Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How do I make a primitive recursive function that does division? I am trying to define a primitive recursive function that does division. I looked at this answer but it seems wrong to me, because according to Wikipedia:
The primitive recursive functions are among the number-theoretic functions, which are functions fro... | Definition by cases is a valid, derived principle of definition for primitive recursive functions. So is subtraction, and so is equality. I will therefore use them freely.
Moreover, it is a good idea to define not just integer division $d(m,n)$ but also the remainder function $r(m,n)$. One can then write
\begin{align}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2290137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Where can I search pairs of primes, of certain gap?
Is there an online database that lets me search prime pairs by the gap among primes?
I know of the primes.utm.edu's search, and I see twin primes (gap = $2$) can be searched by typing "twin", but how can I search any other valid gap, for example; gap of $12$?
I cou... | Here's an online table of prime gaps:
http://www.trnicely.net/gaps/gaplist.html
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2290310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Question on limits in $\; \mathbb R^n $ including norm Let $\;f:\mathbb R \rightarrow \mathbb R^n \;$ a Lipschitz continuous map such that $\; \lim_{x \to x_0} f(x)=0\;$. I want to see the behaviour of the following limit:
$\; \lim_{x \to x_0} \frac{f(x)}{\vert f(x) \vert}\;$ where $\;\vert \;\; \vert \;$ is the Euclid... | This limit must not exist! For example take
\begin{align}
\newcommand{\R}{\mathbb{R}}
f:\mspace{0.3em}
\begin{array}{rcl}
\R &\to &\R^2\\
x &\mapsto& x \left(\begin{array}{cc}\cos x \\ \sin x\end{array}\right)
\end{array}
%
\end{align}
Note that $\|f(x)\|_2=|x|$ and $\frac{x}{|x|} = \operatorname{sgn}(x)$ if $x\neq 0$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2290421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$\sin 9^{\circ}$ or $\tan 8^{\circ} $ which one is bigger? $\sin 9^{\circ}$ or $\tan 8^{\circ} $ which one is bigger ?
someone ask me that , and said without using calculator !!
now my question is ,how to find which is bigger ?
Is there a logical way to find ?
I s there a mathematical method to show which is greater... | I would check the first two terms of the Taylor series. $\sin 9^\circ \approx \frac \pi{20}-\frac {\pi^3}{6 \cdot 20^3}, \tan 8^\circ \approx \frac {2\pi}{45}+\frac {8\pi^3}{3 \cdot 45^3}$, so $$\sin 9^\circ -\tan 8^\circ\approx \frac \pi{20}-\frac {\pi^3}{6 \cdot 20^3}-\frac {2\pi}{45}-\frac {8\pi^3}{3 \cdot 45^3}\\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2290527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 0
} |
Smooth fibration is a submersion in Wikipedia and the following questions: [1], [2] or their respective answers or comments it is said that a smooth fibration is a submersion. To make clear what I mean:
Definition. A smooth map $p\colon E \to B$ is said to satisfy the homotopy lifting property in the smooth category if... | Let $x$ be a point of $B$, consider a chart $f:U\simeq I^n\rightarrow B$ whose domain contains $x$ and $f(0)=x$. Write $Y=\{x\}$ and consider a point $z\in p^{-1}(x)$. You can define $\tilde f:Y\times \{0\}\rightarrow E$ by $\tilde f(x,0)=z$.
Let $F:\{x\}\times I^n\rightarrow B$ defined by $F(x,y)=f(x)$. We have $p\cir... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2290610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Determine the value without solving for a I got this question at the bottom of this page, the last question on the page. I've been taught how to handle this question but I couldn't figure this one out so I'd like to show how the website solved it and I please want someone to explain to me the steps they took to get to ... | hint
use
$$(a+b)^3=a^3+3ab (a+b)+b^3$$
with $b=\frac {1}{3a} $, observe that
$3ab=1$,
and $$27=3^3$$
With $a+b=2$, the result is $$2^3-2=6$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2290812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find $\sin(A)$ and $\cos(A)$ given $\cos^4(A) - \sin^4(A) = \frac{1}{2}$ and $A$ is located in the second quadrant.
Question: Find $\sin(A)$ and $\cos(A)$, given $\cos^4(A)-\sin^4(A)=\frac{1}{2}$ and $A$ is located in the second quadrant.
Using the fundamental trigonometric identity, I was able to find that:
• $\cos^... | Hint
$$\left( \cos(A)+ \sin(A) \right)^2 = 1+2 \sin(A) \cos(A)=\frac{1}{2} \\
\left( \cos(A)- \sin(A) \right)^2 = 1-2 \sin(A) \cos(A)=\frac{3}{2} $$
Take the square roots, and pay attention to the quadrant and the fact that $\cos^4(A) >\sin^4(A)$ to decide is the terms are positive or negative.
Alternate simpler soluti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2290899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Weierstrass factorization theorem for sin Wikipedia says that
$$\sin \pi z = \pi z \prod_{n\neq 0} \left(1-\frac{z}{n}\right)e^{z/n} = \pi z\prod_{n=1}^\infty \left(1-\left(\frac{z}{n}\right)^2\right).$$
Where did the second equality come from?
| The first product is absolutely convergent, so the order of the terms can be changed without changing its value. We pair the $\pm n$ terms, so
$$ \left( 1 - \frac{z}{n} \right)e^{z/n}\left( 1 - \frac{z}{-n} \right)e^{-z/n} = \left( 1 - \frac{z^2}{n^2} \right). $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2291020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that if $G$ is abelian, and $|G| \equiv2 \mod 4$, then the number of elements of order $2$ in $G$ is $1$. I've tried proving it by contradiction, assuming the number of elements is different than one, which, by Sylows $3$, implies that $|G|=2^xm$ with $m$ odd. With that I managed to show that $m\equiv1\mod4$, but ... | Here's an alternative solution without using Sylow's theorems.
Note that $|G|\equiv 2 \pmod 4 \Rightarrow |G|=2(2k+1)$ for some $k\in \mathbb{Z}$.
Since $2$ divides $|G|$, by Cauchy's theorem there exists an element $g\in G$ of order $2$. Now $\langle g\rangle$ is a subgroup of order $2$. Now since $G$ is abelian we ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2291126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Show $PQ$ and $QP$ have the same eigenvalues with density of $GL_n$ There is a wonderful series of lectures on YouTube of Dr. Tadashi Tokieda on Geometry and Topology. In the fourth video in this playlist Tadashi sketches an argument for why if $P$ and $Q$ are $n$ by $n$ matrices then $PQ$ and $QP$ have the same eigenv... | The roots of a complex polynomial are continuous in its coefficients. Hence, if $A_n$ is a sequence of matrices converging to $A$, then the eigenvalues of $A_n$ converge to those of $A$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2291236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
Determining coefficients of an infinite series Here is my question:
Let $s=\sum_{i\ge 0} a_ix^i = (1-4x^2)^{-20}$
$t=\sum_{j\ge 0} b_jx^j = (1+x^5)^{-17}$
Determine $a_i$ and $b_j$ for all $i,j \ge 0$. (The answer will be divided into cases. Ex $a_i$ will depend on whether $i\equiv0\pmod 2$)
Then, determine the coeffic... | It is convenient to use the coefficient of operator $[x^n]$ to denote the coefficient of $x^n$ in a series.
In order to determine the coefficient of $3x^{16}$ in $s(x)t(x)$ we obtain
\begin{align*}
\color{blue}{[x^{16}]3x^4 s(x)t(x)}
&=3[x^{12}]\left(\sum_{k\geq 0}\binom{19+k}{k}4^kx^{2k}\right)
\left(\sum_{l\geq 0}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2291309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find the solution to a PDE with an initial condition
Find the solution to $u_x + y u_y = u$ with initial condition $u(0,y) = \cos(y)$.
Attempted solution - Suppose we parametrize a curve $(x,y)$ by a parameter $\xi$. So that
$$
u=u(x(\xi),y(\xi))
$$
$$
\frac{\mathrm{d}u}{\mathrm{d}\xi}=\frac{\mathrm{d}x}{\mathrm{d}\x... | $$u_x+yu_y=u$$
Set of characteristic ODEs: $\quad \frac{dx}{1}=\frac{dy}{y}=\frac{du}{u}$
First family of characteristics curves, from $\frac{dx}{1}=\frac{dy}{y} \quad\to\quad ye^{-x}=c_1$
Second family of characteristics curves, from $\frac{dy}{y}=\frac{du}{u} \quad\to\quad \frac{u}{y}=c_2$
General solution, with any ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2291391",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Lemma about ordinal arithmetic I'm trying to prove the following lemma:
For any three ordinals $\alpha, \alpha' >0$, $\beta >1$, if $\alpha > \alpha'$, then we have
$$
\beta^\alpha > \beta^{\alpha'} \cdot \delta + \gamma
$$
For $0 < \delta < \beta$ and $\gamma < \beta^{\alpha'}$.
