Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Factoring inequalities on Double Summation (Donald Knuth's Concrete Mathematics) If you have the Concrete mathematics book please refer to page 40 and 41.
So how come this given sum
$$
\sum_{1 \le j < k + j \le n} \frac{1}{k}
$$
becomes
$$
\sum_{1\le k \le n}\sum_{1\le j \le n-k} \frac{1}{k}
$$
?
I do not understand h... | If $1\le j<k+j\le n$, then $1\le j$ and $k+j\le n$; the latter inequality is equivalent to $j\le n-k$, so $1\le j<k+j\le n$ implies that $1\le j\le n-k$, the inequality governing the inner summation. Thus, we want to find some expression such that
$$[1\le j<k+j\le n]=[\text{expression}][1\le j\le n-k]\;.$$
What part(s)... | {
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"url": "https://math.stackexchange.com/questions/1698826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to prove that the sum of the areas of triangles $ABR$ and $ CDR$ triangle is equal to the $ADR$? In the convex quadrilateral $ABCD$, which is not a parallelogram, the line passing through the centers of the diagonals $AC$ and $BD$ intersects the segment $BC$ at $R$. How to prove that the sum of the areas of triangl... | This is tricky.
We shall use an easy fact that if we are given fixed points $Y,Z$, and a variable point $X$ that changes linearly, then $[XYZ]$ changes linearly, where $[\mathcal{F}]$ denotes the oriented area of $\mathcal{F}$.
Let $M,N$ be midpoints of $AC, BD$. Using the fact we know that the function $MN \ni X \ma... | {
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"url": "https://math.stackexchange.com/questions/1698945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Why is $r^2-9=0$ the characteristic equation for $y'-9y=0$ I'm trying to learn how to solve second order differential equations and I don't understand something here:
http://tutorial.math.lamar.edu/Classes/DE/SecondOrderConcepts.aspx
Question says find two solutions to $y'-9y=0$
It then says the characteristic equation... | At first, observe that $y(x) \equiv 0$ solves the equation. Then the usual ansatz for such equations for $y \not \equiv 0$ is $y(x) = Ae^{\lambda x}$ with $\Bbb C \ni A \neq 0$ and $\lambda \in \Bbb C$, since this gives you constant coefficients when you derive. Assuming, that you mean the equation $y''-9y=0$ (since yo... | {
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Transform complex exponential integral to real Question: Transform
$J_n (x)=\frac{1}{2 \pi} \int_{\theta=-\pi}^\pi e^{i(n\theta-x\sin\theta)}$
On answer sheet:
$J_n (x)=\frac{1}{\pi} \int_{\theta=0}^\pi \cos(n\theta-x\sin\theta)$
But how? I am not very good at complex analysis, thanks for any help!
| By definition, the real part:
$$e^{i(n\theta-x\sin\theta)}=\cos(n\theta-x\sin\theta)+i\sin(n\theta-x\sin\theta)\implies$$
$$\implies \text{Re}\,e^{i(n\theta-x\sin\theta)}=\cos(n\theta-x\sin\theta)$$
and as this last one is an even function, we get
$$\frac1{2\pi}\int_{-\pi}^\pi\cos(n\theta-x\sin\theta)=2\left(\frac1{2\p... | {
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"url": "https://math.stackexchange.com/questions/1699117",
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Hamilton paths/cycles in grid graphs Let G be a grid graph with m rows and n columns, i.e. m = 4, n = 7 is shown here:
For what values of m and n does G have a Hamilton path, and for what values of m and n does G have a Hamilton cycle?
So far I've figured out that a grid graph always has a Hamilton path, and has a H... | HINT: You simply need to explain carefully how to produce the desired paths.
*
*There is always a Hamilton path that simply traverses the rows in alternate directions.
*If, say, $m$ is even, as in your example, you can generalize the following idea (which I’ve left slightly incomplete). The evenness of $m$ is wha... | {
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Find the necessary and sufficient condition on $g$ for which $f$ is differentiable at $0$
Suppose two functions $f,g: \mathbb{R} \to \mathbb{R}$ satisfy $f(x) = xg(x)$ for all $x \in \mathbb{R}$. Find the necessary and sufficient condition on $g$ for which $f$ is differentiable at $0$. In such case find $f'(0)$.
We h... | Using the definition we have $\lim_{h\rightarrow 0}\frac{f(0+h)-f(0)}h =\lim_{h\rightarrow 0}\frac{hg(h)-0\cdot g(0)}h =\lim_{h\rightarrow 0}g(h)$ so f'(0) exists iff g is continuous at 0.
| {
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bad explanation in how to solve the cubic The question asked me to solve
$t^3+pt+q=0$ where $p,q$ real and $27q^2+4p^3<0$ using the identity $\cos{3\theta}=4\cos^3{\theta}-3\cos{\theta}$.
Answer goes like this and I have stopped where it just makes no sense at all to me.
Since the discriminant $-(27q^2+4p^3)>0$ impl... | Since $27q^2+4p^3<0$, $p^3<-(27/4)q^2<0$ so $p<0$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Use Cauchy Product to Find Series Representation of $[\ln(1 + x)]^2$ Problem:
Let $f(x) = [\ln(1 + x)]^2$. Use the series for the logarithm to compute that
\begin{align*}
f(x) = [\ln(1 + x)]^2 = \sum_{n = 2}^{\infty}(-1)^n\Bigg(\sum_{k = 1}^{n - 1} \frac{1}{(n - k)k}\Bigg) x^n.
\end{align*}
Use this to evaluate th... | Note that for $-1<x< 1$ we have
$$\begin{align}
\log^2(1+x)&=\sum_{k=1}^\infty\sum_{m=1}^\infty\frac{(-1)^{k+m}x^{k+m}}{k\,m}\tag 1\\\\
&=\sum_{k=1}^\infty\sum_{n=k+1}^\infty\frac{(-1)^{n}x^{n}}{k\,(n-k)} \tag 2\\\\
&=\sum_{n=2}^\infty\sum_{k=1}^{n-1}\frac{(-1)^{n}x^{n}}{k\,(n-k)} \tag 3\\\\
\end{align}$$
as was to be ... | {
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Why is $\int_0^\infty e^{-nx}x^{s-1}dx = \frac {\Gamma(s)}{n^s}$? Why is the following equation true?$$\int_0^\infty e^{-nx}x^{s-1}dx = \frac {\Gamma(s)}{n^s}$$
I know what the Gamma function is, but why does dividing by $n^s$ turn the $e^{-x}$ in the integrand into $e^{-nx}$? I tried writing out both sides in their in... | Note that the definition of the gamma function is:
$$\Gamma(s)=\int_0^{\infty} x^{s-1}e^{-x}dx$$
So, taking $y=nx$ so that $dy=ndx$ and $x=\frac{y}{n}$ yields:
$$\int_0^{\infty}e^{-nx}x^{s-1}=\int_0^{\infty}e^{-y}(\frac{y}{n})^{s-1}ndy=\frac{1}{n^s}\int_0^{\infty} y^{s-1}e^{-y}dy=\frac{\Gamma(s)}{n^s}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Asymptotic relations at infinity I am attempting to show that
If $f(x) - g(x) \ll 1,\, x \to \infty$, then $e^{f(x)}\sim e^{g(x)}, \,x\to \infty$
From the first line, I am able to show that
$$ \lim_{x\to \infty} \frac{f(x) - g(x)}{1} = 0$$
from which it is clear that
$$ \lim_{x\to \infty}\left[f(x) - g(x)\right] = 0$... | As you've shown
$$f(x)-g(x)\to 0,\quad\text{as}\, x\to\infty$$
So
$$\frac{e^{f(x)}}{e^{g(x)}}=e^{f(x)-g(x)}\to 1$$
All you need is
$$e^v\to 1,\quad\text{as}\,v\to 0$$
| {
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Linear system 2 unknowns There are $x$ white and $y$ black pearls and their ratio is $z$. If I add six black and six white pearles, the ratio doubles.
I did the following:
$ \frac{x+6}{y+6} = \frac{2x}{y}$ and then I get
$xy -6(2x-y)=0$
I can find solutions by guessing. Is there any other way?
ADDED:
Now I have to so... | $\frac{x}{y} = z$ and $\frac{x+6}{y+6} = 2z$
$\implies x = zy$ and $\frac{zy+6}{y+6} = 2z$
$\implies 6 = z(y+12) \implies z = \frac{6}{y+12}$
$\implies x = zy = \frac{6y}{y+12}$
Since $x$ and $y$ are the number of pearls, they must be integers. That is
$(y+12) | (6y)$.
