Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Finding the local extreme values of $f(x) = -x^2 + 2x + 9$ over $[-2,\infty)$. I'm tutoring a student, and we were trying to solve the following question:
Find the local extreme values of $f(x) = -x^2 + 2x + 9$ over $[-2,\infty)$.
According to the textbook, the local extreme values are essentially the peaks and the v... | Sometimes just plotting the function cuts through needless distractions:
Clearly at the point $x=-2$, the function is lower than any other values in the specified domain, and hence is a local minimum.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2982073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
The limit $\lim_{x \to 0-} \frac{e^{-x^2}}{\sqrt{\pi}} \int_0^\infty e^{-t^2/4} \frac{e^{2x} \cos t-1}{e^{4x}-2e^{2x} \cos t+1 } dt$ A while back I derived the following expression valid for $x>0$:
$$\sum_{n=1}^\infty e^{-(n+x)^2}= \frac{e^{-x^2}}{\sqrt{\pi}} \int_0^\infty e^{-t^2/4} \frac{e^{2x} \cos t-1}{e^{4x}-2e^{... | We have
\begin{align}
L^-
&=\lim_{x \to 0^-} \frac{e^{-x^2}}{\sqrt{\pi}} \int_0^\infty e^{-t^2/4} \frac{e^{2x} \cos t-1}{e^{4x}-2e^{2x} \cos t+1 } dt\\
&=\lim_{x \to 0^+} \frac{e^{-x^2}}{\sqrt{\pi}} \int_0^\infty e^{-t^2/4} \frac{e^{-2x} \cos t-1}{e^{-4x}-2e^{-2x} \cos t+1 } dt\\
&=\lim_{x \to 0^+} \frac{e^{-x^2}}{\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2982182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Why is it sometimes possible to solve multi variable single equation? Assume you have an equation:
$\ 5x^2+4x+6=ax^2+bx+c $
Now theoretically, this is an equation of 4 variables, and it should not be solvable, but it is very apparent that a,b,c equals 5,4,6, in that order. Something similar happens with complex numbers... | Your question touches on a very important concept in linear algebra, that of linear independence. Some set of vectors $v_1,v_2,\dots,v_n$ are linearly independent if the only solution to $a_1v_1+a_2v_2+\dots+a_nv_n=0$ is when $a_1=a_2=\dots=a_n=0$. For example, $x^2$, $x$, and $1$ are linearly independent because if $f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2982341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Spider Problem Counting Socks and Shoes Problem
A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?
A) 8! (B) $2^8$ (C) $(8!)^2$ (D) $\frac{16!}{2^8}$ (E) 16!
I am havi... | It might be better to put subscripts on the numbers: $L_1$ means the action of putting the sock, and $L_2$ the shoe, on leg $L$. Then we have 16 distinct symbols $1_1,1_2,2_1,2_2,3_1,3_2,\dots,8_1,8_2$ and there are $16!$ ways to permute them without restrictions.
With the sock-before-shoe restriction, for each pair of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2982436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Proof by induction of summation inequality: $1 + 1/2+ 1/3+ 1/4+1/5+⋯+ 1/2^n \leq n + 1$ I have been working on this problem for literally hours, and I can't come up with anything. Please help. I feel like I am about to go insane.
For all n $\in$ N, we have $$1 + \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+\frac{1}{5} +⋯+ ... | For $n \ge 2$, we have $$\frac{1}{n} \le \int_{n-1}^{n} \frac{1}{x}\,dx$$
So, $$\sum_{i=1}^{2^{k}}{\frac{1}{i}} \leq 1+\int_{1}^{2^k} \frac{1}{x}\,dx=1+k \log2 \le 1+k$$ Another proof:as your progress $$\sum_{i=2^k+1}^{2^{k+1}}\frac{1}{i}=\frac{1}{1+2^k}+\frac{1}{2+2^k}+...+\frac{1}{2^k+2^k}\le\frac{2^k}{1+2^k}\le1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2982549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Wedge of aspherical spaces I‘d need a reference for the following fact: the one-point union of (nice) aspherical spaces is aspherical. I.e., from $\pi_kX=0$ and $\pi_kY=0$ follows $\pi_k(X\vee Y)=0$.
EDIT: Let‘s assume that the spaces are nice, e.g. manifolds.
For the wedge of circles this is true because the universa... | To answer my own question (in the setting of CW-complexes and thus also for smooth manifolds):
According to Ganea Link to Ganea‘s paper the homotopy fiber of $X\vee Y\to X\times Y$ is homotopy-equivalent to $\Omega X*\Omega Y$ if $X,Y$ are CW-complexes.
