Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Trigonometric Coincidence I Know that using Taylor Series, the formula of $\sin x$ is
$$x-x^3/3!+x^5/5!-x^7/7!\cdots,$$
and the unit of $x$ is radian (where $\pi/2$ is right angle).
However, the ratio of the circumference and the diameter of a circle is also $\pi$. Is it a coincidence? Or is there a proof?
| The facts you mention are definitely not a coincidence; they are a consequence of how things are defined, and partly a reason why they are defined so. (This is an incredibly vague statement, but the question itself is a little vague).
Some textbooks will define $\sin x$ by the series expansions you just mentioned, and ... | {
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Want to clarify whether I am correct or not, $\Phi(G) \subseteq \Phi(H)$? I Want you to clarify whether I am correct or not regarding following question. I will be thankful to you for telling me if I am wrong:
Let $G$ be finite group and $\Phi(G)$ denotes its frattini subgroup. Let $H$ be a subgroup of $G$ such that $\... | No. Take $G$ to be the dihedral group of order $8$. The Frattini subgroup is the unique normal subgroup of order $2$. But $G$ has two subgroups of order $4$ that are elementary abelian (and as such, they have trivial Frattini subgroup) and yet they contain $\Phi(G)$.
| {
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Three diagonals of a regular 11-sided regular polygon are chosen ; probability of parallelism Could someone help me with this?
Suppose P is an 11-sided regular polygon and S is the set of all lines that contain two distinct vertices of P. If three lines are randomly chosen from S, what is the probability that the chose... | But you are choosing three lines. The probability that you have calculated is for choosing a pair of parallel lines. The question asked is you randomly choose 3 lines and that it contain exactly one parallel pair. I would agree with Steven that the probability space is 55C3. Each edge will have 4 parallel lines and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/480612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Arithmetic progressions with coprime differences Suppose we have finite number $n \geqslant 2$ of arithmetic progressions $\{x \equiv r_1 \pmod {d_1}\}, \ldots ,\{ x \equiv r_n \pmod {d_n}\}$ such that $\gcd(d_1, \ldots, d_n) = 1.$
Is true that some pair of them has nonempty intersection? (I think, it's true).
| Counterexample: the three congruences $x\equiv 0\pmod{6}$, $x\equiv 1\pmod{10}$, $x\equiv 2\pmod{15}$.
One can produce similar counterexamples using any three distinct primes $p,q,r$ instead of $2,3,5$.
Remark: I notice that fedja gave a very similar counterexample in a comment.
| {
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A property of power series and the q-th roots of unity I'm trying to understand why if $ \displaystyle \sum_{n=0}^{\infty} a_{n}x^{n} = f(x) $, then $$ \sum_{n=0}^{\infty} a_{p+nq} x^{p+nq} = \frac{1}{q} \sum_{j=0}^{q-1} \omega^{-jp} f(\omega^{j} x)$$
where $\omega$ is a primitive $q$-th root of unity.
I'm assuming it... | In the right hand side of your expresion, remplace $f(\omega^j x)$ by $\sum_{k=0}^{\infty} a_k \omega^{jk}x^k $.
You will get :
$$\frac{1}{q}\sum_{j=0}^{q-1} \omega^{-jp}\sum_{k} a_k x^k \omega^{jk}$$
By interverting the sums (formally at least it has to be justified : see Jyrki Lahtonen's comment) you get :
$$\sum_{k... | {
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Calculate $ \lim_{x \to 4} \frac{3 -\sqrt{5 -x}}{1 -\sqrt{5 -x}} $ How evaluate the following limit?
$$ \lim_{x \to 4} \frac{3 -\sqrt{5 -x}}{1 -\sqrt{5 -x}} $$
I cannot apply L'Hopital because $ f(x) = 3 -\sqrt{5 -x} \neq 0 $ at $x = 5$
| Let $x=5\cos4y$ where $0\le4y\le\pi$
$x\to4\implies\cos4y\to\dfrac45$
But as $\cos4y=2\cos^22y-1, 2y\to\arccos\dfrac3{\sqrt{10}}=\arcsin\dfrac1{\sqrt{10}}$
$$F=\lim_{x\to4}\frac{3-\sqrt{5+x}}{1-\sqrt{5-x}}=\lim_{2y\to\arcsin\frac1{\sqrt{10}}}\dfrac{\dfrac3{\sqrt{10}}-\cos2y}{\dfrac1{\sqrt{10}}-\sin2y}$$
If we set $\ar... | {
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Applying measure zero definition to Cantor sets I just learned about the concept of measure zero in real analysis, i.e. the definition that a set in $\mathbb{R}^n$ has measure zero if for any $\epsilon$ it can be covered by countably many rectangles whose volume sum to $<\epsilon$.
I'm wondering if knowing only this, I... | Yes, it is enough. If you think about the elimination process to create the Cantor set, after the first stage you see that the Cantor set can be covered by two intervals of length $1/3$ and a bit. After the second stage of middle-third removal, you see that the Cantor set can be covered by four intervals of length $1/... | {
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$\iint_Ex\ dx\ dy$ over $E=\lbrace(x,y)\mid 0\le x, 0\le y\le 1, 1\le x^2+y^2\le 4\rbrace$ (Answer check and curious about alternative methods) $$\iint_Ex\ dx\ dy$$
$$E=\lbrace(x,y)\mid 0\le x, 0\le y\le 1, 1\le x^2+y^2\le 4\rbrace$$
Shape of region
Entirely in first quadrant of xy plane, between two circles, of r=1 an... | Let's use Stoke's Theorem:
\begin{align}
\iint_{E}x\,{\rm d}x\,{\rm d}y
&=
\iint_{E}\,{\partial\left(x^{2}/2\right) \over \partial x}\,{\rm d}x\,{\rm dy}
=
\iint_{E}\nabla\times\left({1 \over 2}\,x^{2}\,\hat{y}\right)\,\cdot\hat{z}
\,{\rm d}x\,{\rm dy}
=
{1 \over 2}\oint x^{2}\,\hat{y}\cdot{\rm d}\vec{r}
\\[3mm]&=
{1 \... | {
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Proof that $3 \mid \left( a^2+b^2 \right)$ iff $3 \mid \gcd \left( a,b\right)$ After a lot of messing around today I curiously observed that $a^2+b^2$ is only divisible by 3 when both $a$ and $b$ contain factors of 3. I am trying to prove it without using modular arithmetic (because that would be way too easy), but fi... | Given that 3 is a factor of A and of B.
If 3 is a factor of A then A is a multiple of 3, say A = 3k, k an integer.Then A²=9k² Analoguously for B² = 9m² So A²+B² = 9k²+9m²=9(k²+m²) and 3 is a factor of 9, k²+m² is another integer.
I think I am seriously overlooking something here? It can't be that easy...
| {
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Power correct notation Ok, I know this may sound dumb, but I am trying to understand which is the correct (most beauty) notation for the power function ${\rm pow}(f(x),n)$.
This is the correct one: $[f(x)]^n$
From trigonometry, where I was used to write $\cos^2x$, we get: $f^n(x)$
And from Bishop's Pattern Recognition ... | The most clear notation is certainly to write $$\left(f(x)\right)^n$$
This is because the notation $f^n(x)$ will frequently refer to the composition $f \circ f \circ ... \circ f$.
| {
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$\limsup$ of indicator function of converging sets Let $\{A_n\}_{n=1}^{\infty}$ be a sequence of bounded sets of $\mathbb{R}^n$, such that $A_n \rightarrow A \subset \mathbb{R}^n$.
Let $\mathbb{1}: \mathbb{R}^n \rightarrow \{0,1\}$ be the indicator function.
1) I am wondering if the following statement holds.
Assume $A... | The post to which you linked and the answer to it contain the answers to your questions: if $\langle A_n:n\in\Bbb N\rangle\to A$, then $\langle 1_{A_n}(x):n\in\Bbb N\rangle\to 1_A(x)$ for each $x\in\Bbb R^n$. It follows that
$$\limsup_{n\to\infty}1_{A_n}(x)=\lim_{n\to\infty}1_{A_n}(x)=1_A(x)$$
for all $x\in\Bbb R^n$. N... | {
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Proof that $\max(x_1,x_2)$ is continuous I haven't done a proof in years and I've become a little stuck on this, I'd appreciate it if somebody could tell me if I've approached the problem correctly...
Question: Prove that the following function $f(x_1; x_2) = \max[x_1; x_2]$, $x_1, x_2 \in \Bbb R$ is continuous over $... | If $x_2>x_1$ then $\max(x_1,x_2)=x_2$ and then it is continuous in that region. Similar analysis in the region $x_1>x_2$.
If $x_2=x_1$ then note that $|\max(y_1,y_2)-\max(x_1,x_2)|\le \max(|y_1-x_1|,|y_2-x_2|)$.
