Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Fixed field of a subgroup of a Galois group
For the Galois group $Gal(\mathbb{Q}(\sqrt2, \sqrt3, \sqrt5)/\mathbb{Q})$, I'm trying to understand how to find the permutations of the roots and how the subgroups of the Galois group are related to their fixed fields.
Take, for example, the permutation that takes $\sqrt3... | By definition, Galois group $\operatorname{Gal}(L/K)$ consists of all $K$-linear automorphisms of $L$, where $K$-linearity is in the sense of linear algebra. For your example, you have $\mathbb Q$-linear maps acting on $L = \mathbb Q(\sqrt 2, \sqrt 3, \sqrt 5)$ and $L$ is $\mathbb Q$-algebra generated by set $\{1,\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1105271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Evaluate $ \int^\infty_0\int^\infty_0 x^a y^{1-a} (1+x)^{-b-1}(1+y)^{-b-1} \exp(-c\frac{x}{y})dxdy $
Evaluate
$$ \int^\infty_0\int^\infty_0 x^a y^{1-a} (1+x)^{-b-1}(1+y)^{-b-1} \exp(-c\frac{x}{y})dxdy $$
under the condition $a>1$, $b>0$, $c>0$. Note that none of $a$, $b$ and $c$ is integer.
Mathematica found the ... | $\int_0^\infty\int_0^\infty x^ay^{1-a}(1+x)^{-b-1}(1+y)^{-b-1}e^{-c\frac{x}{y}}~dx~dy$
$=\Gamma(a+1)\int_0^\infty y^{1-a}(1+y)^{-b-1}U\left(a+1,a-b+1,\dfrac{c}{y}\right)dy$ (according to http://en.wikipedia.org/wiki/Confluent_hypergeometric_function#Integral_representations)
$=\dfrac{\Gamma(a+1)\Gamma(b-a)}{\Gamma(b+1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1105390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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The supremum-like definition of greatest common divisor I have reasonable experience with analysis, but I have just recently begun studying abstract algebra from Dummit and Foote.
I am frightened that I am already getting tripped up on pp.4, with the definition of greatest common divisor.
My intuitive understanding o... | Let $g$ be the gcd of $a$ and $b$. The we can write
$$a=m*g$$
$$b=n*g$$
Where $m$ and $n$ are co-prime ie their gcd is 1. ( This is evident ). Now let us assume $e$ divides both $a$ and $b$. Then $e$ can't be a divisor of $m$ and $n$ because they are co-prime , it means $e$ has to be a divisor of $g$ so that it can div... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1105509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to solve $5000 n \log(n) \leq 2^{n/2}$ I'm trying to solve the following problem:
What is the smallest value of n so that an algorithm with a runtime of $5000 n \log(n)$ runs faster than an algorithm with a runtime of $2^{2/n}$ on the same computer?
So I figured it was just a matter of solving the equation $5000 n ... | Take log of both sides (I'm assuming that "log" here means $\log2$) to get
$$
\log(5000) + \log(n) + \log \log n \le \frac{n}{2}
$$
Since $\log n$ and $\log \log n$ are both small compared to $n$, you can say that $n$ is somewhere around $2\log(5000) \approx 25$. So start with $n = 20$ and $n = 52$, say; for one of the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1105585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Polynomial Diophantine Equation
If $x$,$y$ $\in \mathbb Z$, find all the solutions of
$$y^3=x^3+8x^2-6x+8$$
I have tried factorizing the equation but the polynomial on $\text{R.H.S.}$ doesn't have any integral roots. Further, I deduce that the parity of $x$ and $y$ is same. However, I can't seem to find a way to go f... | Suppose that $x\geq 0$. Then we have $x^3 < x^3 + 8x^2 - 6x + 8 < (x+3)^3$, so $y \in \{x+1,x+2\}$. $y=x+1$ has no integer solutions, and plugging in $y=x+2$ yields the two solutions $x=0$ and $x=9$.
For negative $x$, we can do something similar, though the details are messier:
Suppose $x\leq 0$, and let $u=-x$, $v=-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1105701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Polar coordinate double integral I have to integrate the following integral:
$$
\iint \limits_A sin({x_1}^2 + {x_2}^2) dx_1dx_2
$$
over the set: $A=\{x \in \mathbb{R}^2: 1 \leq {x_1}^2 + {x_2}^2 \leq 9,x_1 \geq -x_2\}$
I understand geometrically what I have to do (i.e. subtract one half of a circle with of radius 1 fr... | Hint: You can use that $1\le r\le 3 \text{ and } -\frac{\pi}{4}\le\theta\le\frac{3\pi}{4}$ to set up the integral.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1105796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that factor modules are isomorphic.
I'm trying to prove (from a previous post) that if $A=k[x,y,z]$ and $I=(x,y)(x,z)$ then $$\dfrac{(x,y)/I}{(x,yz)/I} \cong\dfrac{A}{(x,z)}.$$
I did this by defining the homomorphism $\phi: A \to ((x,y)/I)/((x,yz)/I)$ by $f(x,y,z) \mapsto \dfrac {f(x,y,0)+ I}{(x,yz)/I}$ with th... | $\dfrac{(x,y)/I}{(x,yz)/I} \simeq\dfrac{(x,y)}{(x,yz)}=\dfrac{(x,y)}{(x,y)\cap (x,z)}\simeq\dfrac{(x,y)+(x,z)}{(x,z)}=\dfrac{(x,y,z)}{(x,z)}=\dfrac{(x,z)+(y)}{(x,z)}\simeq$ $\dfrac{(y)}{(x,z)\cap (y)}=\dfrac{(y)}{(xy,zy)}\simeq\dfrac{A}{(x,z)}$.
$\dfrac{(x,yz)}{I}=\dfrac{(x,yz)}{(x^2,xy,xz,yz)}\simeq\dfrac{(x)}{(x^2,xy... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1105868",
"timestamp": "2023-03-29T00:00:00",
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Prove that if $a_{2n} \rightarrow g$ and $a_{2n+1} \rightarrow g$ then $a_n \rightarrow g$ The problem is in the question:
Prove that if sequences $a_{2n} \rightarrow g$ and $a_{2n+1} \rightarrow g$ then $a_n \rightarrow g$.
I don't know how to prove that - it seems obvious when we look at the definition that for suffi... | Yes, for a given $\varepsilon>0$, you will get an $N_1$ from $(a_{2n})$ and an $N_2$ from $(a_{2n+1})$ and you will need to take their maximum or so..
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1105930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Is Fourier transform method suitable for solving equation $\int g(x-t)e^{-t^2} dt = e^{-a|x|}$ Is Fourier transform method suitable for to solve the following equation
\begin{align*}
\int g(x-t)e^{-t^2/2} dt = e^{-a|x|}
\end{align*}
Suppose we take the Fourier transform of the above equation
\begin{align*}
&G(\omega... | The Fourier transform method is useful in showing that there is no solution.
Since $G$ is unbounded, what your argument shows is that there is no solution in $L^1$. It is clear also that $G$ is not in $L^2$, so that the is no solution in $L^2$. Moreover, $G$ is not a tempered distribution, so that there is no solution ... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Any example of strongly convex functions whose gradients are Lipschitz continuous in $\mathbb{R}^N$ Let $f:\mathbb{R}^N\to\mathbb{R}$ be strongly convex and its gradient is Lipschitz continuous, i.e., for some $l>0$ and $L>0$ we have $$f(y)\geq f(x)+\nabla f(x)^T(y-x)+\frac{l}{2}||y-x||^2,$$ and $$||\nabla f(y)-\nabla ... | Here's an example on $\mathbb R: f(x) = x^2-\cos x.$
A way to make lots of examples: Let $f$ be any positive bounded continuous function on $[0,\infty).$ For $x\ge 0,$ set
$$g(x) =\int_0^x \int_0^t f(s)\, ds\,dt.$$
Extend $g$ to an even function on all of $\mathbb R.$ Then $g$ satisfies the requirements.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1106154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Asymptotic of $\sum_{n = 1}^{N}\frac{1}{n}$ versus $\sum_{1 \leq n \leq x}\frac{1}{n}$ Consider the series $\sum_{n = 1}^{N}\frac{1}{n}$. It is well known that we have the asymptotic: $$\sum_{n = 1}^{N}\frac{1}{n} = \log N + \gamma + \frac{1}{2N} + \frac{1}{12N^{2}} + O(N^{-3}).$$
My question: Consider the similar sum ... | Note that $$\log x - \log \lfloor x \rfloor=\log x - \log\left(x - [x]\right)=-\log\left(1-\frac{[x]}{x}\right)=\frac{[x]}{x}+O(x^{-2}),$$
where $[x]$ is the fractional part of $x$, and is clearly $O(1)$. That is, the oscillations around the smooth curve $\log x$ contribute already at the same order as $1/x$. Therefo... | {
"language": "en",
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"question_score": "1",
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Taylor series for $\sqrt{x}$? I'm trying to figure Taylor series for $\sqrt{x}$. Unfortunately all web pages and books show examples for $\sqrt{x+1}$. Is there any particular reason no one shows Taylor series for exactly $\sqrt{x}$?
| if
$$
\sqrt{x}=a_0+a_1x+a_2x^2+\dots
$$
then
$$
x=a_0^2+(a_0a_1+a_1a_0)x+(a_0a_2+a_1a_1+a_2a_0)x^2+\dots
$$
and if you want the identity theorem to hold this is impossible because $a_0=0$ would imply that the coeff of $x$ is zero
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1106344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "44",
"answer_count": 8,
"answer_id": 2
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Limit of complex numbers What would be the limit of following term?
$$\lim_{n \to \infty} \frac{e^{inx}}{2^n}$$
I tried to convert the $e^{inx}$ into trigonometric form and tried to do some simplification but got stuck after that.