My question is, is this even true, and... | Just note that $$\beta^\alpha\geq\beta^{\alpha'+1}=\beta^{\alpha'}\cdot \beta\geq\beta^{\alpha'}\cdot (\delta+1)=\beta^{\alpha'}\cdot\delta+\beta^{\alpha'}>\beta^{\alpha'}\cdot\delta+\gamma.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2291512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
(How to show that something isn't an) Irreducible Polynomial in $\mathbb{Z}_5$
Let $f(x) = x^4+x+p$, where $p$ is a prime. I'm asked to show that if $p \neq -1$ (mod $5$), then $f(x)$ is not irreducible in $\mathbb{Z}_5$.
I guess I could try to show that $f(x)$ is not irreducible for $p=0,1,2,3$, but that seems like ... | Let $f(x) = x^{p-1}+x+q$, $p$ is a prime number, $q$ is any integer, we want to find a root of $f(x)$ in $\mathbb{Z}_p$.
By Fermat's Little Theorem, any $p\nmid a$, $a^{p-1} \equiv 1 \pmod p$, so
$$f(0) = q$$
$$f(1) \equiv 1+1+q\pmod p$$
$$f(2) \equiv 1+2+q\pmod p$$
$$...$$
$$f(p-2) \equiv 1+(p-2)+q\pmod p$$
$$f(p-1) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2291596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
From $a_{n+1}= \frac{3a_n}{(2n+2)(2n+3)}$ to $a_n$, Case 2
Find and prove by induction an explicit formula for $a_n$ if $a_1=1$ and, for $n \geq 1$,
$$P_n: a_{n+1}= \frac{3a_n}{(2n+2)(2n+3)}$$
Checking the pattern:
$$a_1=1 $$
$$a_2= \frac{3}{4 \cdot 5}$$
$$a_3= \frac{3^2} { 4 \cdot 5 \cdot 6 \cdot 7}$$
$$a_4= \frac{... | $a_{n+1} = \frac{3a_n}{(2n+2)(2n+3)} = \frac{3\cdot 3!\cdot 3^{n-1}}{(2n+2)(2n+3)(2(n-1)+3)!} = \frac{3!3^n}{(2n+3)!}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2291696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Difference between units and dimensions Though this question may seem related to Physics, I think that at the very root this is a mathematical question and so I have posted this on math.stackexchange.
Background: Initially I thought that the terms-unit and dimension, refer to the same thing.
Physical quantities are c... | I think there is no difference between dimension and unit in your case. They can be used interchangeably. However, the same word "dimension" is also used in another context, namely, describing the amount of numbers needed to describe a point in your space uniquely. These two use cases should not be confused. They are v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2291784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 7,
"answer_id": 1
} |
Identifying the Probability distribution. If I am certain to receive a phone call in a span of 60 minutes, what distribution does the phone call follow with time, within the 60 minutes period?
Obviously, all instances of time cannot have same probability because if it did, then it would imply that there are chances of ... | The Poisson distribution would allow more than one call in a given hour; it would also allow zero calls with non-zero probability.
I would not recommend this to model a call that is absolutely certain to occur within a given 60-minute interval.
It's unclear why marriage rates should have anything to do with the questio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2291897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving that $ \ln\left(\frac{49}{50}\right)<\sum^{98}_{k=1}\int^{k+1}_{k}\frac{k+1}{x(x+1)}dx<\ln(99)$
If $\displaystyle I = \sum^{98}_{k=1}\int^{k+1}_{k}\frac{k+1}{x(x+1)}dx.$ Then prove that $\displaystyle \ln\left(\frac{49}{50}\right)<I <\ln(99)$
Attempt: $$I = \sum^{98}_{k=1}\int^{k+1}_{k}(k+1)\bigg[\frac{1}{x(x... | Note that, for $x\in[k,k+1]$,
$$ k<x<k+1\Rightarrow\frac{1}{k+1}<\frac{1}{x}<\frac{1}{k}, \frac{1}{k+2}<\frac{1}{x+1}<\frac{1}{k+1}$$
and hence
\begin{eqnarray}
I &=& \sum^{98}_{k=1}\int^{k+1}_{k}\frac{k+1}{x(x+1)}dx\\
&\le&\sum^{98}_{k=1}\int^{k+1}_{k}\frac{1}{x}dx\\
&=&\sum^{98}_{k=1}[\ln(k+1)-\ln(k)]\\
&=&\ln99
\e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2291997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Prove that function $f$ is injective if its Jacobian matrix is positive definite. Assume that $\Omega\in\mathbb{R}^m$ is an open convex set and the vector-valued function $f:\Omega\rightarrow\mathbb{R}^m$ is differentiable. If Jacobian matrix $J_f(x)$ is positive definite for all $x\in\Omega$, prove that $f$ is an inje... | Hint: suppose $x\neq y$ but $f(x)=f(y).\ $ On $\Omega$, define for $0\le t\le 1,\ \gamma (t)=ty+(1-t)x,\ $ which is well-defined because $\Omega$ is convex.
Now, compute $D\langle ((f\circ \gamma)(t)-f(x)), (y-x)\rangle$ and use the Mean Value Theorem and your hypothesis on the Jacobian, to arrive at a contradiction... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2292127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Is my proof for $f(0)=1$ for a specific continuous function correct? Alright, I think I have found a much simpler proof to a question than the one I was provided with, and wanted to hear how it is inevitably incorrect.
Let $f$ be a continuous function that you can always get the derivative of and that is always posit... | The solution is not correct. By definition,
$$f'(0)=\lim_{x\to 0}\frac{f(x)-f(0)}{x}=\lim_{x\to 0}\frac{f(x)-1}{x}$$
so,
$$\lim_{x\to 0}\frac{f(x)-f(0)}{x}-\lim_{x\to 0}\frac{f(x)-1}{x}=0\to \lim_{x\to 0}\frac{f(0)-1}{x}=0\quad (1)$$
but
$$\lim_{x\to 0}\frac{c}{x}=c\cdot \lim_{x\to 0}\frac{1}{x}$$
Doesn't exist if $c\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2292386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Finding the Jordan form for a $4\times4$ matrix $$A:= \begin{bmatrix}4 & -4 & -11 & 11\\3 & -12 & -42 & 42\\ -2 & 12 & 37 & -34 \\ -1 & 7 & 20 & -17 \end{bmatrix}$$
I'm struggling with this matrix: it has $p_A(x) = (x-3)^4 $, yet $\ker(A - 3I)^3 $ is already the whole space $ \mathbb C^4 $. I read on another post tha... | The issue here is that the method you're trying to apply is not quite right. My guess is that the result that is attempted to being applied is that "the algebraic multiplicity of $3$ in the minimal polynomial of your matrix is the size of the largest Jordan block". (you're looking at the geometric multiplicity of your ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2292491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Show that for every integer $n$, $n^3 - n$ is divisible by 3 using modular arithmetic Problem:
Show that for every integer $n$, $n^3 - n$ is divisible by 3 using modular arithmetic
I was also given a hint:
$$n \equiv 0 \pmod3\\n \equiv 1 \pmod3\\n \equiv 2 \pmod3$$
But I'm still not sure how that relates to the questi... | Using the hint is to try the three cases:
Case 1: $n \equiv 0 \mod 3$
Remember if $a \equiv b \mod n$ then $a^m \equiv b^m \mod n$ [$*$]
So $n^3 \equiv 0^3 \equiv 0 \mod 3$
Remember if $a \equiv c \mod n$ and $b \equiv d \mod n$ then $a+b \equiv c + d \mod n$ [$**$]
So $n^3 - n\equiv 0 - 0 \equiv 0 \mod n$.
Case 2: $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2292586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Minimal polynomial of a power of an element over $\mathbb F_{2}$ Let $\beta \in \mathbb F_{16}$ whose minimal polynomial over $\mathbb F_{2}$ is $x^4+x+1$.
What is the minimal polynomial of $\beta^7$ over $\mathbb F_{2}$?
I know from this answer that $\beta$ generates $\mathbb F_{16}$ but I don't know if that helps at ... | Daniel Schepler's suggestion in the comments is a good way to approach this problem. Let me flesh out this method, as well as a few computational tricks/sanity checks.
First, as you note, $\beta$ is a generator for $\mathbb{F}_{16}^{\times}$, which is a cyclic group of order $15$. Since $\gcd(7, 15) = 1$, it follows th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2292662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Moment generating function within another function (a) Let $X$ be an exponential random variable with parameter $\lambda$. Find the moment generating function of $X$.
(b) Suppose a continuous random variable $Y$ has moment generating function
$M_Y(s)= \frac{\lambda^2}{(\lambda-s)^2}$ for $s<\lambda$ and $M_Y(s)+\infty$... | If you add two independent random variables, you convolve their PDF's (i.e. if the densities are $f, g$, their sum has density $h(t) = \int f(s)g(t-s) ds$) or multiply their moment generating functions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2292748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that if $F$ is a splitting Field of $S$ over $K$ and $E$ is an intermediate field then $F$ is a splitting field of $S$ over $E$. This happens to be Hungerford problem 5.3.2. Here $S$ is a set of polynomials in $K[x]$. $F$ is a splitting field of $S$ over $E$ if
*
*Every $f\in S$ splits in $F$
*$F= E(X)$ where... | $E \subseteq F$ because it's an intermediate field and $X \subseteq F$ because $X$ all the roots of polynomials in $S$, since $X \subseteq K(X) = F$, by hypothesis.