If $X,Y$ are aspherical, then their loop spaces are homotopy-equi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2982742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Expected time before Farmer Brown is abducted? Farmer Brown is standing in the middle of his perfectly circular field feeling
very content. It is midnight and there is no moon and unknown to the farmer,
Martian zoologists are landing randomly at points on the circumference of his
field. They land at one minute interval... | So you know that the probability that $k-1$ martians do not hold the farmer is
$$G(k-1)=\frac{k-1}{2^{k-2}}$$
Then the probability that $k$ martians do not hold the farmer is
$$G(k)=\frac{k}{2^{k-1}}$$
and the probability that the $k$-th martian catches the farmer is
$$P(k)=G(k-1)-G(k)=\frac{k-2}{2^{k-1}}$$
Finally
$$E... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2982878",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
geometric hash code or is there a unique affine transformation mapping two 2D points onto (0,0) and (1,1)? How can I compute it? I have a 2D transformation $T$ composed by a scale $\lambda$, a rotation by angle $\theta$ and a translation vector $\begin{bmatrix}t_x\\t_y\end{bmatrix}$:
$$
T=\begin{bmatrix}
\lambd... | Partial solution
$ \begin{cases}
\begin{align*}
0 &= x_{A}\lambda \cos(\theta) - y_{A}\lambda \sin(\theta) + t_{x} \\
0 &= x_{A}\lambda \sin(\theta) + y_{A}\lambda \cos(\theta) + t_{y} \\
0 &= x_{B}\lambda \cos(\theta) - y_{B}\lambda \sin(\theta) + t_{x} - 1\\
0 &= x_{B}\lambda \sin(\theta) + y_{B}\lambda \cos(\theta)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2982975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Evaluating $\lim_{x \to \infty}\frac{1}{x}\int_0^x|\sin(t)|dt$ I would appreciate some help with this problem:
Evaluate:
$$\lim_{x \to \infty}\frac{1}{x}\displaystyle\int_0^x|\sin(t)|dt$$
| Note that $\vert\sin(t)\vert$ is non-negative, periodic with period $\pi$, and that
$$\int_0^\pi\vert \sin(t)\vert dt=2.$$
Let $f(x)$ be the largest integer smaller than or equal to $x/\pi$. Then it holds that
$$\int_0^{f(x)\pi}\vert\sin(t)\vert dt\leq\int_0^x\vert\sin(t)\vert dt\leq\int_0^{[f(x)+1]\pi}\vert\sin(t)\ve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2983075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Octal palindromes with even number digits are all composite numbers? I want to know whether octal palindromes with even number digits (11 or 1221, but not 121) are all composite numbers, and a general proof if so or a counterexample if not.
| let us consider any number in base 8, $a_n 8^n +,\dots, + a_0$ observe that if $n$ is even then $ a^n \equiv 1 \;\text{mod} 9$ and if $n$ is odd then $ a^n \equiv -1\; \text{mod} 9$ then write the number mod 9, it became $-a_n + a_{n-1}+ \dots + a_0$ if $n$ is even ( or with different sign for $n$ odd). In any case we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2983277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Integration by parts $3n$ times. I found the equation below in Repeated integration by parts of a definite integral
\begin{align}
\int_a^b f^{(n)}(x)g(x) dx = (-1)^{(n)}\int_a^b f(x)g^{(n)}(x) dx
\end{align}
which is true if $\left.f^{(k)}(x)g^{(n-k)}(x)\right|_a^b=0 \: \forall k \in [0:n-1]$
So I want to test on the ... | The formula is applicable because when you differentiate $f$ $k$ times with $k <n$ there will be at least one factor of $x^{3}-x^{4}$ left in each term and $x^{3}-x^{4}=0$ for $x=0$ and $x=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2983534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Let $R$ be an integral domain. If $x \in R$ is prime, then $x$ is irreducible. I am trying to understand the proof for the following theorem:
Let $R$ be an integral domain. If $x \in R$ is prime, then $x$ is irreducible.
Here is the proof:
I typed this a while ago and I don't understand the part where if $x | bc$, t... | You have it reversed. What you seek is the inference $(1)\Rightarrow(2)$ below.
Theorem $\,\ (1)\,\Rightarrow\,(2)\!\iff\! (3)\ $ below, $ $ for a nonunit $p\neq 0$
$(1)\ \ \ \color{#c00}{p\ \mid\ ab}\ \Rightarrow\ p\:|\:a\ \ {\rm or}\ \ p\:|\:b\quad$ [Definition of $\:p\:$ is prime]
$(2)\ \ \ \color{#c00}{p=ab}\ \Ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2983631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How to evaluate $\int_{-\infty}^{\infty}\frac{x\arctan\frac1x\ \log(1+x^2)}{1+x^2}dx$ While browsing similar questions on this site I came up with the following integral because I thought I could evaluate it.
$$I=\int_{-\infty}^{\infty}\frac{x\arctan x\ \log(1+x^2)}{1+x^2}dx$$
I've been able to simplify it a bit. We fi... | We may also try attacking the third integral in line $(3)$ in my first answer,
$$L = \int_0^1 \frac{\log^2(1+x)+\log^2(1-x)}{\sqrt{1-x^2}} \, dx$$
by exploiting the generating function of $\dfrac{H_{2k-1}}k$, with $H_k$ the $k^{\rm th}$ harmonic number. Using the Cauchy product, we find
$$-\log(1\pm x) = \sum_{n=1}^\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2983721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
Are $M, P, N $ collinear? Let $\alpha $ a circle of diameters $AB $ and $\beta $ a circle tangent to $AB $ in $ C$ and tangent to $\alpha$ in $T $.