There are only 3 possible $y$ values by giving $x=... | {
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In how many ways can the numbers $1, 2, 3, \ldots, 100$ be arranged around a circular table such that any two adjacent numbers differ by at most $2$? In how many ways can the numbers $1, 2, 3, \ldots, 100$ be arranged around a circular table such that any two adjacent numbers differ by at most $2$?
| First place ‘1’ at a position. On its either side you can place only two numbers (‘2’ and ‘3’). Let us place ‘2’ to left of ‘1’ and ‘3’ to right of ‘1’. Now to the left of ‘2’ you can place only ‘4’ (‘3’ has been already place at right of ‘1’). On proceeding, all even numbers are next to each other forming a half of ne... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $p>3$ and $p+2$ are twin primes then $6\mid p+1$ I have to prove that if $p$ and $p+2$ are twin primes, $p>3$, then $6\ |\ (p+1)$.
I figure that any prime number greater than 3 is odd, and therefore $p+1$ is definitely even, therefore $2\ |\ (p+1)$. And if I can somehow prove $3\ |\ (p+1)$, then I would be done. Bu... | The easy way.
Note that one of $p,p+1,p+2$ must be divisible by $3$, since they are three consecutive numbers, and since $p$ and $p+2$ are prime, that must be $p+1$. We can do the same to show that $p+1$ is divisible by $2$.
Looking modulo $6$.
We can look $\mod 6$. We see that
\begin{align}
6k+0\equiv 0\mod 6&\Righta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1700094",
"timestamp": "2023-03-29T00:00:00",
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Possibility of determining a certain length in a triangle. The figure is on the bottom of the question.
Suppose I have a triangle $ABC$ with an additional point $M$ on the middle of $AB$. Suppose that I know the following three quantitites, where $\ell(\cdot)$ represents the lenth of the line.
*
*$y:=\ell(AM)$
*$k:... | The 2nd and 3rd values determine two hyperbolas (with foci respectively $M,A$ and $M,B$).
The point C is the intersection of these two hyperbolas.
So yes, it's possible.
We first draw $A,B,M$, this we can obviously do. Then we draw the two hyperbolas, we see where they intersect, and that determines $C$ (I do not sa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1700223",
"timestamp": "2023-03-29T00:00:00",
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Find closed formula for $a_{n+1}=(n+1)a_{n}+n!$ $a_{n+1}=(n+1)a_{n}+n!$ where a0=0 and n>=0.
To get the closed form, I'm trying to find an exponential generating function for the above recurrence, but it doesn't seem to be very nice. Am I going about this the wrong way? If so, are there any helpful tricks I can use to ... | Dividing by $(n+1)!$ gives us that $$\frac{a_{n+1}}{(n+1)!}=\frac{a_{n}}{n!}+\frac{1}{n+1}$$
Now substituting $\frac{a_{n}}{n!}=b_{n}$ $$b_{n+1}=b_{n}+\frac{1}{n+1}$$
Thus $b_{n}$ is is the harmonic series.
| {
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"timestamp": "2023-03-29T00:00:00",
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Puzzle: Each entry in a number grid is the average of its neighbors I'm trying to solve the following puzzle:
Each number should be the average of its four neighbors. For example, $x$ should be equal to $\frac{1}{4}(4+10+y+z)$.
I don't know how to make a formula out of it. What's the trick? Can anyone give me a clue ?... | This is a Laplace equation with Dirichlet boundary condition, you can construct a matrix and solve it using Matlab, or you can use Jacobi iteration to solve it manually as the number of unknowns is very small.
You start with all unknowns set to zero. Then starting from the boundary, replace each unknown with the averag... | {
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Is A open or closed? Is it a bounded set? Verify if the given set $A$ it's open or closed. Also verify if it's bounded.
a) $A = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1\}$
I just want the method for verify this things in case of this example, because I have items a,b,c,d,e to do. The other sets are usually points o... | Some highlights: suppose $\;\{(x_n,y_n)\}\subset \Bbb R^2\setminus A\;$ convergent, then we have that
$$\begin{cases}x_n\xrightarrow[n\to\infty]{}x_0\\{}\\y_n\xrightarrow[n\to\infty]{}y_0\end{cases}\;\;\;\text{and}\;\;(x_0,y_0)\in\Bbb R^2\setminus A$$
as we know from limits in the real line, and thus the complement of ... | {
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Irreducible and recurrent Markov chain - theorem notation question In [J. R. Norris] Markov Chains (Cambridge Series in Statistical and Probabilistic Mathematics) (2009), page 35, Theorem 1.7.5 says:
In (ii), does it mean $\gamma^k$ is notation for $\gamma^k_i$ over all $i \in I$ or for some fixed arbitrary $i \in I$?... | Your first interpretation is correct: for example if $I=\{1,2,3\}$, then $\gamma^k$ is short for $(\gamma_1^k,\gamma_2^k,\gamma_3^k)$. Another way this is notated is $(\gamma_i^k)_{i \in I}$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Proving $\{9^n|n\in \mathbb Q\} = \{3^n|n\in \mathbb Q\}$ The statement is: $\{9^n|n\in \mathbb Q\} = \{3^n|n\in \mathbb Q\}$.
I am more familiar with traditional proofs. Do I just split up $9^n$ into $3^{n}\times 3^{n}$?
| Observe that for any $\;n\in\Bbb Q\;$
$$\begin{cases}9^n=3^{2n}\\{}\\3^{n}=9^{n/2}\end{cases}\;\;\;\implies\left\{\,9^n\;:\;n\in\Bbb Q\right\}=\left\{\,3^n\;:\;n\in\Bbb Q\right\}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Winding number of composition of maps If $f,g:S^{1}\rightarrow S^{1}$ maps, show that $N(f\circ g)=N(f)N(g)$, where $N(f)$ is the winding number of $f$.
We defined the winding number of $f$ to be $N(f)=\frac{1}{2\pi}(\tilde{f}(1)- \tilde{f}(0) )$ where $\tilde{f}$ is a lift of $f$.
I tried to look for a lift of $f \ci... | This may be a bit "cheating", but anyway... If $n = N(f)$ and $m = N(g)$, we know from the calculation of $\pi_1(S^1)$ that $f$ is homotopic to $z \mapsto z^n$ and $g$ is homotopic to $z \mapsto z^m$ (two maps with the same winding number are homotopic). But then $f \circ g$ is homotopic to $z \mapsto (z^m)^n = z^{mn}$... | {
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Proof similarity transformations are bijective. I'm trying to prove that similarity transformations from $\mathbb{R}^2$ to $\mathbb{R}^2$ are bijective, I've already proven its injectivity by contradiction, so I'm really just stuck on the surjectivity.
Let's define similarity transformations, we call a function $\psi :... | Hint:
Prove $\psi$ is an affine map, and the associated linear map is injective. Hence, as it is an endomorphism in finite dimension, this linear map is bijective.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1700921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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For what odd prime p is -3 a quadratic residues? Non-residue? For what odd prime p is -3 a quadratic residues? Non-residue?
*
*Having a bit of trouble with this question, we are currently covering a section on quadratic reciprocity and didn't really see anything in my notes that helped me solve this. Any help is gr... | We have $-3$ is a QR of the odd prime $p\ne 3$ in two situations:
(i) $-1$ is a QR of $p$ and $3$ is a QR of $p$ or
(ii) $-1$ is a NR of $p$ and $3$ is a NR of $p$.
Case (i): $-1$ is a QR of $p$ if $p\equiv 1\pmod{4}$. Since $p\equiv 1\pmod{4}$, by Quadratic Reciprocity we have that the Legendre symbol $(3/p)$ is equal... | {
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Sequence of Measurable Sets Converge Pointwise to a Measurable Set I was working on the folllowing exercise from Tao's Measure Theory book:
We say that a sequence $E_n$ of sets in $R^d$ converges pointwise to another set $E$ if the indicator functions $1_{E_n}$ converge pointwise to $1_E$. Show that if $E_n$ are all Le... | This is basically a baby version of the following result (you can skip to bottom).
If $(f_n)$ are measurable and $f_n \to f$ pointwise, then $f$ is measurable.
Why ? Because of pointwise convergence
$$\{x:f(x)> a\}=\cup_{n=1}^{\infty} \{x: f_m(x) > \alpha \mbox{ for all } m \ge n\}.$$
Read the RHS as all $x$ such t... | {
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How to determine if a probability problem has a binomial distribution or not? I'm studying for my Stats midterm and I am confused about the binomial distribution concept.
Among these 2 problems, why does the second question have a binomial distribution and not the first question?