Therefore $F= K(X) \subseteq E(X) \subseteq F$, since $E$ and $X$ both are contained in F therefore the field generated by them.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2292859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$\sin(x) \leq x$ on the interval $[0,1]$ I need to prove that: $\sin(x) \leq x$ on the interval $[0,1]$ using Calculus. First I calculated the area between the graph of the function $x$ and $\sin(x)$ on the interval $[0,1]$, if it is positive then $\sin(x) \leq x$:
$$\int_0^1 x - \sin(x) dx = \frac{x^2}{2} \vert_0^1 ... | Fix $x \in [0,1]$,and write $\sin x = \displaystyle \int_{0}^x \cos t dt, x = \displaystyle \int_{0}^x 1dt\implies x - \sin x = \displaystyle \int_{0}^x (1-\cos t)dt\ge \displaystyle \int_{0}^x 0dt = 0\implies x \ge \sin x$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2292982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Balls and vase $-$ A paradox? Question
I have infinity number of balls and a large enough vase. I define an action to be "put ten balls into the vase, and take one out". Now, I start from 11:59 and do one action, and after 30 seconds I do one action again, and 15 seconds later again, 7.5 seconds, 3.75 seconds...
What i... | What you just discovered is that the cardinality of a set (the number of elements) is not a continuous function, that is, for a convergent sequence $S_n$ of sets you may have
$$\lim_{n\to\infty}\left|S_n\right|\ne \left|\lim_{n\to\infty} S_n\right|$$
where $\left|S_n\right|$ is the cardinality of $S_n$ (e.g. $\left|\{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2293091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 0
} |
Analytic number theory question. Let $n$ be a positive integer, and define $f(n)$ as $n +\lfloor\sqrt{n}\rfloor$, where $\lfloor x\rfloor$ is the greatest positive integer less than or equal to $x$. Prove that the sequence $n, f(n), f(f(n)), f(f(f(n))), \ldots$ contains a perfect square.
| If $n$ is a square, you're done. If $n=m^2+k$ with $0\lt k\lt2m+1$, then in either one or two steps you'll be at a number of the form $(m+1)^2+k'$ with $0\le k'\lt k$. Induction (on $k$) now tells you that eventually you'll land on a square.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2293345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\|UVU^{-1}V^{-1}-I\|\leq 2\|U-I\|\|V-I\|$ $U,V$ are unitary $n\times n$ matrices, and the norm is the operator norm (so we can use $\|UV\|\leq\|U\|\|V\|$).
I've noticed that
\begin{align}
\|UVU^{-1}V^{-1}-I\|&= \|(UV-VU)U^{-1}V^{-1}\|\\
&\leq \|UV-VU\|\|U^{-1}V^{-1}\|
\end{align}
I can bound the first term ... | (I cannot prove the inequality as stated in the exercise, but here is some information that maybe will help you).
Note that $\|UVU^{-1}V^{-1}\|=1$ for all unitaries $U,V$. We would like to show that
$$\tag{1}
\|UVU^{-1}V^{-1}-I\|\leq 2\|U-I\|\,\|V-I\|.
$$
The left-hand-side, as hinted in the exercise, is $\|UV-VU\|$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2293450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find the limit of $(1-\cos x)/(x\sin x)$ as $x \to 0$ Can you please help me solve:
$$\lim_{x \rightarrow 0} \frac{1- \cos x}{x \sin x}$$
Every time I try to calculate it I find another solution and before I get used to bad habits, I'd like to see how it can be solved right, so I'll know how to approach trigonometric l... | We can also use L'Hospital's Rule
$$\lim_{x\to0}\frac{1-\cos x}{x\sin x}=\lim_{x\to0}\frac{\sin x}{x\cos x+\sin x}=\lim_{x\to0}\frac{\cos x}{-x\sin x+2\cos x}=\frac{1}{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2293514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 8,
"answer_id": 3
} |
What is the value of the intersection of X and the set containing X? How to calculate X $\cap$ $\{X\}$ for finite sets to develop an intuition for intersections?
If $X$ = $\{$1,2,3$\}$, then what is $X$ $\cap$ $\{X\}$?
| As far as developing intuition for intersection, the idea of $A \cap B$ are the elements that $A$ and $B$ both have in common. So if we're looking at $X \cap \left\{ X \right\}$ where $X = \left\{ 1,2,3 \right\}$ then it is a matter of
$$X \cap \left\{ X \right\} = \left\{ 1,2,3 \right\} \cap \left\{ \left\{ 1,2,3 \ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2293600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Equality/Equivalence of functions When we say something like:
$$ \frac{dy}{dx} = x$$
we're describing the way a function $y$ changes with respect to $x$.
To solve the differential equation we integrate both sides. Is there a proper way to do this, or is it simply the case that if have a reliable method to get the cor... | Method $2$ would be considered more appropriate and note $\int x dx = \frac{1}{2}x^2 + C$. However, you only need one arbitrary constant in the solution, $y = \frac{1}{2}x^2 + C$ would suffice for a solution since $c_2 - c_1$ would result in a new constant, where we call it $C$.
1) It is okay to integrate both sides o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2293714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Vector valued analytic functions near essential singularity Suppose $f:\mathbb{C}\setminus\{0\}\to X$ is a vector valued analytic function, that has an essential singularity at $0$ ($X$ is some Banach space). It can be easily shown that, in this case, $f$ must be unbounded near $0$. I am intrested whether a stronger co... | This need not be the case.
We could have a function of the form $f(z) = g(z)\cdot x$ where $g \colon \mathbb{C}\setminus \{0\} \to \mathbb{C}$ is holomorphic with an essential singularity at $0$, and $x \in X \setminus \{0\}$.
But it can be the case, e.g. if $f(z) = g(z)\cdot x + h(z)\cdot y$, where $x$ and $y$ are lin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2293824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Statistical Mechanics - Canonical Partition Function - An harmonic Oscillator
*
*The problem statement, all variables and given/known data
With the Hamiltonian here:
Compute the cananonical ensemble partition function given by $\frac{1}{h} \int dq dp \exp^{-\beta(H(p,q)}$
for 1-d , where $h$ is planks constant
*R... | This is my first answer, so I hope I'm doing it right.
As pointed out in an earlier comment, I think you need to start of by getting the limits straight, which will answer a couple of your questions. The integral over $p$ is independent and easily done as you've stated yourself. The integral over $q$ goes from $-\infty... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2293920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to reduce congruence systems with moduli not coprime? I'm given a system of congruences and want to apply the Chinese remainder theorem (CRT) on it, but the greatest common divisor (GCD) of the moduli is not $1$.
Well, I need to reduce the system, such that the GCD is $1$, but I don't know how the "reducing" works.... | Reducing is trivial:
If $a \equiv b \mod n$ then $a \equiv b \mod k$ for all $k|n$.
Just think about it......
=======
$x \equiv 1 \mod 108 \implies x= 108k + 1 = 27(4k) + 1 \implies x \equiv 1 \mod 27$.
$x \equiv 13 \mod 40 \implies x = 40k + 13 = 8(5k) + 13 \implies x \equiv 13 \equiv 5 \mod 8$. etc.
=======
It's goin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2294043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Find the simplified form of $\frac {1}{\cos x + \sin x}$ Find the simplified form of $\dfrac {1}{\cos x + \sin x}$.
a). $\dfrac {\sin (\dfrac {\pi}{4} +x)}{\sqrt {2}}$
b). $\dfrac {\csc (\dfrac {\pi}{4} + x)}{\sqrt {2}}$
c). $\dfrac {\sin (\dfrac {\pi}{4} + x)}{2}$
d). $\dfrac {\csc (\dfrac {\pi}{4} + x)}{2}$
My Attemp... |
This is the graph of the function $\sin x+\cos x$.
Notice that itself looks like a wave.
So, we should be able to deduce that(since only the amplitude and phase seem to have changed),
$$\sin x+\cos x=A\sin( x+\phi)$$
Expanding using the identity for $\sin(A+B)$,
$$\sin x+\cos x=A\sin( x)\cos\phi+A\cos(x)\sin\phi$$
Com... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2294276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Combinatorics struggles I am stuck in combinatorics problem that there must be a solution to but every experiment is leaving me stuck.
I am needing to build a bracket for a some games at a graduation event:
*
*There will be $8$ teams doing eight events. They will complete against each other in each event. The eight ... | For any one team, there is a maximum of $7$ distinct pairings with other teams for the various events. Since you have $8$ events, you must have at least one pairing repeat. Another way of thinking about this would be the fact that there are only 28 distinct team pairings and you are trying to uniquely assign 32 events,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2294390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Compute $gcd\left(1714, 1814\right)$ using Euclidean Algorithm So I know the answer for this is $2$, but based on my own work, I can't get to that solution. I haven't done a gcd before where $b>a$. I thought I could just flip the numbers and use the same method but that didn't seem to work. Here's what I have so far, w... | \begin{align}
GCD(1814,1714)&=(1714,100)\\
&=(100,14)\\
&=(14,2)\\
\end{align}
So, $2$ is the $GCD$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2294451",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
How to find the radix in number system.? I have
$(132)_{10} = (2010)_r$
I have tried the above .. I got the answer that $2(r^3)+r =132 $
From this I am unable to find the value of $r$. Can anyone help me out to solve this problem?
| Our goal is to find a positive integer solution to $2 r^3 + r - 132 = 0$.
