Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Probability of sum of 6 picked integers from [1..36] How we would calculate probability that sum is bigger than 92.5 if we pick 6 random numbers from [1..36] ?
Not putting them back (if took 1, then we can pick [2..36] etc.
I cannot think of how to put this because there are lot of variations of how this can fall. Unle... | Personally I only know about generating function which produces the answer with aid of computer. It may be easier to calculate the answer using script, without using any combinatorical or generatingfunctionological technique. The number of ways to choose six distinct numbers whose sum is equal to or lower than 92 is de... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$\exists \implies \forall$ I want to see some example theorem, when existence implies universality, so $\exists \implies \forall$ is true.
I think matematical induction is a related technique, but I just don't see that induction covers the whole topic. On the other hand, there are some situation, when existence implies... | This's an example from linear algebra:
Let $f\colon\Bbb R^n\rightarrow R^n$ a linear transformation.
these statements are equivalent
*
*$f$ is orthogonal
*$f$ maps an orthonormal basis to an orthonormal basis
*$f$ maps every orthonormal basis to an orthonormal basis
| {
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Show $f$ has a fixed point if $f\simeq c$ I have the following problem:
Show that if $f:S^1\to S^1$ is a continuous map, and $f$ is homotopic to a constant, then $\exists p\in S^1 : f(p)=p$.
My approach is to show that if for all $p, \ $ $f(p)\neq p$, then $f$ is homotopic to $\mathrm {id}_{S^1}$. To prove this I thou... | There is a proposition that you will want to use here which is the following:
Proposition. A map $g\colon S^1\to X$ is null-homotopic if and only if there exists a map $\tilde{g}\colon D^2\to X$ such that the restriction satisfies $\tilde{g}|_{S^1}=g$. That is, $g$ can be extended to a map on the disk.
Now, if $f\col... | {
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Non-integrable systems If a Hamiltonian system in $\mathbb{R}^{2n}$ has $n$ suitable first integrals, then it is called an integrable system, and the Arnold-Liouville theorem tells us all sorts of nice things about the system: In particular, if a flow is compact then the flow takes place on a torus $T^n$.
What means ar... | There is something called Morales–Ramis theory which is (I've been told) the most powerful method for proving nonintegrability. There are preprint versions of various articles and even of a book (Differential Galois Theory and Non-integrability of Hamiltonian Systems) on the webpage of Juan Morales-Ruiz: http://www-ma2... | {
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Procedure for evaluating $\int_{x=\ -1}^1\int_{y=\ -\sqrt{1-x^2}}^{\sqrt{1-x^2}}\frac{x^2+y^2}{\sqrt{{1-x^2-y^2}}}\,dy\,dx$ While solving another problem I have come across this integral which I am unable to evaluate. Can someone please evaluate the following integral? Thank you.
$$\int_{x=\ -1}^1\int_{\large y=\ -\sqr... | Based on the limit of integral $-\sqrt{1-x^2} < y < \sqrt{1-x^2}$ and $-1 < x < 1$, the region of integration is a unit circle in the Cartesian coordinate. See this plot to visualize the region of integration. Using polar coordinate, we have $x^2+y^2=r^2$ and the region of integration will be $0<r<1$ and $0<\theta<2\pi... | {
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$f_n(x):=nx(1-x)^n$ Determine whether the sequence $(f_n)$ converges uniformly on $[0,1]$ I am having a bit of trouble on this revision question.
To determine pointwise convergence: $\lim_{n\rightarrow\infty} = nx(1-x)^n $. For $x=0, x=1$, it's clear that the limit is $0$. How can I determine the limit for $0 \le x \le... | Observe that $$\sup_{x \in [0,1]} |f_n(x)-f(x)| \geq \left|f_n \left(\frac{1}{n} \right) -f\left( \frac{1}{n} \right) \right|=\left(1-\frac{1}{n} \right)^n \to \frac{1}{e}$$
as $n \to \infty$ .So that we can't have $\lim_{n \to \infty} \sup_{x \in [0,1]} |f_n(x)-f(x)|=0$. i.e. there is no uniform convergence.
| {
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Can any function on naturals be interpolated to a smooth function on reals? Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be an arbitrary function from naturals to naturals. Is it always possible to find a function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that
*
*for any $n \in \mathbb{N}$, we have $f(n) = g(n)$, a... | Whittaker–Shannon interpolation, using the $sinc$ function, achieves that. http://en.wikipedia.org/wiki/Whittaker%E2%80%93Shannon_interpolation_formula
$$g(x)=\sum_{n=-\infty}^{+\infty}f_n\frac{\sin\pi(x-n)}{\pi(x-n)}.$$
| {
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Proof of an infinite series formula If $$|x| < 1$$ Prove that
$$\begin{align}\large 1 + 2x + 3x^2 + 4x^3 + \dots = \frac{1}{(1 - x)^2}\end{align}$$
| Another$^2$ approach:
Since
$\sum_{i=0}^{\infty} x^i
= \frac1{1-x}
$,
differentiating both sides
we get
$\sum_{i=1}^{\infty} ix^{i-1}
= \frac1{(1-x)^2}
$.
| {
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Does the sequence $\{\sin^n(n)\}$ converge? Does the sequence $\{\sin^n(n)\}$ converge?
Does the series $\sum\limits_{n=1}^\infty \sin^n(n)$ converge?
| claim. The series $\sum\limits_{n=1}^\infty \sin^n(n)$ diverges.
Lemma. For all number $x$ irrational there exist a rational sequence $\{\frac{p_n}{q_n}\}$ where $\{q_n\}$ is odd such that
$$
\left\vert x-\frac{p_n}{q_n}\right\vert<\frac{1}{q_n^2}
$$
Proof.
Define $x_n=\frac{1}{x_{n-1}-\lfloor x_{n-1}\rfloor}$. Let ... | {
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What is the largest integer with only one representation as a sum of five nonzero squares? It seems to be very well known that $33$ is the largest integer with zero representations as a sum of five nonzero squares. So it seems reasonable to me that as we go higher and higher, numbers have more and more representations ... | You may find it interesting to have a glance at this image: http://oeis.org/A025429/graph (Number of partitions of n into 5 nonzero squares).
-- see also http://oeis.org/A080673 = largest numbers with exactly n representations as sum of five positive squares.
| {
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Why doesn't mathematical induction work backwards or with increments other than $1$? From my understanding of my topic, if a statement is true for $n=1$, and you assume a statement is true for arbitrary integer $k$ and show that the statement is also true for $k+1,$ then you prove that the statement's true for all $n\g... | Consider your first example of counting down:
Suppose property $Q(n)$ holds for $n=1$.
Suppose also that $Q(n)$ implies $Q(n-1)$.
Define a new property $P$ by $P(n)$ if and only if $Q(2-n)$.
We supposed $Q(1)$ so we know $P(1)$.
Suppose $P(n)$. That implies $Q(2-n)$. We deduce $Q(2-n-1)$ which is $Q(2-(n+1))$. And that... | {
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Equivalence Relation using Binary Operations. Question:
Let ∗ be a binary operation on a set A. Assume that ∗ is associative with identity. Let R be the relation on A defined on elements a,b ∈ R as follows: aRb if there exists an invertible element c ∈ A such that c∗b = a∗c. Prove that R is an equivalence relation. Wha... | Reflexivity: $\forall a\in A, aRa$.
If $e$ is the identity element, then $a*e=e*a$, so this holds.
Transitivity: $\forall a,b,c \in A$[$(aRb$ and $bRc) \rightarrow aRc$]
So, let's assume that $aRb$ and that $bRc$. Then $\exists d,f \in A$ such that $a*d=d*b$ (1)
and
$b*f=f*c$. (2)
We multiply $f$ to both sides of equa... | {
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Interchaging $P(\mathrm{limsup})$ with $P(\mathrm{limit})$ for $P$ a probability measure. I have been going through Resnick's 'A Probability Path', and at one point he is trying to prove a version of Fatou's lemma:
$$P(\liminf_{n\rightarrow\infty}A_n)\le\liminf_{n\rightarrow\infty}P(A_n)$$
In the first line of the proo... | For every $n$, let $B_n=\bigcap\limits_{k\geqslant n}A_k$. For every sequence $(A_n)$, the sequence $(B_n)$ is nondecreasing hence has a limit, called $\liminf\limits_{n\rightarrow\infty}A_n$.