1) In a large 2 lb bag of candies, 15% ... | From the wording of the first situation, it sounds like you are not replacing the candy after you pull it out. Therefore you are not repeating the same fixed experiment three times in a row; rather, you are changing the experiment (and the associated probability of success) with each pick.
| {
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"timestamp": "2023-03-29T00:00:00",
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Solve $ 6\sin^2(x)+\sin(x)\cos(x)-\cos^2(x)=5 $
Question: Solve $$ 6\sin^2(x)+\sin(x)\cos(x)-\cos^2(x)=5 $$
$$ 0 \le x \le 360^{\circ} $$
My attempt:
$$ 6\sin^2(x)+\sin(x)\cos(x)-\cos^2(x)=5 $$
$$ 6(\frac{1}{2} - \frac{\cos(2x)}{2}) + \sin(x)\cos(x) -(\frac{1}{2} + \frac{\cos(2x)}{2}) = 5 $$
$$ 3 - 3\cos(2x)+ \s... | HINT
Perhaps the other methods are easier but to continue where you left off,
Realize that $$\sin 2x=7\cos 2x+5$$
Use $$\sin^2 2x+\cos^2 2x=1$$
To make your last equation into a quadratic for $\cos 2x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1701363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Ansätze to solve 2-dimensional nonlinear integral curve I try to solve the initial value problem analytically $X(t_0) = X_0, X \in \mathbb{R}^2$ for the following integral curve:
$x'(t) = -y(t)+x(t) (x(t)^2+y(t)^2-1) \\
y'(t) = x(t)+y(t) (x(t)^2+y(t)^2-1) $
According to the Picard–Lindelöf theorem there should be a sol... | Try the following. We expand both equations by $y$ (first equation) and by $x$ (second equation).
$$x'y=-y^2+xy(x^2+y^2-1)$$
$$y'x=x^2+xy(x^2+y^2-1)$$
Now subtract both equations
$$y'x-x'y=x^2+y^2$$
After deviding by $x^2$, we obtain:
$$\frac{y'x-yx'}{x^2}=1+\frac{y^2}{x^2}$$
Notice that we have the derivative of $y/x$... | {
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Product of $W^{1,p}_0$ functions Let $p>n$, and let $f,g\in W^{1,p}_0(\mathbb{R}^n)$ be two sobolev functions. Prove that $fg\in W^{1,p}_0(\mathbb{R}^n)$.
I was able to prove the Leibniz formula for weak derivativatives, but still I do not understand why the function should be in $L^p$. Probabily it is Morrey inequalit... | Just a slight correction to @Diesirae92's answer, for posteriority's sake:
we can't really write
$$||f(g_n - g)||_{W^{1,p}(\mathbb{R}^n)} \leq ||f||_{\infty} ||(g_n - g)||_{W^{1,p}(\mathbb{R}^n)}$$
as the $W^{1,p}$ also involves the derivative, which may not be bounded.
Instead, what we should write is
\begin{equation}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the probability of dealing all $52$ cards in a standard well shuffled deck getting a single quad? I would like to know if you take a well shuffled deck of $52$ cards and then deal out all of them one at a time without replacement, what is the probability that they will be dealt such that there is exactly one qu... | Joriki provides the elegant solution ... but also shows the value of checking theoretical work via simulation. The discussion got me playing with simulating it in Mathematica, and thought I would post some code.
A pack of cards can be represented by the set:
cards = Table[Range[1, 13], 4] // Flatten
Sampling without ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1701627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Write $x_n=22..244...45$ as sum of $2$ squares I've recently came across this problem and, although I've spent time looking for a solution, I don't have any interesting ideas.
Let the numbers
$$x_1=25$$
$$x_2=2245$$
$$x_3=222445$$
and $x_n=22..244...45$ with $n$ digits $'2'$ and $n-1$ digits $'4'$
Prove that ... | \begin{align*}
x_{n} &= \left( \frac{10^{n}+a}{3} \right)^{2}+
\left( \frac{10^{n}+b}{3} \right)^{2} \\
&= \frac{2(10^{2n})+2(a+b)10^{n}+a^{2}+b^{2}}{9}
\end{align*}
Solving
$$\left \{ \begin{array}{ccc} a+b &=& 1 \\
a^{2}+b^{2} &=& 5 \end{array}
\right.$$
$$(a,b)=(2,-1),(-1,2)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1701703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Evaluate $\binom{m}{i} - \binom{m}{1}\binom {m-1}{ i} + \binom{m}{2}\binom{m - 2}{i} - \ldots + (-1)^{m-i} \binom{m}{m-i}\binom{ i }{i} $ Evaluate the expression
$$\binom{m}{i} - \binom{m}{1}\binom {m-1}{ i} + \binom{m}{2}\binom{m - 2}{i} - \ldots + (-1)^{m-i} \binom{m}{m-i}\binom{ i }{i} $$
I'm really stumped about ... | Here’s a combinatorial version Semiclassical’s computational argument.
You have a box of white balls numbered $1$ through $n$. For some $r$ such that $0\le r\le m-i$ you pick $r$ of the balls and coler them red, and then you pick $i$ of the remaining white balls and color them blue; for a given value of $r$ there are $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1701831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Looking for websites to brush up on algebra skills needed for calculus I've enrolled in an 8 week online Calculus 1 class, we're currently in week 2 and while I understand the calculus concepts (average rate of change, limits) I'm having a hard time on my homework due to not having a strong background in algebra. Are t... | Paul's Online Math Notes may be a good place to check out. You can either look it up on google or go to it here http://tutorial.math.lamar.edu/
It can help out with algebra as well as calculus.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1701917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How does the formula for length of a graph relate to that of parametric description? So I know how to solve for the length of a regular graph and for a parametric description.
Regular graph length formula:
$$L = \int_{x=a}^{x=b}\sqrt[]{1 + (f'(x))^2} \,\,\,dx$$
Parametric description length formula:
$$\int_{t = a}^{t... | They are radically similar ;) you forgot the integral in your first formula (edit: I see you fixed it now)
Notice that you can parametrize the graph of $f$ by $$(t, f(t))$$ Now use the second formula, you will get the first!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1701989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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How to show that $2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2))$ How can we show that:
$2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2))$
for large N.
| $\binom{N}{2}=N(N-1)/2\approx N^2/2$. So $2^{\binom{N}{2}}=\exp(\binom{N}{2}\ln(2))\approx \exp(\frac{N^2}{2}\ln(2))$. Note that $N^2$ is in the exponent, not outside as you have written.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1702081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Could we "invent" a number $h$ such that $h = {{1}\over{0}}$, similarly to the way we "invented" $i=\sqrt{-1}$? $\sqrt{-1}$ was completely undefined in the world before complex numbers. So we came up with $i$.
$1\over0$ is completely undefined in today's world; is there a reason we haven't come up with a new unit to d... | There are the dual numbers, which is another two dimensional associative algebra over the reals like the complex numbers. The basis elements are 1 and h, where we define h as a nonzero number whose square is zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1702197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Can an open ball be closed if the open ball contains infinite points? Consider a metric space $(X,d)$. Is the following statement true?
An non-finite open ball in X with finite radius is never closed. Non-finite in this sense means that the open ball contains an infinite amount of points.
| When $d$ is a metric then $e_1=d/(1+d)$ and $e_2=\max (d,1)$ are metrics equivalent to $d,$ that is, they generate the same topology that $d$ does. Since $\sup e_1\leq 1\geq \sup e_2,$ an open $e_1$-ball or $e_2$-ball of radius $2$ is the whole space, which is both open and closed, regardless of whether it is a finit... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 5
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Let $G$ be a simple graph on $10$ vertices and $38$ edges. Prove that $G$ contains $K_4$ as an induced subgraph. Let $G$ be a simple graph on $10$ vertices and $38$ edges. How do I prove that $K_4$ is the induced subgraph of G?
| Hint: There are only $7$ edges missing from $K_{10}$. Each missing edge can prevent up to $28$ different 4-vertex sets from inducing a $K_4$. How many 4-vertex sets are there?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1702368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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When a graded ring is Cohen-Macaulay? I am trying to solve exercise 19.10 from Eisenbud's Commutative Algebra.
I want to show that if $R=k[x_0,...,x_n]/I$ is a graded ring, then $R$ is Cohen-Macaulay iff $R_{\mathfrak p}$ is Cohen-Macaulay, where $\mathfrak p=(x_0,...,x_n)$.
The hint is to use the graded Auslander-... | This is consequence of
Bruns-Herzog, Exercise 2.1.27 (c): Let $R$ be a Noetherian graded ring and $M$ a finite graded R-module. Suppose in addition that $(R,m)$ is $^*$local. Then $M$ is Cohen-Macaulay if and only if $M_m$ is.
Assuming that we know proofs of former parts, this part can be easily proved:
Let $p$ be a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1702509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Decreasing perpetuity problem A perpetuity pays 1000 immediately. The second payment is 97% of the first payment and is made at the end of the fourth year. Each subsequent payment is 97% of the previous payment and is paid four years after the previous payment. Calculate the present value of this annuity at an annual e... | This is a perpetuity due decreasing in geometric progression and payable less frequently than interest is convertible.
The effective interest rate per period is
$$
i=(1+0.08)^4-1=36.05\%
$$
and the growing rate is $g=-3\%$ (decreasing).
So the perpetuity due has the present value
$$
PV=1000\frac{1+i}{i-g}=3,484.07
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1702665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If a, b, c are three natural numbers with $\gcd(a,b,c) = 1$ such that $\frac{1}{a}+\frac{1}{b}=\frac{1}{c}$ then show that $a+b$ is a square.