To proceed, we use the rational root theorem, which states that any rational solutions $\frac{p}{q}$ must have $p$ dividing $-132$ and $q$ dividing $2$.
Factoring $132$ gives you $2^2 \cdot 3 \cdot 11$. So here are all the possibilities (ignoring... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2294534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Calculating the remainder of the series $1/n!$ (euler number) I am currently studying series and in my textbook and there's an example of calculating the remainder of the series which I don't understand completely
The series in question is:
$$\sum_{i=0}^\infty \frac{1}{n!} = e,$$
so the remainder is
\begin{align}
\sum... | For $k > 1$ we have
$$\frac{1}{n+k}<\frac{1}{n+1}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2294685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Why this time-frequency representation of a pure sine function with Stockwell Transform is different in the boundary? This time-frequency decomposition was made with Stockwell Transform to a pure sine function and graphed with matlab. But as you can see in the boundary of the image the pattern is different. I need to k... | The S-transform is defined using an integral over an infinite time interval. Supposedly your pure sine function is considered zero outside some finite interval? Or it wraps around to the other end, but the phases at the ends do not match? Then it's not so pure at the ends of that interval. There will be some discontinu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2294930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating closed form of $I_n=\int_0^{\pi/2} \underbrace{\cos(\cos(\dots(\cos}_{n \text{ times}}(x))\dots))~dx$ for all $n\in \mathbb{N}$.
I was wondering if there is any way to evaluate a general closed form solution to the following integral for all $n\in \mathbb{N}$.
$$I_n=\int_0^{\pi/2} \underbrace{\cos(\cos(\c... | The mere fact that even the simple case $n=2$ ceases to possess a meaningful closed form in terms of elementary functions, and an entirely new function had to be invented from scratch in order to express its value, should be enough to settle all questions one might have concerning the possibility of finding such a form... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2295034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 1,
"answer_id": 0
} |
Solving the functional equation $\tau \left(\frac{-1}{z}\right) = - \tau(z)$
Let $\mathbb{H} \subset \mathbb{C}$ be the upper half plane. Find $\tau: \mathbb{H} \to \mathbb{C} $, holomorphic and non-constant, satisfying $\tau \left( \frac{-1}{z} \right) = - \tau(z)$.
There is a very good answer already. But since th... | EqWorld is our right partner.
In fact this functional equation belongs to the form of http://eqworld.ipmnet.ru/en/solutions/fe/fe1121.pdf.
The general solution is $\tau(z)=C\left(z~,-\dfrac{1}{z}\right)$ , where $C(u,v)$ is any antisymmetric function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2295129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Suppose G be a non-abelian group and H,K be two abelian subgroups of G. Then must HK be an abelian subgroup of G? Suppose $G$ is a non-abelian group and $H,K$ are two abelian subgroups of $G$. Then must $HK$ be an abelian subgroup of $G$?
I know an example, but I am confused. Thus I just want to check that.
| This is not true. There are even metabelian groups (non-abelian, but $G'$ abelian) which are the product of two abelian subgroups $A$ and $B$, i.e., $G=AB$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2295255",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Prove the perpendicular bisector of chord passes through the centre of the circle
Hello, can someone please give me a simple proof to the following theorem:
"The perpendicular bisector a chord passes through the centre of the circle."
I have attached a diagram of what I mean and web link of a proof that I did not unde... | The proof in the picture is the simplest possible one. All you have to do is write the conditions for congruence thus proving that the triangles in the above picture are congruent. This is the best possible approach to the given question.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2295360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 5
} |
Is $f(x) = x^{10}-x^5+1$ solvable by radicals?
Is $f(x) = x^{10}-x^5+1$ solvable by radicals?
So far I've showed that $f$ is irreducible because if we let $y=x^5$ then $f(y)=y^2-y+1$ which is irreducible because it has a negative discriminant. I also know that $f$ has no real roots so I've concluded that $Gal(L_f/\ma... | Yes, of course.
Let $x+\frac{1}{x}=a$.
Hence,
$$x^{10}-x^5+1=x^{10}+x^7-x^7+x^6-x^5-x^6+1=$$
$$=(x^2-x+1)(x^7(x+1)-x^5-(x^3-1)(x+1))=$$
$$=(x^2-x+1)(x^8+x^7-x^5-x^4-x^3+x+1)=$$
$$=(x^2-x+1)x^4\left(x^4+\frac{1}{x^4}+x^3+\frac{1}{x^3}-x-\frac{1}{x}-1\right)=$$
$$=(x^2-x+1)x^4(a^4-4a^2+2+a^3-3a-a-1)=$$
$$=(x^2-x+1)x^4(a^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2295462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Taking logarithm of sum and products $\newcommand{\cost}{\operatorname{cost}}$My cost-metric is in following form
\begin{equation}
\cost(x,y) = A(x,y_1) \times \sum_{i}b_i B_i(x,y_i)\tag{1}
\end{equation}
where $A$ and $B$'s follow normal distribution. For my computer implementation, I am thinking of taking $\log(\cdot... | No, you can't bring the logarithm inside the summation because that's equivalent to saying that the logarithm of a sum is the sum of logarithms.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2295568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Simplifying a Boolean expression (confused) Could somebody help me make sense of how to use the Boolean rules to simplify this expression?
$$(x'+(yz)')(x + z')'$$
I used distributivity to get
$$(x'+y')(x'+z')(x'+z)$$
I don't know if that was the right path to go down, or where to go from here.
Thanks.
| I'd rather take this from the start:
$$(x'+(yz)')(x+z')'$$
Apply De Morgan's laws to $(yz)'$ and $(x+z')'$:
$$=(x'+y'+z')zx'$$
Distribute:
$$=zx'x'+zx'y'+zxz'$$
$x'x'=x'$ and $zz'=0$:
$$=zx'+zx'y'$$
$zx'$ absorbs $zx'y'$:
$$=zx'$$
This is the simplest form.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2295731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Continued Fraction pattern I have been give then continued fraction
$\dfrac{1}{\dfrac{1}{\dfrac{1}{x-1}-1}-1}$
If I let x=5, I get the following pattern...
$\frac{1}{4}, \, -\frac{4}{3}, \,-\frac{3}{7}, \,-\frac{7}{10}, \,-\frac{10}{17}, \,...$
It appears that (excluding the first case) the previous denominator become... | Rather than starting with $x=5$, it's more enlightening to just write down the sequence in terms of $x$:
$$\frac1{x-1}, -\frac{x-1}{x-2}, -\frac{x-2}{2x-3}, -\frac{2x-3}{3x-5}, -\frac{3x-5}{5x-8}, -\frac{5x-8}{8x-13}, \dots$$
The coefficients in these fractions might look familiar: these are the Fibonacci numbers! More... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2295825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Can we use the defining property of weak derivatives when integrating over general measurable sets? Let $f \in H^1(\mathbb{R}^d)$. Then, we have that for every smooth function $\phi$ with compact support:
$$
\int_{\mathbb{R}^d} f \, \nabla \phi dx= - \int_{\mathbb{R}^d} \nabla f \, \phi dx
$$
where $\nabla f$ denotes t... | No. What you are integrating is actually $\chi_B$ times the integrand over the whole space, i.e.
$$ \int f \chi_B \nabla \phi$$
so what you are asking is wether $f\in H^1 \Rightarrow f\chi_B$ in $H^1$ (which is not true).
Also note that for $B$ with smooth boundary and smooth $f$ there are theorems which tell you that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2295986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Surface integral - spherical I'm trying to calculate the following surface integral $$\int \int_{s_r} \frac{z-R}{(x^2+y^2+(z-R)^2)^{3/2}} dS $$, where $s_r=\{(x,y,z)\in \mathbb{R}^3 : x^2+y^2+z^2=r^2 \}. $
I've switched to spherical coordinates but don't really know how to do it.
| The reason to use spherical coordinates is that the surface over which we integrate takes on a particularly simple form: instead of the surface $x^2+y^2+z^2=r^2$ in Cartesians, or $z^2+\rho^2=r^2$ in cylindricals, the sphere is simply the surface $r'=r$, where $r'$ is the variable spherical coordinate. This means that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2296108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Maximal ideal of $\mathbb{Q}[x,y]$
I want to show that $ \ A =\langle x,y \rangle$, the ideal generated by $x$ and $y$ is maximal in $R = \mathbb{Q}[x,y]$.
I have seen a different solution somewhere but it was kind of longer so I tried something else. I just want to know if it is correct.
Let $A \subseteq B \subsete... | This is almost correct but not quite. You cannot deduce that $1\in B$ from $a_0\in B$ since you only know that $a_0\in\mathbb{Q}[y]$, not that $a_0\in\mathbb{Q}$. What you can say is that $a_0$ has the same constant term as $p$ and hence has nonzero constant term, so you can write $a_0=\sum_{j=0}^m b_jy^j$ for $b_j\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2296334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to (or why cannot) define complex conjugate in the structure $(\mathbb{Z}+i\mathbb{Z},+)$? Let $(\mathbb{Z}+i\mathbb{Z},+)$, where $i$ is the imaginary unit, be a structure with an only operation $+$, the ordinary addition in $\mathbb{Z}$, and with no constant symbols. In this structure, the number zero and the inv... | There is no way to do this.