Thanks. :)
| Hint
Just as you tried $$\frac{e^{inx}}{2^n}=\frac{\cos(nx)+i\sin(nx)}{2^n}$$ Each sine and cosine being between $-1$ and $1$, the numerator is finite and the denominator goes to $\infty$; thus, the limit is $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1106446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Value of $\pi$ by Aryabhata Aryabhata gave accurate approximate value of $\pi$.
He wrote in Aryabhatiya following:
add 4 to 100, multiply by 8 and then add 62,000. The result is approximately the circumference of circle of diameter twenty thousand.
By this rule the relation of the circumference to diameter is given.
Th... | Kim Plofker says that it probably was by computing the circumference of an inscribed polygon with 384 sides. She gives Sarasvati Amma as a source; I have also seen a reference to Florian Cajori with the same claim.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1106543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Conditional Probability - Basic Question On question 2b. below, why does knowing the extra bit of information A increase the probability of D?
| From the question I get the feeling that you want an answer regarding intuition and not so much mathematics. If I'm wrong, please comment and I will change my answer.
From the probabilities given, we see that it is much more likely that the whole shipment consists of forgeries (10%) than it is likely that only 2,3 or ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1106626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Help with Baby Rudin Excercise 3.17. Partial converse to the fact that convergence impiies Cesaro summability. Warning: Pretty involved problem. If $\{s_n\}$ is a complex sequence, define its arithmetic mean $\sigma_n$, by
$$\sigma_n=\frac{s_0+s_1+\cdots+s_n}{n+1}.$$
Put $a_n=s_n-s_{n-1}$ for $n\ge 1$.
Assume $M<\infty... | I agree with most of your work. A couple of notes:
*
*"Choose $m$ and $n$ so that $|\sigma-\sigma_m|<\epsilon$" doesn't really make sense since $m$ is predetermined by your choice of $n$ and $\epsilon$.
*All of the instances of $n$ and $m$ should still be contained within the $\limsup$'s.
So with these small alte... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1106820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Moment generating function of $f(x)=\frac{1-\cos(x)}{\pi x^2}$ does not exist How to prove that $\int_{\mathbb{R}}\exp(tx)\frac{1-\cos(x)}{x^2}dx$ is not finite for any fixed $t$?
Thank you!
| Let $t>0$. Since $1-\cos x\ge0$ we have for any $k\in\mathbb{N}$ we have
$$
\int_{k\pi}^{(k+1)\pi}\exp(t\,x)\frac{1-\cos x}{x^2}\,dx\ge\frac{e^{k\pi t}}{(k+1)^2\pi^2}\int_{k\pi}^{(k+1)\pi}(1-\cos x)\,dx=\frac{e^{k\pi t}}{(k+1)^2\pi}
$$
and
$$
\sum_{k=1}^\infty\frac{e^{k\pi t}}{(k+1)^2}=+\infty.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1106911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$f$ has a local maximum at a point $x \in E$. Prove that $f'(x)=0$ Suppose that $f$ is a differentiable real function in an open set $E \subset \mathbb{R^n}$, and that $f$ has a local maximum at a point $x \in E$. Prove that $f'(x)=0$
| Suppose that at some point $p=(p_1,p_2,\ldots,p_n)$ and for some $i$, we have $\frac{\partial f}{\partial x_i}=c\neq0$.
We can regard $f$ as a differentiable function in a single variable $t$, where you only vary $x_i=t$ and leave $x_j$ constant for $j\neq i$. Denote this as $g(t)$. By definition, $g'(p_i)=c$.
Clearly... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Hahn-Banach theorem exercise Let $X$ be a Banach space (over $\mathbb{R}$) and $u,v\in X$ such that $\|u\|=\|v\|=1$ and $\|2u+v\|=\|u-2v\|=3$. Show that there is $f\in X'$ of unit norm such that $f(u)=f(v)=1$.
My idea is building directly the functional on $Y=\operatorname{Span}\{u,v\}$, something like $f(\alpha u+ \be... | It suffices to prove that $\Vert \alpha u + \beta v\Vert=|\alpha + \beta|$.
Maybe you can proceed through the following steps (I don't know):
*
*Prove that it is true for $\alpha$ and $\beta$ integer numbers.
*Prove that it is true for $\alpha$ and $\beta$ rational numbers, based on the last step.
Finally, since... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Long exact sequence into short exact sequences This question is the categorical version of this question about splitting up long exact sequences of modules into short exact sequence of modules.
I want to understand the general mechanism for abelian categories. Here's the decomposition I managed to get:
But it doesn't ... | You are right, it finally involves cokernels (=coimages of the next arrows).
Your first line is correct: taking cokernel of both sides of $\def\im{{\rm im}\,} \im f=\ker g$, we'll get $\def\coker{{\rm coker}\,} \def\coim{{\rm coim}\,} \coker f=\coim g$. For the converse, take kernel of both sides.
Then, for the objects... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1107157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the right cosets of $H$ in $G$ simple example Question: Let $G$ be a group and $H<G$ a subgroup with $|G:H|=2$
Show that the right cosets of $H$ in $G$ are $H$ and $G\backslash H$
Answer given: There are two right cosets, they are disjoint and their union is $G$. One coset is $H$ and so the other is $G\backslash H... | A nice way to see that distinct right cosets are disjoint is to do the following: Let $H\leq G$, and for $a,b \in G$, define the relation $a \sim b$ if and only if $a \in Hb$. If we can show this is an equivalence relation, we will have right cosets of $H$ as equivalence classes, which shows they partition $G$ into dis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1107249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Can anyone sketch the proof or provide a link that there is always a prime between $n^3$ and $(n+1)^3$ In a recent forum discussion on number theory, it was mentioned that A. E. Ingham had proven that there is always a prime between $n^3$ and $(n+1)^3$.
Does anyone know if there is a link available on the web or knows ... | Back in the mid-80s, when I first opened Apostol's "Introduction to Analytic Number Theory," Apostol stated the theorem that there exists a real $\alpha$ such that $\left\lfloor\alpha^{3^n}\right\rfloor$ was always prime. Apostol noted, though, that the existing proof was non-constructive.
I realized relatively quickly... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1107400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Orthogonal complement of the kernel of $u\in B(H, H')$ Let $H,H'$ be Hilbert spaces and $u \in B(H,H')$. Let $u^\ast$ denote the adjoint.
I know (and can show) that $(\mathrm{im} u)^\bot = \ker u^\ast$.
From this I would deduce that $(\ker u^\ast)^\bot = \mathrm{im} u$. But instead,
$$ (\ker u^\ast)^\bot = \overline{\... | If $X\subseteq H$, then $X^\perp$ is closed.
Let $\{y_n\}$ be any convergent sequence in $X^\perp$ with $y_n\to y$. Then for any $x\in X$,
$$\langle x,y\rangle=\langle x,y-y_n\rangle+\langle x,y_n\rangle=\langle x,y-y_n\rangle.$$
Applying Cauchy-Schwarz, this gives
$$|\langle x,y\rangle|\leq \|x\|\|y-y_n\|$$
which tend... | {
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"url": "https://math.stackexchange.com/questions/1107479",
"timestamp": "2023-03-29T00:00:00",
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ODEs - Seperable differential equation, is an explicit answer necessary in an exam? Find the general solution for the following first order differential equation:
$$\frac{dy}{dt}=(1-y)(3-y)(5-t).$$
I end up getting to
$$\exp(10x-x^2+2C)=(y-3)(y-1).$$
Is there any more simplifying that can be done to express $y$ explici... | i think you made a sign error. separating the equation you get $$\dfrac{dy}{(1-y)(3-y)} = (5-t)dt$$ the partial fraction decomposition gives you $$\dfrac{dy}{y-3} - \dfrac{dy}{y-1} = (10 - 2t)\ dt$$ on integration this gives you
$$\dfrac{y-3}{y-1} = Ce^{10t -t^2}$$
solving the last one for $y$ gives $$y = \dfrac{3-C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1107571",
"timestamp": "2023-03-29T00:00:00",
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Elementary proof of the fact that any orientable 3-manifold is parallelizable A parallelizable manifold $M$ is a smooth manifold such that there exist smooth vector fields $V_1,...,V_n$ where $n$ is the dimension of $M$, such that at any point $p\in M$, the tangent vectors $V_1(p),...,V_n(p)$ provide a basis for the ta... | Kirby gives a nice proof that every orientable 3-manifold is parallelizable at https://math.berkeley.edu/~kirby/papers/Kirby%20-%20The%20topology%20of%204-manifolds%20-%20MR1001966.pdf . (See the unnumbered non-blank page between page 46 and page 47.)