Let $M\in \alpha $ and $N\in CB $ s.t. $MN\perp AB $ and $MN $ is tangent to $\beta $.
Show that $\angle AMC=\angle CMN $.
My idea: Let $BT\cap \beta=$ {P}.
I need t... | Let O and Q be the centers of $\alpha$ and $\beta$ respectively. Note that CNPQ is a square and therefore if PQ is extended to cut $\beta$ at X, then, PQX // AOCNB.
Since $\angle TXP = 0.5 \angle TQP = 0.5\angle TOB = \angle TAB$, TXA is a straight line because the two red shaded angles are on the corresponding angle ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2983813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Is the restriction of a map representing a cohomology class on its Poincare dual null-homotopic? Let $M$ be a 5-manifold (possibly non-orientable),
$g\in H^2(M,\mathbb{Z}_2)$ is represented by a map $\tilde{g}:M\to K(\mathbb{Z}_2,2)$. $\text{PD}(g)$ is the submanifold of $M$ representing the Poincare dual of $g$.
$\t... | It is not true. Think of $M= \mathbb CP^2\times S^1$. $H^2$ is generated by a single element follows from product formula. The element $g$ is the mod-2 reduction of the canonical map $\mathbb CP^2\times S^1 \to \mathbb CP^2\hookrightarrow \mathbb CP^{\infty}$.[since $K(\mathbb Z, 2)= \mathbb CP^{\infty}$] {as Mike obse... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2983927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
complex integral: $\oint dz/z$ I'm trying to determine $$\oint \frac{dz}{z}$$ on a closed path $\mathcal C$, where $\mathcal C$ is the circle $|z|=100$ traversed anticlockwise.
What I did was re-write the function as $$\oint \frac{dz}{z-0}$$ and by Cauchy integration formula, it straight away gives $2\pi i$.
Is this me... | If $f(z)$ is an analytic function inside a simple closed curve $C$,
except for a finite number of isolated singular points $z_i$, $i \in \mathbb{N}$ located inside C, then
\begin{equation}
\oint f(z) dz= 2 \pi i \sum_{j=1}^N r_j
\end{equation}
where $r_j$ are the residue of $f(z)$ at $z = z_j$. This is know as the Cauc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2984008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Recurrence relations with algorithm Let $x \in \mathbb{R}$ and $n \in \mathbb{N}$. $c_0=1,c_1=\cos x$
for $k=1,2,...,n-1$:
$c_{k+1}=2c_1c_k-c_{k-1}$
How to prove that $c_k=\cos kx$?
I tried to show this equality with induction:
$k=1:c_1=\cos x$
$k \mapsto k+1: 2\cos(x)c_{k+1}-c_k$
Here I don't know how to continue. I... | You need strong induction for this problem, where you assume that the $c_n$ formula is true for all $1\le n\le k$:
$$c_{k+1}=2c_1c_k-c_{k-1}=2\cos x\cos kx-\cos(k-1)x$$
We use a product-to-sum identity to get rid of the cosine product:
$$=\cos(kx-x)+\cos(kx+x)-\cos(k-1)x=\cos(k+1)x+\cos(k-1)x-\cos(k-1)x=\cos(k+1)x$$
Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2984164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
length of tangent to a curve passing through another point Let $A$ be a point on the curve
$\mathcal{C} : x^2+y^2-2x-4=0$
If the tangent line to $\mathcal{C}$ at $A$ passes through $P(4,3)$, then what is the length of AP?
Please, include a general method of approaching similar kind of questions.
| Bring the circle into standard form to find our center location and radius.
$$ ( h,k)=(1,0),\, R = \sqrt5 = CT $$
Distance of center to outside point squared
$$ PC^2 = (4-1)^2+ (3-0)^2= 18 $$
$$ PT^2= PC^2-5 = 18-5= 13 \rightarrow PT= \sqrt{13} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2984274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Limit at infinity $\lim_{x\to \infty} x^a a^x=$? $\displaystyle \lim_{x\to \infty} x^a a^x=?$; $0<a<1$
I try to use the property: $a^{\log_a x}= x$ and reescribe the expression
$\displaystyle \lim_{x\to \infty} x^a a^x = \lim_{x\to \infty} \frac{a^{a\log_a x}}{a^{-x}} $ but i can't find the limit yet.
Any suggestion... | Consider $x=ay$; then your expression becomes
$$
x^aa^x=(ay)^aa^{ay}=a^a(ya^y)^a
$$
so you just need to compute
$$
\lim_{y\to\infty}ya^y
$$
For $a\ge1$ the limit is $\infty$. For $0<a<1$ rewrite it as
$$
\lim_{y\to\infty}\frac{y}{a^{-y}}=\lim_{y\to\infty}\frac{1}{-a^{-y}\log a}=0
$$
with a simple application of l'Hôpit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2984346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Finding $\lim_{n\to \infty}\sqrt n \int_0^1 \frac{\,dx}{(1+x^2)^n}$ $$
\lim_{n\to\infty} n^{1/2}
\int_{0}^{1} \frac{1}{(1+x^2)^n}\mathrm{d}x=0
$$
Is my answer correct?