Given $\varepsilon>0$, choose $\delta=\varepsilon$. If $||(y_1,y_2)-(x_1,x_2) ||<\delta$ then we have
$$
|\... | {
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How to solve a system of trigonometric equations This question today appeared in my maths olympiad paper:
If $\cos x + \cos y + \cos z = \sin x + \sin y + \sin z = 0$, then, prove that $\cos 2x + \cos 2y + \cos 2z = \sin 2x + \sin 2y + \sin 2z = 0$.
Can anyone please help me in finding out the solution of this equation... | Putting $a=\cos x+i\sin x$ etc,
we have $a+b+c=0$
and $a^{-1}=\frac1{\cos x+i\sin x}=\cos x-i\sin x$ $\implies a^{-1}+b^{-1}+c^{-1}=0\implies ab+bc+ca=0$
$a^2+b^2+c^2=(a+b+c)^2-2(ab+bc+ca)=0$
Now, $a^2=(\cos x+i\sin x)^2=\cos^2x-\sin^2x+i2\sin x\cos x=\cos2x+i\sin2x$ which is a special case of de Moivre's formula
| {
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What can we conclude about the natural projection maps? In an arbitrary category, we have that even if $X$ and $Y$ have a product $X \times Y$, the natural projections needn't be epimorphisms.
Two questions:
*
*Are there (preferably simple!) conditions we can place on the category such that all the natural projectio... | You might like this theorem:
If $\pi _i:P {\to} A_i$ for $i\in I$ is a product and if $i_0 \in I$ is such that, for each $i \in I$, $hom(A_{i0} ,A_i)$ is not empty
then $\pi _{i0}$ is a retraction.
In general the $\pi _i$ form an extremal mono-source
See:
Abstract and concrete categories: the joy of cats. Proposition... | {
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Why is L'Hôpital's rule giving the wrong answer? Given the following limit:
$$
\lim_{x \rightarrow -1} \frac{x^5+1}{x+1}
$$
The solution using L'Hôpital's rule:
$$
\lim_{x \rightarrow -1} \frac{x^5+1}{x+1} = \begin{pmatrix} \frac{0}{0} \end{pmatrix} \rightarrow \lim_{x \rightarrow -1} \frac{5x^4}{1} = 5 \cdot (-1... | Your application of L'Hôpital's is fine and correct.
The problem is your evaluation/final conclusion...
Recall: $\quad$For $k \in \mathbb N,\;$ $(-1)^n = 1\;$ for (even) $\;n = 2k\,;\;$ $(-1)^n = -1\;$ for (odd) $\;n = 2k+1.\;$
That said, we have: $$5\cdot (-1)^4 = 5\cdot 1 = 5.$$
| {
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Minimal polynomial Let $V$ be the vector space of square matrices of order $n$ over the field $F$. Let $A$ be a fixed square matrix of $n$ and let $T$ be a linear operator on $V$ such that $T(B) = AB$. Show that the minimal polynomial for $T$ is the minimal polynomial for $A$.
Thank you for your time.
| You need to show that every polynomial that kills $A$ kills $T$. But then to show minimality, you need to show that every polynomial that fails to kill $A$ fails to kill $T$.
$$f(A)=0$$
$$f(T)B = \left(\sum_{k=0}^n c_k T^k \right)B = \sum_{k=0}^n c_k (T^k B).\tag1$$
$$
T^k(B) = T^{k-1}(T(B)) = T^{k-1}(AB) = T^{k-2} (A... | {
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Show that a retract of a contractible space is contractible. - Is this proof legit? I am wondering if my proof is correct? Thank you very much.
Show that a retract of a contractible space is contractible.
Given $X$ contracts to $x \in X$, we know there is a family of maps $f_t: X \to X, t \in I$, such that $f_0 = \ma... | There are still some inconsistencies in your text:
Consider a retract on $X$ to $A$, we know there is a map $r:X→A,$ such that
$r(X)=A,$ $r(A)=A.$
This still has to be corrected. It should better say: "$r:X\to A$ such that $r|_A=\Bbb I|_A$"
$\hat f_1(A)=a$ for any $a∈A$
That can be deleted. You already know that... | {
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Meaning of the Rank of a Map of Free Modules? I am reading the section on differentials in Eisenbud's book (Commutative Algebra), and I'm just wondering what he means in sentences like this one:
"Suppose that $J:R^t \rightarrow R^r$ is a map of free modules over a ring $R$ whose rank is less than or equal to $c$, as f... | Presumably it means the largest $k$ such that the induced map $\Lambda^k(J) : \Lambda^k(R^t) \to \Lambda^k(R^r)$ on exterior powers doesn't vanish. (This is a coordinate-free restatement of a condition on vanishing of minors.) At least, that would be my guess. Does the rest of the statement make sense with this interpr... | {
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Fibonacci using proof by induction: $\sum_{i=1}^{n-2}F_i=F_n-2$ everyone.
I have been assigned an induction problem which requires me to use induction with the Fibonacci sequence. The summation states:
$$\sum_{i=1}^{n-2}F_i=F_n-2\;,$$ with $F_0=F_1=1$.
I have tried going through the second one to see if it was right,... | (First an aside: the Fibonacci sequence is usually indexed so that $F_0=0$ and $F_1=1$, and your $F_0$ and $F_1$ are therefore usually $F_1$ and $F_2$.)
The recurrence might be more easily understood if you substituted $m=n-2$, so that $n=m+2$, and wrote it
$$\sum_{i=1}^mF_i=F_{m+2}-2\;.\tag{1}$$
Now see what happens i... | {
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For $N\unlhd G$ , with $C_G(N)\subset N$ we have $G/N$ is abelian Question is that :
let $N\unlhd G$ such that every subgroup of $N$ is Normal in $G$ and $C_G(N)\subset N$.
Prove that $G/N$ is abelian.
what could be the possible first thought (though for me it took some time :)) is to use that $C_G(N)$ is Normal subg... | For any $n\in N$, $g\in G$, there is an integer $k$ s.t. $gng^{-1}=n^k$ (as the subgroup generated by $n$ is normal). That implies that $ghn(gh)^{-1}=hgn(hg)^{-1}$ for all $g,h\in G$, $n\in N$, i.e. that $G/$(the kernel of the conjugation action of $G$ on $N$) is Abelian. The kernel is $C_G(N)$, i.e. $G/C_G(N)$ is Abe... | {
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definition of rectangle I am wondering whether it is fine to define a rectangle with right angles like Wikipedia's page of rectangle.
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral.
Because that the sum of the angles of a qu... | The definition is correct.
It is normal to define objects after a theorem, even if at first it seems counterintuitive. In advanced mathematics it happens all the time that a result (like a theorem) permits you to write a definition that otherwise wouldn't make sense.
There is no ambiguity because:
*
*Quadrilaterals ... | {
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To partition the unity by translating a single function I am trying to show that there exists a (real or complex-valued) function $\psi \in C^\infty(\mathbb{R}^n)$ having the following properties:
*
*The support of $\psi$ is contained in the unit ball $B(0, 1)$. EDIT As Daniel Fischer points out, this condition can... | Start with a continuous partition of unity on $\mathbb{R}$ obtained by translation of a single function, say
$$\psi_0(x) = \begin{cases} 1 &, \lvert x\rvert \leqslant \frac14\\ \frac32 - 2\lvert x\rvert &, \frac14 < \lvert x\rvert \leqslant \frac34\\
0 & \lvert x\rvert > \frac34\end{cases}$$
It is easy to check that $\... | {
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Cumulative distribution function, integration problem Given the continuous Probability density function $f(x)=\begin{cases} 2x-4, & 2\le x\le3 \\ 0 ,& \text{else}\end{cases}$
Find the cumulative distribution function $F(x)$.
The formula is $F(x)=\int _{ -\infty }^{ x }{ f(x) } $
My Solution
The first case is when $... | when x<2, it is $\int_{-\infty}^{x}0du=0$
when x>3, it is obviously 1. You should research the properties of $F(x)$
| {
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A problem on a triangle's inradius and circumradius . I'm trying to solve the following problem :
In $△ABC$, $AB = AC, BC = 48$ and inradius $r = 12$. Find the
circumradius $R$.
Here is a figure that I drew : ( note : it was not given in the question so there may be some mistakes )
I don't know how to solve it , am... | Let $M$ be the midpoint of $BC$, let $P$ be the point where the perpendicular from $O$ meets the side $AB$, and let $|PA|=:x$. Since the two tangent segments from $B$ to the incircle have equal length it follows that $|PB|=24$; therefore $|AB|=24+x$, and $|AO|^2= 12^2+x^2$. It follows that
$$(24+x)^2=24^2+\bigl(12+\sq... | {
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Getting the $x$-intercept of $f(x) = -16x^2 + 80x + 5$ $$f(x) = -16x^2 + 80x + 5$$
I need to find the bigger value of $x$ that makes $f(x) = 0$.
