2$ allows $y = \pm t^3$ is no problem, because $(t^2, -t^3, t^4)$ is also in $B$ (just replace $t$ by $-t$).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Open and closed maps: What good for? I'm wondering what open mappings are actually good for (except for inverse becomes continuous)???
My irritation came since, people stress that an open mapping not necessarily preserves closed sets (well, sure, I mean closed maps are some totally different subject since they don't de... | Open and closed maps become useful when combined with continuity!
Open/Closed Maps:
For a continuous and open/closed map we have:
If it is injective then it is an embedding.
If it is surjective then it is a quotient map.
If it is bijective then it is a homeomorphism.
Note: Neither embeddings nor quotient maps are neces... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/660348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Integration of $\sin(\frac{1}{x})$ How to find the value of the integral?
$$\begin{align}
(1)&&\int_{0}^{1}\sin\left(\frac{1}{x}\right)dx\\\\
(2)&&\int_{0}^{1}\cos\left(\frac{1}{x}\right)dx
\end{align}$$
| The antiderivatives involve the sine of cosine integral functions. From their definition, the antiderivatives are respectively $$x \sin \left(\frac{1}{x}\right)-\text{Ci}\left(\frac{1}{x}\right)$$ and $$\text{Si}\left(\frac{1}{x}\right)+x \cos \left(\frac{1}{x}\right)$$ So, the integrals are $$\sin (1)-\text{Ci}(1)$$ a... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Exercise about linear operator For $X$ Banach, I have to show that if $T\in\mathfrak{L}(X)$ and $||T||_{\mathfrak{L}(X)}<1$ then exists $(I-T)^{-1}$ and
$$
(I-T)^{-1}=\sum_{n=0}^\infty T^n.
$$
For the existence of $(I-T)^{-1}$ I proved that $\ker(I-T)=\{0\}$. But for the second point of the proof I don't know how to ... | Look at this $$(I-T)\circ \sum_{j=0}^{n} T^j =I-T^{n+1} $$ and $$\left(\sum_{j=0}^{n} T^j \right)\circ (I-T)=I-T^{n+1} .$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/660524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Union of ascending ideals is an ideal Could you tell me what I'm doing wrong in proving this proposition?
If $I_1 \subset I_2 \subset ... \subset I_n \subset ...$ is an ascending chain of ideals in $R$, then $I := \bigcup _{n \in \mathbb{N}} I_n$ is also an ideal in $R$.
So we need to prove that the group $(I, +)$ is a... | The following is in the book Algebra, by T. Hungerford.
Let $b \in I_i$ and $c \in I_j$. It follows that $i \leq j$ or $j \leq i$. Say $i \geq j$, then $I_j \subset I_i$. Therefore, $b, c \in I_i$. Since $I_i$ is an ideal, $b-c \in I_i \subset I$.
Similarly, if $r \in R$ and $b \in I$, $b \in I_i$ for some $i$. There... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/660633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A problem of subgroup of a group If $G$ be a finite group of order $pq$ where $p$ & $q$ are prime numbers such that $p>q$. Prove that $G$ has at most one subgroup of order $p$.Hence prove that a group of order 6 has at most one subgroup of order 3.
My try is that i let $G$ has two subgroup $H$ & $K$ of same order $p$.T... | This is correct, though you have not explained that.
To do that, consider the map $H\times K\to G$ defined as $(h,k)\mapsto hk$. What can you say about it if $H\cap K=\{e\}$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/660723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Normal Subgroup Counterexample Im having trouble with the second part of this question,
Let $H$ be a normal subgroup of $G$ with $|G:H| = n$,
i) Prove $g^n \in H$ $\forall g \in G$ (which i have done)
ii) Give an example to show that this all false when $H$ is not normal in $G$.(which I am having trouble with showing)
... | Hint: If $H$ is not normal in $G$, then $G$ is necessarily nonabelian. Consider the smallest nonabelian group you know.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/660832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Constructing sequence of functions I have to construct a sequence of $\{f_i\}$, where $f_i$ belongs to $C[0,1]$ such that:
$$
d(f_i,0) = 1 \\
d(f_i,f_j)=1, \forall i,j \\
\text{Using Sup-Norm metric, i.e.} \mathbb{\|}f\mathbb{\|} = \sup_x{\mathbb{|}f_i (x)\mathbb{|}}
$$
Thanks.