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "37",
"answer_count": 5,
"answer_id": 3
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Understanding the arc length integral formula I believe the proof in my book is slighty more informal than the proof that uses the Mean Value Theorem. Could someone tell me what exactly the difference is, and if there are any mistakes in the proof below? Thanks.
Proof of the arc length integral formula
Divide your int... | Elaboration of my comment:
You want to convert an infinite sum to an integral. As you probably understand that can be interpreted as summing infinitely many rectangles and deciding what the area converges to. But this is not enough! The rectangles have to be infinitely small as well. Take a look at the picture. There a... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Number theory: prove that if $a,b,c$ odd then $2\gcd(a,b,c) = \gcd(a+b,b+c, c+a)$ Please help! Am lost with the following:
Prove that if $a,b,c$ are odd integers, then $2 \gcd(a,b,c) = \gcd( a+b, b+c, c+a)$
Thanks a lot!!
| Let $d=\gcd(a,b,c)$. Then solve $d=ax+by+cz$.
Use that $2a=(a+b)+(a+c)-(b+c)$, $2b=(a+b)+(b+c)-(a+c)$ and $2c=(a+c)+(b+c)-(a+b)$. Then
$$2d=2ax+2by+2cz = (a+b)(x+y-z) + (a+c)(x-y+z) + (b+c)(y+z-x)$$
So we have a solution to:
$$2d = (a+b)X+(a+c)Y + (b+c)Z$$
So we know $\gcd(a+b,a+c,b+c)\mid 2\gcd(a,b,c)$. The other di... | {
"language": "en",
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"source": "stackexchange",
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} |
Invertible "Sigmoid + x" function I need an invertible function that represents a smooth transition between two straight, parallel line segments, like this:
Depicted is $f(x) = -0.3/(1+e^{-10*(x-p)})+0.3/2+x$ (where $p$ is the location of the lowest slope), but it does not seem to be invertible in terms of standard ma... | I think a relatively simple way to obtain an "invertible sigmoid" is to find a suitable cubic polyomial. The method I propose here has the drawback that it does not let you control both the tangency points at the same time.
Let's first simplify the setting. Let's consider the two points $P_1=(-2,-1)$ and $P_2=(2,1)$ an... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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why is $\lim_{x\to -\infty} \frac{3x+7}{\sqrt{x^2}}$=-3? Exercise taken from here: https://mooculus.osu.edu/textbook/mooculus.pdf (page 42, "Exercises for Section 2.2", exercise 4).
Why is $\lim_{x\to -\infty} \frac{3x+7}{\sqrt{x^2}}$=-3*? I always find 3 as the solution. I tried two approaches:
Approach 1:
$$
\lim_{x\... | For $x \to -\infty$, we have that $\sqrt{x^2} = |x| = -x$.
| {
"language": "en",
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Evaluating Integrals using Lebesgue Integration Suppose we are to evaluate:
$$I = \int_{0}^{1} f(x) dx$$
Where
$$f(x)=\begin{cases}1 \space \text{if} \space x\space \text{is rational}, & \newline 0 \space \text{if} \space x \space \text{is irrational} \\ \end{cases}$$
I have been told that this can be done using meas... | Lebesgue integration tells you that the value is zero. Basically, much like how you can split integrals over intervals in Riemann integration, you can split integrals over arbitrary measurable sets in Lebesgue integration. Here we write:
$$\int_{[0,1]} f(x) dx = \int_{[0,1] \cap \mathbb{Q}} 1 dx + \int_{[0,1] \cap \mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1108171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Abelian categories with tensor product Is there a standard notion in the literature of abelian category with tensor product?
The definition ought to be wide enough to encompass all the usual examples of abelian categories with standard `tensor product'. I'd guess something like "symmetric monoidal bi-functor $\otimes ... | Let $(\mathcal{A},\otimes)$ be a monoidal category. It is called abelian monoidal if $\mathcal{A}$ is an abelian category and $\otimes$ is an additive bifunctor (see here, Section 1.5). An example given the linked paper is the category of bimodules over a ring (in fact any closed abelian monoidal category can be exactl... | {
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What's your favorite proof accessible to a general audience? What math statement with proof do you find most beautiful and elegant, where such is accessible to a general audience, meaning you could state, prove, and explain it to a general audience in roughly $5 \pm\epsilon$ minutes.
Let's define 'general audienc... | I would explain the Pigeon Hole principle, or one of its many guises.
In fact, I remember explaining the PHP to a non math student while playing bridge.
I told him that one gets at least $4$ cards of some suit- which is really the PHP.
You can come up with many other interesting "real life" examples, and it's fun.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "121",
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"answer_id": 4
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What's your favorite proof accessible to a general audience? What math statement with proof do you find most beautiful and elegant, where such is accessible to a general audience, meaning you could state, prove, and explain it to a general audience in roughly $5 \pm\epsilon$ minutes.
Let's define 'general audienc... | There's no way to tune a piano in perfect harmony.
There are twelve half-steps in the chromatic scale, twelve notes in each octave of the keyboard. Start at middle "C", and ascend a perfect fifth to "G". That's seven half steps up, with a frequency ratio of 3/2. Drop an octave to the lower "g" -- that's twelve ha... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "121",
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"answer_id": 36
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Considering $ (1+i)^n - (1 - i)^n $, Complex Analysis I have been working on problems from Complex Analysis by Ahlfors, and I got stuck in the following problem:
Evaluate:
$$
(1 + i)^n - (1-i)^n
$$
I have just "reduced" to:
$$
(1 + i)^n - (1-i)^n = \sum_{k=0} ^n i^k(1 - (-1)^k)
$$
by using expansion of each term.
Th... | There are a number of spiffy techniques one could use on this problem, but Ahlfors doesn't get to conjugation and modulus until 1.3 and geometry of the complex plane until Section 2 (of Chapter 1), while this is still in 1.1. [I dug up my copy of the third edition to see how much was discussed to that point.]
abel sh... | {
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Why do some other people use dek and el rather than letters as the eleventh and twelfth digits in the dozenal or duodecimal system? I've noticed on a YouTube video titled Base $12$ - Numberphile that some other people who use the duodecimal system use dek and el for the eleventh and twelfth digits. I know for one thin... | As mentioned in the comments, It works fine if you use a's and b's, but they tried to use names for "ten" and "eleven" which suggest their meaning. "dek" from "decem", coming from the latin word for 10, and "el" from "eleven", coming from the english word eleven.
It was a stylistic choice and nothing more.
| {
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$\sin \left( {5x} \right) = 2\sin \left( {3x} \right)\sin \left( {4x} \right)$ ask gentlemen to help solve the equation
Where the real number
$$ x \in \mathbb{R}: \sin \left( {5x} \right) = 2\sin \left( {3x} \right)\sin \left( {4x} \right);
$$
I notice that
$$x = k\pi \quad ;k \in \mathbb{Z}$$
are solutions
| Expressing everything in terms of $\sin(x)= s$ and $\cos(x) = c$, the equation says
$$ -64\,{c}^{5}{s}^{2}+16\,{c}^{4}s+48\,{c}^{3}{s}^{2}-12\,{c}^{2}s-8\,c{
s}^{2}+s = 0$$
Taking the resultant of the left side and $c^2 + s^2 - 1$ with respect to $c$, we get
$$ {s}^{2} \left( 4096\,{s}^{12}-14336\,{s}^{10}+19968\,{s}^... | {
"language": "en",
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Why is a linear transformation a $(1,1)$ tensor? Wikipedia says that a linear transformation is a $(1,1)$ tensor. Is this restricting it to transformations from $V$ to $V$ or is a transformation from $V$ to $W$ also a $(1,1)$ tensor? (where $V$ and $W$ are both vector spaces). I think it must be the first case since it... | It's very common in tensor analysis to associate endomorphisms on a vector space with (1,1) tensors. Namely because there exists an isomorphism between the two sets.
Define $E(V)$ to be the set of endomorphisms on $V$.
Let $A\in E(V)$ and define the map $\Theta:E(V)\rightarrow T^1_1(V)$ by
\begin{align*}
(\Theta A)(\o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1108842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
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If $G$ is a group whereby $(a\cdot b)^{i} =a^i\cdot b^i$ for three consecutive integers $i$ for all $a, b \in G$, show $G$ is abelian. If $G$ is a group in which $(a\cdot b)^{i} =a^i\cdot b^i$ for three consecutive integers $i$ for all $a, b \in G$, show that $G$ is abelian.
Proof: Let $x$ be the smallest of the 3 cons... | Use $\backslash$overbrace{below}^{above}, as in (right click and select to see LaTeX commands):
$$
\overbrace{a\ldots a}^{27}
$$
The proof looks fine to me.
| {
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"source": "stackexchange",
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Which arrangement produces the largest number? I learnt that the power tower $2\uparrow3\uparrow4\uparrow...\uparrow n$ is larger
than any power tower with a different order of the numbers $2,3,4,...,n$.