Naturally, I thought to do:
$$0=-16x^2+80x+5$$
and I applied the quadratic formula
$$0=\frac{-80\pm\sqrt{6080}}{-32}$$
but the answer doesn't seem like it would be correct. D... | First of all, please remember that the $x$-intercept is where the graph $y = \operatorname{f}(x)$ meets the $x$-axis. If you're not plotting a graph then it doesn't make sense to talk about $x$- and $y$-intercepts.
You're looking for the solutions to the equation $\operatorname{f}(x)=0$.
If $\operatorname{f}(x)=-16x^2+... | {
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Number of divisors of $9!$ which are of the form $3m+2$ Total number of divisors of $9!$ which are is in the form of $3m+2$, where $m\in \mathbb{N}$
My Try: Let $ N = 9! = 1\times 2 \times 3 \times 2^2 \times 5 \times 2 \times 3 \times 7 \times 2^3 \times 3^2 = 2^7 \times 3^4 \times 5 \times 7$
Now If Here $N$ must b... | Hint: If the factor is of the form $3m+2$, then the prime factorization must be of the form
$$ 2^a \times 3^0 \times 5^b \times 7^c, $$
where $a+b \equiv 1 \pmod{2} $ and $ c= 0$ or $1$.
Count the number of possibilities.
| {
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Easy way to find the streamlines In a textbook, this problem appears:
Find the streamlines of the vector field
$\mathbf{F}=(x^2+y^2)^{-1}(-y\hat{x}+x\hat{y})$.
The system we need to solve, I suppose, is:
$\dfrac{dx}{d\tau}=\dfrac{-y}{x^2+y^2}$
$\dfrac{dy}{d\tau}=\dfrac{x}{x^2+y^2}$
This is a text which just introd... | Hint: divide side by side the two equations (for example the second by the first), obtaing$$\frac {dy}{dx}=-\frac x{y}$$The solutions are$$x^2+y^2=c \quad (c>0)$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Given mean and standard deviation, find the probability Lets say that you know the mean and the standard deviation of a regularly distributed dataset. How do you find the probability that a random sample of n datapoints results in a sample mean less than some x?
Example- Lets say the population mean is 12, and the stan... | If you mean "normally distributed", then the distribution of the sample mean is normal with the same expected value as the population mean, namely $12$, and with standard deviation equal to the standard deviation of the population divided by $\sqrt{40}$. Thus it is $4/\sqrt{40}\approx0.6324555\ldots$. The number $10$ ... | {
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"url": "https://math.stackexchange.com/questions/482530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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New & interesting uses of Differential equations for undergraduates? I'm teaching an elementary DE's module to some engineering students. Now, every book out there, and every set of online notes, trots out two things:
*
*DE's are super-important, vital, can't live without 'em, applications in every possible branch ... | You may find this interesting that the ODE theory is getting involved well in studying Avalanches. See here, here and here for example.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/482659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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”figure 8” space embedded in S2 Let M3 be the 3-manifold defined as the quotient space of I × S2 by the
identification {0} × {x} s {1} × {Tx}, where T : S2 → S2 is a reflection
through a plane in R3. Find π1(M) and π2(M).
| The universal cover is $\tilde M=\mathbb R\times S^2$: your manifold is $(\mathbb R\times S^2)/\mathbb Z$, with $n\in\mathbb Z$ acting by $n\cdot (t,x)=(t+n,T^nx)$. As the result, $\pi_1(M)=\mathbb Z$, $\pi_2(M)=\pi_2(\tilde M)=\mathbb Z$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to calc $\log1+\log2+\log3+\log4+...+\log N$? How to calculate $\log1+\log2+\log3+\log4+...+\log N= log(N!)$? Someone told me that it's equal to $N\log N,$ but I have no idea why.
| A small $caveat$: "someone" is wrong: try, for example, with $N=2$; then
$$2\log 2=\log 2^2,$$
while
$$\log1+\log 2=\log 2!=\log 2.$$
For more details on the relationship between the 2 logarithms, I refer to the comments under the OP.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove $m^*(A) = m^*(A + t)$ Define $m^*(A) = \inf Z_A$ as the outer measure of $A \subseteq \mathbb{R}$ where
$$Z_A = \left\{\sum_{n=1}^{\infty}|I_n| : I_n \text{ are intervals}, A \subseteq \bigcup_{n=1}^{\infty}I_n\right\} $$
We want to show $m^*(A) = m^*(A + t)$. It will suffice to show $Z_A = Z_{A + t}$. So, p... | You may want to explain why it suffices to show $Z_A = Z_{A+t}$.
The argument for $Z_A \subseteq Z_{A+t}$ looks good (you may want to use the subset notation to make what you're doing clearer, but I don't think it's necessary).
The argument for $Z_{A+t} \subseteq Z_A$ uses both $J_n$ and $I_n$; I think you should just ... | {
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Numbers divisible by the square of their largest prime factor Let $p(n)$ be greatest prime factor of $n$, denote $A=\{n\mid p^2(n)\mid n,n\in \mathbb N\}.$
$A=\{4,8,9,16,18,25,27,32,36,49,50,\cdots\},$ see also A070003.
Define $f(x)=\sum_{\substack{n\leq x\\n\in A}}1.$ Erdős proved that
$$f(x)=x \cdot e^{-(1 + o(1))\sq... | What follows is self contained proof that $$f(x)=\sum_{\begin{array}{c}
n\leq x\\
n\in A
\end{array}}1 \ll x e^{-c\sqrt{\log x}}.$$ If you work more carefully with the friable integers, you can recover Erdős' result.
Proof. We may write $$\sum_{\begin{array}{c}
n\leq x\\
n\in A
\end{array}}1=\sum_{\begin{array}{c}
n... | {
"language": "en",
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Differential equations: solving separable equation Solve the separable equation $y' = (x-8)e^{-2y}$ satisfying the initial condition $y(8)=\ln(8)$. I can not figure this out I am not sure what I am doing wrong.
| We have $$\frac{dy}{dx}=(x-8)e^{-2y}\implies e^{2y}dy=(x-8)dx$$
Integrating either sides
$$\frac{e^{2y}}2=\frac{x^2}2-8x+C\implies e^{2y}=x^2-16x+2C$$ where $C$ is an arbitrary constant
Putting $x=8, y=\ln8, e^{2\ln 8}=8^2-16\cdot8+2C$
$2C-64=(e^{\ln 8})^2=8^2\implies 2C=64+64$
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove that roots are real I am stuck with this equation, I need to prove the roots are real when $a, b, c \in R$
The equation is $$(a+b-c)x^2+ 2(a+b)x + (a+b+c) = 0$$
If someone could tell me the right way to go about this, so I can attempt it.
Thank you
EDIT: I have made an error in the question. I have now corrected... | We look at the discriminant of the the polynomial, which for a quadratic $ax^2 +bx +c$ is $b^2 -4ac$, plugging the values in for our polynomial gives
$$\Delta = 4(a+b)^2-4(a+b-c)(a+b+c)\\
= 4[(a+b)^2 - (a+b)^2+c^2]\\
= 4c^2$$
Since the square of a real number is positive, we know that the roots must be real, by lookin... | {
"language": "en",
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How to prove $n ≡ n_0 + n_1 + \dots +n_k \pmod{b-1}$ I am trying to prove this statement, where $n$ has base $b$ representation, which can be understood easily using this example:
In base $10$, mod $9$ of any number can be found by adding up its digits and doing the mod $9$ of that sum.
It's been a while since I've d... | Note that $n = \sum_{i=0}^kn_ib^i$ and $b \equiv 1 \operatorname{mod} (b - 1)$. Now use the fact that if $a \equiv b \operatorname{mod} m$ and $c \equiv d \operatorname{mod} m$, then $a + c \equiv b + d \operatorname{mod} m$ and $ac \equiv bd \operatorname{mod} m$; in particular, $a^n \equiv b^n \operatorname{mod} m$.
| {
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Derivative $\Delta x$ and $dx$ difference This may seems like a dummy question but I need to ask it.