But I am not sure of method by which I have done.
| $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2984468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 6,
"answer_id": 1
} |
Bounding the difference of rank-1 matrices Let $\|\cdot\|$ denote the Frobenius norm and $x,y \in \mathbb{R}^n$
I need a bound of the form
$$
\|x \cdot x^\top - y \cdot y^\top \| \leq C \|x-y\|_2 \quad (*),
$$
where $C>0$ does not depend on $x$ or $y$.
This seems to work when $\|x\|_2=\|y\|_2 = 1$. (Then I get $C=2$ ... | For the Frobenius norm, we have
$$
\|xy^T\|_F^2 = \sum_{i,j}(x_i y_j)^2 = \|x\|_2^2 \|y\|_2^2.
$$
This implies
$$
\|xx^T-yy^T\|_F= \|(x-y)x^T+y(x-y)^T\|_F \le
\|(x-y)x^T\|_F+\|y(x-y)^T\|_F
\le (\|x\|_2 + \|y\|_2)(\|x-y\|_2).
$$
The constant cannot be independent of $x,y$, because the mapping $x\mapsto xx^T$ is 'quadra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2984627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is a set in between two sets of equal measure measurable? The Lebesgue sigma algebra is complete with respect to Lebesgue measure, which means that if $A$ is a Lebesgue measurable set with Lebesgue measure $0$ and $B$ is a subset of $A$, then $B$ is Lebesgue measurable as well. But I'd like to know if something strong... | If $A$ and $C$ are each Lebesgue measurable, then so is $C\setminus A$ and so by finite additivity it has measure zero.
But then $B\setminus A$ also has measure zero and hence is measurable, so $$B=A\cup(B\setminus A)$$ is the union of two measurable sets, hence is measurable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2984760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
On existence of positive integer solution of $\binom{x+y}{2}=ax+by$ How can I prove this?
Prove that for any two positive integers $a,b$ there are two positive integers $x,y$ satisfying the following equation:
$$\binom{x+y}{2}=ax+by$$
My idea was that $\binom{x+y}{2}=\dfrac{x+2y-1}{2}+\dfrac{y(y-1)}{2}$ and choose... | If $a=b$ then let $(x,y)=(a,a+1)$.
Otherwise, w.l.g. suppose $a>b$ and let $x+y=2t(a-b)$ for some positive integer $t$. Then
$$t(a-b)\Big(2t(a-b)-1\Big) =ax+by=(a-b)x+2bt(a-b)$$
$\text{Therefore } x=t\Big(2t(a-b)-1\Big)-2bt=t\Big(2t(a-b)-(2b+1)\Big)$.
$x$ will be a positive integer providing $t>\frac{2b+1}{2(a-b)}.$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2984918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 3
} |
$GL(2,R) / SL(2,R)$ isomorphic to R* I am needing to write a prove showing that $GL(2,\mathbb{R}) / SL(2,\mathbb{R}) $ is isomorphic to $\mathbb{R}^*$.
I know that $SL(2,\mathbb{R})$ is a normal subgroup of $GL(2,\mathbb{R})$ but I'm not sure how to use that or where I should start.
Any help would be appreciated. Thank... | The homomorphism $GL(2,\Bbb R)\to\Bbb R^*$, $A\mapsto \det A$ has $SL(2,\Bbb R)$ as kernel.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2985047",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
If $A$ is a square matrix that satisfies $A^2-A+2I=0$, show that $A+I$ is invertible If $A$ is a square matrix that satisfies $A^2-A+2I=0$, show that $A+I$ is invertible. I understand how to find if $A$ is invertible but I don't know how to solve for the $A+I$ version.
| $(A+I)(A-2I)=A^2-2A+A-2I=A^2-A-2I=-4I$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2985212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Gamma Distribution Moments Show that for X ~ Gamma($\alpha$, $\beta$), for positive constant $\nu$,
$E[X^\nu] = \dfrac{\beta^\nu*\Gamma(\nu + \alpha)}{\Gamma(\alpha)}$.
I have the following solution:
Solution
However, I don't understand how we get that $\dfrac{1}{\Gamma(\alpha)*\beta^\alpha}*\int_{0}^{\infty}x^{(\nu+\a... | One way to understand the calculation is to recall that for a gamma distribution with shape $\alpha$ and scale $\beta$, $$f_X(x) = \frac{x^{\alpha-1} e^{-x/\beta}}{\beta^\alpha \Gamma(\alpha)}, \quad x > 0.$$ The denominator, being independent of $x$, suggests that $1/(\beta^\alpha \Gamma(\alpha))$ is the required mul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2985359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
A martingale bounded from below is $L^1$ bounded
Let $(X_n,\mathcal F_n)$ be a martingale bounded from below i.e. $X_n\geq M$ for some $M\in\mathbb R$. Then show that $\sup_n E|X_n|<\infty$.