Using the sets $C_n=\bigcup\limits_{k\geqslant n}A_k$, a similar result holds for $\limsup\limits_{n\rightarrow\infty}A_n$.
| {
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Prove that $\sum_{n=1}^{\infty} \frac{a_n}{n^z}$ converges absolutely and uniformly
Let $(a_n)$ be a sequence in $\mathbb{C}$. Assume that
$\sum_{n=1}^{\infty} \frac{a_n}{n^z}$ converges absolutely for some
$z= z_0 \in \mathbb{C}$. Prove that the series converges absolutely
and uniformly on $\{z \in \mathbb{C}:... | Things you need to change
First
$$|n^{ib}|=|\exp{i\cdot b\cdot ln(n)}|=|cos(b\cdot ln(n))+i\sin(b\cdot ln(n))|=\sqrt{cos(b\cdot ln(n))^2+sin(b\cdot ln(n))^2}=1$$
Second
You need to say $|a_n|\geq a_n$ and $|n^z|= n^a$ so $\sum_{n=1}^{\infty} \frac{a_n}{n^{\alpha}}$ converge uniformly on $]z_o,\infty[$
Third
Just forget... | {
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How do you find the domain and range without having to graph? Like, is their an algebraic method? For example if I am asked to find the domain of $g(t) = \sqrt{t^2 + 6t}$ , how do I determine the range of this?
Is their a universal algebraic method that I don't know about?
| $t^2 + 6t \geq 0$. Thus $g(t) \geq 0$. This gives the range $[0, +\infty)$. For more "details" about the range, take a non-negative real number $r \geq 0$, then show that:
you can find an $t$ with $t \leq -6$ or $t \geq 0$ such that: $g(t) = r$. This translates to the equation:
$\sqrt{t^2 + 6t} = r \Rightarrow t^2 + 6t... | {
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Introduction to discrete subgroups of the euclidean group I am looking for a general introduction to discrete subgroups of the euclidean group (= group of isometries in euclidean space).
Even though I searched quite a bit, I was unable to find a good introduction.
Any hints for which book or survey to look?
| If you are interested in dimensions 2 and 3, consider reading "Geometries and Groups" by Nikulin and Shafarevich.
If you are interested in higher dimensional groups as well, Wolf's "Spaces of constant curvature," covers basics (and much more).
| {
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How many different bases in $\mathbb{Z}/p\mathbb{Z}$ Let $K = \mathbb{Z}_p$, for some prime $p$, and $\text{dim}\:V = n$. $V$ is a vector space over $K$.
I need to find out how many different bases are in $V$.
Now I know the answer is the product of all
$$\frac{1}{n!}\prod\limits_{i=0}^{n-1} (p^n-p^i)$$
The solution s... | $-1$ because a basis can't contain the zero vector. ;) More precisely, a family of vectors which contains the zero vector $\mathbf{0}$ can't be free, because then you have a linear combination with non-zero coefficients which yields the zero vector (e.g., $1\cdot \mathbf{0} = \mathbf{0}$ ).
| {
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How to tell whether a representation of a group is faithful or unfaithful? From just the character table and the basis functions of the irreducible representations, how do I know whether a representation is faithful or unfaithful?
For the 1-D representation it is trivial to know the answer, of course, so I am only talk... | The numbers in the right hand section of the table are called the character values. A representation is faithful if and only if the number in the $E$ column of that row only appears once in that row. So $\Gamma_1$ is not faithful, since all the columns have the same value as $E$ (namely, $1$). For nearly the same reaso... | {
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Coordinates of tilted circle. The original question is as follows:
Imagine a wire located at the intersection of $x^2+y^2+z^2=1$ and $x+y+z=0$, whose density depends on position according to $\rho({\bf x})=x^2$ per unit length. Show that the mass of the wire is $\frac{2}{3}\pi$.
I am thinking to parametrize the interse... | If you really want to do it the hard way, you could use the knowledge that the intersection is a circle and points on it are orthogonal to the unit vector $\hat{\eta} = (\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$.
To find a parametrization of the circle, you need a point on the circle as a starting poi... | {
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Prove that system of equation implies statement How to prove that
$$
\begin{cases}
x_1 + x_2 + x_3 & = 0 \\
x_1x_2 + x_2x_3 + x_3x_1 & = p \\
x_1x_2x_3 & = -q \\
x_1 & = 1/x_2 + 1/x_3
\end{cases}
$$
implies
$$
q^3 + pq + q = 0\,\,?
$$
| Denote the equations by $(1),\ldots ,(4)$. Then $(4)$ says $x_1x_2x_3=x_2+x_3$ and $(3)$ says $x_1x_2x_3=-q$. This gives $x_3=-x_2-q$. Substitute this into $(1)$. This gives
$x_1=q$.
Then $q\cdot (2)-(3)$ gives $-q(p+q^2+1)=0$, or
$$q^3+pq+q=0.$$
| {
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How did Fourier arrive at the following regarding his series and coefficients? I am reading Karen Saxe's "Beginning Functional Analysis." Perhaps it is poor exposition on her part, but she states:
...Fourier begins with an arbitrary function $f$ on the interval from
$-\pi$ to $\pi$ and states that if we can write $$... | If
$$
f(x) = \frac{a_0}{2} + \sum_{k = 1}^\infty a_k\cos(kx) + b_k\sin(kx)
$$
then
$$
\int_{-\pi}^\pi f(x)\cos (nx)\,dx = \int_{-\pi}^\pi \quad \underbrace{\left\{ \frac{a_0}{2} + \sum_{k = 1}^\infty a_k\cos(kx) + b_k\sin(kx) \right\}} \quad \cos(nx)\,dx
$$
because the part over the $\underbrace{\text{underbrace}}$ is... | {
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How can I calculate this limit? ( I tried l'Hopital and failed ) I have to calculate this : $$ \lim_{x\to 0}\frac{2-x}{x^3}e^{(x-1)/x^2} $$ Can somebody help me?
| Letting $w=1/x$, we have
$$
\lim_{x\downarrow 0}\frac{2-x}{x^3}e^{(x-1)/x^2} = \lim_{w\to+\infty} \left(2 - \frac 1 w \right) w^3 e^{w^2\left(\frac 1 w - 1\right)} = \lim_{w\to+\infty} (2w^3 - w^2) e^{w-w^2}
$$
$$
= \lim_{w\to+\infty} \frac{2w^3-w^2}{e^{w^2-w}}.
$$
L'Hopital should handle that.
Maybe I'll post somethin... | {
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Do all distributions of R.V.s have a singular part and a continuous part? Consider the probability distribution of a real-valued R.V. as the equivalence class of generalized PDFs where the integral over each measurable set in $\mathbb{R}$ is the same in each PDF.
1) Can any R.V.'s distribution be represented as the su... | There exist distributions which are neither discrete nor continuous. For example, the Lebesgue-Stieltjes measure generated by the Cantor function.
| {
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Three consecutive integers which are power of prime but not prime Does there exist three consecutive positive integers such that each of them is the power of a prime i.e., is there exist $n \in \mathbb{N}$, such that $n=p^i$, $n+1 = q^j$ and $n+2 = r^k$, where $p$, $q$ and $r$ are primes and $i,j,k >1$.
| No. Catalan conjecture...................
| {
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Why does it follows that $\alpha \in (\mathbb Z_p^*)^2 \iff \alpha^{(p-1)/2} = 1$ from proved results? Suppose I
've proved the following, where $(\mathbb Z_p^*)^2$ denotes the set of unit residue classes modulo $p$.
Why does it then follows that $\alpha \in (\mathbb Z_p^*)^2 \iff \alpha^{(p-1)/2} = 1$ and $\alpha \no... | The direction $(\Rightarrow)$ is $\rm(ii),(iii)$ in the theorem. The opposite direction follows because the set of nonzero squares and nonsquares form a partition of $\,\Bbb Z_p^*.\,$ Thus if $\,\alpha^{(p-1)/2} = 1\,$ then $\,\alpha\,$ is either a square or a nonsquare, but it cannot be a nonsquare since those map to ... | {
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I don't know how to interpret this strange $\prod$ I have got a $\prod$ that is exactly as follows:
$$\prod\limits_{k=0, k \ne k}^n \frac{x-c_k}{c_k-c_k}$$
I am not sure how to interpret this. My guesses are that it equals either $0, or ,1, or ,x$. But perhaps it isn't defined?
| Strictly as written the product is $1$. There is no $k$ for which $k \neq k$, the product is thus empty (no $k$ fulfills the condition) and thus $1$.