Consider the definition of derivative:
$$\frac{d}{dx}F(x) = \lim_{\Delta x->0}\frac{F(x+\Delta x) - F(x)}{\Delta x} = f(x)$$
Also:
$$f(x)\Delta x = F(x+\Delta x) - F(x) \tag{When $\Delta x$ gets closer to $0$}$$
I can a... | Your question is very good. There's something called the "non-standard" numbers. Trying to define them, we would have the set $$\{ \alpha, \text{such that } 0 < \alpha < x, \forall x \in \mathbb{R}\}$$
What happens, is that $\mathrm{d}x$ is in that set, while $\Delta x$ isn't. For instance, let's differentiate $y = f(... | {
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If $x$ is real and $x + \frac1x$ is rational, show by strong induction that $x^n + \frac{1}{x^n}$ is rational for all $n$. Suppose that $x$ subset of Real numbers such that $x + \frac{1}{x}$ is subset of Rational numbers. Using strong induction, show that for each $n$ subset of Natural numbers, $A_n = x^n + \frac{1}{x^... | If you need to use strong induction: Using the hint $$ A_1 A_n = x^{n+1}+\frac{1}{x^{n+1}} +x^{n-1}+\frac{1}{x^{n-1}} = A_{n+1} + A_{n-1}. $$ You want to rearrange this for an expression in $A_{n+1}$, the answer should follow from strong induction.
| {
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Prove that if $\theta$ is an angle with $\cos(2\theta)$ irrational then $\cos \theta$ is also irrational Prove that if $\theta$ is an angle with $\cos(2\theta)$ irrational then $\cos \theta$ is also irrational. (hint: recall that $\cos(2\theta)=2\cos^2(\theta)-1$ )
| If $x$ is some rational number, what can you say about $2x^2 - 1$?
Using this reasoning, suppose $\cos \theta$ were rational (even though it isn't). Then what would you know about $\cos 2\theta$? Why would this be a problem?
Conclude that since $\cos \theta$ being rational results in a problem, $\cos \theta$ must be ... | {
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If $A \subseteq B$, is it true that $C \cap A \subseteq C \cap B$? If $A \subseteq B \implies C \cap A \subseteq C \cap B$ ?
Let $x \in C \cap A \implies x \in C$ and $x \in A$ $\implies$ $x \in C$ and $x \in B$ $\implies x \in C \cap B$
is this valid?
| Not quite, since your definition of intersection isn't correct.
If $x \in C \cap A$, then $x \in C$ and $x \in A$. If $x \in A$, then $x \in B$, so $x \in C$ and $x \in B$, so $x \in C \cap B$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is $D^{n+1}/S^{n} = S^{n+1}$ true? I went to my first lecture in Algebraic Topology and managed to get really confused. It seems like they assumed that the following statement was "obvious":
$D^{n+1}/S^{n} = S^{n+1}$
Where $D^{n}$ is the unit disk/ball in $\mathbb{R}^{n}$ and $S^{n}$ is the unit sphere in $\mathbb{... | Here's one way to set up a concrete homeomorphism.
The sphere $S^{n+1}$ is homeomorphic to the $1$-point compactification of $\mathbb{R}^{n+1}$. This is witnessed by the stereographic projection maps. Take the disc $D^{n+1} \subset \mathbb{R}^{n+1}$, map the inner open disc to $\mathbb{R}^{n+1}$ (you have probably seen... | {
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positive characteristic and multiple roots I can't understand a proof in Milne, proposition 2.12 at pag 29. In particular, i can't prove the implication $c)\Rightarrow d)$ where:
c) $F$ has characteristic $p\neq 0$ and $f$ is a polynomial in $X^p$ ;
d) all the roots of $f$ are multiple
Suppose $f(X)=g(X^p)$ and $g(X)=... | If they aren't already in $K$ (as when $F$ hence$ K$ are finite), the $\alpha_i$ are in some extension $L$ or $K$.
For example, pick $F = \Bbb F_p(X^p)$. The polynomial $Y^p - X^p$ in $F[Y]$ has $p$ repeated roots in the extension $L = \Bbb F_p(X)$ of $F$ since $Y^p - X^p = (Y-X)^p$
| {
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Assuming there exist infinite prime twins does $\prod_i (1+\frac{1}{p_i})$ diverge? Assume there are an infinite amount of prime twins. Let $p_i$ be the smallest of the $i$ th prime twin. Does that imply that $\prod_i (1+\frac{1}{p_i})$ diverges ?
| Based on André Nicolas hint I realized :
$\prod_{i=1}^k (1+\dfrac{1}{p_i}) < (\sum_{i=1}^k (1+\dfrac{1}{p_i}))^2$
And by Brun's theorem it follows the product converges.
Q.E.D.
mick
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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understanding IVT vs MVT I know that if the MVT was applied to physics then it would roughly translate to saying that the average velocity = instantaneous velocity.
But suppose that my average velocity on $[0,T]$ was $10$. Then $\frac{f(T)-f(0)}{T-0}=10.$ Assuming that $\int f' \,dt = f$, then $\frac{f(T)-f(0)}{T-0} =... | Hopefully this isn't awkward, but following anorton's suggestion I've copied my comment to an answer:
That is correct. If $f′(t)$ is continuous, then you can apply the IVT as you do. Lots of books don't require this hypothesis for the MVT, though. All you need for the MVT is $f$ continuous on $[0,T]$ and differentiable... | {
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Proving that the limit of a sequence is $> 0$ Let $u$ be the complex sequence defined as follows :
$u_0=i$ and
$ \forall n \in \mathbb N, u_{n+1}=u_n + \frac {n+1-u_n}{|n+1-u_n|} $ .
Consider $w_n$ defined by $\forall n \in \mathbb N,w_n=|u_n-n|$ .
I have to prove that $w_n$ has a limit $> 0$.
Here is what I've prove... | Well the other proofs are quite long...
I guess this is shorter.
A simple computation proves that the sequence $n-Re(u_n)$ is increasing.
Now by contradiction, if $l=0$ then $|Re(u_n-n)|=n-Re(u_n)$ converges to $0$
But $n-Re(u_n)$ is increasing.
Hence $\forall n \in \mathbb N, n-Re(u_n) = 0$
That's a contradiction.
I'd... | {
"language": "en",
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"source": "stackexchange",
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What is the simplest $\Bbb{R}\to\Bbb{R}$ function with two peaks and a valley? What is the simplest $\Bbb{R}\to\Bbb{R}$ function with two peaks and a valley?
I have a set of points in $\Bbb{R^2}$ and I would like to fit a curve to the points, the points approximately lie on a curve like the one depicted in the followin... | When you say that you "think that the function should smoothly go to zero to the left of the left peak and to the right of the right peak", notice that it does not have to. Moreover, both $a$ and $b$ can be chosen arbitrarily.
The easiest way to do this would be to take a polynomial. If you want the simplest polynomia... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/484117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How many $5$-element subgroups does $S_7$ have?
How many $5$-element subgroups are there in $S_7$, the group of permutations on $7$ elements?
Let $H$ be a $5$-element subgroup of $S_7$. We have $\mbox{ord}(H) = 5$ and $5\mid 7!$.
But I don't have any idea how can I find 5-element subgroups.
| Hint:
A subgroup of $S_7$ of order $5$ must be cyclic (all groups of prime order are cyclic), and therefore is generated by an element of order $5$. The only elements of order $5$ in $S_7$ are $5$-cycles (why?), but each subgroup contains $4$ such cycles. This reduces the problem to one of a combinatorial flavor.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A clarification regarding partial derivatives Let us suppose the $i^{th}$ partial derivative of $f:\Bbb{R}^n\to \Bbb{R}$ exists at $P$; i.e. if $P=(x_1,x_2,\dots,x^n)$, $$\frac{f(x_1,x_2,\dots,x_n+\Delta x_n)-f(x_1,x_2,\dots,x_n)}{\Delta x_n}=f'_n (P)$$
My book says this implies that $$f(x_1,x_2,\dots,x_n+\Delta x_n)-f... | Compare the definition of the derivative to your ratio when $f:\mathbb R\to\mathbb R$, $x\mapsto x^2$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Matrix generating $\operatorname{SL}_n(\mathbb{R})$ How do I show that the following matrices generate $\operatorname{SL}_2(\mathbb{R})$
$\begin{pmatrix}
1 & a \\
0 & 1 \\
\end{pmatrix}$ or $\begin{pmatrix}
1 & 0 \\
a & 1 \\
\end{pmatrix}$
| Let $G$ be the span of the matrices
$$
\begin{pmatrix}
1 & a \\ 0 & 1
\end{pmatrix}
\quad\text{and}\quad
\begin{pmatrix}
1 & 0 \\ a & 1
\end{pmatrix}
$$
with $a\in \mathbb R$.
We have
$$
\begin{pmatrix}
1 & 1 \\ 0 & 1
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\ -1 & 1
\end{pmatrix}
\begin{pmatrix}
1 & 1 \\ 0 & 1
\end{pmatri... | {
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"source": "stackexchange",
"question_score": "2",
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Why do you add +1 in counting test questions? Here's an example question from the SAT question of the day:
On the last day of a one-week sale, customers numbered 149 through 201 were waited on. How many customers were waited on that day?
Possible answers: 51, 52, 53, 152, 153.