It is easy to observe that $X_n$ converges almost surely to some $X\in L^1$ as $X_n-M$ is a non-negative martingale. I can conc... | $X_n^{-} \leq M^{-}$ so $EX_n^{-}$ is bounded. Since $X_n=EX_1$ for all $n$ (by martinagle property) we also know that $EX_n^{+}-EX_n^{-}=EX_n$ is bounded. This makes $E|X_n|=EX_n^{+}+EX_n^{-}$ bounded.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2985482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Basic questions about pythagorean triples and "n-lets" I've had some difficulties finding answer to the two following questions:
1) Given one of natural numbers $a,b$ where $b$ is even and $a^2+b^2=c^2$ is there only one such a pythagorean triple?
2)How about a sum of $n$ squares of natural numbers that is equal to a ... | To construct n-lets, we can begin with this function to find values of $(m,n)$ for Euclid's formula:
$$n=\sqrt{m^2-A}\text{ where m varies from }\lceil\sqrt{A}\rceil\text{ to }\frac{A+1}{2}$$
This will let us find a triple with a matching side A, if it exists, for any $m$ that yields a positive integer $n$.
Let's begin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2985610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
$A + B = \{\,x + y \mid x ∈ A,\, y ∈ B\,\}$ is closed in $ [0,\infty)$ for any closed $A, B \subseteq [0,∞)$ If $A, B$ are closed subsets of $[0,∞)$, then
$$A + B = \{\,x + y \mid x ∈ A,\, y ∈ B\,\}$$
is closed in $ [0,\infty)$.
If $A ,B$ are closed sets in $\mathbb R$ , then i know counterexamples but considering only... | Take a sequence $(x_n)\subseteq A+B$ convergent to $x\in [0,\infty)$. We want to show that $x\in A+B$.
It follows that $x_n=a_n+b_n$ for some sequences $(a_n)\subseteq A$ and $(b_n)\subseteq B$. Since $A,B$ are subsets of $[0,\infty)$ then $$0\leq a_n\leq x_n$$
$$0\leq b_n\leq x_n$$
In particular, since $x_n$ is bounde... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2985787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Little-o notation for $\ln(x)$ I want to Show the following:
$\ln(x) = o (e^{\sqrt{\ln x}})$ (Little-o Notation).
So I need to Show that: $\displaystyle{\lim_{x \rightarrow \infty} \frac{\ln(x)}{e^{\sqrt{\ln x}}}=0}.$
Can you help me, please?
| As gimusi noted, you want to prove $\lim_{y\to\infty}y^2\exp -y=0$. The function is continuous and non-negative for $y\ge 0$ with only one turning point, and famously integrates to the finite value $2$, which implies the limit.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2985926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
The relationship between the differential and the directional derivative of a function I am currently studying differential manifolds (from John M. Lee 's book), and have a question concerning the difference between what is defined as the $\textbf{differential of a function F}$, and the $\textbf{directional derivative ... | The relationship between the differential and directional derivative is the same in differential manifolds as in Euclidean space. The derivative is a linear function. Linear functions take in vectors and output vectors. When the input vector is a unit vector, the output is called the directional derivative. That is bec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2986070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Finding sum of a finite series. Consider the series
$$\frac {q_1} {p_1} + \frac {q_1 q_2} {p_1 p_2} + \cdots + \frac {q_1q_2 \cdots q_n} {p_1 p_2 \cdots p_n}$$ where $p_i + q_i = 1$ and $0 < p_i < 1$ and $0 < q_i < 1$ for all $i=1,2, \cdots , n$.
How can I find the sum of this series? Please help me in this regard.... | Hint:
\begin{align}
\frac{q_1}{p_1}+\frac{q_1 q_2}{p_1 p_2} &=
\frac{q_1}{p_1} \left( 1+\frac{q_2}{p_2} \right) \\
&= \frac{q_1}{p_1 p_2}
\end{align}
Updates for further thoughts:
The sum is not as trivial as we think at first glance. First of all, we write the sum into Horner's form
$$S_n=\frac{q_1}{p_1}
\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2986177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Can some group $G$ have an infinite number of cosets? My textbook states the number of cosets of $H$ in a finite group $G$ is $|G|/|H|$, however Is it possible for a group $G$ to have a subgroup $H$ such that $gH$ has infinite cardinality? I can't think of any examples.
Edit, a more concise way of putting it is,
is the... | An interesting fact about the Prüfer $p$-group $\mathbb{Z}(p^\infty)$: if $H$ is a proper subgroup of $\mathbb{Z}(p^\infty)$, then
$$
\mathbb{Z}(p^\infty)/H\cong \mathbb{Z}(p^\infty)
$$
However, all proper subgroups of $\mathbb{Z}(p^\infty)$ are finite.