If a, b, c are three natural numbers with $\gcd(a,b,c) = 1$ such that $$\frac{1}{a} + \frac{1}{b}= \frac{1}{c}$$ then show that $a+b$ is a perfect square.
This can be simplifie... | To show this, we note that $c(a+b)=ab$. Now let $g$ be the gcd of $a$ and $b$, which need not necessarily be $1$. Denote $a=a'g$ and $b=b'g$ so that we get $c(a'+b') = a'b'g$.
Because $a' + b'$ is relatively prime to both $a'$ and $b'$, it follows that it divides $g$. But g also divides $c(a'+b')$. Further, note that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1702758",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Probability that one soccer player will have his favourite t-shirt in the dressing room.
In a soccer team there are $11$ players and $11$ t-shirts numbered
from $1$ to $11$.The players go in the dressing room one at a
time, casually.
Each one, once he arrives, takes a random t-shirt, except
Danilo who prefers... | $P(Danielo\; selects\; his\; prefered\; 8\; number\; Tshirt)=1$
$P(Danielo \; enters \; at\; any\; position )=\frac{1}{11}$
How will Danielo get his prefered T shirt??
This can happen if Danielo goes first and picks his tshirt OR goes Second and picks his tshirt but keeping in mind that the person before him selects a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1702860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Looking for an example of an infinite metric space $X$ such that there exist a continuous bijection $f: X \to X$ which is not a homeomorphism I am looking for an example of an infinite metric space $X$ such that there exists a continuous bijection $f: X \to X$ which is not a homeomorphism. Please help. Thanks in advanc... | The Cantor set minus one point.
Indeed, in general let $u:U\to V$ is a non-open continuous bijection between topological spaces. Define $X=U\times\mathbf{N}^c\sqcup V\times\mathbf{N}$ (I assume $0\in\mathbf{N}$ and I mean the complement in $\mathbf{Z}$). Define $f(x,n)=(x,n+1)$ for $n\neq 0$ and $f(x,-1)=(u(x),0)$. The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1702979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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explanation for multiplication rule for independent events with|A| > 2 This is my assumption:
if A is the set of events:
$ A = \{A1, A2, A3\} $
And i want to find out if they are mutually independent, all i have to do is check that the two following conditions hold true:
$ r1: P(A1 \cap A2) = P(A1)P(A2) $
$ \land $
$ r... | A finite set of events is mutually independent if and only if every event is independent of any intersection of the other events — that is, if and only if for every $n$-element subset ${A_i}$,
$$
\mathrm{P}\left(\bigcap_{i=1}^n A_i\right)=\prod_{i=1}^n \mathrm{P}(A_i)
$$
(see here). Hence, we also have to check that $P... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If the tangent at the point $P$ of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the major axis and minor axis at $T$ and $t$ respectively If the tangent at the point $P$ of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the major axis and minor axis at $T$ and $t$ respectively and $CY$ is perpendicular o... | Equation of $Tt$ is $bx\cos \theta + ay\sin \theta - ab = 0$. Hence $C{Y^2} = \frac{{ab}}{{{b^2}{{\cos }^2}\theta + {a^2}{{\sin }^2}\theta }}$ and $C{P^2} = {a^2}{\cos ^2}\theta + {b^2}{\sin ^2}\theta $. Now,
$$P{Y^2} = C{P^2} - C{Y^2} = \frac{{ab}}{{{b^2}{{\cos }^2}\theta + {a^2}{{\sin }^2}\theta }} - \left( {{a^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1703222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Definite Integral - Question about notation For an arbitrary function $f(t)$ we define:
\begin{equation}
\bar{f}(t) =\int^{t}_{0} f(\tau) d\tau
\end{equation}
Is it true the following?
For an arbitrary function $f(x,t)$ we define:
\begin{equation}
\bar{f}(x,t) =\int^{t}_{0} f(x) dx
\end{equation}
I get confused becau... | No, the right-hand-side does not depend on x, since you integrate over it.
You might define a function like
$$\bar{f}(x,t)=\int_0^t f(x,y)dy$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Morphisms in the category of rings We know that in the category of (unitary) rings, $\mathbb{Z}$ is the itinial object, i.e. it is the only ring such that for each ring $A,$ there exists a unique ring homomorphism $f:\mathbb{Z} \to A$.
This means, in particular, that $\mathbb{R}$ does not satisfy this property, so for ... | In fact, there is a universal ring with two distinct morphisms from $\mathbb{R}$, the ring $\mathbb{R}\otimes_\mathbb{Z}\mathbb{R}$.
I am unsure of whether it has nicely-presented quotients with the same property.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1703553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 7,
"answer_id": 2
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Fourier analysis — Proving an equality given $f, g \in L^1[0, 2\pi]$ and $g$ bounded We were given a challenge by our Real Analysis professor and I've been stuck on it for a while now. Here's the problem:
Consider the $2\pi$-periodic functions $f, g \in L^1[0, 2\pi]$. If $g$ is bounded show that
$$
\lim_{n \to +\in... | Let $\epsilon > 0$ be given.
Let $M$ be a bound for $|g|$, which is assumed to exist. Define $f_{N}(t)=f(t)\chi_{\{ x : |f(x)| \le N\}}(t)$. Let $g_{\delta}$ be a standard mollification of $g$. Then $g_{\delta}\in\mathcal{C}^{\infty}(\mathbb{R})$ is $2\pi$-periodic, $|g_{\delta}| \le M$, and
$$
\lim_{\delta\right... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is there a function for every possible line? I'm currently in Pre-Calculus (High School), and I have a relatively simple question that crossed my mind earlier today. If I were to take a graph and draw a random line of any finite length, in which no two points along this line had the same $x$ coordinate, would there be ... | The only straight lines in the $x$-$y$ plane that are not functions are those that are perfectly vertical. Those are of the form $x=c$, where $c$ is a constant.
All other lines can be expressed in the form $y =f (x)= mx + b$ where $m $ is the slope of the line and $b $ is the $y$-intercept-- the $y$ value when $x$ i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Is $\sum_{n=0}^\infty (a \cdot r^n)$ equivalent to $\lim_{n \to \infty}\sum_{k=0}^n (a \cdot r^k)$? In other words, when writing down an infinite sum, are we always implying that it's actually the limit of that series as the number of terms approaches infinity, or is there some subtle difference?
| The sum
$$\sum_{n=0}^\infty$$
is defined to be
$$\lim_{k\to\infty}\sum_{n=0}^k$$
so yes they are the same. Of course this is an abuse of notation, since $\infty$ is not a number. In the same way, it is not "proper" to write the interval $[0,\infty)$, but we all understand what it means.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1703847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve the differential equation $y'' + y' + a(x e^{-x} - 1 + e^{-x}) = 0$ I am a bit stuck on the particular solution for this system. According to Wolfram, the general solution is
$$\frac{1}{2} a e^{-x} x^2 + 2 a x e^{-x} + \frac 12 x + \frac 12 c_1 e^{-x} + c_2$$
The last two terms are from the homogenous solution, ... | Observe that $xe^{-x}+e^{-x}-1$ is sum of "product of polynomial and exponential" and polynomial. Intiutively, we can suppose that some particular solutions is also "product of polynomial and exponential" and polynomial. Let $y=p(x)e^{-x}+ \alpha x^2+\beta x+\gamma$, where $p(x)$ is a polynomial. Then
\begin{align}
y'... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1703981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How can I calculate this limit $\lim_{(x,y)\to(0,0)} \frac{xy(1-cos(x^2+y^2))}{(x^2+y^2)^{\frac{5}{2}}}$? How can I calculate this limit $$\lim_{(x,y)\to(0,0)} \dfrac{xy(1-cos(x^2+y^2))}{(x^2+y^2)^{\frac{5}{2}}}$$ at the origin?
I tried to use the substitution $x ^2 + y^2=t$ but how can I evaluate the value of $xy$? I... | Outline: Note that $(x-y)^2\ge 0$, so $|xy|\le \frac{1}{2}(x^2+y^2)$. One can also get this from polar coordinates, for $|xy|=r^2|\cos\theta\sin\theta|=\frac{1}{2}r^2|\sin(2\theta)|\le \frac{r^2}{2}$.
Now you can comfortably let $t=x^2+y^2$. You will need to look at the behaviour of $1-\cos t$ near $0$. This can be ta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1704075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Does the following sequence converge, if so find the limit, if not prove that no limit exists: $a_n=(-1)^nn^2$ $$a_n=(-1)^nn^2$$
I know that this sequence does not converge and hence does not have a limit. I have tried proving that this sequence does not have a limit by contradiction.
I assumed that the limit 'a' exis... | There are different ways to see that it does not converge. One of the simplest ways is to note that your sequence $a_n$ is unbounded: It tends to $\infty$ along all even numbers $2n$, and tends to $-\infty$ along all odd numbers $2n+1$. Since a convergent sequence is bounded, this gives the result.