This seems like some sort of "trick question" or a typo (one of the $k$ should be something else), like
$$\prod\limits_{k=0, \kappa \ne k}^n \frac{x-c_{\kappa}}{c_k-c_{... | {
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Is there a mistake in this question: $\forall a\in A: |\{x\in A:x\le a \}|=|\{ y\in B :y\le a \}|$?
Two ordered sets $(A,\le_A), (B,\le_B)$ and there's an isomorphic function $f:A\to B$
Prove $\forall a\in A: |\{x\in A:x\le a \}|=|\{ y\in B :y\le a \}|$
I think there's a mistake in this question, how can you compare... | Yes, you are right, there should be $f(a)$ instead of $a$.
| {
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Integer solutions to $x^x=122+231y$ How can I find the integer solutions to the following equation (without a script or trial and error)?
$$x^x=122+231y$$
| The function $f_a:\Bbb N\to\Bbb Z_p$ defined as $f_a(x)=a^x$ where $a\in\Bbb Z_p^*$ is periodic and its period is a divisor of $p-1$. Then the modular equation $x^x\equiv 2 \pmod 3$ has only to be checked for $x\in\{1,\ldots,6\}$. And
$$1^1\equiv 1\pmod 3$$
$$2^2\equiv 1\pmod 3$$
$$3^3\equiv 0\pmod 3$$
$$4^4\equiv 1\pm... | {
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True or false: for all $x$ there is a $y$ such that if $x$ is non negative then $y^2 = x$
For all $x$ there is a $y$ such that if $x$ is non negative then $y^2 = x$
Is my logic correct in proving that statement is true ? Can provide an explanation of how to test this proof ?
$x=2$ then $y^2 = 2$
$x=2$ then $y = \sqrt... | You cannot prove the statement by using only one value $x = 2$ to show that for that particular value $x$, there exists a $y$ such that $y^2 = x$.
Why not?
You can't stop at showing it's true for one $x$, or two $x$, or even a million values of $x$, because the statement is a claim about all $x\geq 0$. Proving a "for ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can Zeno's dichotomy paradox be disproved using mathematics? A brief description of the paradox taken from Wikipedia:
Suppose Sam wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter,... | We know that if Sam runs fast enough and long enough, he will eventually catch up to the bus. If both are moving at a constant speed, there is no need to decompose their motion into infinitely many, ever decreasing intervals. A simple application of the speed-distance-time formula will tell us that Sam will catch up to... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 9,
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Is the series $\frac{1}{(n+1)^p}-\frac{1}{(n-1)^p}$ where 0Sorry for my bad English.
I really suspect it is convergent. But I can't prove it.
Since ${x^p}$ is not derivable at x=0, I can't using taylor expansion to find the order of infinitesimal, thus nth-term test cannot be used. I tried other test but they seem to ... | Notice that if
$$a_n = \frac{1}{n^p} $$
The series you're asking about it equivalent to
$$ \sum_{n=2}^\infty a_{n+1} - a_{n-1} $$
Hint: It's telescoping.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is $\{12,18\} \cup ([-6,6] \setminus(-2,2))$ a compact set? Is the following set of real numbers compact?
$$\{12,18\} \cup ([-6,6] \setminus(-2,2))$$
It is obviously bounded (upper bound is $18$, lower bound is $-6$) but is it closed? I am not so familiar with topologic terms so please apologize if this question may s... | Yes, it's closed. Your set can be written as $\{12,18\} \cup [-6,-2]\cup [2,6]$ and thus it is closed as finite union of closed sets.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/814034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Applying L'Hôpital's rule infinitely I tried to prove that $\int\limits_0^\infty t^{x-1} e^{-t} \, \mathrm{d}t$ satisfies the functional equation of the gamma function $\Gamma(x+1)=x\Gamma(x)$, so I partially integrated $\Gamma(x+1)$, yielding $\left[-e^{-t}\,t^x\right]_0^\infty+x \Gamma(x)$.
It is obvious to me that
$... | The question of how to find $\displaystyle\lim_{a\to\infty}\frac{a^x}{e^a}$ or limits similar to it seems to come up often here.
And L'Hopital's rule, when it finds the answer, gives little or no insight.
Every time $a$ increases by $1$, the fraction $\dfrac{a^x}{e^a}$ is multiplied by $\dfrac{(a+1)^x/a^x}{e}<\dfrac{2}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Explain why if $u=\sqrt{i+2}$ is in $\mathbb{Q}(i)$, an extension of the rational numbers, there exists b... Explain why if $u=\sqrt{i+2}$ is in $\mathbb{Q}(i)$, an extension of the rational numbers, there exists $b \in \mathbb{Q}(i)$ which is a root of $a(x)=-1+8x^2+4x^4$.
I have looked at the minimum polynomial for $... | Well, if $u\in\Bbb Q(i),$ then $u=x+iy$ for some $x,y\in\Bbb Q,$ yes? Now, squaring both sides, we have $$i+2=x^2-y^2+2xyi.$$ Hence, $x^2-y^2=2$ and $y=\frac1{2x},$ so....
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $\overline{\mathrm{Int}\,(\overline{\mathrm{Int}\,A})} = \overline{\mathrm{Int}\,A}$ I need to proof this statement, and I don't know where to start.
In every topological sapce, we have that $\overline{\mathrm{Int}\,(\overline{\mathrm{Int}\,A})} = \overline{\mathrm{Int}\,A}$
I tried to show that $\mathrm{Int... | Hint: One inclusion is fairly straightforward (use $\operatorname{Int}B \subseteq B$). For the other, show that $\operatorname{Int}A \subseteq \operatorname{Int}(\overline{\operatorname{Int}A})$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that $abc=[ab,bc,ca]*(a,b,c)=(ab,bc,ca)*[a,b,c]$ Let $a,b \in \mathbb N$. Show that
$$ abc=[ab,bc,ca](a,b,c)=(ab,bc,ca)[a,b,c] .$$
How would I prove this?
| Fix a prime p. Show that the prime power p^k that divides each side is the prime power that divides abc.
Hence, we can even conclude that the expression is equal to abc
| {
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Proof of the identity of a Boolean equation $Y+X'Z+XY' = X+Y+Z$ How to prove the following the identity of a Boolean equation?
$$ Y+X'Z+XY'=X+Y+Z $$
I have tried :
$
\space\space\space\space\space
Y+X'Z+XY'\\
=X'Z+XY'+Y\\
=X'Z+XY'+Y(X+X')\\
=X'Z+XY'+XY+X'Y\\
=X'(Z+Y)+X(Y'+Y)\\
=X'(Z+Y)+X‧1\\
=X'(Z+Y)+X\\
$
Then, ... | Logically,
*
* Y+X'Z+XY' is true whenever Y is true, irrespective of X and Z.
* Removing all inputs where Y is false, X'Z+XY' is true, when X is True (we know Y is false), irrespective of Z.
* And finally, when both X and Y are false, X'Z is true, hence the expression is true.
In all three cases, we were only c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/814579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Numerical integration fails I am doing something wrong.
This is my algorithm to evaluate the integral $$\int_0^1 \frac{1}{1+x}dx= \log(2).$$
with the Newton Cotes algorithm (Simpson and 3/8).
Both give me that for large n (number of subintervals of $[0,1]$ that I take, where I apply each Newton-Cotes algorithm separate... | Your programming error, misusing the limit n instead of the index i, inside of your loop was only part of the problem. Simpson's method, just as Splines, share endpoints between the curve segments, so your use of 3*n x values was entirely off. The looping below will give you n segments with n-1 knots, a total of 2*n ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Subgroups of $\mathrm{PSL}(2,q)$ of order $2q$ Let $q\equiv 1\pmod 4$. Is it true that $\mathrm{PSL}(2,q)$ has a unique class of conjugate subgroups of order $2q$?
I looked at the references that appear in this MO question, the only relevant refernce there is Oliver King's notes where he cites the classification given... | Yes, it is true. Let $B$ denote the normalizer of a Sylow $p$-subgroup of $G = {\rm PSL}(2,q)$, where $q = p^{a}$ for the odd prime $p.$ Then $B$ may be taken as the image (mod $\pm I$ of the group of upper triangular matrices of determinant $1.$ Let $M$ be a subgroup of $G$ of order $2q$. Then $M$ has a normal $2$-com... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integral involving a logarithm and a linear rational function $$\int_{0}^{1} \frac{\log x}{x-1}dx$$
I was wondering: is it possible to evaluate this integral with real methods? Playing around with a series expansion I was able to recognize that the integral is equal to $\zeta(2)$, but since I don't know how to evaluate... | Another way: it's actually easier to expand the denominator: $-\frac{1}{1-x} = \sum_{k=0}^{\infty} x^k$ and since the bounds on the integral are $0$ and $1$, $x^k$ converges uniformly, and you can interchange summation and integration. You get an integral of the form
$$
I_k = - \int_{0}^{1}x^k \log x dx
$$
which are e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/814856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Proofs involving Well-Defined and One-to-One
Chartrand, 3rd Ed, P224-225: Define a relation $R$ as a relation from A to B.