For other examples, consider $V$ a two-dimensional vector space ov... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2986374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Find an orthononormal basis of $U^{\bot}$ given $U$ Given $U =$ span$\lbrace u_1 = (6,2,-2,-2),u_2 = (-1,1,-1,-1)\rbrace$ find an orthonormal basis for $U$ and $U^\bot$
What I've done so far: $(<u_1,u_2>) = 0 \implies u_1\bot u_2 \implies$ orthonormal basis given by $e_1 = \frac{u_1}{||u_1||},e_2 = \frac{u_2}{||u_2||}$... | As you've noted, $e_1,e_2$ is an orthonormal basis for $U$. Now, that basis can be expanded with some $e_3,e_4$ to an orthonormal basis of $\Bbb R^4$ (I am assuming that we are working in $\Bbb R^4$). Then $e_3$ and $e_4$ must be an orthonormal basis for $U^\perp$. Let's show this. First, $e_3$ and $e_4$ are linearly ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2986498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What is the remainder when $4^{10}+6^{10}$ is divided by $25$? Without using calculator, how to decide? Must go with last two digits of $4^{10}+6^{10}$, can tell the last digit is $2$. How to tell the tenth digit of the sum?
Thanks!
| The last digit of $4^x$ is $4,6,4,6,4,6\dots \implies 4^{10}$ has last digit $6$
The last digit of $6^x$ is always $6 \implies 6^{10}$ has last digit $6$
Therefore $4^{10} + 6^{10}$ has last digit $2$.
Any multiple of $25$ has last digit $0$ or $5\implies
25$ cannot be a divisor!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2986592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Evaluating $\lim_{x\to3}\frac{\sqrt[3]{x+5}-2}{\sqrt[4]{x-2}-1}$ without L'Hopital I have the following limit question, where different indices of roots appear in the numerator and the denominator
$$\lim_{x\to3}\frac{\sqrt[3]{x+5}-2}{\sqrt[4]{x-2}-1}.$$
As we not allowed to use L'Hopital, I want to learn how we can pro... | It is rather unpopular to use the standard limit $$\lim_{x\to a} \frac{x^n-a^n} {x-a} =na^{n-1}$$ for evaluating limits of algebraic functions, but it is my preferred approach.
Dividing the numerator and denominator of the given expression by $x-3$ and putting $x+5=u$ in numerator and $v=x-2$ in denominator we can see ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2986763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 4
} |
The interval [0,1] with additional modulo 1 The interval $[0, 1]$ is an abelian group with addition modulo $1$.
Let $H$ be a proper subgroup of $[0, 1]$, which is closed as subset of $[0,1]$.
Show $H$ is finite.
I assumed $H$ is infinite: since $H$ is closed, all limit points of $H$ are in $H$.
Then let $x$ be in $[0, ... | If you did not know you are asking about the subgroups of the circle $S^1$ in the complex plane $\mathbb{C}$. The exponential $e^{ 2\pi i}\colon [0, 1]_{/\mathbb{Z}}\rightarrow S^1$ defines a group isomorphism that is actually an homeomorphism of spaces.
The problem is widely explained in other questions, for example t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2986965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
how to show that random variable is almost surely finite? Coin toss problem Consider the Coin toss problem, i.e. let $Z_{i} : \Omega \rightarrow \{0,1\}$ with
$$
Z_{i}\left(\omega\right) =
\begin{cases}
1 & \text{if }\omega = H\\
0 & \text{if } \omega = T
\end{cases}
$$
be the outcome of the $i$-th coin toss with $\Om... | Hints:
*
*Show that $$A_k := \{Z_{4k}=0, Z_{4k+1}=1, Z_{4k+2}=0, Z_{4k+3}=1\}$$ satisfies $\mathbb{P}(A_k) = 1/16$ for each $k \in \mathbb{N}$.
*Show that the events $A_k$, $k \geq 1$, are independent.
*It follows from Step 1 that $$\sum_{k \geq 1} \mathbb{P}(A_k) = \infty.$$ Apply the Borel Cantelli lemma (using ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2987309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to show these three-regular graphs on 10 vertices are non isomorphic?
The number of vertices and edges are same, with each vertex having the same degree and the degree sequence of the graph is also the same. I have even tried finding a bipartite graph in any one of them even that seems to fail.
Question- How to sh... | The first graph is the only one not containing $4$-cycles, so it's not isomorphic to the other two. The second and third graphs are not isomorphic because the third is planar and the second contains a subdivision of $K_{3,3}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2987449",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
The function from the $\{\}$ to an any other set? In the $\mathcal{SET}$ - category of sets and maps between them - there is an initial object - the $\{\}$. It means that there is unique map from the $\{\}$ to an any other set (object of $\mathcal{SET}$).
Not sure I understand it completely. A map $\{\} \mapsto S \in O... | Think of it this way: we say two functions $f$ and $g$ on the same domain are different if there exists an $x$ in that domain so that $f(x) \neq g(x)$. In the case where the domain is empty, there obviously cannot be such an $x$, so $f$ and $g$ are automatically not different.