Is this also true for conway-chains and for bowers array notation ?
Are $$2\rightarrow 3\rightarrow 4\rightarrow...... | No for both.
For example, for $n = 4$ we have $2 \rightarrow 3 \rightarrow 4 = 2 \rightarrow 4 \rightarrow 3 = 2 \uparrow \uparrow 65536$, whereas $3 \rightarrow 2 \rightarrow 4 = 3 \uparrow \uparrow 3^{27}$.
We can show by induction that $2 \rightarrow 3 \rightarrow n < 3 \rightarrow 2 \rightarrow n$, as
$$
2 \rightar... | {
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Example of Parseval's Theorem In the textbook "Mathematics for Physics" of Stone and Goldbart the following example for an illustration of Parseval's Theorem is given:
Until 2.42 I understand everything but I don't understand the statement:
" Finally, as $\sin^2(\pi(\zeta-n))=\sin^2(\pi \zeta)$ "
Can you explain me w... | $\sin{\pi n} = 0$ when $n \in \mathbb{Z}$. Thus, because $\cos{\pi n} = (-1)^n$, we have
$$\sin{\pi (\zeta - n)} = \sin{\pi \zeta} \cos{\pi n} - \sin{\pi n} \cos{\pi \zeta} = (-1)^n \sin{\pi \zeta} $$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is there a proof for the fact that, if you perform the same operation on both sides of an equality, then the equality holds? Is there a proof for this, or is it just taken for granted?
Does one need to prove it for every separate case (multiplication, addition, etc.), or only when you are operating with different elem... | No, it needs no proof. In general one has that if $x=y$ and $f$ is a function that has $x$ and $y$ in its domain then $f(x)=f(y)$. Since $a+b$ is just another way of writing $+(a,b)$, one has that if $a=c$ and $b=d$ then $a+b=c+d$ (same for multiplication).
| {
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"url": "https://math.stackexchange.com/questions/1109319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Why does $\lim_{x\rightarrow 0}\frac{\sin(x)}x=1$? I am learning about the derivative function of $\frac{d}{dx}[\sin(x)] = \cos(x)$.
The proof stated: From $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$...
I realized I don't know why, so I wanted to learn why part is true first before moving on. But unfortunately I don't have... | hint use that $$\cos(\theta)\le \frac{\sin(\theta)}{\theta}\le 1$$ and take the limit for $\theta$ goes to zero
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Embeddings of a subfield of $ \mathbb{C} $ I'm trying to understand / solve the following problem:
Let $ L \subset \mathbb{C} $ be a field and $ L \subset L_1 $ its finite extension ($ [L_1 : L] = m $). Prove that there are exactly $ m $ disintct embeddings
$$ \sigma_1, \dots, \sigma_m: L_1 \hookrightarrow \mathbb{C} $... | Let $f(x)$ be the minimal polynomial of $a$ over $L$, it then has to be of degree $m$.
This means, we can express $a^m$ by $L$-linear combination of lower powers $a^k$ ($k<m$), and by definition, $a$ doesn't satisfy any other algebraic equations that doesn't follow from $f(a)=0$.
Now, if we replace the root $a$ by an i... | {
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How to solve this Linear Algebra problem involving a system of linear equations?
The following is what I have so far.
I'm not sure how to use my echelon matrix to find out which values for the variables can provide an answer to the question or how to prove it. I was thinking of plugging in arbitrary numbers for $x_3... | Expanded coefficients matrix and its reduction:
$$\begin{pmatrix}1&\!\!-2&1&y_1\\
2&1&q&y_2\\
0&5&\!\!-1&y_3\end{pmatrix}\stackrel{R_2-2R_1}\rightarrow\begin{pmatrix}1&\!\!-2&1&y_1\\
0&5&q-2&y_2-2y_1\\
0&5&\!\!-1&y_3\end{pmatrix}\stackrel{R_3-R_2}\rightarrow$$$${}$$
$$\rightarrow\begin{pmatrix}1&\!\!-2&1&y_1\\
0&5&q-2&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1109558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $\frac{(n-a)^2}{n}$ can be written as $1-\left(\frac{n}{a}\right)^2\cdot\frac{n}{(n/a)^2}$ \left(\frac{n}{a}\right)^2\cdot\frac{n}{(n/a)^2}$.
I have got so far to $(a^2/n)-2a+n$
But I can not see how to proceed. Can anyone help please?
| $$\frac{(n-a)^2}{n} =(n-a)^2\left(\frac{1}{n}\right) =(n-a)^2\left(\frac{1}{n}\right)\frac{a^2}{a^2} =\left(\frac{n-a}{a}\right)^2 \frac{a^2}{n}= \left(\frac{n}{a}-1\right)^2 \frac{a^2}{n} = \left(\frac{n}{a}-1\right)^2 \frac{n^2}{n^2} \frac{a^2}{n} = (-1)^2\left(1-\frac{n}{a}\right)^2 \frac{n}{(\frac{n}{a})^2}=\left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1109659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Vectors of force and their angles This is a homework problem which has me stumped. I have just begun Calculus 3 and this is the first section introducing vectors. .
The embedded picture states the problem and shows some of my hen's scratching in an effort to solve it. After studying the example problem from the same... | Using Lami's there
$\frac{F_2}{sin (90+\alpha)}=2F_1=\frac{50}{sin (120-\alpha)}$
$\frac{F_2}{cos \alpha}=2F_1=\frac{50}{sin (120-\alpha)}$
$\frac{F_2}{cos \alpha}=\frac{50}{sin 120cos\alpha-cos 120sin\alpha}$
$\frac{F_2}{cos \alpha}=\frac{50}{\frac{\sqrt3}{2}cos\alpha+\frac{1}{2}sin\alpha}$
$\frac{F_2}{cos \alpha}=\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1109837",
"timestamp": "2023-03-29T00:00:00",
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On adding terms to limits Is it always possible to add terms into limits, like in the following example? (Or must certain conditions be fulfilled first, such as for example the numerator by itself must converge etc)
$\lim_{h \to 0} {f(x)} = \lim_{h \to 0} \frac{e^xf(x)}{e^x}$
| It's not clear to me what "add terms into limits" is supposed to mean in general. In your example, you're not really doing anything: $f(x)$ and $\dfrac{e^x f(x)}{e^x}$ are equal, so any well-defined operation on them produces the same results.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finite cyclic subgroups of $GL_{2} (\mathbb{Z})$ How could we prove that any element of $GL_2(\mathbb{Z})$ of finite order has order 1, 2, 3, 4, or 6?
I am aware of the proof supplied here at this link: https://www.maa.org/sites/default/files/George_Mackiw20823.pdf. But I am curious if there are any other proofs.
| Suppose that $A$ has finite order in ${\rm GL}(2,\Bbb Q)$. Then we know there is $k$ such that $X^k-1$ annihilates $A$. If $m_A$ is the minimal polynomial of $A$, $\deg m_A\leqslant 2$. Moreover, $m_A$ divides $X^k-1$ so $m_A$ has as its irreducible factors the irreducible factors of $X^k-1$. Since everything is monic,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1110011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Summation of an infinite series The sum is as follows:
$$
\sum_{n=1}^{\infty} n \left ( \frac{1}{6}\right ) \left ( \frac{5}{6} \right )^{n-1}\\
$$
This is how I started:
$$
= \frac{1}{6}\sum_{n=1}^{\infty} n \left ( \frac{5}{6} \right )^{n-1} \\
= \frac{1}{5}\sum_{n=1}^{\infty} n \left ( \frac{5}{6} \right )^{n}\\\\... | hint: differentiate the identity
$$\sum_{k=0}^{\infty} x^k = \frac{1}{1-x} $$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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prove the given inequality (for series ) For any given $n \in \Bbb N,$ prove that, $$1+{1\over 2^3}+\cdots+{1\over n^3} <{3\over 2}.$$
| Hint: Notice that as $f(x)=\frac{1}{x^3}$ is decreasing for every $x>0$ then the following expression holds
$$\sum_{k=1}^n \dfrac{1}{k^3}<\int_0^n\dfrac{1}{x^3}d x$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1110138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Basin of attraction of the fixed map $f(x) = x-x^3$ Prove that the interval $(-\sqrt 2 ,\sqrt 2 )$ is the basin of attraction of the fixed point $0$ of the map $f(x)=x-x^3$, for $x \in \mathbb{R}$.