$R$ is well-defined means: $(a,b), (a,c) \in R \implies b = c$.
P220: A function $f: A \to B$ is one-to-one means:
For all $x, y \in A$, if $f(x) = f(y)$, then $x = y$.
I've observed that in the ... | Let's represent function $f:A\to B$ as a set of ordered pairs $f=\{(a_1,b_1), (a_2,b_2)\ldots\}$.
If this is a one-to-one function, then:
*
*If $(a,b)\in f$ and $(c,b)\in f$, then $a=c$.
*For all $a\in A$, there is exactly one $b\in B$ such that $(a,b)\in f$.
You have correctly noticed that if we reverse all the or... | {
"language": "en",
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Determine which Fibonacci numbers are even (a) Determine which Fibonacci numbers are even. Use a form of mathematical induction to prove your conjecture.
(b) Determine which Fibonacci numbers are divisible by 3. Use a form of mathematical induction to prove your conjecture
I understand that for part a that all multiple... | Part A:
Base case:
$F(0) = 0$, 0 is even.
$F(3) = 2$, 2 is even.
Inductive Hypothesis:
Assume $F(k)$ is even for some arbitrary positive integer $k$ that is divisible by 3.
Want to prove: That $F(k+3)$ is even given the inductive hypothesis.
\begin{align*}
F(k+3) &= F(k+2) + F(k+1)\\
&= F(k+1) + F(k) + F(k+1)\\
&= F(... | {
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"timestamp": "2023-03-29T00:00:00",
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How can I define $\mathbb{N}$ if I postulate existence of a Dedekind-infinite set rather than existence of an inductive set? Suppose in the axioms of $\sf ZF$ we replaced the Axiom of infinity
There exists an inductive set.
with the Axiom of Dedekind-infinite set
There exists a set equipollent with its proper subs... | Suppose that $A$ is a Dedekind-infinite set. First consider $T=\operatorname{TC}(A)$, the transitive closure of $A$. Now consider the function $f(x)=\operatorname{rank}(x)$, whose domain is $T$.
By the axiom of replacement the range of $f$ is a set, and it is not hard to prove that it has to be an ordinal.
Finally, pro... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Dynkin Diagram $SU(n)$ The goal is to give the Dynkin diagram of $SU(n)$. One can show that the complexification of the Lie algebra $\mathfrak{g}$ of $G$ is given by $\mathfrak{G}_{\mathbb{C}}=\mathfrak{sl}(n,\mathbb{C})$ (thus the traceless matrices). Moreover one can show that if $\mathfrak{t}$ is the set of all diag... | The Dynkin diagram of $SU(n+1)$ is the diagram of type $A_n$, because the Dynkin diagram of the Lie algebra $\frak{sl}(n+1)$ is the Dynkin diagram of the Lie algebra $\frak{su}(n+1)$. For details see for example here, or section $9.10.1$ here.
| {
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"source": "stackexchange",
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Ordered tuples of proper classes From time to time I encounter notation like this:
A triple $\langle \mathbf{No}, \mathrm{<}, b \rangle$ is a surreal number system if and only if ...
The confusing part is that a proper class is used as a component of an ordered tuple (that eventually supposed to have some encoding vi... | Let $\langle a,b \rangle=\{\{a\},\{a,b\}\}$ denote a Kuratowski pair. Suppose we have $n$ classes (some of them can be proper classes):$$C_i=\{x\mid\phi_i(x)\},\ 1\le i\le n.$$
Our goal is to find a class that could represent an ordered tuple of these. We can use the following class for this purpose:
$$\langle\!\langle... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that the binomial coefficient is congruent to 0 mod p. Let $p$ be a prime number, and let $k$ be an integer such that $0<k<p$. Prove that the binomial coefficient ${p\choose k}\equiv 0\pmod p$.
How would I prove this?
| There are $p$ chairs arranged uniformly around a circular table. We want to choose $k$ of them.
We say that two such choices $A$ and $B$ are equivalent if $B$ is obtainable from $A$ by a rotation. Since $p$ is prime, if $k$ is different from $0$ or $p$, there are precisely $p$ choices that are equivalent to $A$.
Thu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/815393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
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middle school question on geometry
As you can see from the picture, the angle $A$ is $90^\circ$, and the segments $BD$ and $CE$ (which intersect at $F$) are angle bisectors of the angles $B$ and $C$, respectively. When the length of $CF$ is $\frac72$ and and the quadrilateral $BCDE$ has an area of $14$, what is the le... | This is not a middle school level answer. But perhaps a complicated answer is better than no answer at all. Start by choosing coordinates as follows:
$$
A=\begin{pmatrix}0\\0\end{pmatrix}\quad
B=\begin{pmatrix}b\\0\end{pmatrix}\quad
C=\begin{pmatrix}0\\c\end{pmatrix}\quad
D=\begin{pmatrix}0\\d\end{pmatrix}\quad
E=\begi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/815487",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to calculate the range of $x\sin\frac{1}{x}$? I want to find the range of $f(x)=x\sin\frac{1}{x}$. It is clearly that its upper boundary is $$\lim_{x\to\infty}x\sin\frac{1}{x}=1$$ but what is its lower boundary?
I used software to obtain the result $y\in[0.217234, 1]$ and the figure is
How to calculate the value ... | It might be easier to replace $x$ by ${1 \over x}$... then your goal is to find the minimum of ${\sin x \over x}$. Taking derivatives, this occurs at an $x$ for which ${\cos x \over x} - {\sin x \over x^2} = 0$, or equivalently where $\tan x = x$. According to Wolfram Alpha, the first such minimum occurs at $x = 4.4934... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/815580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How find this sum $S(x)=\sum_{k=1}^{\infty}\frac{\cos{(2kx\pi)}}{k}$ Find this sum
$$S(x)=\sum_{k=1}^{\infty}\dfrac{\cos{(2kx\pi)}}{k},x\in R$$
my idea: since
$$S'(x)=2x\pi\cdot\sum_{k=1}^{\infty}\sin{(2kx\pi)}$$
then I can't.
| $\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\dow... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Hardy-Littlewood maximal theorem (Marcinkiewicz) I have two pages from a book called "Garnett" and I will present Hardy-Littlewood maximal theorem in class on Wednessday.
The theorem is stated:
if $f\in L^p(\mathbb{R}), 1 \leq p \leq \infty,$ then $Mf(t)$is finite a.e.
b) if $f\in L^p(\mathbb{R}), 1 < p \leq \infty,$ t... | I know this post is old, but here is the solution if you were still wondering. The idea is to use Fubini's theorem. Let $\chi_A$ be the characteristic function of $A \subset \mathbb{R}$.
$||Tf||_p^p = \int_{\mathbb{R}}|Tf(x)|^p dx = \int_\mathbb{R} \int_0^{Tf} p\lambda^{p-1} d\lambda dx = \int_\mathbb{R} \int_0^\infty ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Supremum of convex lipschitz functions. Let $f_i:K\to R, i\in I$ be a family of convex, equi-Lipschitz functions on some compact subset $K$ of $\Bbb R^n$. Is it true that $\sup f_i$ is also Lipschitz continuous(assuming that the sup exists)? Thank you
| That is true. The $\sup$ even exists everywhere, if it exists at at least one point $x_0 \in K$.
Let us first derive for $x,y \in K$ arbitrary:
$$f_i(x) = f_i(y) + f_i(x) - f_i(y) \leq f_i(y) + L \cdot |x-y|,$$
where $L$ is the joint(!) Lipschitz constant for the $f_i$.
Taking the supremum yields
$$\sup_{i \in I} f_i(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/815854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Compute $\pi^n(S^1\times S^{n+1})$. What is the space of homotopy classes of maps $S^1\times S^{n+1}\to S^n$? Is there a simple way to compute it, if we know $[S^{n+1}, S^n]\simeq\mathbb{Z}^2$ (resp. $\mathbb{Z}$ for $n=2$)?
| Here's an attempt assuming you are interested in unbased homotopy classes of maps.