| Set
$$
f_n(x)=\left\{
\begin{array}{lll}
0 & \text{if}& x\le \frac{1}{n+1}, \\
n(n+1)x-n& \text{if} & x\left[\frac{1}{n+1},\frac{1}{n}\right], \\
1 & \text{if}& x\ge \frac{1}{n}.
\end{array}
\right.
$$
Clearly,
$$
\sup_{x\in[0,1]}|f_m(x)-f_n(x)|=1,
$$
if $m\ne n$.
| {
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"url": "https://math.stackexchange.com/questions/660904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Limits of trigonometric function I know my answer is correct, but are my steps correct?
$$
\begin{align}
& \lim_{t \to 0} \frac{\tan(2t)}{t}\\[8pt]
& = \lim_{t \to 0} \frac{1}{t} \tan(2t)\\[8pt]
& = \lim_{t \to 0} \frac{\sin(2t)}{t \cdot \cos(2t)} \cdot \frac{2}{2}\\[8pt]
& = \lim_{t \to 0} \frac{2 \cdot \sin(2t)}{2t \... | Yes exactly is correct congratulation
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/661014",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Decide whether the following properties of metric spaces are topological or not I have to decide whether the following properties of metric spaces are 'topological' or not,
(a) every continuous function on M is bounded
(b) $(\forall x \in M)(\forall y \in M) d(x; y) > 0$
(c) $(\forall x \in M)(\forall y \in M) d(x; y) ... | In short, a property of a metric space is topological if it does not (really) depend on the metric. That is, if you have a different metric that produces the same notions of convergence, continuous functions, etc. (or even if you only have the notions of convergence, contnuou functions, etc.) then the property does not... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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If $mv < pv < 0$, is $v > 0?$ (1) $m < p$ (2) $m < 0$ Is there a way to simplify this equation and not rely on testing numbers via trial and error?
If $mv < pv < 0, is v > 0$?
(1) $m < p$
(2) $m < 0$
We have to figure out if statement 1 by itself is sufficient to answer this question, or if statement 2 is sufficient by... | You can try the two cases separately.
Assume $v$ is negative and then divide through by $v$ giving
$$ m > p > 0\;\;\; (*)$$
This is contradicted by both statements (1) and (2). Hence by contradiction if (1) or (2) are true, then $v$ cannot be negative.
On the other hand if you assume $v$ is positive and then divide by... | {
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Probability distribution of dot product? Sorry if this is a basic question. I don't know much about statistics and the closest thing I found involved unit vectors, a case I don't think is easily generalizable to this problem.
I have a reference vector $\mathbf V$ in some $\mathbb R^n$.
I have another vector in $\mathbb... | V.X will be Normal, as linear combination of normal rvs is again normal. Check this.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/661246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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solve equation in positive integers Can anybody help me with this equation?
Solve in $\mathbb{N}$:
$$3x^2 - 7y^2 +1=0$$
One solution is the pair $(3,2)$, and i think this is the only pair of positive integers that can be a solution. Any idea?
| There are infinitely many solutions in positive integers. $7y^2-3x^2=1$ is an example of a "Pell equation", and there are standard methods for finding solutions to Pell equations.
For example, the fact that $(x,y)=(2,3)$ is a solution to $7y^2-3x^2=1$ is equivalent to noting that $(2\sqrt7+3\sqrt3)(2\sqrt7-3\sqrt3)=1$.... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Construct polyhedron from edge lengths I'm interested in the following problem: I am given the combinatorial structure (vertices, edges, faces) and edge lengths of a polyhedron. From this I'd like to infer the vertex positions.