Remember that for a function $f: \mathcal{M}\rightarrow\mathcal{M}$ (or indeed any function or relation in general) to be definable in $\mathcal{M}$, it must be fixed by automorphisms: if $\alpha$ is an automorphism of $\mathcal{M}$, we must have that $f(\alpha(m))=\alpha(f(m))$ for all $m\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2296422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Counterexamples for uniqueness of viscosity solutions Recall a classical comparison result for viscosity solutions:
Let $H:[0,T]\times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$
*
*continuous
satisfying
*$\vert H(t,x,p)-H(t,x,q) \vert \le C \vert p-q \vert$
*$\vert H(t,x,p) - H(s,y,p)\... | I assume you are interpreting viscosity solutions via Ishii's notion of solution when $H$ is discontinuous (where you replace $H$ by its upper and lower semicontinous envelopes in the super and subsolution properties, respectively). In this case consider the PDE (or ODE rather)
$$u'(x) = f(x)$$
where $f(x)=1$ for $x$ r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2296539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Inverse or approximation to the inverse of a sum of block diagonal and diagonal matrix I need to calculate $(A+B)^{-1}$, where $A$ and $B$ are two square, very sparse and very large. $A$ is block diagonal, real symmetric and positive definite, and I have access to $A^{-1}$ (which in this case is also sparse, and block ... | Let $A_1,A_2,\cdots,A_q$ be the diagonal blocks of $A$, and $a_{1,1},a_{1,2},\cdots,a_{1,n_1},a_{2,1},a_{2,2},\cdots,a_{2,n_2},\cdots,a_{q,1},a_{q,2},\cdots,a_{1,n_q}$ the diagonal elements of $B$, then the inverse of the sum would simply be a diagonal block matrix with blocks: ${(A_i+diag(a_{i,1},\cdots,a_{i,n_i}))}^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2296724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Two individuals are walking around a cylindrical tower. What is the probability that they can see each other? It'd be of the greatest interest to have not only a rigorous solution, but also an intuitive insight onto this simple yet very difficult problem:
Let there exist some tower which has the shape of a cylinder an... | Hint:
By symmetry, one may assume that the first individual is located on the horizontal radius on the left.
The surface he can see is the portion of the lane cut by the tangents to the tower. The area of this surface, $R(r)$, can be computed with a little bit of trigonometry, as the sum of an annular sector, two right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2296854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 2,
"answer_id": 1
} |
Are there infinite cardinals $\kappa$, $\lambda$ with $\kappa^\lambda = \kappa$? Going through an exam question for revision.
I need to prove the following:
Are there infinite cardinals $\kappa$, $\lambda$ with $\kappa^\lambda = \kappa$?
I really am not sure, though intuition says no.
I am then asked to state and pr... | Yes. Try $\kappa=2^{\aleph_0}$ and $\lambda=\aleph_0$.
We need to biject sequences of 0-1 sequences with 0-1 sequences. But sequences of sequences are just a 2-dimensional array, and this can be treated wit Cantor's zigzag enumeration.
If $\kappa$ is finite, then $2^\kappa$ is finite. If $\kappa\ge\aleph_0$, then $2^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2296915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Understanding the proof of convergence criterion for infinite products via the relation of series? In the text "Functions of one Complex Variable" I'm having trouble understanding the proof for convergence criteria of an infinite product via it's relation to infinite series as seen in Corollary $(8.1.4)$
$Corollary \, ... | Note that $1 + |a_{n+1}| \geqslant 1$ for all $n$. Hence, $P_{n+1} = P_n(1 + |a_{n+1})\geqslant P_n$.
Since the sequence $(P_n)$ is nondecreasing and bounded above by $\exp(\sum_{n=1}^\infty |a_n|)$, it converges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2297021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Is there a way to see this geometrically? In my answer to this question -
Finding the no. of possible right angled triangle.
- I derived this result:
If a right triangle
has integer sides
$a, b, c$
and integer inradius $r$,
then all possible values
of $a$ and $b$
can be gotten in terms of $r$
as follows:
For every poss... | As shown in this answer by using Heron's formula, if a generic triangle has integer inradius $r$ and integer sides $a$, $b$, $c$, then its sides can be written as $a=x+y$, $b=x+z$, $c=y+z$, where positive integers $x$, $y$, $z$ satisfy
$$
r^2(x+y+z)=xyz.
$$
If the triangle is rectangle, Pythagoras' constraint $a^2+b^2=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2297169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
$C^\infty_c (K)$ is separable for $K$ compact $C^\infty_c (K)$ is the space of smooth functions supported on $K$ a compact subset of $\mathbb{R}^d$. For simplicity assume $K$ is just a ball centered at the origin. This has the smooth topology in which convergence is uniform convergence of the functon and all derivative... | For simplicity let's assume $K=$ the closed Unit ball in $R^n.$ Let $\lambda_n$ be a sequence of positive number such that $\lambda_n \nearrow 1.$
Define
$$A_{n} = \{P : K \overset{smooth}{\longrightarrow} R ~| P ~\text{is a rational polynomials on}~ B_ {\lambda_n} \text{and } P=0 ~\text{on}~ K\setminus B_{\lambd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2297275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Show that if $(x_n) \rightarrow x$ then $(\sqrt{x_n}) \rightarrow \sqrt{x}$. Let $x_n \ge 0$ for all $n \in \mathbf{N}$ and $x>0$, show that if $(x_n) \rightarrow x$ then $(\sqrt{x_n}) \rightarrow \sqrt{x}$.
My textbook does the following proof:
Let $\epsilon >0$, we must find an $N$ such that $n \ge N$ implies $|\sqrt... | Because $\sqrt{x_n}+\sqrt{x}\geq \sqrt x$ and thus $$\frac{1}{\sqrt{x_n}+\sqrt x}\leq \frac{1}{\sqrt x}.$$
When you'll see continuous function, such a proof is easier using the continuity of $x\longmapsto \sqrt x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2297395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find $\lim_{x\to 1}\frac{p}{1-x^p}-\frac{q}{1-x^q}$ Find $\lim_{x\to 1}(\frac{p}{1-x^p}-\frac{q}{1-x^q})$
My attempt:
I took LCM and applied lhospital but not getting the answer.Please help
| Hint: write your term in the form
$$\frac{p(1-x^q)-q(1-x^p)}{(1-x^p)(1-x^q)}$$ and use L'Hospital. the result is $$\frac{1}{2}(p-q)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2297504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
How to solve the recurrence relation $f(1,n)=n+2$? Given the following information:
$f(0,n)= n+1$ $ \ \ $ $\forall n$
$f(m,0)=f(m-1,1)$ when $m>0$
$f(m,n)=f(m-1, f(m,n-1))$ when $m>0$ and $n>0$
I have worked out that $f(1,n)=f(0,f(1,n-1))= f(1,n-1) + 1$
But I am unsure of how to get $f(1,n) =n+2$ from this stage ?
| You already have:
$$f(1,n)=f(1,n-1) + 1$$
Notice $f(0,1)=f(1,0)=2$, and this is because
$f(0,n)= n+1$, and you let $n=1$, you get $f(0,1)= 2$
$f(m,0)=f(m-1,1)$, and you let $m=1$, you get $f(1,0)=f(0,1)$
Thus $f(1,0)=f(0,1)=2$
Let $a_n=f(1,n)$
So you have $a_n=a_{n-1}+1$, and $a_0=2$
Then
$$a_n=(a_n-a_{n-1}) + (a_{n-1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2297588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Limit of function calculations I must solve limit of next function:
$$\lim_{x\to \infty}\frac{2x^3+x-2}{3x^3-x^2-x+1}$$
Does my calculations are proper? If not where is my mistake?
$$=\lim_{x\to \infty}\frac{x^3\left(2+\frac{1}{x^2}-\frac{2}{x^3}\right)}{x^3\left(3-\frac{1}{x}-\frac{1}{x^2}+\frac{1}{x^3}\right)} \\
\ =... | You are correct, if you have the ratio of two polynomial of the same degree $n$ then
$$\lim_{x\to +\infty}\frac{a_nx^n+a_{n-1}x^{n-1}+\dots +a_0}{b_nx^n+b_{n-1}x^{n-1}+\dots +b_0}=\lim_{x\to +\infty}\frac{x^n(a_n+\frac{a_{n-1}}{x}+\dots +\frac{a_0}{x^n})}{x^n(b_n+\frac{b_{n-1}}{x}+\dots +\frac{b_0}{x^n})}\\=\lim_{x\to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2297710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Find all the values of a for a system that has a) no solution b) 1 solution c) infinitely many solutions Solving for a system for a with a matrix
x + ay - z = 1
-2x - ay + 3z = -4
-x -ay + ay = a + 1
This is the solution I found:
1 a -1 1
0 a 1 -2
0 0 a-1 a+2
And reduced it even further
1 0 -2 3
0 1 ... | You can only perform your final reduction if $a\ne0$; so you need to
split off $a=0$ as a separate case and investigate it individually.