The correct answer here is 53, which i... | Let $C_n$ be the customer numbered $n$. List the customers in question:
$$C_{149},C_{150},C_{151},\ldots,C_{200},C_{201}\;.$$
Now write their numbers in the form $148+\text{something}$:
$$C_{148+\underline1},C_{148+\underline2},C_{148+\underline3},\ldots,C_{148+\underline{52}},C_{148+\underline{53}}\;.$$
In this form ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/484393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Help verify $\lim_{x\to 7} \frac {x^2+7x+49}{x^2+7x-98}$ . So my question is "Evaluate the limit" $\displaystyle \lim_{x\to 7} \frac {x^2+7x+49}{x^2+7x-98}$
I know you can't factor the numerator but you can for denominator. But either way you can't divide by $0$. So I say my answer is D.N.E.
If anyone can verify that... | Observe $x^2+7x+49=x^2+7x-98+147$ hence:$$\lim_{x\to7}\frac{x^2+7x+49}{x^2+7x-98}=\lim_{x\to7}\frac{x^2+7x-98+147}{x^2+7x-98}=\lim_{x\to7}\left(1+\frac{147}{x^2+7x-98}\right)$$Now note $x^2+7x-98=x^2-7x+14x-98=(x+14)(x-7)$ hence as $x\to7$ our limit tends to $\pm\infty$.
| {
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Prove that $mn < 0$ if and only if exactly one of $m,n$ is positive I need to prove that $mn < 0$ if and only if $m > 0$ and $n < 0$ or $m < 0$ and $n > 0$.
So I need to prove two cases:
1. If $m < 0$ and $n > 0$ or, in the alternative, if $m > 0$ and $n < 0$, then $mn < 0$.
2. If $mn < 0$, then $m < 0$ and $n > 0$ or... | It looks OK up until the last sentence. You say that if $m$ is negative and $n$ is positive or $m$ is positive and $n$ is negative $(-m)n < 0$ and $m(-n) < 0$. This is false; these quantities are positive, not negative. I think I know what you meant, but you need to rewrite this part.
Note, if $m$ is negative, don't wr... | {
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On the Divergence of $s_n=\cos{\frac{\pi}{3}n}$: A Proof Question:
Show that $s_n=\cos{\frac{\pi}{3}n}$ is divergent.
Attempt:
Suppose that $\lim_{n\rightarrow \infty}(\cos{\frac{\pi}{3}n})=s$, then given an $\epsilon$, say $\epsilon=1$, we can find an $N\in\mathbb{N}$ so that $$\begin{vmatrix} (\cos{\frac{\pi}{3}n}... | You are correct in your work.
However, as suggested above, an easier way is to just show that there are two subsequences converging to different limits.
In your case, $s_{6n+1}$ converges to $0.5$ and $s_{6n+3}$ converges to $-1$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Understanding 'root' in its context
For which of the following primes p, does the polynomial $x^4+x+6$ have a root of multiplicity$> 1$ over a field of characteristic $p$? $p=2/3/5/7$.
My book solves it using the concepts of modern algebra, which I am not very comfortable with.
I wonder if there is an intuition base... | The question is slightly ambiguous, because a polynomial may only have its roots in an extension of the field where it's defined, for instance $x^2+1$ has no roots in $\mathbb{R}$, but it has roots in $\mathbb{C}$.
However, having a multiple root is equivalent to be divisible by $(x-a)^2$, where $a$ is in the field whe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/484699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $ \frac{a^2}{a+b}+ \frac{b^2}{b+c} \geq \frac{3a+2b-c}{4} $
Prove that:
$$ \frac{a^2}{a+b} + \frac{b^2}{b+c} \geq \frac{3a+2b-c}{4} : (a, b, c)\in \mathbb{R}^+$$
This is just one of these questions where you just have no idea how to start. First impressions, I don't see how any known inequality can be us... | $$\begin{align}
\frac{a^2}{a+b} + \frac{b^2}{b+c} &\geqslant \frac{3a + 2b - c}{4}\\
\iff \frac{a^2}{a+b} - a + \frac{b^2}{b+c} - b &\geqslant - \frac{a + 2b + c}{4}\\
\iff -\frac{ab}{a+b} - \frac{bc}{b+c} &\geqslant - \frac{a+b}{4} - \frac{b+c}{4}\\
\iff \frac{(a+b)^2 - 4ab}{4(a+b)} + \frac{(b+c)^2 - 4bc}{4(b+c)} &\ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/484761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Finding the conditional extrema | Correct Method Find the conditional extrema of $ f(x,y) = x^2 + y $ subject to $x^2 + y^2 = 4$
Do you I use the Lagrange multipliers for this?
Or do I use the partial differential method and figure out what side of the zero-point this equation falls under:
$$ f_{xx}f_{yy} - {f_{xy}}^2 ... | We have:
$$\tag 1 f(x,y) = x^2 + y ~~ \mbox{subject to} ~~\phi(x) = x^2 + y^2 = 4$$
A 3D plot shows:
If we draw a contour plot of the two functions, we get:
From this plot, we see four points of interest, so we will use Lagrange Multipliers to find those and then we just need classify them as local or global min or m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/484835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to solve a complex polynomial?
*
*Solve:
$$ z^3 - 3z^2 + 6z - 4 = 0$$
How do I solve this?
Can I do it by basically letting $ z = x + iy$ such that $ i = \sqrt{-1}$ and $ x, y \in \mathbf R $ and then substitute that into the equation and get a crazy long equation? If I did that I suspect I wouldn't be able to ... | The easiest thing is just try to guest a root of the polynomial first. In this case, for
$$p(z) = z^3 - 3z^2 + 6z - 4,$$
we have that $p(1) = 0$.
Therefore, you can factorize it further and get
$$z^3 - 3z^2 + 6z - 4 = (z-1)(z^2 - 2z + 4)$$
$$= (z-1)((z-1)^2 + 3).$$
Their roots are just $$z_{1} = 1, \hspace{10pt}z_{2} =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/484906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 3
} |
How to find the inverse of 70 (mod 27) The question pertains to decrypting a Hill Cipher, but I am stuck on the part where I find the inverse of $70 \pmod{ 27}$. Does the problem lie in $70$ being larger than $27$?
I've tried Gauss's Method:
$\frac{1}{70} = \frac{1}{16} ^{\times2}_{\times2} = \frac{2}{32} = \frac{2}{5}... | A couple of ideas (working all the time modulo $\,27\,$):
$$\begin{align*}\bullet&\;\;70=14\cdot 5\\
\bullet&\;\;14\cdot2=28=1\implies& 14^{-1}=2\\
\bullet&\;\;5\cdot 11=55=2\cdot 27+1=1\implies&5^{-1}=11\end{align*}$$
Thus, finally, we get
$$70^{-1}=14^{-1}\cdot 5^{-1}=2\cdot 11=22$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/484990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 6,
"answer_id": 1
} |
Evaluate $\lim_{x\to49} \frac{x-49}{\sqrt{x}-7}$ Evaluate $\lim_{x\to 49} \frac{x-49}{\sqrt{x}-7}$
I'm guessing the answer is 7 but again that is only a guess. I don't know how to solve this type of problem.
Please help.
| $$
\lim_{x \to 49} \frac {x - 49}{\sqrt x - 7} = \lim_{x \to 49} \frac {(\sqrt x + 7)(\sqrt x - 7)}{\sqrt x - 7} = \lim_{x \to 49} (\sqrt x + 7) = 14
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/485158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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Undergraduate math competition problem: find $\lim \limits_{n \to \infty} \int \limits^{2006}_{1385}f(nx)\, \mathrm dx$ Suppose $f\colon [0, +\infty) \to \mathbb{R}$ is a continuous function and $\displaystyle \lim \limits_{x \to +\infty} f(x) = 1$. Find the following limit:
$$\large\displaystyle \lim \limits_{n \to \i... | Using the substitution $t=nx$, we get $I_n = \int^{2006}_{1385}f(nx)dx = \frac{1}{n}\int_{1385n}^{2006n} f(t) dt$. Let $I=2006-1385$.
Now let $\epsilon>0$, and choose $L>0$ such that if $t\ge L$, then $-\frac{\epsilon}{I} < f(t)-1 < \frac{\epsilon}{I}$.
Now choose $N\ge \frac{L}{1385}$. Then if $n \ge N$ and $t \in [... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/485267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 1
} |
Period of the function $\sin (8\pi \{x\})$ My question is to to find the period of
$$\sin(8\pi\{x\}),$$
where $\{\cdot\}$-is the fractional part of function.