How one would prove this? In the examples I've seen so far they usually prove that a fixed point has a certain basin of at... | Let $F(x)=\left|f(x)\right|$. I claim that $F(x)<\left|x\right|$ for all $x\in\left(-\sqrt{2},\sqrt{2}\right)$. This is quite easy to see from a graph:
It can be proved using the factorization
$$F(x) = \left|x-x^3\right| = \left|x\right|\,\left|1-x^2\right|.$$
Now, the claim is trivial for $0 < x \leq 1$ since then $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Integral $\int \frac{x+2}{x^3-x} dx$ I need to solve this integral but I get stuck, let me show what I did:
$$\int \frac{x+2}{x^3-x} dx$$
then:
$$\int \frac{x}{x^3-x} + \int \frac{2}{x^3-x}$$
$$\int \frac{x}{x(x^2-1)} + 2\int \frac{1}{x^3-x}$$
$$\int \frac{1}{x^2-1} + 2\int \frac{1}{x^3-x}$$
now I need to resolve one... | The Heaviside cover-up method for solving partial fraction decompositions deserves to be more widely known. We want to find $A,B,C$ in this equation:
$$\frac{x+2}{x(x-1)(x+1)}=\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x+1}$$
To find $A$, multiply by $x$:
$$\frac{x+2}{(x-1)(x+1)}=A+\frac{Bx}{x-1}+\frac{Cx}{x+1}$$
Now put $x=0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1110311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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Generating a data set with given mean and variance Suppose we have to create n integers in a given range say between 1 and 1000 with given mean and variance .My question is :Is there an algorythm that can tell us whether such a data set exists and how to create it if it exists?Thanks in advance for any help.
| Let $p_1, p_2, ..., p_n$ be unknown reals, say $n=1000$, and let $M\ge0$, and $s^2$ be the required mean and variance, respectively.
The following conditions are to be met:
$$\sum_{k=1}^n p_i=1$$
$$\ \ \ \sum_{k=1}^n ip_i=M$$
$$\ \ \ \sum_{k=1}^n (i-M)^2p_i =s^2$$
with the restriction that
$$\ 0\le p_i \le 1$$ for all ... | {
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Proving that $BI$, $AE$ and $CF$ are concurrent?
Let $ABC$ be a triangle, and $BD$ be the angle bisector of $\angle B$. Let $DF$ and $DE$ be altitudes of $\triangle ADB$ and $\triangle CDB$ respectively, and $BI$ is an altitude of $\triangle ABC$. Prove that $AE$, $CF$ and $BI$ are concurrent.
I saw that $BEDF$ is a... | Consider that:
$$\frac {AF}{BF}=\frac{FA}{FD}\cdot \frac{FD}{FB}=\cot A\cdot \tan \frac{B}{2}$$
$$\frac {BE}{EC}=\frac{BE}{DE}\cdot \frac{DE}{CE}=\cot \frac{B}{2}\cdot \tan C$$
$$\frac{IC}{IA}=\frac{IC}{IB}\cdot \frac{IB}{IA}=\cot C\cdot \tan A$$
Multiply those equalities side by side, $AE,BF,CI$ are concurrent by Cev... | {
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"url": "https://math.stackexchange.com/questions/1110630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that the edge coloring number is smaller than or equal to two times the maximum degree Let G be a graph with maximum degree ∆(G) and χ’(G) the edge coloring number. Prove that χ’(G) ≤ 2∆(G) without using Vizing's theorem.
I really don't have a clue on how to tackle this problem. Can anybody push me in the right d... | For any graph G with maximum $\Delta(G)=0, \chi'(G)=0$ and for any graph G with maximum $\Delta(G)=1, \chi'(G)=1$.
Assume as an inductive hypothesis that $\chi'(H) \leq 2\Delta(H)$ for all graphs $H$ with $\Delta(H)<\Delta(G)$ or with $\Delta(H)=\Delta(G)$ and $E(H)<E(G)$.
Now if we remove a vertex of maximum degree $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1110769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find the real and imaginary part of the following I'm having trouble finding the real and imaginary part of $z/(z+1)$ given that z=x+iy. I tried substituting that in but its seems to get really complicated and I'm not so sure how to reduce it down. Can anyone give me some advice?
| $$\begin{align}
&\frac{z}{z+1}\\
=& \frac{x+iy}{x+iy+1}\\
=&\frac{x+iy}{x+iy+1}\times \frac{x-iy+1}{x-iy+1}
\end{align}$$
I'm sure you can take the rest from here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1110876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Angular momentum in Cylindrical Coordinates How to calculate the angular momentum of a particle in a cylindrical coordinates system
$$x_1 = r \cos{\theta}$$
$$x_2 = r \sin{\theta}$$
$$x_3 = z$$
Thanks.
| I assume you are seeking the infinitesimal deformation operators generating rotations, as in quantum mechanics, so, then, $\vec{p}\sim \nabla$.
In that case, since
$$
\vec{R}= z \hat z + r \hat r ,\\
\nabla = \hat z \partial_z + \hat r \partial_r + \hat \theta \frac{1}{r} \partial _\theta,
$$
$$
\vec L = \vec{R} \tim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1110982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Forming equations for exponential growth/decay questions Problem
Dry cleaners use a cleaning fluid that is purified by evaporation and condensation after each cleaning cycle. Every time the fluid is purified, 2.1% of it is lost. The fluid has to be topped up when half of
the original fluid remains.
a) Create a model w... | Let the initial volume of the container be $V_0$ and the density be $\rho$.
Let the evaporation and condensation be uniform and that 2.1% of the volume is lost everytime the purifying process is over.
Thus the model is $${\rho\times(\dot V_0 - \dot V_1)} = 0.021*\rho\times\dot V_0$$
Cancelling $\rho$, and converting th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1111063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Example: convergence in distributions Give an example $X _n \rightarrow X$ in distribution, $Y _n \rightarrow Y$ in distribution, but $X_n + Y_n$ does not converge to $X+Y$ in distribution.
I got a trivial one. $X_n$ is $\mathcal N(0,1)$ $\forall n$, $Y_n=-X_n$, $X$ and $Y$ are also $\mathcal N(0,1)$, then $X _n \righ... | Notice that if $(X_n)$ and $(Y_n)$ are weakly convergent, then the sequence $(X_n+Y_n)_{n\geqslant 1}$ is tight, hence we can extract a weakly convergent subsequence. Therefore, the problem may come from the non-uniqueness of the potential limiting distributions.
Consider $(\xi_i)_{i\geqslant 1}$ a sequence of i.i.d. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1111276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How can I prove that two recursion equations are equivalent? I have two recursion equations that seem to be equivalent. I need a method to show the equivalence relation between them. The equations calculate number of ones in binary representation of the number.
The equations are given below:
1)
$$
f(0) = 0
$$
$$
f(n) ... | You can do it with induction I believe. The induction hypotheses will have to be chosen cleverly to simplify the expression for $g(n)$. My suggestion would be to do induction steps on an exponential scale. That is, assume that $g(n) = f(n)$ for all $n$ less than or equal to $2^m$. Then, prove that $g(n) = f(n)$ for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1111382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to solve the recurrence relation $t_n=(1+c q^{n-1})p~t_{n-1}+a +nbq$? How to solve $$t_n=(1+c q^{n-1})p~t_{n-1}+a +nbq,\quad n\ge 2.$$ given that $$t_1=b+(1+c~p)(a~q^{-1}+b),\qquad p+q=1.$$
N.B- Some misprints in the question I corrected. Sorry for the misprint $an+b$ is actually $a+nbq$.
| let
$$\dfrac{h_{n-1}}{h_{n}}=(1+cq^{n-1})p$$
so we have
$$\dfrac{h_{n-2}}{h_{n-1}}=(1+cq^{n-2})p$$
$$\cdots\cdots$$
$$\dfrac{h_{1}}{h_{2}}=(1+cq)p$$
$$\dfrac{h_{1}}{h_{n}}=\prod_{k=1}^{n-1}(1+cq^k)p\Longrightarrow h_{n}=\dfrac{h_{1}}{\prod_{k=1}^{n-1}(1+cq^k)p}$$
then we have
$$t_{n}=\dfrac{h_{n-1}}{h_{n}}t_{n-1}+an+b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1111476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Proof with orthogonal vectors in real analysis If $x,y \in \mathbb{R}^n$, then $x$ and $y$ are called perpendicular (or orthogonal) if $\langle x,y\rangle =0$. If $x$ and $y$ are perpendicular, prove that $|x+y|^2=|x|^2+|y|^2$.
Seems pretty basic, but I'm missing something.
| $$\big|x+y\big|^2= \langle x+y,x+y \rangle=\langle x,x+y\rangle+y\langle x+y\rangle$$ $$= \langle x,x \rangle + \langle x,y \rangle + \langle y,x\rangle+ \langle y,y\rangle$$ $$=\big|x\big|^2+\big|y\big|^2$$
as $\langle x,y \rangle = \langle y,x \rangle =0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1111572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Number of circular permutation of word 'CIRCULAR' Hey please help me with this question...
Find the number of circular permutation of the word 'CIRCULAR'.
Number of circular permutaion is (n-1)!
| An essential point to note is that all orbits under cyclic rearrangement have $8$ distinct elements, since there are some letters that occur only once and can serve as marker. Now one can just count the number of permutations of the letters, and divide by $8$, namely $\frac{8!}{2!2!1!1!1!1!}/8=1260$.