You also point out the cases $a=1$ and $a=-2$. Why don't you investigate
these too? For instance when $a=1$ you get the final row $0\ 0\ 0\ 3$
which is the equation $0x+0y+0z=3$: impo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2297934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Question about statement of axiom of choice Here is a statement of axiom of choice given in Folland's book Real Analysis.
If $\{X_\alpha\}$ is a nonempty collection of nonempty sets, then, $\Pi X_\alpha$ is empty.
What does it mean to say "a nonempty collection"? Isn't it just enough to say that a collection of nonemp... |
What does it mean to say "a nonempty collection"?
It means that the collection $\{X_\alpha\}_{\alpha \in A}$ itself is non-empty, i.e., that the indexing set $A$ is non-empty.
Isn't it just enough to say that a collection of nonempty sets?
No, because $\emptyset$ is a collection of non-empty sets.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2298040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Proving $\mathcal{F}^{-1}\left\{\frac{i}{2t}\hat{f}(t)\right\}=-\int_{-\infty}^tf(s)\,ds$ in the sense of distributions Let $f\in C^\infty_0(\mathbb{R})$ with $\operatorname{supp}f\subset(0,\infty)$. I would like to prove that
$$\mathcal{F}^{-1}\left\{\frac{i}{2t}\hat{f}(t)\right\}=-\int_{-\infty}^tf(s)\,ds,\qquad t\i... | As definition of the Fourier transform we take
$$\hat\phi(\xi) = \int \phi(x) e^{-i\xi x} \, dx$$
Then we have $\hat\delta = 1$ since $\langle \hat\delta(t), \phi(t) \rangle = \langle \delta(t), \hat\phi(t) \rangle = \hat\phi(0) = \int \phi(x) 1 \, dx = \langle 1(t), \phi(t) \rangle.$
Now, $H = \frac12 (1 + \theta),$ w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2298155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Why does the Category Theory Definition of $\mathbf{Set}$ Product Not Define a Subset of a Product? 1 Definition of Product
From the Wikipedia Definition of a Categorical Product (in the simple binary case):
2 Question
Now let us focus our attention on $\mathbf{Set}$.
$$
Y = \{1, 2 \} \\
X_1 = \{a, b \} \\
X_2 = \{c, ... | I think you're not being precise enough with the crucial question:
wouldn't we have that $Y$ maps to $\{ (a, c), (b, d) \}$
Well, certainly there does exist a function $g:Y\to \{(a,c),(b,d)\}$ such that $g(1)=(a,c)$ and $g(2)=(b,d)$. But this $g$ is not the product of $f_1$ and $f_2$, precisely because of the definit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2298275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Finding solutions to $2^x+17=y^2$
Find all positive integer solutions $(x,y)$ of the following equation:
$$2^x+17=y^2.$$
If $x = 2k$, then we can rewrite the equation as $(y - 2^k)(y + 2^k) = 17$, so the factors must be $1$ and $17$, and we must have $x = 6, y = 9$.
However, this approach doesn't work when $x$ is ... | It looks like I need to spell out the details for insipidintegrator.
If $x$ is even, the prime $17$ is the product of $y+2^{x/2}$ and $y-2^{x/2}.$ Averaging, we find $y=9$ whence $x=6.$
If $x$ is odd, write $y-2^{x/2}=\frac{17}{y+2^{x/2}}$. Letting $x=2n+1$, we have
$\Big|\frac{y}{2^n}-\sqrt{2}\Big|=\frac{17}{2^n(y+2^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2298402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 5,
"answer_id": 0
} |
Dedekind complete model Prove that there is no first order theory $T$ in $ L=\{<\}$ such that for every linearly order set $A$, $A$ is a model of $T$ iff $A$ is Dedekind complete.
| Here is a slight variation of Wore's argument: Suppose that $T$ would characterize Dedekind complete linear orders.
Let $T^*$ be $T$ together with $\operatorname{DLONE}$ - the theory of dense linear orders without endpoints. Now $T^*$ characterizes Dedekind complete dense linear orders without endpoints and is consiste... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2298497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Is this a valid proof to prove that if $a_n$ converges, then $a_{n+1}-a_{n}$ converges to $0$? by definition I saw someone's comment on another website on this proof and they presented this (by definition):
now, by the definition of a convergent sequence,
for all $\epsilon _1 >0$ there is in fact an $n> N_1$ such ... | Yes. This is correct. Using the triangular inequality: $|a_{n+1} - a_n| = |a_{n+1} -L + L - a_n| \le |a_{n+1} - L| + |a_n - L| < \epsilon$, therefore when you take the limit you get $a_{n+1} - a_n = 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2298612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Is there a systematic way of determining the "correct" asymptotic approximation? Consider these two quadratic equations,
$$\text{i)} \quad x^2+4x-5-\epsilon$$
$$\text{ii)} \quad \quad x^2+(4+\epsilon)x+4-\epsilon = 0$$
If we attempt to find an asymptotic approximation of the form
$$x = x_0 + \epsilon x_1+...$$
for i) ... | Compute the differential of the equation.
*
*$(2x+4)dx-d\epsilon=0$. This shows that $\dfrac{dx}{d\epsilon}$ is finite a the roots.
*$(2x+4+\epsilon)dx+(x-1)d\epsilon=0$. At the double root $x=-2$, $\dfrac{dx}{d\epsilon}$ is infinite so that an "ordinary" approximation (entire function) cannot work.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2298696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Derivatives of trigonometric functions: $y= \frac{x \sin(x)}{1+\cos(x)}$
I'm trying to find the derivative of:
$$y= \frac{x \sin(x)}{1+\cos(x)}$$
I've tried but I can't achieve the simplified form -
Here's my try-
$$y' = \left(\frac{x \sin(x)}{1+\cos(x)}\right)'$$
$$y' = \frac{x\sin^2(x) + (\cos(x)+1 )(\sin(x)+x\cos... | Note that $x\sin^2 x = x(1 - \cos^2 x)$. So we can rewrite the numerator as $$x-x\cos^2 x + x\cos^2 x+(1+\cos x)\sin x +x\cos x = (1+\cos x)(x+\sin x)$$ so $$y'=\frac{(1+\cos x)(x+\sin x)}{(1+\cos x)(1+\cos x)} = \frac{x+\sin x}{1+\cos x}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2298811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Showing that $\sum_{k=1}^n\dfrac{1}{2^{k-1}}$ for $n \geq 2$ is not an integer . Suppose $n\geq2 ,s(n)=\sum_{k=1}^n\dfrac{1}{2^{k-1}} $
$$s(2)=1+\frac 12=1.5\\s(3)=1+\frac12+\frac14=1.75 ,\\\vdots$$
Is an elementary proof to $s(n)$ can never be an integer number ?
As honestly as possible : One of my students( k-12) as... | $2^{n-1} s(n)$ is odd and $\frac{\text{odd}}{\text{even}\neq 0}$ cannot be an integer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2298890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Solution to this integral? could anyone solve this integral ?
$$\int_0^\infty \frac{e^{-x}\sin(x)\cos(ax)}x~\mathrm dx$$
well i have tried opening up the sin*cos using trigonometric identities but that didn't help so much
| Note that we can write
$$\begin{align}
e^{-x}\sin(x)\cos(ax)&=\text{Re}\left(e^{-x}e^{iax}\sin(x)\right)\\\\
&=\text{Re}\left(\frac{e^{i(a+1+i)x}-e^{i(a-1+i)x}}{2i}\right)
\end{align}$$
Hence, we have from the Generalized Frullani's Theorem
$$\begin{align}
\int_0^\infty \frac{e^{-x}\sin(x)\cos(ax)}{x}\,dx&=\text{Re}\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2299268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 2
} |
How many terms (summands) are in the sum? I realize there are similar stacks to my question such as:
How many summands are there
Although I require further clarifications to understand.
Consider the sequence: 4 + 11 + 18 + 25 + ... + 249.
1) How many summands are in the sum.
2) Compute the sum.
| By simple investigation, you can observe that the sequence you're dealing with is an arithmetic progression, hence if you name it $a_n$, then you're interested in summing the sequence $a_n$ defined by:
$$
a_n=4+7n, n\in\mathbb N
$$
And to know how many summands there are, yoi solve for $n$ such that:
$$
a_n=249=4+7n
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2299359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
periodical function $f$ with period $1$ satisfying $f(x)+f(x+1/2)=f(2x)$ is zero. Let $f$ be continuously differentiable on $\Bbb R$, periodical with period $1$, and $f(x)+f(x+1/2)=f(2x)$ for all $x\in\Bbb R$. Show that $f\equiv 0$.