I know that the period of $\{\cdot\}$ is 1 and the period of $\sin(8\pi x)$ is $1/4$. But how to find the overall period of the given function?
| $\sin(8 \pi \{x\})
= \sin(8 \pi ( x - \lfloor x \rfloor))
= \sin(8 \pi x - 8 \pi \lfloor x \rfloor)
= \sin(8 \pi x)$,
hence $\sin(8 \pi \{x\})$ has period $\frac{1}{4}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/485340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Counter-Example (or Proof) to $\int_{0}^{1}f_{n}\;dx\to0$ Implies $f_{n}\to0$ a.e. $x$ Whenever $f_{n}\geq0$. I am dealing with a problem at the moment where the hypothesis can be restated as $\int_{0}^{1}f_{n}\;dx\to0$ and $f_{n}\geq0$. Under these conditions, I want to conclude that $\lim f_{n}$ exists and is $0$ fo... | The canonical counterexample is to take the indicator functions of $[0,1]$; $[0,1/2]$,$[1/2,1]$, $[0,1/4]$, $[1/4,1/2]$, $[1/2,3/4]$, $[3/4,1]$ &c.
(If the pattern is not evident: break $[0,1]$ into $2^k$ intervals let $f_n$ be the sequence of indicator functions of each $2^k$ intervals obtained at each step from left ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/485404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why can't all subsets of sample space be considered as events? My textbook is Probability and random processes by Grimmett & Stirzaker and the first chapter does not explain this," for reasons beyond the scope of the book". The authors introduce the reader to sample spaces and to events and then go on to say that event... | google vitali sets......you cannot measure all the sets if you assume axiom of choice to be correct...however recently there was a paper which showed that you can measure almost all sets if you do not assume axiom of choice
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/485461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Proving a function $f(m,n)$ which satisfies two conditions is a constant I found the following question in a book only with one sentence. "This question can be solved by an elementary way. Note that the following two are false: (1) If a function is bounded from below, then it has minimum value. (2) A monotone decreasin... | A probabilistic proof is given at the bottom of this page.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/485527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
On verifying Proj S is a scheme In Hartshorne II Prop 2.5, it says $D_+{(f)}$ is homeomorphic to $\text{Spec}(S_{(f)})$, but I cannot prove it. Since $D_+{(f)}$ homeomorphic to $S_f$, I have to show $\text{Spec}(S_{(f)})$ homeomorphic to $\text{Spec}(S_f)$.
Take $p \in \text{Spec} S$. I cannot show $S_f(pS_f \cap pS_(... | You already defined the (homeo)morphism $\phi:D_+(f)\to\textrm{Spec}\,S_{(f)}$ by $\mathfrak p\mapsto \mathfrak pS_f\cap S_{(f)}$. Its inverse $\psi$ sends $\mathfrak q\mapsto \ell^{-1}(\mathfrak qS_f)$, where $\ell:S\to S_f$ is localization at $f$. (I write $\ell^{-1}(-)$ instead of $S\cap -$ as I do not have Hartshor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/485641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $\sum_n^{\infty} \prod_{k=0}^n \dfrac{1}{x+k} = e \sum_ n^{\infty} \dfrac{(-1)^n}{(x+n)n!}$ Let $$f_n(x) = \prod_{k=0}^n \dfrac{1}{x+k}.$$ Show that $$\sum_{n=0}^{\infty} f_n(x) = e \sum_ {n=0}^{\infty} \dfrac{(-1)^n}{(x+n)n!}.$$
| Using the partial fraction identity that was proved by a straightforward induction technique at this MSE post, we have that
$$\prod_{k=0}^n \frac{1}{x+k} =
\frac{1}{n!} \sum_{k=0}^n (-1)^k \binom{n}{k} \frac{1}{x+k}.$$
Now to compute $$\sum_{n=0}^\infty \prod_{k=0}^n \frac{1}{x+k}$$ we ask about the coefficient of $$\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/485715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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Find all functions $f(x+y)=f(x^{2}+y^{2})$ for positive $x,y$ Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}$ such that for any $x,y\in \mathbb{R}^{+}$ the following holds:
$$f(x+y)=f(x^{2}+y^{2}).$$
| If
$$f(A(x,y))=f(B(x,y))$$
on a connected open set of $(x,y)$ where $A$ and $B$ are functionally independent (have nonzero Jacobian determinant), the function $f$ is constant on that set .
Locally one has both $xy$-coordinates and $AB$-coordinates. From $(A,B)$ you can get to all close enough $(A + \epsilon_1, B +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/485774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 4,
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Why is compactness so important? I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel theorem, and a proof continuous functions on R from closed subintervals of R are bounded. It ... | Compactness does for continuous functions what finiteness does for functions in general.
If a set $A$ is finite, then every function $f:A\to \mathbb R$ has a max and a min, and every function $f:A\to\mathbb R^n$ is bounded. If $A$ is compact, then every continuous function from $A$ to $\mathbb R$ has a max and a min a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/485822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "209",
"answer_count": 13,
"answer_id": 11
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coordinate of shorter line If I have a line segment with endpoints AB,CD. The length of the line is 5 units.
If I make the line shorter (eg. 3 units), and one of the endpoints is still AB, how do I figure out what the new CD is?
Thanks for the help
| We assume that your A, B, C, D are coordinates.
So we will more conventionally call them $(a,b)$ and $(c,d)$.
For your particular case, the coordinates of the new endpoint are $(x,y)$, where
$$x=a+\frac{3}{5}(c-a)\qquad\text{and} \qquad y=b+\frac{3}{5}(d-b).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/485888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Greatest possible integer value of x+y? I found a interesting question in one exam.
If 5 < x < 10 and y = x + 5, what is the greatest possible integer value of x + y ?
(A) 18
(B) 20
(C) 23
(D) 24
(E) 25
MySol: For max value of x+y , x should be 9.
So x+y = 9+14 = 23
But this is not correct.
Can someone explain.
| Note that $x+y=2x+5$. The greatest possible integer value of $2x$ occurs at $x=9.5$.
Remark: Unfortunately, a bit of a trick question. Not nice! One of my many objections to multiple choice questions is that they are too often designed to fool people into giving the "wrong" answer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/485965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Integration of $x^2 \sin(x)$ by parts How would I integrate the following?
$$\int_0^{\pi/2} x^2\sin(x)\,dx$$
I did $u=x^2$ and $dv=\sin(x)$
I got
$x^2-\cos(x)+2\int x\cos(x)\,dx.\quad$ I then used $u=x$ and $dv=\cos(x).$
I got
$$x^2-\cos(x)+2[x-\sin(x)-\int\sin(x)]$$
then
$x^2-\cos(x)+-2 \sin(x)(x)-\cos(x)\Big|_0^{\pi... | You need to multiply $u$ and $v$, then subtract the subsequent integral:
So you should have $$\begin{align} \int_0^{\pi/2} x^2\sin(x)\,dx & = -x^2\cos(x)+2\int x\cos(x)\,dx \\ \\
& = -x^2 \cos x + 2\Big[x \sin x - \int \sin x\,dx\Big]\\ \\
& = -x^2\cos x + 2x \sin x - (-2\cos x)\\ \\
& = -x^2 \cos x + 2x \sin x + 2\co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/486062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Proving Cantor's theorem $\DeclareMathOperator{\card}{card}$From Problem-Solvers Topology
Prove the following:
CANTOR'S THEOREM
If $A$ is a set, then $$\card A < \card \mathcal{P}(A)$$
where $\card A$ stands for the cardinality of set $A$.
My Answer
If $A = \emptyset$, then $\card A =0$ and $\card \mathcal{P}(A)=1$.
... | $\DeclareMathOperator{\card}{card}$No. Your answer is not correct. Between any two sets (both having two elements or more) there is a non-surjective map. Your task is to show that every set $A$ and every function from $A$ to $\mathcal P(A)$ is not surjective.
To clarify the point you're missing, the argument in your pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/486126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Trigonometric problem using basic trigonometry
If $x$ is a solution of the equation: $$\tan^3 x = \cos^2 x - \sin^2 x$$
Then what is the value of $\tan^2 x$?
This is the problem you are supposed to do it just with highschool trigonometry , but i can't manage to do it please help
Here are the possible answers:
$$a) ... | $\tan^2 x=\cos^2 x-\sin^2 x$
$\sin^2 x=\cos^4 x -\sin^2 x \cos^2 x$
$0=\cos^4 x -\sin^2 x \cos^2 x -\sin^2 x$
$0=\cos^4 x - \sin^2 x(1+\cos^2 x)$
$0=\cos^4 x - (1-\cos^2 x)(1+\cos^2 x)$
$0=\cos^4 x -(1- \cos^4 x)$
$0=2 \cos^4 x -1$
$\cos^2 x=\frac {\sqrt 2}{2}$
$\large \frac {1}{\cos^2 x}=\sqrt 2$
$\tan^2 x=\sqrt 2 \si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/486194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Book Searching in Stability Theory. Can anyone recommend me a book on Stability Theory with an intuitive approach?