Just to show the c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1111664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Equation of circle touching a parabola Suppose we have a parabola $y^2=4x$ . Now, how to write equation of circle touching parabola at $(4,4)$ and passing thru focus?
I know that for this parabola focus will lie at $(1,0)$ so we may assume general equation of circle and satisfy the points in it . hence
$$x^2 +y^2 + 2g... | the tangent to the parabola at $(4, 4)$ has slope $1/2$ so the radius has slope $-2.$ let the center of the circle touching the parabola $y^2 = 4x$ at $(4,4)$ be
$x = 4 + t, y = 4 - 2t$. now equating the radius $$5t^2 = (3+t)^2 + (4-2t)^2$$
you can find $t$ which will give you the center and the radius of the circle.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1111749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Does $\int_0^\infty \sin(x^{2/3}) dx$ converges? My Try:
We substitute $y = x^{2/3}$. Therefore, $x = y^{3/2}$ and $\frac{dx}{dy} = \frac{2}{3}\frac{dy}{y^{1/3}}$
Hence, the integral after substitution is:
$$ \frac{3}{2} \int_0^\infty \sin(y)\sqrt{y} dy$$
Let's look at:
$$\int_0^\infty \left|\sin(y)\sqrt{y} \right| d... | $\sin x^{2/3}$ remains above $1/2$ for $x$ between $[(2n+1/6)\pi]^{3/2}$ and $[2n+5/6]^{3/2}$, so the integral rises by more than $\left([2n+5/6]^{3/2}-[2n+1/6]^{3/2}\right)\pi^{3/2}/2$ during that time.
$$[2n+5/6]^{3/2}-[2n+1/6]^{3/2}=\frac{[2n+5/6]^3-[2n+1/6]^3}{[2n+5/6]^{3/2}+[2n+1/6]^{3/2}}\\
>\frac{8n^2}{2[2n+1]^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1111952",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
} |
Getting the standard deviation from the pdf
A normally distributed random variable with mean $\mu$ has a probability density function given by $\dfrac{\gamma}{\sqrt{2\pi\sigma}}$ $\exp(-\dfrac{\gamma ^2}{\sigma} \dfrac{(x-\mu)^2}{2}) $
So the standard deviation is the square root of the variance, which is $E[(x-\mu)... | Usually one writes
$$
\frac1{\sigma\sqrt{2\pi}}\exp\left(-\frac 1 2 \left( \frac{x-\mu}\sigma \right)^2\right). \tag 1
$$
If one must add that extra parameter $\gamma$, then one has
$$
\frac\gamma{\sigma\sqrt{2\pi}}\exp\left(-\frac 1 2 \gamma^2\left( \frac{x-\mu}\sigma \right)^2\right). \tag 2
$$
This amounts to just p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1112020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
inequality symbols question (beginning algebra) Please help me with this problem:
"In each of the following exercises $x$ and $y$ represent any two whole numbers. As you know, for these numbers exactly one of the statements $x < y, x = y$, or $x > y$ is true.
Which of these is the true statement for each of the followi... | Your reasoning is correct, but your reasoning actually does not contradict the answers suggested in the book.
As voldemort pointed out in a comment, $x>y$ is logically equivalent to $y<x$ [e.g., $3>2$ and $2<3$]. This is due to the symmetric nature of the relations $<$ and $>$. That is, $x>y$ and $y<x$ are logically t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1112142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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The cardinality of the set of open subsets and related proofs. I have been going around in circles trying to prove this things for the last week, I would really appreciate any ideas on any of the next proofs.
Let C be the linear continuum with no endpoints and D a dense countable subset of C.Let O be the set of all th... | These are a handful of questions that have been asked before. Let me provide a few helpful hints:
*
*One direction is trivial; for the other direction show that every open set is the union of open intervals, and every open interval is the union of open intervals with endpoints in $D$.
*Use the fact that if $U$ is o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1112254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Solving augmented linear system $\left(\begin{smallmatrix}1&0&-2&~~~&0\\0&1&0&&0\\0&0&0&&0\end{smallmatrix}\right)$ $$\left(\begin{smallmatrix}1&0&-2&~~~&0\\0&1&0&&0\\0&0&0&&0\end{smallmatrix}\right)\to \mathbb{L}=\langle\left(\begin{smallmatrix}2\\0\\1\end{smallmatrix}\right)\rangle$$
I have seen a lot ot tutorials, b... | You're trying to solve the matrix equation
$$\left(\begin{matrix}
1 & 0 & -2\\
0 & 1 & 0\\
0 & 0 & 0
\end{matrix}\right)
\left(\begin{matrix}
x \\
y \\
z
\end{matrix}\right)
=\left(\begin{matrix}
0 \\
0 \\
0
\end{matrix}\right)
$$
Subsituting
$$\left(\begin{matrix}
2 \\
0 \\
1
\end{matrix}\right)$$
for
$$\left(\begin{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1112326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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(Infinite) Nested radical equation, how to get the right solution? I've been tasked with coming up with exam questions for a high school math contest to be hosted at my university. I offer the following equation,
$$\sqrt{x+\sqrt{x-\sqrt{x+\sqrt{x-\cdots}}}}=2$$
and ask for the solution for $x$.
Here's what I attempted ... | You know that:
$$\sqrt{x+\sqrt{x-\sqrt{x+\sqrt{x-\cdots}}}}=A = 2$$
and hence
$$A = \sqrt{x+\sqrt{x-A}} = 2$$
or equivalently
$$\sqrt{x+\sqrt{x-2}} = 2$$
Clearly, $\sqrt{x-2}$ is well defined when $$x \geq 2. ~~~(1)$$
Then, squaring both side, you get:
$$x+ \sqrt{x-2} = 4 \Rightarrow \sqrt{x-2} = 4-x ~~~(2).$$
Since $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1112441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Proving Set Operations I'm trying to prove that if $A$ is a subset of $B$ then $A \cup B = B$, but I am having trouble trying to proves this mathematically. I know that since $A$ is a subset, then $A$ has an element $x$ which is in $B$. So when $A \cup B$, some of the elements in $A$ are already in $B$; so the result o... | The first thing you need to do is make sure you know what the definitions of union, intersection, and subset really mean.
Union: $A\cup B = \{x : x\in A \space\text{or}\space x\in B\}$.
Intersection: $A\cap B = \{x : x\in A \space\text{and}\space x\in B\}$.
Subset: We say that $A$ is a subset of $B$, written $A\subset... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1112545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Solution to Differential Equation I'm looking for a solution to the following differential equation:
$$ y'' = \frac{c_1}{y} - \frac{c_2}{y^2} $$
where $c_1$ and $c_2$ are non-zero constants, and y is always positive.
The resulting function should be periodic. Any help is appreciated.
| $$y' y'' = \frac12 \frac{d}{dx} (y'^2) = c_1 \frac{y'}{y} - c_2 \frac{y'}{y^2} $$
Integrate both sides to get
$$\frac12 y'^2 = c_1 \log{y} + \frac{c_2}{y} + K_1$$
where $K_1$ is a constant of integration. Now take the square root of both sides and integrate to get
$$\pm x+K_2 = \frac1{\sqrt{2}}\int \frac{dy}{\sqrt{K_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1112622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How would the intersection of two uncountable sets form a countably infinite set? This is based off my last question How would the intersection of two uncountable sets be finite?
Here is the problem(from Discrete Mathematics and its Applications)
The book's definition on countable
And the definition of having the sam... | How about $[0, 1] \bigcup \{2, 3, 4, 5, \dots \}$ and $[5, 6] \bigcup \{7, 8, 9, 10, \dots\}$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1112721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Linear transformation of variable under the integral sign. Easy change of variables question I realize this might be a basic question, but I need a sanity check.
Let $f(\vec{x})$ be a function that takes $n$-dimensional vectors and returns a real number. Suppose the goal is to compute
$$\int_{\mathbb{R}^n} f(\vec{x})\,... | The other answer uses the same name for the transformed variable, which can be confusing for some. Let's call $\vec{y} = T \vec{x}$. Then
\begin{align*}
\int_{\mathbb{R}^n} f(\vec{x}) d \vec{x} &= \int_{\mathbb{R}^n} f(T^{-1}\vec{y}) \frac{1}{|\det T | } d \vec{y}
\end{align*}
and also
\begin{align*}
\int_{\ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1112800",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Unconditional probability I understand the tree diagram but my answer is wrong.
Urn I contains three red chips and one white chip. Urn II contains two red chips and two white chips. One chip is drawn from each urn and transferred to the other urn. Then a chip is drawn from the first urn.What is the probability that th... | Here is a tree diagram for the scenario.