Now, I know that the above information does not uniquely determine the polyhedron. For insta... |
You might extend this into three dimensions as a prism.
| {
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"url": "https://math.stackexchange.com/questions/661422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Solving for the determinant of a matrix? I know the rules of solving for the determinants of a matrix and I totally thought I was doing this right, but the answer output is incorrect.
I took the approach that I was adding the $a, b$ and $c$, so nothing would change. However, I notice I am multiplying the second row by ... | If ${\rm det}\ A=1$, can you calculate
$$ {\rm det}\ (\left( \begin{matrix} 1&0&0\\ 1 &-4 &0 \\ 0&0&1 \end{matrix} \right) A) $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/661500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Evaluating $\int_\gamma |z|\,|dz|$ I'm updating my question to reflect changes that have occurred.
I'm kind of stuck trying to figure out this integral:
$$\int_\gamma |z|\,|dz|=\int_{0}^{\pi}|t||1+t||dt|$$
where $\gamma(t)=te^{it}$ where $0 \leq t \leq \pi$.
Since I have reduced this to $t$, do I approach this for $-t$... | I understand the question as asking for
$$I = \int_\gamma |z| |dz|$$
where the curve $\gamma$ is defined to be the locus of $z(t)$ from $t=0$ to $t=\pi$ with $z(t) = t e^{it}$.
Then $|z| = t$ and $|dz| = |e^{it} + ite^{it}| = |1+it| = \sqrt{1+t^2}$, so
$$\begin{eqnarray}
I & = & \int_0^\pi t\sqrt{1+t^2} dt \\
& = & \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/661573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
How to solve $\log \sqrt[3]{x} = \sqrt{\log x} ?$ How to solve $$\log \sqrt[3]{x} = \sqrt{\log x} $$
| Using $$m\log a=\log(a^m)$$ when both logs are defined
$$\log\sqrt[3] x=\sqrt{\log x}\implies\frac13 \log x=\sqrt{\log x}$$
$$\sqrt{\log x}(\sqrt{\log x}-3)=0$$
$$\sqrt{\log x}=0\iff \log x=0\iff x=1$$
$$\sqrt{\log x}-3=0\iff \log x=9$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/661614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $\frac{1}{1*3}+\frac{1}{3*5}+\frac{1}{5*7}+...+\frac{1}{(2n-1)(2n+1)}=\frac{n}{2n+1}$ Trying to prove that above stated question for $n \geq 1$. A hint given is that you should use $\frac{1}{(2k-1)(2k+1)}=\frac{1}{2}(\frac{1}{2k-1}-\frac{1}{2k+1})$. Using this, I think I reduced it to $\frac{1}{2}(\frac{1}{n... | $$\frac{1}{(2k-1)(2k+1)}=\frac{1}{2}(\frac{1}{2k-1}-\frac{1}{2k+1})
$$This implies that the sum is
$$\frac{1}{2}\big\{\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-...-\frac{1}{2n-1}+\frac{1}{2n-1}-\frac{1}{2n+1}\big\}$$ cancel terms and complete
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/661701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A closed ball in a metric space is a closed set
Prove that a closed ball in a metric space is a closed set
My attempt: Suppose $D(x_0, r)$ is a closed ball. We show that $X \setminus D $ is open. In other words, we need to find an open ball contained in $X \setminus D$.