More formally, a function is formally defi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2987549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Properties of Lucas sequence I want to prove the following properties of Lucas sequence:
*
*$3\mid L_m \iff m\equiv 2\pmod 4$
*$L_k\equiv 3\pmod 4$, where $2\mid k$ and $3\nmid k$.
$$$$
For the first property do we use induction?
Does the second property follow from the first one?
| In both cases, it helps to consider the Lucas numbers modulo $m$. For example, modulo $3$, the Lucas numbers (zero-indexed) begin $$2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, \dots$$ and you may be able to spot a periodic pattern here: the sequence $2,1,0,1,1,2,0,2$ repeats over and over.
If you prove this periodic p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2987649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Equivalence of two methods for generating random numbers that sum to 1 I want a vector $\vec{v}$ of $N$ non-negative random numbers that sum to 1.
Let $X(a)$ be the (continuous) uniform distribution over interval $[0, a]$.
Let $S(n) = \sum_{i = 1}^{n} v_{i}$ be the partial sum of the elements of $\vec{v}$
Method 1
*
... | These methods are certainly not equivalent. It is easy to see that in the first method, $\mathbb{E}[v_n]=\frac 1n$, while in the second method, $\mathbb{E}[v_k]=\frac 1{2^k}$ (you can prove this with induction and linearity of expectation), and so $\mathbb{E}[v_n]=\frac 1{2^n}$.
In response to a comment on another ans... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2987792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Proof of Isoperimetric Inequality The authors of "Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Inequality" give several proofs of the classical isoperimetric inequality in the plane in the first chapter. There is one I cannot understand.
Let us assume $\Omega \subset \mathbb{C}$ has a $C^1... | It is Stoke's theorem after swapping the order of integration.
As differentials on the $z$-plane (i.e., fixed $\zeta$),
\begin{align*}
\mathrm{d}\left(\frac{\bar{\zeta}-\bar{z}}{\zeta-z}\mathrm{d}z\right)
&=\frac{\partial}{\partial\bar{z}}\left(\frac{\bar{\zeta}-\bar{z}}{\zeta-z}\right)\,\mathrm{d}\bar{z}\wedge\mathrm{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2987943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Evaluating the integral $\int_0^{\infty}\frac{dx}{\sqrt[4]{x}(1+x^2)}$ using Residue Theorem I need to evaluate the integral $$\int_0^{\infty}\frac{dx}{\sqrt[4]{x}(1+x^2)}$$
I've been given the hint to use the keyhole contour. So I would first choose the principal branch of $\sqrt[4]{\cdot}$, then I have the "keyhole" ... | Keeping it simple we introduce
$$f(z) = \exp(-(1/4)\mathrm{Log}(z)) \frac{1}{1+z^2}$$
with the branch cut of the logarithm on the positive real axis and
argument from $0$ to $2\pi.$ The slot of the keyhole rests on the
positive real axis and the contour is traversed counter-clockwise. Let
the segment above t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2988050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Eigenvalues of matrix $A$ $n \times n$, knowing that $A^2+A-6I_n=O_n$ How can I find ALL eigenvalues of matrix $A$, $n \times n$, $n \ge 3$, knowing that
$$A^2+A-6I_n=O_n.$$
I applied the Cayley-Hamilton theorem, and obviously two of the eigenvalues are $-3$ and $2$.
However, how do I find the rest of them?
| I will use a small result related to minimal polynomial.
Result: The characteristic polynomial and the minimal polynomial have the same roots, possibly with different multiplicities.
Now, $x^2+x-6=(x+3)(x-2)$ annihilates $A$. So, the minimal polynomial of $A$ divides $(x+3)(x-2)$. So, we have three options for the mi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2988219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
For which polynomials $p(x)$ is $p(p(x))+p(x)$=$x^4+3x^2+3$, for all $x \in \mathbb{R}$ For which polynomials $p(x)$ is $p(p(x))+p(x)$=$x^4+3x^2+3$, for all $x \in \mathbb{R}$
Since the power of the right hand side is 4, $p(x)$ has to be 2.