Let $[X,Y]$ denote based homotopy classes of maps, then what we are looking for is the space $[S^1_+ \wedge S_+^{n+1},S^{n}]$
\begin{align}
[S^1_+ \wedge S_+^{n+1},S^{n}] &= [S^{n+1}_+,Maps(S^1_+,S^n)]
\end{align}
$Maps(S^1_+,S^n)$ is t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/815919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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arctan maps the unit disk onto a band around the imaginary axis Let $D\subseteq\mathbb{C}$ be the unit disk; that is, $D=\{z\in\mathbb{C}:\ |z|<1\}$. Let $B\subseteq \mathbb{C}$ be some band around the imaginary axis: $B=\{z\in\mathbb{C}:\ |\text{Re}(z)|<\pi/4\}$.
Why does it hold that the principal branch of $\arctan$... | Here's an attempt:
let $U=\left\{z\in\mathbb{C}:\ \text{Re}(z)>0\right\}$. Define
$g:V\to U$ as follows:
\begin{equation*}
g(z) = g(x+yi) = \exp(-2y + 2xi).
\end{equation*}
We have that $g$ maps $V$ conformally to $U$. Now define $h:U\to D$ as
follows:
\begin{equation*}
h(z) = \frac{i(1-z)}{1+z}
\end{equation*}
We ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/816018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Balkan MO problem Let $S = \{A_1,A_2,\ldots ,A_k\}$ be a collection of subsets of an $n$-element set $A$. If for any two elements $x, y \in A$ there is a subset $A_i \in S$ containing exactly one of the two elements $x, y$, prove that $2^k\geq n$.
This is a question from Balkan MO 1997, and I did not quite understand... | You have a set $A$ with $n$ elements.
Then there is a collection of $k$ sets $\{A_1,\ldots,A_k\}$, each of which is a subset of $A$, that is, $A_i \subseteq A$ for $1 \leq i \leq k$. You need to prove that
If for any pair $x,y \in A$ there is a set $A_i$ that distinguishes between $x$ and $y$ (that is, contains exact... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/816151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
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Kernel of linear transformation in $\Bbb R^3$ Let $T: \Bbb R^3 \rightarrow \Bbb R^3$ be a linear transformation satisfying
\begin{align*}
T(0,1,1) =& (-1,1,1) \\
T(1,0,1) =& (1,-1,1) \\
T(1,1,0) =& (1,-1,0) .
\end{align*}
Is it necessary true that $\ker(T) = \operatorname{Sp}\{(1,-1,1)\}$ ?
Well, I tried to say that w... | Let the basis for the domain be $B=\{v_1,v_2,v_3\}=\{(0,1,1),\;(1,0,1),\;(1,1,0)\}$. Let $w_1,w_2,w_3$ be the respective images of $v_i's$ under $T$.
A simple observation shows that: the set $\{w_1, w_2\}$ is linearly independent (as they are not multiples of each other) whereas $\{w_1,w_2,w_3\}$ is a dependent set bec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/816217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How do I find the sum of the infinite geometric series? $$2/3-2/9+2/27-2/81+\cdots$$
The formula is $$\mathrm{sum}= \frac{A_g}{1-r}\,.$$
To find the ratio, I did the following:
$$r=\frac29\Big/\frac23$$
Then got:
$$\frac29 \cdot \frac32= \frac13=r$$
and $$A_g= \frac23$$
Then I plug it all in and get:
$$\begin{align*}
\... | $$
\frac23 - \frac29 + \frac2{27} - \frac2{81} + \dots
=
\frac23\left(1 + (-\frac13) + (-\frac13)^2 + (-\frac13)^3 + ...\right)
$$
Now just use the formula:
$$1 + x + x^2 + x^3 + \dots = \frac1{1-x}$$
| {
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"timestamp": "2023-03-29T00:00:00",
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How to solve $\int \frac{\,dx}{(x^3 + x + 1)^3}$?
How to solve $$\int \frac{\,dx}{(x^3 + x + 1)^3}$$ ?
Wolfram Alpha gives me something I am not familiar with. I thought that the idea was using partial fractions because $x^3$ and $x$ are bijections, there must be a real root but it seems that Wolfram Alpha is using... | Hint: the polynomial $x^3+x+1$ has one real root, say $\alpha$. Then $x^3+x+1=(x-\alpha)(x^2+\alpha x+\alpha^2+1)$ and then apply integration techniques of rational expressions of polynomials with repeated factors, see for example https://math.la.asu.edu/~surgent/mat271/parfrac.pdf
| {
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Finite groups of which the centralizer of each element is normal. Recently I noticed that if $G$ is a finite group and $g \in G$ for which the centralizer $C_G(g)$ is a normal subgroup, all of the elements of the conjugacy class $g^G$ commute with each other, and hence their product is a element of the center $Z(G)$ of... | I believe the comment by James is correct, these groups are precisely the $2$-Engel groups.
Claim: The following statements are equivalent for a group $G$.
*
*Every centralizer in $G$ is a normal subgroup.
*Any two conjugate elements in $G$ commute, ie. $x^g x = x x^g$ for all $x, g \in G$.
*$G$ is a $2$-Engel gro... | {
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If Y dominates X and Y is a CW complex, then X has the homotopy type of a CW complex
Let $f\colon X \to Y$ and $g \colon Y \to X$ be maps such that $g \circ f \simeq \mathrm{id}_X$, and suppose $Y$ ix a CW complex. Then show that $X$ has the homotopy type of a CW complex
This is an exercise in May's Concise Course i... | See Theorem 3.6.3 here. The proof is quite long (4.5 pages) and would not be appropriate for reproduction at MSE. (I am well-aware of and, in general, agree with, the policy that one should not provide "link only" answer. However, in this case, link-only seems to be the only reasonable option.)
Edit: See also Proposit... | {
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Coproduct of $(0,1)$-Algebras I am trying to find the coproduct of $(\mathbb {Z},0,+1) $ with itself in the category of $(0,1) $-Algebras. Finding $\mathbb {N}\sqcup\mathbb {N} $ was easy, since $\mathbb{N} $ is initial. But I don't know how this coproduct looks in general.
| Coproducts can be computed by means of generators and relations. In this case, it is not hard to see that the coproduct in question is two copies of $\mathbb{Z}$ glued along the non-negative integers, i.e. the algebra whose underlying set is $(\mathbb{Z} \times \{ 0, 1 \}) / \sim$ where $(n, m) \sim (n', m')$ if and on... | {
"language": "en",
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Some infinite dimensional linear algebra, kernels of linear maps I'm studying functional analysis (namely weak convergence) and need to prove the following result: if $f,f_1,\ldots f_n$ are some linear maps $X\to \mathbb{C}$, where $X$ is a vector space over $\mathbb{C}$ then the inclusion $\bigcap\mathrm{Ker}f_i\subse... | The map $(f_1,\dots,f_n):V\to\mathbb C^n$ factors through the injective map $F:V/\bigcap\mathrm{Ker}f_i\to\mathbb C^n$. Since $\bigcap\mathrm{Ker}f_i\subset \mathrm{Ker}f$, the map $f:V\to\mathbb C$ factors through $\tilde f:V/\bigcap\mathrm{Ker}f_i\to\mathbb C$. Since $F$ is injective, we can extend the linear form $\... | {
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Stalk is a local object of a sheaf Let $X$ be a topological space. Let $\mathcal{F}$ be a sheaf on $X$. The stalk is the direct limit
$$ \mathcal{F}_x = \lim_{\underset{V \ni x}{\longrightarrow}} \mathcal{F}(V) $$
Let $U \subset X$ be an open subset that contains $x \in X$. Then $$(\mathcal{F}|_U)_x \cong \mathcal{F}_x... | First, it's confusing that you use $U$ twice, so I'm replacing your second $U$ with $V$. You can use the Yoneda lemma: maps out of the limit are the same as maps out of all $F(U)$ with compatibilities. This is by definition of limit if you like. The functor I mean here is the functor taking an object $X$ to the set ... | {
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How to work out the formula that connects several numbers I have an interesting problem. Say I have lots of datasets like this:
a = 21
b = 23
c = 58
d = 498
etc (lots of other values)
X = 85
I need to find the formula that derives X from a, b, c, d etc, with the added complication that I don't know whether all of the... | If you think there is a linear relationship between the $a, b, c$, etc., and $x$, then you could find the least-squares solution to the system of equations $\mathbf {Ay = X}$. The matrix $\mathbf A$ will consist of rows of the form $[a_i\ b_i\ c_i \ldots]$, and $\mathbf X$ is a column vector containing the values $x_i$... | {
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Simple (not for me) combinatorics question There are four balls of different colors, and four boxes of colors, same as those of the balls.What are the number of ways, in which, the balls, one each in a box, could be placed, such that a ball does not go to a box of its own color.