A natural attempt is to use Fourire series. Let $f(x)=a_0/2+\sum_{n=1}^\infty(a_n\cos ... | You can inductively prove that
$$ f(x)
= \sum_{k=0}^{2^n - 1} f\left(\frac{x}{2^n} + \frac{k}{2^n}\right) $$
holds for all $x \in \Bbb{R}$ and $n \geq 1$. Since $f \in C^1$, this implies
$$ f'(x) = \sum_{k=0}^{2^n - 1} f'\left(\frac{x}{2^n} + \frac{k}{2^n}\right) \frac{1}{2^n} \xrightarrow[n\to\infty]{} \int_{0}^{1} f'... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2299535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Find the eigenvalues and a basis for an eigenspace of matrix A
Find the eigenvalues and a basis for each eigenspace of matrix A:
\begin{bmatrix}
1 & -3 & 3 \\
2 & -2 & 2 \\
2 & 0 & 0 \\
\end{bmatrix}
I found the eigenvalues by computing $|A-\lambda I|$:
$\lambda_1 = 0,$
$\lambda_2 = 1,$
$\lambda_3 = -2$
H... | Correct, you subtract each eigenvalue from the main diagonal and use row reduction to find the eigenvectors. You may also want to spend a minute or two trying to spot the eigenvectors because that tends to be faster than row reduction. For instance in your case the second and third columns are scalar multiples of each ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2299628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Show that the integral of $1/(1-x^x)$ from $0$ to $1/e$ is divergent How can you show that $\int_0^{1/e}\frac{dx}{1-x^x}$ diverges? Do you have to substitute $x = \frac1u$?
| Using the inequality: $e^{-y} > 1 - y, 0 < y < 1$. Put $u = x\ln x$. Observe that $x \to 0^{+} \implies u \to 0^{-}$. Thus you can write $u = -y, 0 < y < 1$. The inequality $e^{-y} > 1 - y$ can be proved easily on $y \in (0,1)$. Thus $\dfrac{1}{1- x^x} = \dfrac{1}{1-e^{x\ln x}} = \dfrac{1}{1-e^{-y}} > \dfrac{1}{y} = -\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2299775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Neglect $1/2 \ln(2\pi n)$ in Stirlings approximation formula, but this term is not bounded or gets smaller, but larger One form of Stirlings approximation reads
$$
\ln(n!) \approx n\ln(n) - n + \frac{1}{2} \ln(2\pi n)
$$
another
$$
\ln(n!) \approx n\ln n - n.
$$
But thats makes me wonder, for the difference of both i... | This is called an asymptotic expansion.
Since you have $$\frac{\ln(n!)}{n\ln(n)}\xrightarrow[n\to\infty]{} 1,$$
$n\ln(n)$ is a valid approximation for $\ln(n!)$.
You actually have that the error goes to $0$ at the speed $\frac 1{\ln(n)}$ which is very slow.
So if you want a better approximation, you can notice that
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2299881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
A infinite series expansion for $e^e$. How can $e^e$ be expressed in an infinite series with as much simplification as possible.
*
*I wrote the series of $e^x$ by keeping $x$ as $e$ and from there I also expanded every $e$ in this expansion now I was thing about expanding it further by binomial theorem but I am not ... | Let $f(x)=e^x$
Then $$f^{(1)}(x)=f^{(2)}(x)=...=f^{(n)}(x)=...=e^x, \forall x\in\Bbb R.$$
Then applying Taylor's theorem we get -
$$f(x)=f(0)+xf^{(1)}(0)+\frac {x^2}{2!} f^{(2)}(0)+...+\frac {x^n}{n!} f^{(n)}(\phi), 0\lt\phi\lt1.$$
Then you will get
$$e^x=1+x+\frac{x^2}{2!}+...$$
i.e. $$e^x=\sum_{n=0}^{\infty}\frac {x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2299994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Converting decision tree into a logical expression
I need to convert this decision tree into a logical expression by using "and", "or" and "not" logical operators. I have been trying to solve this for 3 days. Any help would be appreciated.
| It's
$$(\neg F \wedge \neg H) \vee (\neg F \wedge H \wedge J) \vee (F \wedge G) \vee (F \wedge \neg G \wedge K)$$
(Here '$\wedge$' means 'and', '$\vee$' means 'or' and '$\neg$' means 'not'.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2300372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Reason for order of multiplication of ordinals I understand that addition and multiplication on ordinal numbers cannot be commutative and I know why $1+\omega$ and $2\times\omega$ must be different from $\omega+1$ and $\omega\times2$, respectively. For addition, I see why $1+\omega$ is just $\omega$, because it represe... | Well, it is quite arbitrary. But one soft reason to define it this way is that it makes ordinal arithmetic left distributive, i.e. for all ordinals $\alpha, \beta, \gamma$
$$\alpha \cdot (\beta + \gamma) = \alpha \cdot \beta + \alpha \cdot \gamma$$
and allows for left cancellation
$$
\alpha > 0 \wedge \alpha \cdot \bet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2300478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How to find power series of $f(z)=\frac{e^z}{1-z}$ at $z_0=0$? I tried to calculate few derivatives, but I cant get $f^{(n)}(z)$ from them. Any other way?
$$f(z)=\frac{e^z}{1-z}\text{ at }z_0=0$$
| Hint:
$$\frac1{1-z}=\sum_{n=0}^\infty z^n$$
$$e^z=\sum_{n=0}^\infty\frac{z^n}{n!}$$
Now apply Cauchy products to see that
$$\frac{e^z}{1-z}=\sum_{n=0}^\infty z^n\sum_{k=0}^n\frac1{k!}=\sum_{n=0}^\infty e_n(1)z^n$$
where $e_n(x)$ is the exponential sum formula.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2300613",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Degenerate eigenvalues problem for a 4x4 system In summary, my question is whether or not I'm allowed to have the zero vector as my generalised eigenvector. I'm given the system $$x'=\begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -2 & 2 & -3 & 1 \\ 2 & -2 & 1 & -3 \end{bmatrix}x$$ I also found two eigenvalues: 0 & -... | Let's name your matrix A.
The matrix $(A + 2E) = \begin{pmatrix} 2 & 0 & 1 & 0 \\ 0 & 2 & 0 & 1 \\ -2 & 2 & -1 & 1 \\ 2 & -2 & 1 & -1 \end{pmatrix}$ has two linearly independent ordinary eigenvectors with eigenvalue 0: $\begin{pmatrix} -1 \\ 0 \\ 2 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ -1 \\ 0 \\ 2 \end{pmatri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2300725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Calculating $\sum_{k=1}^\infty \frac{k^2}{2^k}=\frac{1}{2}+\frac{4}{4}+\frac{9}{8}+\frac{16}{16}+\frac{25}{32}+\cdots+\frac{k^2}{2^k}+\cdots$ I want to know the value of $$\sum_{k=1}^\infty \frac{k^2}{2^k}=\frac{1}{2}+\frac{4}{4}+\frac{9}{8}+\frac{16}{16}+\frac{25}{32}+\cdots+\frac{k^2}{2^k}+\cdots$$
I added up to $k=5... | If we start with the power series
$$ \sum_{k=0}^{\infty}x^k=\frac{1}{1-x} $$
(valid for $|x|<1$) and differentiate then multiply by $x$, we get
$$ \sum_{k=1}^{\infty}kx^k=\frac{x}{(1-x)^2}$$
If we once again differentiate then multiply by $x$, the result is
$$ \sum_{k=1}^{\infty}k^2x^k=\frac{x(x+1)}{(1-x)^3}$$
and sett... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2300889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Solve the Inequality $x+\frac{x}{\sqrt{x^2-1}} \gt \frac{35}{12}$ Solve the Inequality $$x+\frac{x}{\sqrt{x^2-1}} \gt \frac{35}{12}$$
First of all the Domain of LHS is $(-\infty \:\: -1) \cup (1 \:\: \infty)$
So i assumed $x=\sec y$ since Range of $\sec y$ is $(-\infty \:\: -1) \cup (1 \:\: \infty)$
So
$$\sec y+ |\cs... | HINT:
Clearly, we need $x>0$ so will be $\sec y,\csc y\implies0< y<\dfrac\pi2$
Now $\sec y+\csc y>\dfrac{35}{12}$
$\iff\left(\dfrac{35}{12}\right)^2<\sec^2y+\csc^2y+2\sec y\csc y=\sec^2y\csc^2y+2\sec y\csc y$ as $\sec^2y\csc^2y=\sec^2y+\csc^2y$
Set $\sec y\csc y=u$ to find $$u^2+2u>\left(\dfrac{35}{12}\right)^2\iff(u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2301181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Suppose $h:M \rightarrow E$ is a homeomorphism onto its image . Show that $h(M)$ is a $G_{\delta}$-set. Let $M$ and $E$ be complete metric spaces.
Suppose $h:M \rightarrow E$ is a homeomorphism onto its image (i.e. $h$ is a continuous one-to-one map, and $h^{-1}|_{h(M)}$ is continuous).
Show that $h(M)$ is a ... | Let $B_E = \{ e \in E: \| e \| \leq 1 \}$ be the unit ball centered at $e$ with radius $1.$
Let $(\varepsilon_n)_{n \in \mathbb{N}}$ be a sequence of positive real numbers such that $\varepsilon_n \rightarrow 0$ as $n \rightarrow \infty.$
We claim that
$$h(M) = \bigcap_{n \in \mathbb{N}}\bigcup_{y \in h(M)}(y+ \vareps... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2301302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why is $a^4+b^4$ factorable in two different ways, and why are the solutions not the same? I am working on factoring $a^4+b^4$ and I have found two different solutions to this.