A reminder on how to use the tree diagram, on each branch the probability of traveling along that branch from the previous branching point is written. For example, to travel along the topmost branch on the left corresponds to the event of pulling a red from the first urn and a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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can probability be negative I am solving a question that says:
Given that $$P(B)=P(A\cap B)=\frac{1}{2}$$ and $$P(A)=\frac{3}{8}$$
Find
$$P(A\cap {B}^{c})$$
My answer is
$$P(A\cap {B}^{c})=P(A)-P(A\cap B)=(-)\frac{1}{8}$$
| It's because the two conditions contradict each other. The first says that $P(B)$ is $\frac{1}{2}$, and also that whenever $B$ happens $A$ happens, so already we know that $P(A) \ge \frac{1}{2}$. But the second says that $P(A) \lt \frac{1}{2}$. The derivation you have carried out is another way to say the same thing... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Discrete Time Fourier Transform of the signal represented by $x[n] = n^2 a^n u[n]$ I have a homework problem that I am just not sure where to start with. I have to take the Discrete Time Fourier Transform of a signal represented by:
$$x[n] = n^2 a^n u[n]$$
given that $|a| < 1$, $\Omega_0 < \pi$, and u[n] being the un... | To include this as the answer that most helped me, since it was just as a comment to my original post. The most straightforward solution seemed to be by using the property of the Discrete Time Fourier Transform for when you are multiplying by n in the time domain, that corresponds to a derivation in the Fourier/freque... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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understanding a statement in Gill, Murray and Wright "Practical Optimization" Hi: I'm reading the book "Practical Optimization" and there's a part in Chapter 3 that
I can't prove to myself but I'm sure it's true. On page 64, they define the Taylor expansion of $F$ about $x^{*}$:
(3.3) $F(x^{*} + \epsilon p) = F(x^{*}) ... | Divide the inequality by $\epsilon$. Then for $\epsilon\to0$ the left-hand side tends to $p^Tg(x^*)<0$. (Assuming that $G$ is continuous).
Thus, there is $\bar\epsilon$ such that the expression is negative for all $\epsilon\in(0,\bar\epsilon)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1113153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Computing lim sup and lim inf of $\exp(n\sin(\frac{n\pi}{2}))+\exp(\frac{1}{n}\cos(\frac{n\pi}{2}))$ and $\cosh(n\sin(\frac{n²+1}{n}\frac{\pi}{2}))$? It's the first time I encounter lim sup and lim inf and I only just know about their definitions. I have difficulties finding out about lim sup and lim inf of the followi... | Note: After submitting the answer I realized that I've made a mistake. The conclusions that I draw are to be referred to $\sup$ and $\inf$ of the sequences, and not $\limsup$ and $\liminf$. I apologize for the error. I wanted to delete the answer, but then reconsidered it and I'll leave it here in case someone will fin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1113258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Showing coercivity of a function I am well attuned to the definition for a function to be coerce,
which is that $\lim_{\|x\| \to \infty}f(x) = \infty$
ie the values of $f$ go to infinity as the norm goes to infinity.
So
Ex.1 $f(x_1,x_2) = x_1^4 + x_2^4 - 3x_1x_2$
I thought this was immediately obvious because
$x_1^4$... | Ex.1 No, your answer is not rigorous. It is true, but you need to prove it. My suggestion is to show that $$\lim_{\|(x_1,x_2)\| \to +\infty} \frac{x_1 x_2}{x_1^4+x_2^4}=0.$$
Ex.2 If $x \perp a$, then $f(x)=0$. And since on the subspace $\{a\}^\perp$ there are vectors of arbitrarily large norm...
Ex. 3 If $x_1=x_2=t$, t... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding the nth term in a recursive coupled equation. I'm probably missing something simple, but if I have the recursive sequence:
$$
a_{i+1} = \delta a_i+\lambda_1 b_i
$$
$$
b_{i+1} = \lambda_2 a_i + \delta b_i
$$
how would I find a formula for $a_n$, $b_n$, or even $\frac{a_n}{b_n}$, given, for example, $a_0 = 1$, $b... | Let $X_i:=[a_i,b_i]^T$ then $$X_{i+1}=AX_i$$ where $$A:=\begin{bmatrix}\delta&\lambda_1\\\lambda_2&\delta\end{bmatrix}$$
Then it is clear that $$X_i=A^{i}X_0$$
To get a nice formula for $A^i$ first write it as $P^{-1}JP$, where $J$ is its Jordan form. Notice that if $\lambda_1\lambda_2\neq0$ $J$ is going to be diagonal... | {
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Prove, using the mean value theorem, that $x+1 \lt e^x \lt xe^x+1$ for $x \gt 0$ Prove, using the mean value theorem, that $x+1 \lt e^x \lt xe^x+1$ for $x \gt 0$.
What I tried so far: Let there be a function $f(x)=xe^x$. So $f'(x)=xe^x+e^x$. Let there be arbitrary interval $(0,b)$. So by the mean value theorem there i... | Hint: $$x+1 \lt e^x \lt xe^x+1 \iff 1 \lt \frac{e^x-e^0}{x - 0} \lt e^x$$
Now can you apply the Mean-Value Theorem?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding an isomorphism of groups
Let $R$ be a commutative ring, $R_U$ be the group of units of R. Show that
(i)
$\mathbb{C}_U \cong \left(\left(\mathbb{R},+\right)/\mathbb{Z}\right)\times\left(\mathbb{R},+\right)$
(ii)
$(\mathbb{Z}[x] / (x^2-x))_U \cong \mathbb{Z}_2 \times \mathbb{Z}_2$
where $\cong$ stands for ... | $\mathbb C_U=\mathbb C^{\times}\simeq(\mathbb R/\mathbb Z,+)\times(\mathbb R_{> 0},\cdot)$ by using the isomorphism you suggested. Moreover, $(\mathbb R_{>0},\cdot)\simeq(\mathbb R,+)$ by using logarithm.
$\mathbb{Z}[x] / (x^2-x)\simeq\mathbb Z[x]/(x-1)\times\mathbb Z[x]/(x)$ by CRT. Moreover, $\mathbb Z[x]/(x-1)\simeq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1113743",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is there a formula of coefficients of Newton-Cotes Method in numerical intergation? We know the coefficients of Newton-Cotes method in numerical integration are:
2-points $ 0.5$ , $0.5$
3-points $ 1/6$, $2/3$, $ 1/6$
4-points $1/8$, $3/8$, $3/8$, $1/8$
and so on
I ask if there ... | Maybe read: http://people.clas.ufl.edu/kees/files/NewtonCotes.pdf
It presents a method easily convertible to MATLAB that can be used to generate the coefficients:
function q = NewtonCotesCoefficients(degree)
q = 0:degree;
m = fliplr(vander(q));
for ii = 0: degree
b(ii + 1) = degree^(ii + 1)/(ii + 1);
end
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1113826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Field homomorphisms of $\mathbb{R}$ How to prove that $\mathrm{Hom}(\mathbb{R})=\mathrm{Aut}(\mathbb{R})$ ?
(We treat it as field homomorphisms. )
I know that $\mathrm{Aut}(\mathbb{R})=\{\mathrm{id}\}$ and $\mathrm{Mon}(\mathbb{R})=\mathrm{Aut}(\mathbb{R})$.
| Hint: Let $\phi: \mathbb R \to \mathbb R$ is a ring homomorphism. Prove the followings: (1). $\phi(r) = r, \forall r \in \mathbb Q.$ (2). $\phi(x) > 0, \forall x >0.$ (3). $|x-y|<\frac{1}{m} \Rightarrow |\phi(x) - \phi(y)|<\frac{1}{m}.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1113934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Simplify $\arctan (\frac{1}{2}\tan (2A)) + \arctan (\cot (A)) + \arctan (\cot ^{3}(A)) $ How to simplify $$\arctan \left(\frac{1}{2}\tan (2A)\right) + \arctan (\cot (A)) + \arctan (\cot ^{3}(A)) $$ for $0< A< \pi /4$?
This is one of the problems in a book I'm using. It is actually an objective question , with 4 optio... | As $0<A<\dfrac\pi4\implies\cot A>1\implies\cot^3A>1$
Like showing $\arctan(\frac{2}{3}) = \frac{1}{2} \arctan(\frac{12}{5})$,
$\arctan(\cot A)+\arctan(\cot^3A)=\pi+\arctan\left(\dfrac{\cot A+\cot^3A}{1-\cot A\cdot\cot^3A}\right)$
Now $\dfrac{\cot A+\cot^3A}{1-\cot A\cdot\cot^3A}=\dfrac{\tan^3A+\tan A}{\tan^4A-1}=\dfrac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1114007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Heat Equation, possible solutions NOTE: This is a homework problem. Please do not solve.
I was given a problem that asked me to find a function of the form $u_n(x,t)=\chi_n(x) \cdot T_n(t) $ that solves the heat equation with the following conditions:
$u_t = u_{xx}\\
u(0,t)=0\\
u_x(1,t)=0$
That is all of the informatio... | *
*Plug $u(x,t)=T(t)\,X(x)$ into the heat equation.
*Obtain an equation where on the left hand side you have $T$ and $T'$ and on the right hand side $X$ and $X'$.
*If a function that depends only on $t$ is equal to a function that depends only on $x$, what type of functions can they be?
*Obtain a second order ordin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1114172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Proving with Big O Notations Is there a way I can prove that $O(3^{2n})$ does NOT equal $10^n$? How would that be done? Also, is it okay to simplify $O(3^{2n})$ to $O(9^n)$ to do so?
| It seems that you want to show that $10^n\notin O(3^{2n})$.