I name the balls A,B,C,D.
Corresponding... | What you're looking for is called a derangement. There exists a general formula for derangement of $n$ objects. Here's how you get it:
First determine all the possibilities. In this case, $$T=n!$$
Now determine how many choices have atleast $1$ object going into its designated spot. It's $${n \choose 1}(n-1)!$$
Now su... | {
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Probability - Conditional statements with union and intersection There are two flowers, $A$ and $B$.
The probability that each one is pollinated is $0.8$.
The probability that $B$ is pollinated given $A$ is pollinated is $0.9$.
What is the probability that:
a) both flowers are pollinated?
b) one or the other or both i... | Note that $A$ and $B$ are not independent so $P(A\cap B)\not=P(A) P(B)$.
Rather, $P(A\cap B)=P(A) P(B\vert A)$.
This should give you (a) and then (b) and (c) just need to be corrected accordingly. The same reasoning applies to (d).
| {
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Is there a plane that passes through a pair of lines that have no points in common? I'm reading a book on geometry in Spanish by Ana Berenice Guerrero (see here, p. 18,19). So, there's this theorem that says that given a pair of lines with no points in common there's only one plane that have both of them.
I have read ... | Well. It is only possible to construct a View where the two lines appear to be parallel, and there exists only one such view. So there is some truth in it .. However if you were to have a PLANE.. it will have only 2 points as the two lines are skewed ... so in conclusion, view = possible, Plane = impossible.
| {
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Rigorous method for showing this limit
Prove the following limit; $$\lim_{x \to +\infty}\dfrac{\exp(x^2)}{10^{|x|}}$$
The limit of this is $=\infty$
But what is the best method to show this:
L'Hospital doesn't seem very helpful here.
i.e as $x\to\infty$ the function becomes $\dfrac{\exp(x^2)}{10^{x}}$.
This is a $\... | Hint: Note that $a^b=e^{b\ln a}$, so $10^{|x|}=e^{|x|\ln 10}$. Thus,
$$\lim_{x\to+\infty}\dfrac{e^{x^2}}{10^{|x|}}=\exp\left(\lim_{x\to+\infty}\left(x^2-|x|\ln 10\right)\right)$$
| {
"language": "en",
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Cosets for lie groups I am looking for a general way of determining cosets for $(G\times H)/H$, where $G$ and $H$ are Lie groups.
For example what are the cosets $(SU(3)\times SU(2))/SU(2)$. Is there a general method of determining it? (I am actually trying to use it to find the triviality of a fiber bundle whose base ... | The way the question is phrased is a little ambiguous. How does $H$ sit inside $G\times H$ as a subgroup? If it sits inside it in the canonical way as $\{1\}\times H$,
then the space of cosets is canonically isomorphic to $G$ and each coset is simply
$G \times \{g\}$ for $g$ an element of $H$. I.e., for each element... | {
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Why does $\frac{(x^2 + x-6)}{x-2} = \frac{(x+3)(x-2)}{x-2}$? I'm not the best at algebra and would be grateful if someone could explain how you can get from,
$$\frac{x^2 + x-6}{x-2}$$
to,
$$\frac{(x+3)(x-2)}{x-2}$$
| Key fact: Knowing the roots of a polynomial (where the polynomial equals zero), let us factor it.
So if $n$ and $m$ are two roots of the quadratic $ax^2+bx+c$, then we can factor it as $$ax^2+bx+c=a(x-n)(x-m).$$ The roots of a quadratic can be determined using the quadratic formula: $$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}... | {
"language": "en",
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Counting when there are two inclusive conditions.
How many 8-bit sequences begin with $101$ OR (inclusive) have a $1$ as
their fourth bit?
For the first condition, we need only to decide the values of the $5$ other bits, so there are
$$2^5$$
sequences starting with $101$.
For the second condition, we have to decide... | The big problem here: if $A_1$ is the set of sequences which satisfy the first property and $A_2$ is the set of sequences which satisfy the second, then (exactly as you suggest), $\lvert A_1\rvert+\lvert A_2\rvert$ over-counts.
However, it over-counts in a very predictable way: namely, any sequence which is in exactly ... | {
"language": "en",
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proof of special trig limits I'm trying to prove a special trig limit, which is...
$$\lim_{x \rightarrow 0} \frac{1 - \cos{x}}{x}=0$$
So far, this is what I have (and I'll explain where I'm confused)
Using the squeeze theorem,
$h(x) \leq f(x) \leq g(x)$
$$-x^2 + 1 \leq \cos{x} \leq 1 $$
$$-x^2 + 1 - 1 \leq \cos x - 1 \... | It's not exactly correct to go from
$$0\le1-\cos x\le x^2$$
to
$$0\le{1-\cos x\over x}\le x$$
because dividing through by $x$ reverses the inequalities if $x$ is negative. What is OK is to conclude
$$0\le\left|{1-\cos x\over x}\right|\le |x|$$
The squeeze theorem still applies.
However, where did the opening inequalit... | {
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Integration by parts, natural logarithm I am having A LOT of problems with this one equation, could anyone help me? I know the answer, I just don't understand how to get there.
$$\int x^3 e^{x^2} dx$$
There's the equation, and the answer is:
$$\int e^{x^2} x^3 dx = \frac 1 2 e^{x^2} (x^2 - 1) + \text{ constant} $$
I ... | Let $u=x^2$ and $dv=xe^{x^2}\,dx$. Then $du=2x\,dx$ and $v$ can be taken to be $\frac{1}{2}e^{x^2}$. So we arrive at
$$\frac{1}{2}x^2e^{x^2} -\int xe^{x^2}\,dx.$$
This last integral is straightforward, indeed has already been done.
Remark: Even though integration by parts works directly, the preliminary substitution $... | {
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How to find the following sum? $\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{4n + 9}} - \frac{1}{{4n + 7}}} \right)} $ I want to calculate the sum with complex analysis (residue)
$$
1 - \frac{1}{7} + \frac{1}{9} - \frac{1}{{15}} + \frac{1}{{17}} - ...
$$ $$
1 + \sum\limits_{n = 0}^\infty {\left( {\frac{1}{{4n + 9}} -... | Here is a way to evaluate your series with the method of residues.
$$\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{4n + 9}} - \frac{1}{{4n + 7}}} \right)} = \sum_{n=2}^{\infty} \left(\frac{1}{4n+1} - \frac{1}{4n-1}\right) =\sum_{n=2}^{\infty} \frac{-2}{(4n)^2 - 1} = f(n)$$
Consider a function
$$ f(z)= \frac{-2}{(4z)^... | {
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Normal Abelian Subgroup does not imply Abelian Quotient Group I'm a bit confused and just need some clarification about what I am missing in this:
I have $S_4$ with normal subgroup $N=\lbrace(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\rbrace$.
I know that $N$ is normal (and abelian), which means $gN=Ng, \forall g\in G$, so to... | Recall that the group operation on $\frac{G}{N}$ is $(g_1N)(g_2N) = g_1g_2N$ if you use left cosets or $(Ng_1)(Ng_2) = Ng_1g_2$ if you use right cosets. Knowing that $gN = Ng$ does not imply $g_1g_2N = g_2g_1N$ but rather that $g_1g_2N = Ng_1g_2$.
| {
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Is it true that $X\simeq S^2\vee S^2$? Let $X$ be the quotient space of $S^2$ under the identifications $x\sim -x$ for every $x$ in the equator $S^1$. Is it true that $X\simeq S^2\vee S^2$, that is, $X$ is homeomorphic to $S^2\vee S^2$?
| You can consider the cellular homology with $\Bbbk=\Bbb Z/2\Bbb Z$-coefficients. Both spaces are CW complexes, the quotient space $X=S^2/\sim$ has a CW complex structure with one $0$-cell, one $1$-cell and two $2$-cells attached to the one skeleton (a circle) by degree $2$ maps, while the wedge sum $Y=S^2\vee S^2$ has ... | {
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Write this surd in its simplest form.
Express $\dfrac{1}{2+ \sqrt3}$ in its simplest form.
NB: The textbook has the answer as $2 - \sqrt3$ but I can't see how that was achieved.