First, I have factored it to $$a^4+b^4=(a+b)[a^3-a^2b+ab^2+b^3]-2ab^3$$
But then I also found that the equation is factorable to $$a^4+b^4=(a^2... | Contrary to what's being said in the comments, neither of those expressions is wrong because both yield $a^4 + b^4$ when expanded. It is, however, incorrect to call this a factorization:
$$a^4+b^4=(a+b)[a^3-a^2b+ab^2+b^3]-2ab^3$$
When you factor an expression, say $F(x)$, you break it down into two or more factors, su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2301484",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Prove that $\left[\mathbb{Q}(\sqrt[3]{5}+\sqrt{2}):\mathbb Q\right]=6$
Prove that $\left[\mathbb{Q}(\sqrt[3]{5}+\sqrt{2}):\mathbb Q\right]=6$
My idea was to find the minimal polynomial of $\sqrt[3]{5}+\sqrt{2}$ over $\mathbb{Q}$ and to show that $\deg p(x)=6$
Attempt:
Let $u:=\sqrt[3]{5}+\sqrt{2}\\
u-\sqrt[3]{5}=\sqr... | First, $p:=x^3-5$ and $q:=x^2-2$ are the minimal polynomials of $\sqrt[3]{5}$ and $\sqrt{2}$ over $\mathbb{Q}$ since they are monic and using Eisenstein's criterion, they are irreducible over $\mathbb{Q}$. Then, the following is an annihilator polynomial with rational coefficients of their sum:
$$\textrm{res}_y(p(y),q(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2301573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
What does Liu mean by "topological open/closed immersion" in his book "Algebraic Geometry and Arithmetic Curves"? In his book "Algebraic Geometry and Arithmetic Curves", Liu defines open/closed immersions of locally ringed spaces in terms of topological open/closed immersions:
What does he mean by the terms "topologic... | Yes, that's a correct definition. Yours (1.) is also equivalent to 2. below.
*
*$f(X)$ is open (closed) and $f$ is a homeomorphism on its image
*$f$ is open (closed) and a homeomorphism on its image
If we then define an immersion to be a homeomorphism on its image, then an open (closed) immersion really is an imm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2301719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 1,
"answer_id": 0
} |
Prove fact about polynomial with coefficients from $\mathbb{Z}$ Let $f \in \mathbb{Z}[x]$.
And for more than 3 (i.e $\geqslant 4$) distinct $a \in \mathbb{Z}\ f(a) = 1$.
Prove that $\forall a \in \mathbb{Z} \ f(a)\neq -1$.
I have clearly no idea how to tackle this.
The only thing I've noticed (I'm pretty sure it is ... | Let
$$
f(x)=(x-1)(x-3)(x-5)(x-7)+1.
$$
Then $f(a)=1$ for $a=1,3,5,7$. Nevertheless we have $f(6)=-14<-1$.
As for the first claim, compare with this question, for an idea how to tackle this.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2301807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Heegaard Splittings of Non-orientable 3 manifolds A well known and oft-utilized fact from 3-manifold topology is that all closed, orientable 3-manifolds admit Heegaard splittings.
I am trying to understand what the appropriate notion of Heegaard splitting for a closed, nonorientable 3-manifold should be, assuming I wa... | Considering question 2) let me exemplificate a 3-manifold which can be splitted along an orientable surface:
Let $N_3$ be the nonorientable genus three surface.
Take $E=N_3\times S^1$. Since $N_3=T_0\cup_C M\ddot{o}$, where $T_0$ is a puntured 2-torus, $M\ddot{o}$ is a Möbiusband and
$C=\partial T_0=\partial M\ddot{o}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2301919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Given that $f(E)$ is compact if and only if $E$ is, can we deduce the continuity of $f$? Let $(X_1,d_1)$ and $(X_2,d_2)$ be metric spaces. A criterion for the global continuity of some $f:X_1\to X_2$ is that for all closed $E\subseteq X_2$, $f^{-1}(E)$ is closed. This is a corollary of the analogous theorem for counter... | Yes, $f$ must be continuous. For metric spaces, continuity is equivalent to sequential continuity (this requires some choice, but topology without choice is very strange, so we assume choice anyway).
So suppose that $X_1, X_2$ are metric spaces, and $f \colon X_1 \to X_2$ is not continuous at $p$. Then there is a seque... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2302024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Normed vector space inequality $|\|x\|^2 - \|y\|^2| \le \|x-y\|\|x+y\|$ I'm looking at an old qualifying exam, and one question is to prove the following inequality in any normed vector space:
$$ |\|x\|^2 - \|y\|^2| \le \|x-y\|\|x+y\| $$
My initial thought was that
$$ |\|x\|^2 - \|y\|^2| = |(\|x\|+\|y\|)(\|x\|-\|y\|)|=... | We may assume w.l.o.g. that $\|x\|^2 \geq \|y\|^2$. Write $x = u + v$ and $y = u - v$. Now the inequality can be rewritten as
$$
\|u + v\|^2 \leq 4 \|u\| \|v\| + \|u - v\|^2.
$$
But this is the inequality one gets by combining $\|u + v\|^2 \leq (\|u\| + \|v\|)^2$ and $|\|u\| - \|v\||^2 \leq \|u - v\|^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2302122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 1
} |
Bayes Theorem Coin Problem There are 3 coins. One is regular (both head and tail) and the other two only have head sides. Now, flip one coin and get head. The question is what is the probability of this coin is regular one?
Or Twisted
There are 3 coins. One is regular (both head and tail) and the other two only have h... | you are using $C$ for the regular coin. So you should be calculating $P(C|H)$, which is
$$P(C|H)=\frac{P(H|C)*P(C)}{P(H)}=\frac{\frac{1}{6}}{\frac{5}{6}}=\frac{1}{5}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2302222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How to find $\lim\limits_ {n\to\infty}n^5\int_n^{n+2}\frac{{x}^2}{{ {2+x^7}}}\ dx$? How to find $$\displaystyle\lim_ {n\to\infty}n^5\int_n^{n+2}\dfrac{{x}^2}{{ {2+x^7}}}\ dx$$Can I use Mean Value Theorem? Someone suggested I should use Lagrange but I don't know how it would help.
| $$
\begin{align}
2n^5\frac{n^2}{2+(n+2)^7}&\le n^5\int_n^{n+2}\frac{x^2}{2+x^7}\,\mathrm{d}x\le2n^5\frac{(n+2)^2}{2+n^7}\\[12pt]
\frac2{\frac2{n^7}+\left(1+\frac2n\right)^7}&\le n^5\int_n^{n+2}\frac{x^2}{2+x^7}\,\mathrm{d}x\le\frac{2\left(1+\frac2n\right)^2}{\frac2{n^7}+1}
\end{align}
$$
Apply the Squeeze Theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2302388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
If $A$ is densely defined symmetric operator, $\lambda-A$ is onto $X \implies \lambda \in \rho(A)$ I want to understand the following statement:
If $\lambda \in \mathbb{C}\setminus\mathbb{R}$ and $A$ is densely defined symmetric operator ($A:D(A)\to X$), $\lambda-A$ is onto $X \implies \lambda \in \rho(A)$
I think th... | One does not need closedness. Let $z=x+iy$, then we compute
$$ \Vert (A-z)\phi \Vert^2 = \Vert (A-x)\phi\Vert^2 + y^2 \Vert \phi \Vert^2 + \langle (A-x)\phi , -iy \phi \rangle + \langle -i y \phi , (A-x) \phi \rangle $$
Now use the fact that $A$ is symmetric to show the last two terms cancel. Then you end up with
$$ \V... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2302527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
The approximation function of $\frac{x}{y}$ Is there a approximation function of $$\frac{x}{y},$$ and the approximation function is in the form of $f(x) + f(y)$ or $f(x) - f(y)$. That's to say the approximation function can split $x$ and $y$.
| Though the question is unspecific about what constitutes an "approximation", the answer appears to be "no".
As lulu notes in the comments, an approximation $\frac{x}{y} \approx f(x) + f(y)$ leads (for $x = y$) to
$$
1 = \frac{x}{x} \approx f(x) + f(x) = 2f(x)\quad\text{for all $x$.}
$$
Similarly, an approximation $\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2302661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Find the range of $f(x) = 3x^4 - 16x^3 + 18x^2 + 5$ without applying differential calculus
Find the range of $f(x) = 3x^4 - 16x^3 + 18x^2 + 5$ without applying differential calculus.
I tried to express $$f(x)=3x^4-16x^3+18x^2+5=A(ax^2+bx+c)^2+B(ax^2+bx+c)+C $$ which is a quadratic in $ax^2+bx+c$ which itself is quadr... | The range is $[k,+\infty)$ where $k$ is the minimum value such that the inequality
$$
3x^4-16x^3+18x^2+5\ge k
$$
is true for any $x \in \mathbb{R}$ and this is the minimum value $k$ such that the equation
$$
3x^4-16x^3+18x^2+5- k=0
$$
has a double solution, that is the discriminant of $3x^4-16x^3+18x^2+5- k$ is null.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2302781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.