Pick $$t \in X-D \implies d(t,x_0) > r \implie... | No I do not think this is correct. The idea seems correct, but the execution was poor. You should specify that $y\in X\backslash D$. I am also not sure how you justify your last inequality. If $t$ is arbitrary in $X\backslash D$ we cannot conclude $d(z,t)<r_1$ and $-d(z,t)>-r_1.$
Here is how I would solve the problem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/661759",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 2,
"answer_id": 1
} |
Uniform convergence of Taylor series I am trying to show:
If $f: B(0,R) \to \mathbb C$ is analytic then the Taylor series of $f$ at $0$ converges uniformly to $f$ in $B(0,r)$ for all $r\in (0,R)$
But I got stuck with my proof. Please can somebody help me? Here is what I have so far:
From here I have
$$ f(z) = \sum_{n=... | We have to show, that $\sum_{n=0}^\infty z^n c_n$ is unformly convergent for $|z|<r$. The trick is, that $C\subset B(0,R)$ can be any smooth closed curve rounding the origin. So take $\varrho$ such that $r<\varrho<R$ and let $C:=\{z\,|\,|z|=\varrho\}=\{\varrho e^{it}\,|\,0\leq t\leq 2\pi\}$. Then
$$
|c_n| =\left|{1\ov... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/661846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How to solve the following limit? $$\lim_{n\to\infty}{\bigg(1-\cfrac{1}{2^2}\bigg)\bigg(1-\cfrac{1}{3^2}\bigg) \cdots \bigg(1-\cfrac{1}{n^2}\bigg)}$$
This simplifies to $\prod_{n=1}^{\infty}{\cfrac{n(n+2)}{(n+1)^2}}$.
Besides partial fractions and telescope, how else can we solve? Thank you!
| \begin{align}
L&=\lim_{n\to\infty}{\bigg(1-\cfrac{1}{2^2}\bigg)\bigg(1-\cfrac{1}{3^2}\bigg) \cdots \bigg(1-\cfrac{1}{n^2}\bigg)}\\
&=\lim_{n\to\infty}{\bigg(1-\cfrac{1}{2}\bigg)\bigg(1-\cfrac{1}{3}\bigg) \cdots \bigg(1-\cfrac{1}{n}\bigg)\bigg(1+\cfrac{1}{2}\bigg)\bigg(1+\cfrac{1}{3}\bigg) \cdots \bigg(1+\cfrac{1}{n}\bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/661879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Area of a triangle The following problem in elementary geometry was proposed to me. As a mathematical analyst, I confess that I can't solve it. And I have no idea of what I could do. Here it is: pick a triangle, and draw the three mediana (i.e. the segments that join a vertex with the midpoint of the opposite side). Us... | Use vectors.It will be really helpful.Define sides of triangle ABC as A=0, B= b vector and C= c vector.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/662053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 8,
"answer_id": 2
} |
Why is the notation for differentiation like this? Consider the notation for denoting the differentiation of a function $f(x)$.
$$\frac{d[f(x)]}{dx}$$
I mean, this notation doesn't make any sense. $dx$ means a vanishingly small $x$, which can be understood, if we take $x$ to mean $\Delta x$ in a loose sense. But what ... | The $d[f(x)]$ indicates the vanishingly small change in $f(x)$ corresponding to the vanishingly small change in $x$ indicated by $dx$. Their ratio is the derivative.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/662180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Continuous function between metric spaces Let $(X , d)$ and $(Y , \rho)$ be two metric spaces. Let $S = \{x_1, x_2, x_3, ...\}$ be a countable dense subset of $X$. Let $f : X \rightarrow Y$ be a continuous function. The prove that;
For a closed set $F$ in $Y$, $f(x)$ belongs to the closed set $F$ if and only if for e... | "$\Rightarrow$" Let $f(x) \in F$, i.e. $x \in f^{-1}(F)$. Let $n \in \mathbb{N}^+$. Since $f$ is continuous at $x$, we find some $\delta>0$ such that $d(x,y) < \delta $ implies $d(f(x),f(y))< \frac{1}{n}$. Since $S$ is dense, there is some $s \in S$ such that $d(x,s) < \min(\delta,\frac{1}{n})$. Hence, $d(x,s) < \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/662315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proof that the number of diagonals of a polygon is $\frac{n(n-3)}{2} $ For $n \geq 3$ proof that the number of diagonals of a polygon is $\frac{n(n-3)}{2} $ using induction.