I tried $\dfrac{1}{2} + \dfrac{1}{\sqrt3}$ and multiplying the top and bottom by $\sqrt3 $ to get $\dfrac{1}{2} + \dfrac{\sqrt3}{3}$ so far.... | Any time you are simplifying an expression like $$\frac{c}{a \pm \sqrt{b}},$$ multiply it with $$\frac{a\mp \sqrt{b}}{a\mp \sqrt{b}}$$ which gives you $$\frac{c(a\mp \sqrt{b})}{(a\pm \sqrt{b})(a\mp \sqrt{b})} = \frac{c(a\mp \sqrt{b})}{a^2 - b}$$ which has no more square roots in the denominator.
| {
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Reduction of structure group of real vector bundles I'm trying to show that the structure group of real vector bundles can be reduced to the orthogonal group. This is an exercise in Differential Forms in Algebraic Topology by Bott and Tu. The book gives a hint by asking to show that the general linear group is the dire... | What you are looking at is actually the Gram–Schmidt process, this gives the desired decomposition of a matrix $g \in GL(n)$ into $\lambda u \lambda^{-1}$, where $u \in O(n)$ in a natural way, i.e. the correspondence does not depend on choices and is smooth for a smooth family of matrices $g: U \to GL(n)$.
| {
"language": "en",
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Question about implication with antecedent $P(x)$ of $x$ that is false for all values of $x$. Suppose $x \in R^+$ and we want to prove the implication $x < 0 \Rightarrow P(x)$, where $P(x)$ is some statement of $x$.
How should one tackle this situtation ?
Normally one should prove the implication in the case that the ... | Since $x\in \mathbb R^+$, $$x \lt 0 \rightarrow P(x)$$ is always true, because an implication is always true when the antecedent if false.
Remember that the only situation in which an implication $a \rightarrow b$ is false is when $a$ (antecedent) is true AND $b$ (consequent) is false. If needed, refer to the truth-ta... | {
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Simple equation for $x$ but getting no proof.
Show that there is at least one real value of $x$ for which
$$x^{1/3} + x^{1/2} = 1$$
I did draw the graphs of $x^{1/3}$ and $1-x^{1/2}$ and showed that they met at a point, but I don't think it's a good algebraic proof.
How should I proceed after substituting $x$ for $z^... | Hint:
Consider the function:
$f(x)=x^{1/3}+x^{1/2}$
$f(0)=?$
$f(1)=?$
| {
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Converting to a partial fraction. I'm trying to do an inverse Laplace operation on $I(s)$ shown below but I'm struggling on finding what $A$ & $C$ are on the partial fraction and how to do it. I calculated what $B$ equals by making $s=0$.
$$I(s)=\frac{1}{s^2(R+L)}=\frac{A}{s}+\frac{B}{s^2}+\frac{C}{R+Ls} \\
1=As(R+Ls)+... | Set $s$ to $-\frac{R}{L}$, eliminating $A$ and $B$ to find $C$ so that
$$C\frac{R^2}{L^2}=1\Rightarrow C=\left(\frac{L}{R}\right)^2$$
Note that the coefficient of $s^2$ is zero in your second equation:-
$$AL+C=0\Rightarrow A=-\frac{C}{L}=-\frac{L}{R^2}$$
| {
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Puzzle about 3 boxes with 2 balls inside (black or white) with mixed labels on them We have 3 boxes. In every one there are 2 balls. One of them has 2 black balls, second 2 white balls, third black and white ball.
On every box is a right plate(label): BB,WW,BW. Unfortunatelly somebody mixed the plates and now only NON... | Easy. Pick from BW box, what ever color you get (let's say W) that box had to hold 2 of, so it WW. You know that the BB box must hold the the BW balls because it can't hold the BB balls. Only thing left is is the WW box and the BB balls.
| {
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"timestamp": "2023-03-29T00:00:00",
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deriving second order transfer function from spring mass damper system.. I am having a hard time understanding how a differential equation based on a spring mass damper system $$ m\ddot{x} + b\dot{x} + kx = 0$$ can be described as an second order transfer function for an inpulse response, which looks something like thi... | If you want to derive the transfer function out of a differential equation, first you need to select "input" and "output" of the system. In your system I believe the equation is
$$ m \ddot{x} + b\dot{x} + kx = ku $$
where $u$ is the input and $x$ is the output. If you select all initial conditions as $0$, then you can ... | {
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Can I compute this integral analytically? I will give a small background and explain the variables and the system first. I have two images which are observed and are constant and we can treat them as continuous functions and I will call them $r$ and $f$. In my problem, I am trying to find a continuous transform (which ... | Yes, it can be solved using double integration. For simplicity, we integrate $\int_{ - \infty }^{\infty} {{e^{ - {x^2}}}dx}$. Consider the circular disc ${D_b}:{x^2} + {y^2} \le {b^2}$ with polar coordinates $(r,\theta)$ in the set $\Gamma :0 \le \theta \le 2\pi ,0 \le r \le b
$. Therefore,
\begin{align}
\int_{{D_b}}... | {
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Visualizing the square root of 2 A junior high school student I am tutoring asked me a question that stumped me - I was wondering if anyone could shed some light on it here.
We were talking about how the square root of 2 is an irrational number, and that means you can't write that value as the ratio of two integers. Th... | Here is another way of approximating the square root of two by rational numbers which doesn't depend on the decimal system.
Suppose $p^2-2q^2=\pm 1$ so that $\left(\cfrac pq\right)^2=2\pm\cfrac 1{q^2}$, then the larger we can make $q$ the closer $\cfrac pq$ is to $\sqrt 2$.
Consider now $(p+2q)^2-2(p+q)^2=p^2+4pq+4q^2-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/818845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 10,
"answer_id": 1
} |
limits of function without using L'Hopital's Rule $\mathop {\lim }\limits_{x \to 1} \frac{{x - 1 - \ln x}}{{x\ln x+ 1 - x}} = 1$ Good morning.
I want to show that without L'Hopital's rule :
$\mathop {\lim }\limits_{x \to 1} \frac{{x - 1 - \ln x}}{{x\ln x + 1 - x}} = 1$
I did the steps
$
\begin{array}{l}
\mathop {\lim... | $
\displaylines{
\left\{ \begin{array}{l}
t = 1 + u \\
u \cong \ln t \\
\end{array} \right. \cr
\Rightarrow \cr
\mathop {\lim }\limits_{t \to 1} \left[ {\frac{{\left( {1 + t} \right)\ln t}}{{t\ln t - t + 1}}} \right] = \mathop {\lim }\limits_{u \to 2} \left[ {\frac{{\left( {2 + u} \right)u}}{{\left( {1 +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/818908",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Show that $f:\mathbb{R}^+ \longrightarrow \mathbb{C}^\times$ defined by $f(x)=e^{ix}$ is a homomorphism Can someone please verify my proof?
Show that $f:\mathbb{R}^+ \longrightarrow \mathbb{C}^\times$ defined by $f(x)=e^{ix}$ is a homomorphism, and determine its kernel and image.
Let $x$ and $y$ be arbitrary elements... | This question probably originates from Ex 2.4.6 of the book Algebra by Michael Artin. I believe the notation there
$f:\mathbb{R}^+ \longrightarrow \mathbb{C}^\times$
means
$f:(\mathbb{R},+) \longrightarrow (\mathbb{C},\times)$
In particular, it doesn't mean the domain is limited to only positive real numbers.
Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/818984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Vector Cross Products And Position Vectors I just realised that i've made a silly mistake on the past few practice exam papers, so I would really appreciate it if you could take a look at how i'm solving this kind of problem so that I can be sure I have sorted it. Essentially, the mistake I had made was solving the pro... | Make sure you completely understand dot products and cross products of vectors.
Your solution for part A is incorrect because you used incorrect definitions.
$\vec{BA}\cdot \vec{BC}$ is equal to $|\vec{BA}| |\vec{BC}|\cos\theta$, not $|\vec{BA}| |\vec{BC}|\sin\theta$.
I haven't done the calculation for part B, but your... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/819085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Algebra Difference in Roots Question.
Let D be the absolute value of the difference of the 2 roots of the equation 3x^2-10x-201=0. Find [D]. [x] denotes the greatest integer less than or equal to x.