I have some course notes on that subject, but it's really abstract and theoretical. I really want to understand it, ex:
Stable by Lyapunov/Asymptotically Stable/Globally Asymptotically Stable/ Lyapunov'... | I like the book written by Jorge Sotomayor.
Teoria Qualitativa das Equações Diferenciais.
Thats a good one!
Or
Perko L. Differential equations and dynamical systems (Springer, 1991)(K)(T)(208s)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/486265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
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How can all 3 of these be true?
*
*Most numbers are composite.
*If you choose a random whole number there is a 50/50 chance that it's even or odd.
*If you take 2 random whole numbers and multiply them there is a 75% chance the result is even and a 25% chance it is odd.
(That is even*even=even, odd*even=even, even*o... | You're question #3 almost answer themselves, if there are 4 possible options, and 3 out of 4 of the possible options give us an even product, than there is a 75% chance the product is even and a 25% chance that the product is odd. The first 2 questions are general proporties which can be shown by taking any 2 consecuti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/486306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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positive series convergent properties So I have a test in the next weak and I found myself struggled in an "easy" question.
Given two series $(a_n)_{n=1}^{\infty} \ $ so that $\forall n \in \mathbb{N} \ a_n > 0$ then:
$ \sum _{n=1}^{ \infty} a_n$ convergent iff
$ \sum _{n=1}^{ \infty} \frac{a_n}{a_n +1}$ convergent.
S... | Hint: $$\frac{1}{a_n+1} = 1 - \frac{a_n}{a_n+1}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/486383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Solving a radical equation $\sqrt{x+1} - \sqrt{x-1} = \sqrt{4x-1}$ $$
\sqrt{x+1} - \sqrt{x-1} = \sqrt{4x-1}
$$
How many solutions does it have for $x \in \mathbb{R}$?
I squared it once, then rearranges terms to isolate the radical, then squared again.
I got a linear equation, which yielded $x = \frac54$, but when I put... | $$x+1+x-1-2\sqrt{x+1}\sqrt{x-1}=4x-1\implies(2x-1)^2=4(x^2-1)\implies$$
$$4x^2-4x+1=4x^2-4\implies 4x=5\implies x=\frac54$$
But, indeed
$$\sqrt{\frac54+1}-\sqrt{\frac54-1}\stackrel ?=\sqrt{5-1}\iff\frac32-\frac12=2$$
which is false, thus no solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/486484",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
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Every quotient of a number ring is finite Let $K$ be a number field, i.e. a subfield of $\mathbb{C}$ of finite degree over $\mathbb{Q}$. Let $\mathscr{O}_K$ be the ring of integers of $K$, i.e. algebraic integers which are in $K$. Let $I$ be an ideal of $\mathscr{O}_K$. I read many times that the quotient $\mathscr{O}_... | I guess you know that, as an Abelian group, $O_K\cong\mathbb Z^k$, where $k=[K:\mathbb Q]$. Now if $0\neq a\in J$ then $(a)\subset J$, and as $a$ divides its norm $Na\in\mathbb Z$, also $(Na)\subset(a)$. And $\mathbb Z^k/(Na)\mathbb Z^k$ is certainly finite.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/486591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
If $E$ is a measurable set, how to prove that there are Borel sets $A$ and $B$ such that $A\subset E$, $E\subset B$ and $m(A)=m(E)=m(B)$? If $E$ is a measurable set, then how to prove that there are Borel sets $A$ and $B$ such that $A$ is a subset of $E$, $E$ is a subset of $B$ and $m(A)=m(E)=m(B)$?
| I assume by $m$ you mean Lebesgue measure on $\mathbb R^n$. Use that this measure is regular. This gives us that, if $m(E)<\infty$, then for any $n$ there are a compact set $K_n$ and and open set $U_n$ with $K_n\subset E\subset U_n$, and $m(E)-1/n<m(K_n)$ and $m(U_n)<m(E)+1/n$.
This implies that $A=\bigcup_n K_n$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/486642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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using induction to prove $(n+1)^2 < 2n^2$ (Im not English and just started doing maths in English so my termiology is still way off)
So the title for $n\ge 3$
*
*First I use calculate both sides with $3$, which is true
*I make my induction. $(k+1)^2 < 2k^2$
then I replace $N$ with $k+1$: $(k+2)^2 < 2(k+1)^2$
Now... | HINT: $(k+2)^2=\big((k+1)+1\big)^2=(k+1)^2+2(k+1)+1$; now apply the induction hypothesis that $(k+1)^2<2k^2$. (There will still be a bit of work to do; in particular, you’ll have to use the fact that $k\ge 1$.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/486705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
How to relate the valuation of x/y (For a minimal Weierstrass equation) I'm reading an article about elliptic curves, but since I'm not very experienced on this subject, I ended up getting stuck. The problem starts as:
"Let $K/\mathbb{Q}$ be a number field and $E/K$ an elliptic curve defined over $K$. Let $v\in M_K$ be... | If $y$ is a unit, then (1) says $v(x^{-1})<0$ iff $v(x) < 0$ which is a contradiction unless $v(x)=0$. So we can look for counterexample by finding a point where $v(y)=0$ but $v(x) \ne 0$.
Consider the curve $y^2=x^3-4$ over $\mathbb{Q}$. Its discriminant is $-2^8 3^3$, hence it is a global minimal Weierstrass equa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/486772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Problem with differentiation as a concept. I don't understand quiet good something here, for example if we want to find the derivative of the function $\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(h)}{h} $ and if we compute it from the function: $ f(x) = 12 + 7x $
We get that the derivative of $f(x)$ is equal to... | The whole point of the limit operation is that it avoids any bad behaviour of a function around the given point. We don't care what the function value is, nor whether it's even defined at a given point.
In your case, so long as $h \ne 0$, we can cancel to find that $\frac {7h}{h} = 7$; it doesn't matter that $\frac{7h}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/486842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 1
} |
What does it mean for a number to be in a set? Frustratingly my book gives me several examples of a number in a set but offers no explanation at all.
Anyways what is going on here? According to the book $2$ is not an element of these sets:
$$\{\{2\},\{\{2\}\}\}$$
$$\{\{2\},\{2,\{2\}\}\}$$
$$\{\{\{2\}\}\}$$
What is goin... | You can think of 'is an element of' as stripping off a single layer of set braces.
$2$ is not an element of $\{\{2\}\}$, because removing one layer of braces, you get $\{2\}$, the set containing $2$, which is different from $2$.
Also, if this wasn't the case, then there would be no way to distinguish between, for examp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/486865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 11,
"answer_id": 0
} |
Simple approximation to a sum involving Stirling numbers? I have also posted this question at https://mathoverflow.net/questions/141552/simple-approximation-to-a-sum-involving-stirling-numbers#141552. I have an exact answer to a problem, which is the function:
$f(x,y)=\frac{1}{y^x}\sum_{i=1}^{x-1} [i\binom{y}{x-i}(x-i)... | Note that $\binom{y}{x-i}(x-i)!=(y)_{x-i}$ is the Pochhammer symbol or "falling factorial" and the Stirling numbers of the second kind relate falling factorials to monomials by this formula,
$$
\sum_{i=0}^x(y)_i\,\begin{Bmatrix}x\\i\end{Bmatrix}=y^x\tag{1}
$$
The recurrence relation for Stirling numbers of the second k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/486917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Is powers of rationals dense in $\mathbb R$ Consider $\mathbb {\tilde {Q}} = \{ x^n : x \in \mathbb Q \} $ $n$ is fixed odd integer.
I have two questions here.
*
*Is this set dense in $\mathbb R$ and
*Is there any bijection exists between $\mathbb Q$ and $\mathbb {\tilde Q}$
For the first question, I think the se... | Here is a simpler proof of density. Consider the map $f(x)=x^n$, $n>0$ is odd, $f: {\mathbb R}\to {\mathbb R}$. This map is clearly continuous. The intermediate value theorem implies that this map is surjective. The set of rational numbers is dense in ${\mathbb R}$. Therefore, its image under the continuous map $f$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/486978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
If $\lim_{n\to\infty}\int_0^1 f_n(x)dx=0$, are there points $x_0\in[0,1]$ such that $\lim_{n\to\infty}f_n(x_0)=0$? This is part of an old qual problem at my school.
Assume $\{f_n\}$ is a sequence of nonnegative continuous functions on $[0,1]$ such that $\lim_{n\to\infty}\int_0^1 f_n(x)dx=0$. Is it necessarily true th... | No. The standard counterexample would be indicator functions of $[0, 1]$, $[0, 1/2]$, $[1/2, 1]$, $[0, 1/3]$, $[1/3, 2/3]$, and so on.