I don't know how to start this problem, can you give me a hint?
| From any vertex we cannot draw a diagonal to its $2$ adjacent sides and the vertex itself.
So for a polygon containing $n$ sides we cannot draw $3$ diagonals, it means we can draw $n-3$ diagonals.
For $n$ vertices we can draw $n(n-3)$ diagonals.But each diagonal has two ends, so this would count each one twice. So t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/662503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 5
} |
If $X$ and $Y$ are uniformly distributed on $(0,1)$, what is the distribution of $\max(X,Y)/\min(X,Y)$? Suppose that $X$ and $Y$ are chosen randomly and independently according to the uniform distribution from $(0,1)$. Define $$ Z=\frac{\max(X,Y)}{\min(X,Y)}.$$
Compute the probability distribution function of $Z$.
Can ... | We find the cdf $F_Z(x)$ of the random variable $Z$. By symmetry, it is enough to find $\Pr(Z\le z|Y\gt X)$, that is,
$$\frac{\Pr((Z\le z) \cap (Y\gt X))}{\Pr(Y\gt X)}.$$ The denominator is $\frac{1}{2}$. so we concentrate on the numerator.
The rest of the argument is purely geometric, and will require a diagram. Draw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/662601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
About the roots of a quadratic equation Let $m_1$ and $m_2$ the real and diferent roots of the quadratic equation $ax^2+bx+c=0$. Do you know some way to write $m_1^k + m_2^k$ in a simplest form (linear, for example) using just $a,b,c,m_1$ and $m_2$?
Thanks for the attention!
| You can use Newton's polynomial to calculate any value of $m_1^k + m_2^k$, without actually calculating the roots and using just the coefficients in front of the variable. Actually it works for any degree polynomial. Let $s_k = m_1^k + m_2^k$ and $m_1$ and $m_2$ be roots of $ax^2 + bx + c=0$. Then we have:
$$as_1 + b =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/662684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
What's the diference between $A<\infty$ and $A<\aleph_0$? In my topology class the teacher gave some examples of topologies, and I'm trying to prove that they really are topologies. If $X$ is a set then:
*
*$\mathcal C=\{A:\# (X-A)<\infty\}$ is a topology in $X$.
*$\mathcal C'=\{A:\# (X-A)<\aleph_0\}$ is a topology... | There must have been some kind of misunderstanding (either on your part or your instructor's). $\aleph_0$ is the cardinality of the natural numbers which [assuming the Axiom of Choice...] is the smallest infinite cardinal. Thus the second expression is just a more formal way of writing the first one.
Maybe what was i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/662840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proving a summation involving binomial coefficients. I need to prove the following inductively: (http://upload.wikimedia.org/math/9/e/5/9e57871ba17c1ad48e01beb7e1bb3bb9.png)
$$\sum_{i=1}^{n} i{n \choose i} = n2^{n-1}$$
And for the life of me I can't figure out how to express the formula with n+1 in terms of n.
| ** I was going to withdraw this answer as it does not use induction, but decided to leave it anyway and take the criticism. Thanks lab bhattacharjee for setting me right**
Start with
$$
(x+1)^n = \sum _i {n \choose i} x^i
$$
and differentiate with respect to $x$
$$
n(x+1)^{n-1} = \sum _i {n \choose i} i x^{i-1}
$$
N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/662921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Coordinates rotation by $120$ degree If I have a point on a standard grid with coordinates say:
$A_1=(1000,0)$
$A_2=(707,707)$
Is there a easy way to transfer this points to $\pm 120$ degrees from the origin $(0,0)$, and keeping the same distance?
So for $A_1$, the result should be something like:
$B_1=(800,-2... | In general, the rotation matrix of angle $\theta$ is $$r_\theta=\left(\begin{array}{cc}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{array}\right).$$
In your case, $\theta=\pm120°$, so $\cos\theta=-1/2$ and $\sin\theta=\pm\frac{\sqrt 3}2$.
You just have to apply the matrix to the vector to get the image of the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/663064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
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