I came across this question in a Math Competition and I am not sure how to solve it without using a calculator, since c... | $$3x^2-10x-201=0\\ \iff x^2-\frac{10}3x-67=0$$
Assuming the quadratic formula is available to use,
$$x=\frac{10}6\pm\frac{\sqrt{\frac{100}9+4\cdot 67}}2\\=\frac 53\pm\sqrt{\frac{25}9+67}$$
So the square root term is greater than $\sqrt{64}$ but less than $\sqrt{81}$ and is therefore between $8$ and $9$ in value, and th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/819149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Why do negative exponents work the way they do? Why is a value with a negative exponent equal to the multiplicative inverse but with a positive exponent?
$$a^{-b} = \frac{1}{a^b}$$
| Think of it this way: exponentiation is equivalent to repeated multiplication, in the sense that, for example, $3^4=3\times3\times3\times3$; so a multiplication repeated a negative number of times should use the multiplicative inverse, division. Therefore, a negative exponentiation could be represented as a repeated di... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/819218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 8,
"answer_id": 7
} |
The order of elements in the quotient group Let $G$ be a group, $N$ a normal subgroup of $G$, $a \in G$, and let $k = o(a)$.
I don't understand why the order of an element in $G/N$ is not necessarily equal to the order of the "corresponding" element in $G$ (i.e, why it might be that $o(a) \neq o(aN)$).
My reasoning is ... | The best example to illustrate this is, I believe, the infinite cyclic group $\mathbb{Z}$. Here, every non-trivial element has infinite order. Now, consider a quotient group, for example $\mathbb{Z}/3\mathbb{Z}$. This quotient group is cyclic, of order three. Thus, every element has order three.
In your proof your issu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/819316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Constructing a sequence Given two distinct, positive real numbers, how can I use these two numbers (and their non-zero integer linear combinations) to construct a sequence converges to zero? The sequence can only be of the two original positive numbers, or their non-zero integer linear combinations.
| I misread the question, and thought the asker was just trying to show the sequence exists. Still, I'll leave the answer here since it is not totally trivial to show.
Call the two numbers $a$ and $b$. Let $c = \inf\{r: r > 0, r = ka + lb$ for some integers $k$ and $l\}$. It suffices to show that $c = 0$. Suppose $c$ we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/819408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Is this proof of $\sum_{i = 1}^n a_i^k \leq (\sum_{i = 1}^n a_i)^k$ correct? I came across the following proof, and although I believe the result, something seems fishy and I can't put my finger on it. The base case might not be enough, or we might have to consider various $k$ somewhere...or maybe I'm just paranoid ! ... | Alternately let $x_i = \dfrac{a_i}{\displaystyle \sum_{j=1}^n a_j}$, then $0 < x_i < 1$, and $\displaystyle \sum_{i=1}^n x_i = 1$. Thus we have:
$\displaystyle \sum_{i=1}^n x_i^k \leq \displaystyle \sum_{i=1}^n x_i = 1$ since $0 < x_i^k \leq x_i < 1$ for $\forall k \geq 1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/819478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$f$ is bounded $\iff$ $F/\log$ where $F(x)= \int_{[1,x]}f(t)/t \,dt$ Hi everyone I'm stuck with one exercise. This says the following:
Let $F(x)= \int_{[1,x]}f(t)/t \,dt$ where $f$ is a non-decreasing function. Show that $f$ is bounded $\iff$ $F/\log$ is also bounded on $[1,\infty)$.
($\Rightarrow$) Let $M$ be a n... | Show the contrapositive. If $f$ is not bounded, then for every $K \in (0,\infty)$, there is an $x\in [1,\infty)$ with $F(x) > K\log x$.
Given $K$, since $f$ is nondecreasing and unbounded, there is an $x_0 \in (1,\infty)$ such that $f(x) > 2K$ for all $x \geqslant x_0$. Now, for $x > x_0$, we have
$$F(x) = \int_1^x \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/819528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
associativity on elliptic curves -- Milne's proof In the proof that the group law on an Elliptic curve is associative, Milne (http://www.jmilne.org/math/Books/ectext5.pdf, page 28) sets up 3 cubics,
and claims that they all contain the $8$ points $O,P,Q,R,PQ,QR,P+Q,Q+R$ where $AB$ denotes the third point of intersectio... | I agree that this looks like a typo – or even two. Consider the illustration on that same page:
Apparently the last cubic should be
$$L(P,Q\color{red}{+}R)\cdot L(Q,R)\cdot L(P\color{red}{Q},O)=0$$
or something along these lines. It corresponds to the three horizontal lines in that illustration, just like the second c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/819624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Are numbers of the form $n^2+n+17$ always prime Someone claimed that a number, multiplied by the number after it plus 17 is always prime, and showed several cases. I'm not a complete amateur in Number Theory, and I know that $17*18+17=17*19$, so it does not work for $n\equiv0(\mod17)$ but does it always work for other ... | There are plenty of numbers besides multiples of $17$ that fail to give primes in that formula.
Even with "handicaps" like the one you give, there's just no polynomial that always gives primes. according to Mathworld, Legendre proved this long, long ago: http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html
See ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/819706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 3
} |
Differentiation Matrix for central-difference scheme? Central-difference scheme is defined to be:
$f'(x) = \frac{f(x+d(x)) - f(x-d(x)))} {2*d(x)} + O(d(x)^2)$
Assume periodic boundary conditions, so that: $f(n+1)=f(1)$
I understand how to find all the center values of the matrix, but what I don't get is the first and l... | $$
A =
\pmatrix{ 0 & 1 & 0 & & \dots & -1 \cr
-1 & 0 & 1 & & & \cr
0 & -1 & 0 & 1 & \ddots & \vdots \cr
\vdots & \ddots & & \ddots & & \cr
& & & -1 & 0 & 1 \cr
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/819800",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A fierce differential-delay equation: df/dx = f(f(x)) Consider the following set of equations:
$$
\begin{array}{l}
y = f(x) \\
\frac{dy}{dx} = f(y)
\end{array}$$
These can be written as finding some differentiable function $f(x)$ such that
$$
f^{\prime} = f(f(x))
$$
For example, say $y(0) = 1$. Then $\left. \frac{dy}... | Just playing around a bit
with JJacquelin's answer.
Regarding
$f(x)
=\left(\frac{1}{2}+ i\frac{\sqrt{3}}{2}\right)^{\frac{1}{2}- i\frac{\sqrt{3}}{2}}x^{\frac{1}{2}+ i\frac{\sqrt{3}}{2}}
$,
since
$\frac{1}{2}+ i\frac{\sqrt{3}}{2}
=e^{i\pi/3}
$
and
$\frac{1}{2}- i\frac{\sqrt{3}}{2}
=e^{-i\pi/3}
$,
this becomes
$\begin{ar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/819888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
} |
Show that no application $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, of $C^k$ class, $k \geq 1$ can be injective How can I proof this:
Show that no application $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, of $C^k$ class, $k \geq 1$ can be injective, i.e., there are $A,B \in \mathbb{R}^2$ such that $A \neq B$ and $f(A) = f(... | If $df =0$ everywhere, then $f(a, b)$ is constant and hence cannot be injective. On the other hand, if at some point $(a_0, b_0) \in \Bbb R$ we have $df(a_0, b_0) \ne 0$, we must also have
$f_a(a_0, b_0) = \dfrac{\partial f}{\partial a}(a_0, b_0) \ne 0 \tag{1}$
or
$f_b(a_0, b_0) = \dfrac{\partial f}{\partial b}(a_0, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/819977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Getting a "straight" in dice rolls Suppose that you have $k$ dice, each with $N$ sides, where $k\geq N$. The definition of a straight is when all $k$ dice are rolled, there is at least one die revealing each number from $1$ to $N$.
Given the pair $(k,N)$, what is the probability that any particular roll will give a ... | What about simply $\dfrac{^k\text{P}_n(k-n)!}{n^k}$, since $k \ge n$. If $k=n$ then we simply get $\dfrac{n!}{n^k}$?
Since there are only $n$ sides to the dice and if $k=n$, we are asking the total number of permutations of $n$ different values which is $^n\text{P}_n=n!$ divided by all the possible outcomes of $n^k$. I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/820015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
What is the best way to calculate log without a calculator? As the title states, I need to be able to calculate logs (base $10$) on paper without a calculator.
For example, how would I calculate $\log(25)$?
| In case anybody wondered why the algorithm given by Ezui actually works, here is a little algebra to explain why:
The algorithm states, that to find the base $10$ logarithm of $x$ one should repeatedly carry out the following three steps:
*
*$d=\max(n\mid 10^n\leq x)$, store $d$ as the next digit
*$y=x/10^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/820094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
"answer_count": 5,
"answer_id": 2
} |
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