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove that there are infinitely many integers n such that $4n^2+1$ is divisible by both $13$ and $5$
Prove that there are infinitely many integers n such that $4n^2+1$ is divisible by both $13$ and $5$.
This can be written as:
$$65k = (2n)^2 + 1$$
It's clear that $k$ will always be odd.
Now I am stuck. I wrote a prog... | $$4n^2+1\equiv0\pmod{13}\iff4n^2\equiv-1+65$$
As $(4,13)=1,4n^2\equiv64\iff n^2\equiv16\iff n\equiv\pm4\pmod{13} $
Similarly, $n\equiv\pm4\pmod5$
Now use Chinese Remainder Theorem
| {
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"url": "https://math.stackexchange.com/questions/1704353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 10,
"answer_id": 7
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Show $F_1$ is a continuous linear functional Let $(X,||\cdot ||)=(C[0,1],||\cdot ||_{\infty})$ and $F_1 :X \rightarrow \mathbb R$ be defined by $$F_1 (f)=\int _{1/2}^{3/4} f(t) dt$$ Show $F_1$ is a continuous linear functional.
So we need to show it is a linear functional first and then show it is continuous. By our l... | For any $\;\epsilon>0\;$ choose $\;\delta=4\epsilon\;$ , so that
$$||f-0||_\infty\iff ||f||_\infty<\delta\implies\left\|\int_{1/2}^{3/4}f(t)dt\right\|\le\int_{1/2}^{3/4}||f(t)||dt\le\delta\int_{1/2}^{3/4}dt=\frac14\delta=\epsilon$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1704431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Contour integral of $\frac{x^{p-1}}{1+x}$ I am trying to find the integral $$\int_0^\infty\frac{x^{p-1}}{1+x}\;\mathbb{d}x$$
I know that this is easily expressible in terms of beta function. But i need to prove that it's value is $\dfrac{\pi}{\sin{p\pi}}$ using a good contour I guess.
I tried of taking a counter like t... | There is no contour for the function as it is in the question, since
$p-1$ is a real number in the interval $(-1,0).$ If $p-1$ is rational we
have a multivalue function with as many values as the denominator of $p-1$ and this implies to create some brunch cuts on the contour. If $p-1$ is
irrational then there are an in... | {
"language": "en",
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"answer_id": 2
} |
Prove $x^2+y^4=1994$ Let $x$ and $y$ positive integers with $y>3$, and $$x^2+y^4=2(x-6)^2+2(y+1)^2$$
Prove that $x^2+y^4=1994$.
I've tried finding an upper bound on the value of $x$ or $y$, but without sucess. Can anyone help me prove this problem? Note that $x^2+y^4=1994$ is the result we are trying to prove, not an a... | Hint replacing the $x^2,y^4$ with given condition we get $(x-6)^2+(y+1)^2=992$ so thats equal to the equation of a circle located at $h,k$ ie(6,-1) so it got only $4$ integer points which can be proved by using symmetry and at $x,y$ axis as its radius is approximately $31.5$ but out of those $4$ integer points only $37... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1704660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
How can addition be non-recursive? Tennenbaum's theorem says neither addition nor multiplication can be recursive in a non-standard model of arithmetic. I assume recursive means computable and computable means computable by a Turing machine (TM).
The 3-state TM described below takes the unary representation of two natu... | If you take a countable nonstandard model, you first enumerate that model in $\omega$. That is, you consider the domain of the model to be a set $\{ a_0, a_1, \ldots \}$.
In your algorithm, the input $n$ represents the actual number $n$, and not the element $a_n$ of the model. But this is not what the question is askin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1704855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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tricky GCD Question I am trying to show that the $ gcd (\frac {n}{gcd(n,d)} , \frac {d}{gcd(n,d)} ) = 1 $.
My steps let $gcd(n,d) = K = xn+yd$ then I need to find some linear combination of $\frac {n} {K} $ and $\frac {d} {K} $ that gives 1. Any hints would be appreciated.
| We have
$$ x \cdot \frac nK + y \cdot \frac dK = \frac{xn + yd}K = 1. $$
Hence $\gcd(\frac nK, \frac dK) = 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1704962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How can I determine if this function is continuous at x=1? Is 2/0 a discontinuity or infinity for a function?
For the question: Given the function $ (x^2+1)/(x-1) $, is the function continuous at x=1?
When I took the right hand and left limits, I got infinity in both cases and f(1) would be 2/0 which I was informed e... | For a function $f(x)$ to be continuous at some point $c$ of its domain, it has to satisfy the following three conditions:
*
*$f$ has to be defined at $c$
*$\lim\limits_{x \to c} f(x)$ has to exist
*the value of the limit must equal to $c$
In your case, the function $\frac{x^2+1}{x-1}$ is not defined at $x=1$, so... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1705148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Expansion and factorisation I have a little problems with a few questions here and I need help.. Thanks ...
*
*Factorise completely
$$9x^4 - 4x^2 - 9x^2y^2 + 4y^2 $$
My workings ..
$$ (3x^2+2x)(3x^2-2x) - y^2 (9x^2-4) = (3x^2 + 2x)(3x^2 -2x) - y^2 (3x+2)(3x-2) $$
*Factorise $3x^2 + 11x - 20$ and , hence Factori... | *
*$$9x^4 - 9x^2y^2 - 4x^2 + 4y^2$$
Group in paris such that; $9x^2(x^2-y^2) - 4(x^2 - y^2) = (x^2 - y^2)(9x^2 - 4)$
Then using the difference of 2 squares we get: $$(x-y)(x+y)(3x-2)(3x+2)$$
2.$3x^2 + 11x - 20$, factorise to $(x-5)(3x-4)$. As @mathlove pointed out $11(a-b) - 20 + 3(?)^2$, where $ ? = a-b$. Which is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1705373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Finding modular inverse (wrong approach) I'm trying to find the modular inverse of $$30 \pmod{7} $$ I have tried using the Euclidean algorithm and it gave me the right answer, which is $x \equiv 6 \pmod{7} $. However, I tried using another approach that I thought would be simpler, but it resulted in a wrong answer. The... | First, write $\;30=2\pmod 7\;$ , and now use the Euclidean algorithm with this, which is way easier.
By the way, the answer indeed is $\;4\;$ , since $\;30\cdot4=120=1+17\cdot7\;$ , or simpler: $\;2\cdot4=1+7\;$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1705459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Using properties of limits, calculate $\lim_{n\to \infty}\left(\frac1{n^2}+\frac1{(n+1)^2}+\cdots+\frac1{(2n)^2}\right)$ Using the properties of limits, calculate the following limits, if they exist. If not, prove they do not exist:
$$\lim_{n\to \infty}\left(\frac1{n^2}+\frac1{(n+1)^2}+\frac1{(n+2)^2}+\cdots+\frac1{(2n... | Let F(n): $$\frac{1}{n^2}+\frac{1}{(n+1)^2}+\cdots+\frac{1}{(2n)^2}$$
Consider two series:
$$G(n): \frac{1}{n^2}+\frac{1}{n^2}+\frac{1}{n^2}+\cdots +\frac{1}{n^2}$$
and
$$H(n):\frac{1}{(2n)^2}+\frac{1}{(2n)^2}+\frac{1}{(2n)^2}+\cdots+\frac{1}{(2n)^2}$$
Notice that
$$H(n)<F(n)<G(n)$$
Now, $$\lim_{n \to \infty} G(n) = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1705543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Does this apply: $\sin^2(\pi t-\pi) + \cos^2(\pi t-\pi) = 1$ So I'm solving for the length of the following parametric description:
$$
\Bigg[x(t)=3\cos(\pi t-\pi)\,\,\,\,\,\,\,y(t)=3\sin(\pi t - \pi)$$
I applied the formula for solving for length, namely:
$$\int_{0}^{2} \sqrt{(-3\pi \sin(\pi t - \pi))^2+(3\pi \cos(\pi ... | Hint you can verify using formulae $$\sin(a-b)=\sin(a)\cos(b)-\cos(a)\sin(b)$$ and $$cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1705606",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Show that $R(k,l) = R(l,k)$ Let $R(k,l)$ denote the Ramsey number.We proved in class a theorem that says
$$R(k,l) \leq {k+l-2\choose{k-1}} $$
And supposedly we can use this to show that $R(k,l) = R(l,k)$ for all $k,l \in \mathbb{N}$. However I am not seeing it. I feel like I use should induction, but I'm not sure how.... | $${k+l-2\choose{k-1}}=\frac{(k+l-2)!}{(k-1)!(k+l-2-(k-1))!}=\frac{(k+l-2)!}{(k+l-2-(l-1))!(l-1)!}={k+l-2\choose{l-1}}$$
Thus, $k$ and $l$ are interchangable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1705707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Show that $f(x)\equiv 0$ if $ \int_0^1x^nf(x)\,dx=0$ Let $f \in C[0,1]$. If for each integer $n \ge 0$ we have
$$ \int_0^1x^nf(x)\,dx=0$$ show that $f(x) \equiv 0$
| The equality implies that $f$ is orthogonal to every polynomial in $L_2[0,1]$, so in particular it is orthogonal to the Legendre polynomials (which form a basis in $L_2[0,1]$). This implies that $f$ must be $0$ a.e., and Since $f$ is continuous, we must conclude that $f \equiv0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1705836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Solving equation with Complex Numbers and two unknowns. I have the following track:
If $z=a+bi$, with the imaginary number $i$, solve the following equation
$$z^2+|z|^2-18=0$$
Well, first of all I replaced $z$ with $a+bi$ and then $|a+bi|^2$ with $(a^2+b^2)$.