In order to make these continuous, add in line segments on either end with very large slope.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/487049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Proving a point inside a triangle is no further away than the longest side divided by $\sqrt{3}$ Problem:
In a triangle $T$ , all the angles are less than 90 degrees, and the longest side has length $s$. Show that for every point $p$ in $T$ we can pick a corner $h$ in $T$ such that the distance from $p$ to $h$ is less... | Rephrasing Christian Blatter: for any point interior to an acute triangle, measure the distance to each of the four vertices, and find the smallest value. Now find the point where the this minimum value is largest. Almost obviously, this is the circumcenter. Connect the circumcenter to each of the vertices and drop per... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/487121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
How many ways to choose $k$ out of $n$ numbers with exactly/at least $m$ consecutive numbers? How many ways to choose $k$ out of $n$ numbers is a standard problem in undergraduate probability theory that has the binomial coefficient as its solution. An example would be lottery games were you have $13983816$ ways to cho... | After some further googling I found the following reference which gives a general formula and a derivation:
Lottery combinatorics by McPherson & Hodson
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/487207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Neighborhood of the Origin in the Weak Topology I am in difficulty with the following question. Let $H$ be a real infinite-dimensional Hilbert space and $u\in H\setminus \{0\}$. Let $V$ be an any neighborhood of $0$ in the weak topology on $H$. Is there a vector $v\ne 0$ such that $tv\in V$ for all $t\in \mathbb{R}$ an... | Base on the solution of Daniel Fischer, choosing $V=\{x\in H: |\langle u, x\rangle|<1\}$.
Then $V$ is a neighborhood of $0$ in the weak topology. Let $v\ne 0$ such that $tv\in V$ for all $t\in \mathbb{R}$. Then $\langle u, tv\rangle=t\langle u, v\rangle<1$ for all $t\in \mathbb{R}$. Hence $\langle u, v\rangle=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/487311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding the kernel of an action on conjugate subgroups I'm trying to solve the following problem:
Let $G$ be a group of order 12. Assume the 3-Sylow subgroups of $G$ are not normal. Prove that $G\cong A_4$.
Here's my attempt: let $\mathscr S$ be the set of 3-Sylow subgroups of $G$. Since the elements of $\mathscr S$... | Let $S_1$ and $S_2$ be two of the Sylow $3$-subgroups. If $a \in \ker \phi$, then in particular
$$aS_1a^{-1} = S_1 \Rightarrow a \in N_G(S_1).$$
The same holds for $S_2$, so
$$\ker \phi \subset N_G(S_1) \cap N_G(S_2).$$
Now, since the Sylow $3$-subgroups aren't normal, what is $N_G(S_i)$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/487380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Exponential algebra problem We need to solve for x:
$$54\cdot 2^{2x}=72^x\cdot\sqrt{0.5}$$
My proposed solution is below.
| I think this is a more systemactic way:
$$54\cdot 2^{2x}=72^x\cdot\sqrt{0.5}$$
Apply logarithm on both sides of the equation. For now the base of the logaritm does no really matter
$$\log{(54\cdot 2^{2x})}=\log{(72^x\cdot\sqrt{0.5})}$$
and simplify by applying the laws of logarithm for products and powers
$$\log{(54... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/487458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
General misconception about $\sqrt x$ I noticed a large portion of general public (who knows what square root is) has a different concept regarding the surd of a positive number, $\sqrt\cdot$, or the principal square root function.
It seems to me a lot of people would say, for example, $\sqrt 4 = \pm 2$, instead of $\s... | The square root of $x$ is a number which when squared gives $x$. For $16$ there are two such numbers, so there are two square roots of $16$. For $0$, there is one and for any negative number there is none.
Now, simply call the non-negative square root of a number, the principal square root. There is only one such numbe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/487509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 2
} |
Seeking for a proof on the relation between Euler totient and Möbius function Can someone help me prove the relation
$\varphi\left(n\right)={\displaystyle \sum_{d|n}}d\mu\left(n/d\right)$,
where $\mu$ is the Möbius function defined by
$$
\mu\left(n\right)=\begin{cases}
1 & \mbox{if }n=1\\
\left(-1\right)^{t} & \mbox{if... | Let $n = p_1^{e_1}p_2^{e_2}\ldots p_m^{e_k}$ for some primes $\{p_1, \ldots, p_m\}$. By definition $\phi(n)$ equals the number of elements in the set $\{0,1,\ldots,n-1\}$ that have no common divisor with $n$. We will count the number of elements that will be excluded in this setup, i.e. the number of elements of the se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/487599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 2
} |
Let $M_1,M_2,M_3,M_4$ be the suprema of $|f|$ on the edges of a square. Show that $|f(0)|\le \sqrt[4]{M_1M_2M_3M_4}$ Let $G$ denote the interior of the square with vertices $1,i,-1,-i$. Suppose $f$ is holomorphic on $G$ extends continuously to $\overline{G}$, and $M_1,M_2,M_3,M_4$ are the suprema of $|f|$ on the edges ... | Consider the function
$$g(z) = f(z)\cdot f(iz)\cdot f(-z) \cdot f(-iz).$$
$g$ is holomorphic on $G$ and extends continuously to $\overline{G}$, and the maximum of $g$ on each of the edges is at most $M_1\cdot M_2\cdot M_3\cdot M_4$.
$g(0) = f(0)^4$. The maximum modulus principle does the rest.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/487675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Calculus: The tangent line intersects a curve at two points. Find the other point. The line tangent to $y = -x^3 + 2x + 1$ when $x = 1$ intersects the curve in another point. Find the coordinates of the other point.
This was never taught in class, and I have a test on this tomorrow. This question came off of my test ... | Your statement that "This was never taught in class" might astonish your instructor. But even if not, it is very unreasonable to expect to be required to do only that which someone has shown you how to do. And this is so close to the beaten path that it's not a good example of something you might not have been shown ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/487740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 2
} |
Additive group of rationals has no minimal generating set In a comment to Arturo Magidin's answer to this question, Jack Schmidt says that the additive group of the rationals has no minimal generating set.
Why does $(\mathbb{Q},+)$ have no minimal generating set?
| Let $S\subseteq\mathbb Q$ be such that $\langle S\rangle=\mathbb Q$. Fix $a\in S$, and put $T=S\setminus\{a\}$, let us see that also $\langle T\rangle=\mathbb Q$. We have $$\frac{a}{2}=a\cdot k_0+\sum_{i=1}^na_i\cdot k_i,$$
for some $k_i\in\mathbb Z$ and $a_i\in T$. Then $$a=a\cdot (2k_0)+\sum_{i=1}^na_i\cdot (2k_i),$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/487820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 0
} |
Closed form solution to simple recurrence I have this recurrence :
$$f(i) = \begin{cases}
0 &i=0\\
1 &i=M\\
\frac{f(i-1) + f(i+1)} 2& 0 < i < M
\end{cases}$$
I have guessed that
$$f(i) = \frac i M$$
and proved it via induction.
What is the right way of solving it without guessing ?
Later Edit:
Thank you very much for ... | If you look at the third form, you should recognize it as an arithmetic mean. $f(i)$ is the mean of $f(i-1)$ and $f(i+1)$, which tells you immediately that you're looking for a linear equation. The endpoints then give you slope and intercept.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/487884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Unexpected Practical Applications of Calculus Calculus shows up in a lot of places in the world. Specifically, here are three areas where I see it used the most:
*
*Optimization problems.
*Anything involving rates of change (e.g. velocity $\rightarrow$ acceleration).
*Anything involving "averages" (e.g. surface a... | Ryan, perhaps a bit unexpected is the application of calculus in the human heart. More precise, cardiac output. The definition of cardiac output is the volume of blood pumped by the heart per unit time. The formula for this turns out to be a Riemann sum which in turn becomes an integral. And I find that unexpected in t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/487985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Converting between $T_1$ and $T_2$ Given that every $T_2$ space is a $T_1$ space, is it possible to start with a $T_1$ space and to specify in terms of its sets a family of additional sets sufficient to make that $T_1$ space into a $T_2$ space? If so, can this be done for any $T_1$ space or only particular ones?
| I suspect that every way of doing this is somehow either arbitrary or trivial.
Suppose we have an operation $\mathcal{H}$, such that if $(X, T)$ is a T1 space
then $T \subset \mathcal{H}(T)$ and $(X, \mathcal{H}(T))$ is a T2 space.
Suppose furthermore that this operation preserves symmetry, in the sense that if
$f: X \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/488075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Meaning of signed volume I want to understand the definition of the determinant of a $n\times n$ real matrix $A$ as the signed volume of the image of the unit cube $C'$ under the linear transformation given by $A$, i.e. $x\to Ax$. However I am failing to make sense of the words signed volume. What will be the precise d... | In this context, signed volume is simply a term that carries slightly more information than volume alone. It's analogous to speed and velocity.
The magnitude of the determinant of a linear transformation is the number that it scales volumes in the space by. We only need to consider the unit ball however, because if yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/488163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 3
} |
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