To prove that $10^n$ is not $O(3^{2n})$ it is enough to show that for any $M$ there is $n$ such that $10^n > M\cdot3^{2n}$, in particular
\begin{align}
10^n &> M\cdot 9^n\\
\frac{10^n}{9^n} &> M \\
n &> \log_{\frac{10}{9}} M
\end{align}
so $n = \left\lfloor\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1114284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Double Factorial I am having trouble proving/understanding this question.
Let $n=2k$ be even, and $X$ a set of $n$ elements. Define a factor to be a partition of
$X$ into $k$ sets of size $2$. Show that the number of factors is equal to $1 \dot\ 3 \dot\ 5 \cdots\ (2k-1)$.
Lets suppose our set $X=\{x_1,x_2,...,x_n\... | You can also partition the set as $\{x_1,x_3\}$ and its complement and as $\{x_1,x_4\}$ and its complement. Giving three factors in total.
In general there are always $2k-1$ two element sets containing some fixed element. This observation also suggest a way towards a proof.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1114394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
I can't remember a fallacious proof involving integrals and trigonometric identities. My calc professor once taught us a fallacious proof. I'm hoping someone here can help me remember it.
Here's what I know about it:
*
*The end result was some variation of 0=1 or 1=2.
*It involved (indefinite?) integrals.
*It was... | The simplest one I have is not actually 0=1 but $\pi=0$. This is one of my favourites,the most shortest and has confused a lot of people.
$\int \frac{dx}{\sqrt{1-x^2}} = sin^{-1}x$
But we also know that $\int - \frac{dx}{\sqrt{1-x^2}} = cos^{-1}x$
So therefore $sin^{-1}x=-cos^{-1}x$
But also, $sin^{-1}x+cos^{-1}x=\pi/2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1114605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 4,
"answer_id": 0
} |
A sequence of Continuous Functions Converges Uniformly over $\mathbb{R}$ if it Converges Uniformly over $\mathbb{Q}$ I'm trying to show that if ${f_n}$ is a sequence of real functions that is continuous over all of $\mathbb{R}$ and that converges uniformly to $f$ over $\mathbb{Q}$, then it converges uniformly to $f$ ov... | let $r\in \mathbb Q$ then by definition $|f_n(r)-f_m(r)|<\frac{\epsilon}{2}\forall n,m\geq p$
Since $\mathbb Q$ is dense in $\mathbb R$ given $x\in \mathbb R\exists r\in \mathbb Q$ such that $|x-r|<\delta $
Since $f_n$ is continuous $|x-r|<\delta \implies |f_n(x)-f_n(r)|\epsilon \forall n\geq p$
thus for $x\in \mathbb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1114690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
The maximum of $\binom{n}{x+1}-\binom{n}{x}$ The following question comes from an American Olympiad problem. The reason why I am posting it here is that, although it seems really easy, it allows for some different and really interesting solutions. Do you want to give a try?
Let $n$ be one million. Find the maximum valu... | $$t_x=\binom n{x+1}-\binom nx=\frac{(n-2x-1)}{(x+1)}\binom nx$$
$$\frac{t_x}{t_{x-1}}=\frac{n-x+1}{(x+1)(n-2x+1)(n-2x)}=z$$
If $z\ge1$:
$$\frac{n-x+1}{(x+1)(n-2x+1)(n-2x)}\ge1\implies x\le z_0(\text{say})$$
Then maximum occurs at :
$$x=\lfloor z_0\rfloor$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1114783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Transition probability matrix of Markov chain
Given that
$g(x)=\begin{cases} 1/3 \quad\text{for } x=0\\ 1/3 \quad \text{for }
x=1\\ 1/3 \quad \text{for } x=2\end{cases}$
Explain why independent draws $X_1,X_2,\dots$ from $g(x)$ give rise to
a Markov chain. What is the state space and what is the transition
proba... | Markov Chains are stochastic processes $\{X_n\}$ such that $X_n$ depends only on $X_{n-1}$, in the sense that $\mathbb{P}(X_n = a_n \mid X_{n-1} = a_{n-1}, \ldots, X_1 = a_1) = \mathbb{P}(X_n = a_n \mid X_{n-1} = a_{n-1})$. This still holds here, as both sides equal $1/3$, although the "dependence" is in this case is r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1114891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Functions - finding the domain Question:
Consider the function: $$f(x) = \log(2x + 1) - \log(x - 3)$$
What will be the domain of this function?
I used two approaches to solve this question. Both approaches got me different answers. Consider that we do not merge the two $log$ together. As we know that the value insi... | Consider whether the domain of this contains "-1":
$$y=\sqrt x-\sqrt x$$
Or if the following contains "2":
$$\frac{(x+1)(x-2)}{x-2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1114992",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Limit of logarithm function, $\lim\limits_{n\to\infty}\frac{n}{\log \left((n+1)!\right)}$ Determine $$\lim_{n \to \infty} \frac{n}{\log \left((n+1)!\right)}.$$
Now, I know that $\log x < \sqrt{x} < x$ and trying to apply comparison test, but it does not work. Please help.
| We have that
$$
\lim_{n\to\infty}\frac n{\log(n+1)!}=\lim_{n\to\infty}\frac n{\log 2+\log3+\ldots+\log(n+1)}.
$$
Since the logarithm is a monotone function, we can estimate the sum with integrals from below and above in the following way
$$
\int_1^{n+1}\log x\mathrm dx\le\sum_{i=2}^{n+1}\log i\le\int_2^{n+2}\log x\math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1115104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
factor the following expression $25x^2 +5xy -6y^2$ How to factor
$$25x^2 +5xy -6y^2$$
I tried with $5x(5x+y)-6y^2$. I'm stuck here.
I can't continue.
| The trick here is to manipulate the expression $25x^2+5xy-6y^2$. Try the following:
\begin{align}
25x^2+5xy-6y^2 &= 25x^2+15xy-10xy-6y^2\tag{manipulate}\\[0.5em]
&= 5x(5x+3y)-2y(5x+3y)\tag{factoring}\\[0.5em]
&= (5x-2y)(5x+3y)\tag{group}
\end{align}
Is this clear now?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1115172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 2
} |
Can a non-periodic function have a Fourier series? Consider two periodic functions. Assume their sum is not periodic.
The periodic functions can be represented by a Fourier series. If you add up the Fourier series, you get a series that represents their sum. But their sum is not periodic, yet you have described it usin... | A Fourier series means the amplitude of the different harmonics, who are an integer multiple of a base frequency.
It is easy to see, that this base frequency simply doesn't exist in your case.
Although a Fourier transform of a such function of course exist, which is trivially
$$F(s)=\delta(t-\omega_1)+\delta(t-\omega_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1115240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 6,
"answer_id": 0
} |
Given $S \subset \Bbb{R}$, show $\textbf{int}(S)+\textbf{ext}(S)+\partial S =\Bbb{R}$ The way I proved it is that we knwo R is open so intR=R.
For any point in IntS is inside of IntR and any point in ExtS is inside of IntR.
any point that is neither intS nor extS is still inside of IntR
So intR is collection of all int... | Since $\text{ext}(S), \space \text{int}(S), \space \partial S \subseteq \Bbb{R}$ it follows that the union $$\text{ext}(S) \cup \text{int}(S) \cup \partial S \subseteq \Bbb{R}$$ Now you need to show the reverse inclusion. Recall: $$\text{ext}(S) = \Bbb{R} \setminus \overline{S} \\ \text{int}(S) =\Bbb{R} \setminus \ove... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1115294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find the least number b for divisibility What is the smallest positive integer $b$ so that 2014 divides $5991b + 289$?
I just need hints--I am thinking modular arithmetic?
This question was supposed to be solvable in 10 minutes...
| Using the Extended Euclidean Algorithm as implemented in the Euclid-Wallis Algorithm:
$$
\begin{array}{r}
&&2&1&38&2&25\\\hline
1&0&1&-1&39&-79&2014\\
0&1&-2&3&-116&235&-5991\\
5991&2014&1963&51&25&1&0\\
\end{array}
$$
Therefore, $2014\cdot235-5991\cdot79=1\implies5991\cdot79+1\equiv0\pmod{2014}$.
Multiply the last equ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1115400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Finding the volume using cylindrical shells about the x-axis So I have spent about a hour on this problem and figured it was time to ask for some advice. The problem is to find the volume using cylindrical shells by rotating the region bounded
by
$$8y = x^3,\qquad y = 8,\qquad x = 0$$
about the x-axis.
I changed the o... | I agree with you. The volume is
\begin{align*}
\int_0^8 2\pi y x\,dy &= \int_0^8 2\pi y \left(2\sqrt[3]{y}\right)\,dy \\
&= 4\pi \int_0^8y^{4/3} \,dy \\
&= 4\pi \left[\frac{3}{7}y^{7/3}\right]^8_0 = \frac{12\pi}{7} \cdot 2^7 = \frac{1536\pi}{7}
\end{align*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1115572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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