After, I got $a$ as a function of $b$, but in this case ... | Assume that $z$ is a solution, then
$$
z^2 = 18-|z|^2
$$
and hence
$$
|z|^2 = |18 - |z|^2|
$$
Put $t = |z|$. Thus $t^2 = 18-t^2$ or $t^2 = t^2 - 18$, but the second possibility clearly can't occur. Hence $2t^2 = 18$, so $t = \pm 3$, but since $t$ is positive, we must have $t=3$.
Returning to the original equation, it n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1705925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that finding a cliques is NP-Complete? I presented my proof bellow, so is it complete (formal) proof.
Proof :
1- We can verify the solution in Polynomial time.
2- 3-SAT (NP-Complete) can be polynomialy reduced to clique! (as the following):
*
*We assume a formula 3-sat (F) as an example :
F= (X1 + X2 + X3).(... | Your proof is correct and your steps are clear.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1706043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove $\frac{\sec(x) - \csc(x)}{\tan(x) - \cot(x)}$ $=$ $\frac{\tan(x) + \cot(x)}{\sec(x) + \csc(x)}$
Question: Prove $\frac{\sec(x) - \csc(x)}{\tan(x) - \cot(x)}$ $=$ $\frac{\tan(x) + \cot(x)}{\sec(x) + \csc(x)}$
My attempt:
$$\frac{\sec(x) - \csc(x)}{\tan(x) - \cot(x)}$$
$$ \frac{\frac {1}{\cos(x)} - \frac{1}{\s... | multiply and divide $ \frac{1}{\sin{x}+\cos{x}}$ by $ \frac{1}{\sin{x}\cos{x}}$ then in the numerator substitute $1$ by $sin^2{x} + cos^2{x}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1706184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 2
} |
What is the mean value of $\max\limits_{j}X_j$? Let $X_j$ be a random variable that is $1$ with probability $x_j^*$, and $0$ with probability $1-x_j^*$. The random variables $X_j$ are independent and $j$ belongs to $\{1,\ldots,n\}$ for some positive integer $n$. I would like to calculate the mean value of $\max\limits_... | The max is either $0$ or $1$.
The probability that it's $0$ is the probability that all (independent) $X_j$ are $0$, that is
$$P(max X_j =0)=\prod_j P(X_j=0)=\prod_j (1-x^*_j)$$
And of course,
$$P(max X_j =1)=1-\prod_j (1-x^*_j)$$
Thus $E(max X_j)=1-\prod_j (1-x^*_j)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1706298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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How to find the unknown in this log inequality?? Find all values of the parameter a $\in\Bbb R$ for which the following inequality is valid for all x $\in\Bbb R$.
$$ 1+\log_5(x^2+1)\ge \log_5(ax^2+4x+a) $$
I'm lost when I got to this stage: $ 5x^2-4x+5\ge ax^2+a$
I did this but still don't know how to proceed: $ (5-a)... | Hint: To find where a quadratic function $f$ satisfies $f(x)\geq 0$, first find where $f(x)=0$. On each of the remaining intervals, the sign of $f$ must be constant. The same method is useful for any such inequality ($<,>,\leq,\geq$).
This method works with any polynomial $f$, not just quadratics.
It also extends easil... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1706398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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For what simple piecewise smooth loops is this integral zero? I'm trying to solve the following problem:
For what simple piecewise smooth loops $\gamma$ does the following equation hold:
$$\int\limits_\gamma \frac{1}{z^2 + z + 1} \, \mathrm{d}z= 0$$
I'm allowed to appeal to Cauchy's integral theorem, and I have d... | The roots of $z^{3}-1$ are $e^{k(2\pi i/3)}$ for $k=0,1,2$ and
$$
z^3-1 = (z-1)(z^2+z+1).
$$
So $(z-e^{2\pi i/3})(z-e^{-2\pi i/3})=z^2+z+1$, and
$$
\frac{1}{(z-e^{2\pi i/3})(z-e^{-2\pi i/3})}=\frac{A}{z-e^{2\pi i/3}}+\frac{B}{z-e^{-2\pi i/3}},
$$
where $A$, $B$ are easily seen to satisfy $A=-B$.
Assumi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1706524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Show that the set of all even permutations in $G$ forms a subgroup of $G$.
Show that if $G$ is any group of permutations, then the set of all even permutations in $G$ forms a subgroup of $G$.
I know that I need to show the closure, identity, and inverses properties hold. So I need to prove:
1) If $p$ and $q$ are even... | Let $E$ be the set of even permutations in $G$ (which is presumably a group of permutations).
Let $p$ and $q$ be elements of $E$.
Check to see if $pq^{-1}$ is also an element of $E$. (Note: this checks all three conditions simultaneously).
A permutation is called an even permutation if its expression as a product of d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1706634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Volume of a curved cone Apparently, the volume of this cone is (1/16)π(r^2)h. My question is why this is the case, can someone please geometrically explain the reason behind the 1/16 bit. The radius is supposed to be proportional to the square of its height. Thanks.
| \begin{equation}
r=a \cdot h^2
\tag{01}
\end{equation}
\begin{equation}
dV=\pi r^{2} dh = \pi a^{2}h^{4}dh
\tag{02}
\end{equation}
\begin{equation}
V=\int_{h=0}^{h=H}\pi a^{2}h^{4}dh=\dfrac{\pi a^{2}H^{5}}{5}=\dfrac{\pi R^{2}H}{5}
\tag{03}
\end{equation}
If instead of eq.(01) your curve was
\begin{equation}
r=a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1706732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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The meaning of bounding curve of a surface I am reading Stokes Theorem. I am not able to understand the meaning of bounding curve of a surface. What is the definition of boundary curve of a surface $$z=f(x,y)$$
In particular I am trying to figure out the boundary of the portion of the surface $x^2 + y^2 + z^2 =25$ belo... | There is no boundary for $z = f(x,y)$ unless you specify a domain for $(x,y)$. In your particular example, the boundary is the circle where the plane $z=4$ intersects the sphere $x^2 + y^2 + z^2 = 25$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1706829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
continuous and uniformly continuous I remember proving $x^3$ is not uniformly continuous on $\mathbb{R}$.
Then I read the proof of a theorem: Suppose $D$ is compact. Function $f: D \rightarrow \mathbb{R}$ is continuous on $D$ if and only if f is uniformly continuous.
It's obvious that $f(x)=x^3$ is continuous at $x$ f... | Since your function is a polynomial and hence derivable.Thus using the fact if the derivative exists and is bounded then it is is uniformly continuous.
$f(x)=x^3$, $f'(x)=3x^2$ and it is bounded on your interval
$|f'(x)|\le 3b^2$ for all $x$ in $[a,b]$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1706910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
"Natural" example of cosets Do you know natural/concrete/appealing examples of right/left cosets in group theory ?
This notion is a powerful tool but also a very abstract one for beginners so this is why I'm looking for friendly examples.
| While finite group examples may be easier to first digest, cosets naturally come up in calculus as a way to say what indefinite integrals are: the indefinite integral of an integrable function $f$ is the coset $ \{ F + c : c \in \mathbb R \} = F + \mathbb R$, where $F$ is some antiderivative of $f$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1706973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "57",
"answer_count": 14,
"answer_id": 12
} |
How to simplify this surd: $\sqrt{1+\frac{\sqrt{3}}{2}}+\sqrt{1-\frac{\sqrt{3}}{2}}$ $$\sqrt{1+\frac{\sqrt{3}}{2}}+\sqrt{1-\frac{\sqrt{3}}{2}} = x$$
We have to find the value of $x$.
Taking the terms to other side and squaring is increasing the power of $x$ rapidly, and it becomes unsolvable mess.
I think the answer l... | Hint: square the surd and note the cross term may be simplified.
Also note that $\sqrt{3}/2 = \cos{(\pi/6)}$ and you may use the double angle formula $1 +\cos{t} = 2 \cos^2{(t/2)}$, etc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1707100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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The concept of "almost everywhere" wrt to zero-dimensional measure $\mathcal H^0$ From my understanding, a function $u$ is defined $\mathcal H^0$-a.e. on an interval $I$ means that $u$ is only not "defined" on a set of points $A \subset I$ such that $\mathcal H^0(A)=0$
However, I think $\mathcal H^0(A)=0$ means that $A... | Yes, the zero-dimensional Hausdorff measure $\mathcal H^0$ is simply the counting measure, and the only null set for this measure is the empty set.
Example of usage: if $A\subset \mathbb{R}^2$ is a Borel set of zero length (meaning $\mathcal H^1(A)=0$), then for almost every $x\in\mathbb{R}$ the slice
$$A_x = \{y\in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1707220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Necessity of Galerkin Approximation in Existence Proof for Parabolic PDEs I'm studying the proof of the existence of solutions to linear parabolic PDEs, e.g. the heat equation $u_t = \Delta u + f$. The standard technique seems to be to do a Galerkin approximation, see e.g. Section 6.5 in these notes.
My question is: I... | I think I've found the answer. The proof is based on showing that the map
$$
\Phi(u) := u_0 + \int_0^t \Delta u + f \, dt
$$
is a contraction and hence has a fixed point. In the infinite-dimensional problem, this definition of $\Phi$ doesn't make sense since $\Delta : H^1_0 \to H^{-1}$ and thus it is not clear what the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1707285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Deck transformation of $p : Y \to X : z \mapsto z^3 - 3z$ Let $X = \mathbb{C} \setminus \{ \pm 2 \}$ and $Y = \mathbb{C} \setminus \{ \pm 1, \pm 2 \}$. The map
$$
p : Y \to X : z \mapsto z^3 - 3z
$$
is a 3-branched covering.
Problem: Find $\operatorname{Deck}(Y/X)$, the group of Deck transformations of $Y$.
My try: My ... | Too long for a comment. This is an idea for a solution. It seems that the covering is not normal.
Let $a$, $b$, $c$, $d$ be loops around $-2,-1,1,2$ which generate $\pi_1(Y)=F(a,b,c,d)$ and $u,v$ be loops around $-2,2$ which generate $\pi_1(X)=F(u,v)$ then $p_{*}(a)=u^3$, $p_*(b)=uv^2$, $p_{*}(c)=u^2v$, $p_{*}(d)=v^3$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1707382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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For which parameter values does the series $\sum 1/(n^\beta + (-1)^n n^\alpha)$ converge? Given that $0 < \beta < \alpha \le 1$, for which $\alpha,\beta$ does the series
$\sum^\infty_{n=1}\frac {1}{n^\beta + (-1)^n n^\alpha}$
converge ?
| Write $a_n$ for the general term of the series. Also, let $\delta = a-b$.
Set
$$b_n = a_n - \frac{(-1)^n}{ n^{a}} = \frac{(-1)^n}{n^a (1+ (-1)^n n^{-\delta})}-\frac{(-1)^n}{n^a}=\frac{(-1)^n}{n^a} \times \left (1- \frac{1}{1+ (-1)^n n^{-\delta}}\right)=(*).$$
Now $$(*)= \frac{(-1)^n}{n^a} \times \frac{(-1)^n n ^{-\de... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1707516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Unknown Inequality $$ \left( \sqrt{3}\sqrt{y(x+y+z)}+\sqrt{xz}\right)\left( \sqrt{3}\sqrt{x(x+y+z)}+\sqrt{yz}\right)\left( \sqrt{3}\sqrt{z(x+y+z)}+\sqrt{xy}\right) \leq 8(y+x)(x+z)(y+z)$$
I can prove this inequality, but i need know if this inequaliy is known...
| We need to prove that
$$\prod\limits_{cyc}(a\sqrt{3(a^2+b^2+c^2)}+bc)\leq8\prod\limits_{cyc}(a^2+b^2)$$
which is true even for all reals $a$, $b$ and $c$.
Indeed, let $a+b+c=3u$, $ab+ac+bc=3v^2$, where $v^2$ can be negative, and $abc=w^3$.
Hence, it's obvious that the last inequality is equivalent yo $f(w^3)\leq0$, whe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1707614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How to determine if the following series converge or not? $\Sigma_{n=1}^{\infty} a_n $ where:
*
*$ a_n = \frac{1}{\ln(n)^{\ln(n)}}$
*$a_n = \frac{1}{n }-\ln\left( 1+\frac{1}{n}\right)$
in the first case, I really have no idea
in the second case, is it correct to say that for $ \frac{1}{n }-\ln\left( 1+\frac{1}{n}\r... | For
2.
$\begin{array}\\
a_n
&= \frac{1}{n }-\ln\left( 1+\frac{1}{n}\right)\\
&= \frac{1}{n }-\int_1^{1+1/n} \frac{dx}{x}\\
&= \frac{1}{n }-\int_0^{1/n} \frac{dx}{1+x}\\
&= \int_0^{1/n} (1-\frac{1}{1+x})dx\\
&= \int_0^{1/n} (\frac{x}{1+x})dx\\
&< \int_0^{1/n} x\,dx\\
&= \frac{x^2}{2}|_0^{1/n}\\
&= \frac{1}{2n^2}\\
\end... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1707695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Distance in Poincaré disk from origin to a point given Let $C$ circle $x^2+y^2=1$ find the distance (Poincaré disk) from $O=(0,0)$ to $(x,y)$
The distance in Poincaré is $d=ln(AB,PQ)$ where AB are a segment of the curve and P and Q are points in the limits of Poincaré disk. Then $A=(0,0)$ and $B=(x,y)$ but I dont know ... | See, metric entry in the Poincaré Disk Model :
$$|u| = \sqrt{x^2 + y^2}$$
$$\delta(u, v) = 2 * \frac{|u-v]^2}{(1 - |u|^2)(1 - |v|^2)}$$
In your case, $v = O$ and $|v| = 0$ so :
$$\delta(u, O) = 2 * \frac{|u|^2}{(1 - |u|^2)}$$
$$argcosh(1 + \delta(u, O))$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1707798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Some proofs about the infinite products; the series and the analyticity Definition A.1.4. The infinite product $\prod_{n=1}^{\infty}(1+a_{n}(x))$, where $x$ is a real or complex variable in a domain, is uniformly convergent if $p_{n}(x)=\prod_{m=k}^{n}(1+a_{n}(x))$ converges uniformly in that domain, for each $k$.
Theo... | No, that's totally wrong. For the series to be convergent, you certainly need $M_n \to 0$. But $1 + |a_n| > 1$, which is certainly not bounded by $2 M_n$.
Hint: What you need to do is use logarithms:
$$\prod_{n=1}^N (1 + a_n(x)) = \exp\left(\sum_{n=1}^N \log(1+a_n(x))\right)$$
As for the corollary, that's just the f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1707914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Can a bounded function always attain its least upper bound on a bounded rectangle in $R^n$? Suppose we have a rectangle $Q$, and $Q\subset R^n$. Then $Q$ is bounded by the definition of higher dimensional rectangles. Suppose $f$ is a bounded function defined on $Q$. Since $f$ is bounded, we can produce its infimum $\in... | Consider the rectangle $Q$ given by $(0,1)^n\subset \mathbb R^n$ with coordinates $x_1\cdots x_n$. Then the function $f(x_1\cdots x_n)=x_1$ is bounded on $Q$ but attains neither its supremum on $Q$ (which is $1$) nor its infimum on $Q$ (which is $0$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1708056",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Counterexample Question Let $f:X\rightarrow Y$ be a morphism of varieties. If $f(X)$ is dense in $Y$, then $\tilde{f}:\Gamma(Y)\rightarrow \Gamma(X)$ is injective, where $\tilde{f}$ is the homomorphism induced by $f$. In fact, if $X$ and $Y$ are affine, then we have if and only if. Can we relax the prerequisites a bit... | Take $X$ to be an embedding of a closed point into $Y=\mathbb P^1$. Then $\Gamma(Y)\to\Gamma(X)$ is an isomorphism.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1708153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
How do I write $F_1+F_3+F_5+\ldots+F_{2n-1}$ in summation notation? How do I write $F_1+F_3+F_5+\ldots+F_{2n-1}$ in summation notation?
$F_i$ represents the Fibonacci sequence. I can't figure out how to write this in summation notation. Clear steps would be marvelous.
| Since the index is odd, ie $\{1,3,5,7,...\}$ then $2n+1$ gives the odd numbers for $n \geq 0$. In summation form it is seen to be
$$\sum_{n=0}^{m-1} F_{2n+1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1708232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Functional Limits and Continuity so here is a problem I have been working on:
Assume g is defined and continuous on all of $\mathbb{R}$.
If $g(x_0) > 0$ for a single point $x_0 \in \mathbb{R}$, then $g(x)$ is in fact strictly positive for uncountably many points.
So far I've come up with the following:
(I've already ... | A possible proof is this one :
Let $\epsilon = \frac{g(x_0)}{2}$, then by continuity, there exist $\delta > 0$ such that $\forall x\in ]x_0-\delta, x_0+\delta[, 0<g(x_0)-\epsilon \leq g(x) \leq g(x_0)+\epsilon$
So $\forall x\in ]x_0-\delta, x_0+\delta[, g(x >0)$ and as there are uncountably points in this interval, we ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1708328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proof By Induction With Integration Problem I am required to prove this formula by induction$$ \int x^k e^{\lambda x} = \frac{(-1)^{k+1}k!}{\lambda^{k+1}} + \sum_{i=0}^k \frac{(-1)^i k^\underline{i}}{\lambda^{i+1}}x^{k-i}e^{\lambda x}$$
where $k^{\underline i}$ is a falling factorial $k(k-1) \cdots (k-i+1)$ (assuming t... | According to your definition of the operator $\int$, the LHS for $k=0$ becomes $$\int e^{\lambda x}=\int_0^x e^{\lambda\xi}\; d\xi = \left[\frac{e^{\lambda\xi}}{\lambda}\right]_{\xi = 0}^{\xi = x}=\frac{e^{\lambda x}}{\lambda}-\frac1\lambda$$
And working out the right-hand side for $k=0$ will give you the same result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1708407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Contradictory equations Question: Find whether the solution set of $$\begin{cases}2x = 1\\ y + 5 = x\\ x = y + 3\end{cases}$$ is a singleton.
My attempt: Rewriting the first equation will give us $x = \frac{1}{2}$.
The other two equations can be written as $x - y = 5$ and $x - y = 3$.
Now, the solution of these equatio... | A vertical line ( first equation ) cuts two parallel lines ( last two ) transversely, creating two points of intersection or two solutions.
If there were three (non-concurrent) lines then there would be three points of intersection enclosing a triangle.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1708509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Root test is stronger than ratio test? I am a little bit confused regarding the meaning of the phrase :" Root test is stronger than ratio test", and was hoping you will be able to help me figure it out. As far as I can see here: https://www.maa.org/sites/default/files/0025570x33450.di021200.02p0190s.pdf
The limit from ... | Consider the example of series
$$\sum 3^{-n-(-1)^n}$$
root test establishs the convergance but ratio test fails
onother example series with nth term
$a_n=2^{-n}$ if n is odd
$a_n=2^{-n+2}$ if n is even
for second series
when n is odd or even and tends to $\infty$
${a_n}^{\frac{1}{n}}=\frac{1}{2}$
Hence by cauchys root ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1708595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Use this sequence to prove that there are infinitely many prime numbers.
Problem:
By considering this sequence of numbers
$$2^1 + 1,\:\: 2^2 + 1,\:\: 2^4 + 1,\:\: 2^8 +1,\:\: 2^{16} +1,\:\: 2^{32}+1,\ldots$$
prove that there are infinitely many prime numbers.
I am thinking that if I can show that every pair of numbe... | If $2^{2^n}\equiv -1\pmod p$, then show that $2^{2^{m}}\not\equiv-1\pmod p$ for any $m<n$.
This is because if $2^{2^m}\equiv -1\pmod p$ then: $$-1\equiv 2^{2^n}=\left(2^{2^{m+1}}\right)^{2^{n-m-1}}\equiv 1\pmod p$$
So $p=2$. But $p$ can't be $2$.
So $2^{2^n}+1$ and $2^{2^m}+1$ can't have any prime factors in common if... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1708687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 0
} |
joint exponential distribution range problem So the question asks:
Let $X, Y$ be random variables, with the following joint probability density function:
$$f_{X,Y}(x,y) = \left\{ \begin{array}{ 1 l } kye^{-y} & \mbox{if $0≤|x|≤y$}\\ 0 & \mbox{ otherwise} \end{array} \right.$$
Find the value of the constant $k$.
So ... | Your order of integration is incorrect, which is why your computation doesn't work. The outermost integral cannot be a function of any other variables of integration. If you write $$\int_{x=-y}^y \int_{y=0}^\infty f(x,y) \, dy \, dx,$$ then what you have essentially done is integrated a function $f$ with respect to $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1708794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to show that $f(x; \alpha, \beta) = \frac{1}{\beta^{\alpha} \Gamma(\alpha)}x^{\alpha-1}e^{-x/\beta}$is a pdf? I have some problems trying to prove the following problem:
A continuous random variable $X$ is said to have a gamma distribution with parameters $\alpha > 0$ and $\beta > 0$ if it has a pdf given by:
$$... | $f(x)$ is a pdf if:
$f(x) \geq 0$ for all x. And,
$\int_{-\infty}^{\infty} f(x) dx = 1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1708922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Duality of $L^p$ spaces Let $p,q\in(1,\infty)$ be such that $1/p+1/q=1$ and let $(\Omega, \mathcal A,\mu)$ be a $\sigma$-finite measure space.
Claim: The map $$\phi:L^q(\Omega)\to \left(L^p(\Omega) \right)^*,\quad \phi(g)(f)=\int_\Omega fgd\mu$$ is an isometric isomorphism.
Proving that $\phi$ is well-defined, linear... | Question 1 : choose $f=g^{q-1}\cdot sign(f)$
Question 2 : Notice that $\phi \in \Big(L^q(\Omega) \Big)^{*}$ we ca define $$\nu (A)=\phi( \mathbb{1} _{A})$$, $\nu$ ia a measure and absolutly continuos acording to $\mu$ so find with Radon-Niodim Theorem we get $g$ that works for $f$ an indicator. We imediatly conclude th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1709035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Finding number of roots using Rolle's Theorem, and depending on parameter I need to count the number of real solutions for $ f(x) = 0 $ but I have an $m$ in there.
$$ f(x) = x^3+3x^2-mx+5 $$
I know I need to study $m$ to get the number of roots, but I don't know where to begin. Any suggestions?
| You can use the discriminant of the cubic function to find the nature of the roots. The general cubic equation is
$$ax^3+bx^2+cx+d=0$$
so in your case
$$a=1,\ b=3,\ c=-m,\ d=5$$
In general, the discriminant is
$$\Delta=18abcd-4b^3d+b^2c^2-4ac^3-27a^2d^2$$
You can substitute in your values and get a cubic expression in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1709129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
required knowledge for solving this equations Hi everyone there is some kind of functions which they're confusing to me, I already studied whole function knowledge requirements but they wasn't enough for solving new type of questions at all, I studied IB diploma math either. I will write down my questions and I will be... | The functional equation $f(x)+f(x+1)=2x+4$ has many solutions, unless you require additional properties. You can choose $f(\frac{1}{2})$ arbitrary!
Choose, for example, $f(\frac{1}{2})=42$. Then the equation $f(\frac{1}{2})+f(\frac{3}{2})=2\times\frac{1}{2}+4$ implies $f(\frac{3}{2})=5-42=-37$. Further, you have $f(\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1709233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Deciding whether a complex function has a primitive in any open subset of $\mathbb{C}$ I need to decide whether $g(z)=\bar{z}$ has a primitive in any open subset of $\mathbb{C}$. I'm struggling to decide where to start. I was thinking of using Cauchy's integral theorem i.e. if $g(z)$ is holomorphic then $$\int_C g(z) \... | $\overline{z}$ is not even holomorphic. Let $\gamma$ be the unit circle in the positive orientation, then
$$ \int_\gamma \overline{z}\,dz = i \int_0^{2\pi} \overline{e^{i\phi}} e^{i\phi}\,d\phi = i\int_0^{2\pi} \,d\phi = 2\pi i
$$
By Goursat's theorem, $\overline{z}$ cannot be holomorphic in the unit disk. You can expa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1709324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Stuck on basic limit problem: $\lim_{x \to 0} \frac{\sin(\tan x)}{\sin x}$ Consider $\lim_{x \to 0} \frac{\sin(\tan x)}{\sin x}$. The answer is $1$. This is clear intuitively since $\tan x ≈ x$ for small $x$. How do you show this rigorously? In general, it does not hold that $\lim_{x \to p} \frac{f(g(x))}{f(x)} = 1$ if... | Recall that $\tan x = \frac{\sin x}{\cos x}$ and that $\cos x = \sqrt{1 - \sin^2 x}$. Let $u = \sin x$
\begin{align}
\lim_{x \to 0} \frac{\sin(\frac{\sin x}{\cos x})}{\sin x} &= \lim_{x \to 0} \frac{\sin(\frac{\sin x}{\sqrt{1 - \sin^2 x}})}{\sin x}\\
&= \lim_{u \to 0} \frac{\sin \frac{u}{\sqrt{1-u^2}}}{\frac{u}{\sqrt{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1709415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 2
} |
Are $\mathbb{R}$ and $\mathbb{R}^n$ isomorphic for $n > 1$? My linear algebra class recently covered isomorphisms between vector spaces. Much emphasis was placed on the fact that $\mathbb{R}^m$ is isomorphic to particular subsets of $\mathbb{R}^n$ for positive $m < n$. For example, $\mathbb{R}^2$ is isomorphic to every... | I have absolutely no doubt that your book covers this point. Two vector spaces are isomorphic if and only if they have the same dimension. So $\mathbb{R}$ (dimension $1$) cannot be isomorphic to any $\mathbb{R}^n$ (dimension $n$) if $n>1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1709480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
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