So I assumed a solution of: $p(x)=ax^2+bx+c$ and then i put it in $p(p(x))+p(x)... | Identifying the coefficient of $x^4$ one gets $a=1$ and then the coefficient of $x^3$ you get $b=0$ hence $c=1$ or $c=-3$. Then check that $x^2+1$ is indeed a solution while $x^2-3$ is not.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2988298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Solving a Markov Chain Let the distribution on variables $(X_t)$ for $t \in N$ satisfy a Markov chain. Each variable can take the values $\{1, 2\}$. We are given the pmfs
$$p(X_1=i) = 0.5$$
for $i=1,2$ and
$$p(X_{t+1} = j\mid X_t = i) = p_{i,j}$$
where $p_{i,j}$ is the $(i, j)$-th element of the matrix
$$P=\begin{pmatr... | Hints: By the law of total probability:
$$P(X_3=2)=\sum_{i,j\in \{1,2\}}P(X_3=2\mid X_1=i,X_2=j)P(X_1=i,X_2=j)$$ But by the Markov property, $X_3$ is independent of $X_1$, hence the inner term simplifies to $$P(X_3=2)=\sum_{i,j\in \{1,2\}}P(X_3=2\mid X_2=j)P(X_1=i,X_2=j)$$ Now, $P(X_3=2\mid X_2=j)$ is easy to compute ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2988376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
How do I prove that $\cos\left(2x\right)=1-2\sin^2\left(x\right)$? While trying to solve the equation $\sin\left(x\right)=\cos\left(2x\right)$, a user on this forum suggested that I turn the equation into a quadratic form by converting $\cos(2x)$ using the identity $\cos\left(2x\right)=1-2\sin^2\left(x\right)$.
What is... | write $$\cos (2x)=\cos(x+x)$$
you know $$\cos(x+y)=\cos x\cdot \cos y-\sin x\cdot \sin y$$
So, $$\cos(2x)=\cos x\cdot \cos x-\sin x\cdot \sin x$$
Or, $$\cos(2x)=\cos^2x-\sin^2x$$
write $\cos^2x$ as $1-\sin^2x$
So, $$\cos(2x)=1-\sin^2x-\sin^2x$$
you get$$\cos(2x)=1-2\sin^2x$$
Similarly get $$\cos(2x)=\cos^2x-\sin^2x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2988504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 4
} |
Inverse of operator with kernel $K(x,y)=(1-xy)^{-1}$ on the unit interval Suppose $$f(x)=\int_0^1 \frac{g(y)}{1-xy}dy.$$ Is there a $G(x,y)$ such that
$$g(y)=\int G(x,y)f(x)dx?$$ (with some integration contour)
| So we have an operator $T$ bringing $g(x)$ into $(Tg)(x)=\int_{0}^{1}\frac{g(y)}{1-xy}\,dy$ and we want an explicit representation for $T^{-1}$, fine. We may notice that $[x^n](Tg)(x)=\int_{0}^{1}y^n g(y)\,dy$, so the reconstruction of $g$ from $Tg$ is equivalent to solving the moment problem. Under suitable regularity... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2988546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Simple proof on set difference
Prove if $A$ is a subset of $B$, then $A\backslash B= \emptyset$.
I used a proof by contrapostive.
The assumption ($A\backslash B$ not being equal to the empty set) means there exists an $x$ such that $x\in A\backslash B$ implies $x\in A$ and $x\not\in B$. This implies that $A$ is not a... | Your proof is correct. You could have avoided using the contrapositive and just proved it directly:
Suppose that $A \subseteq B$. If there is some $x \in A \setminus B$, then $x \in A$ and $x \not\in B$, which contradicts the assumption that $A \subseteq B$; hence $A \setminus B$ has no elements, so is empty.
It does... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2988817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Orthonormal basis: Countable $\infty$ vs. Uncountable $\infty$ My doubt is the following, when you create an orthonormal basis for a space, the number of coefficients in each vector, and the number of vectors is equal to the dimension of the space (at least in finite dimensional spaces).
For example the standard basis ... | You can indeed specify a member of ${\mathcal L}_2(-\pi,\pi)$ with countably many real numbers. That doesn't say you have to: you can also choose to specify
the values $f(t)$ for all $t \in (-\pi,\pi)$ (modulo the non-uniqueness due to the fact that these are really
equivalence classes of functions rather than functi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2988914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Consequences of $f \neq 0, f^2 = 0$ I have been given the following problem.
Given $f: V \to V$ linear. $f \neq 0$, $f^2 = 0$, demonstrate that $(f(u_1), f(u_2) ... f(u_m))$ is linearly independent, where $(u_1, u_2...u_m)$ is a basis of the supplementary subspace of the kernel ($U \oplus \ker(f) = V$)
My reasoning ... | Hint:
You don't really need the hypothesis $f^2=0$. Consider the restriction of $f$ to the supplementary subspace $U$. Which properties does it have?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2989038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Suppose $f:G\longrightarrow H$ is a group homomorphism with $H$, $Ker(f)$ finite. Is $G$ finite? Let $f:G\longrightarrow H$ be a group homomorphism with $G$ not necessarily a finite group, but $H$ is a finite group. By the first isomorphism theorem we have:
$\frac{G}{Ker(f)}\cong Im(f)$.
Suppose further that we know th... | The kernel is one of the cosets in the quotient group and all cosets are the same size. Since the image is finite, there are a finite number of cosets. A finite number of cosets, each of a finite size implies that there are a finite number of elements in total.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2989138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Proving $e$ is irrational using a Beukers-like integral We know the following, for some integers $a_n,b_n$ and $n$ where $n\geq0$:
\begin{align}
&I_n = \int_0^1 x^n(1-2x)^n e^x dx = a_ne+b_n \\
&|I_n|= \left\lvert \int_0^1 x^n(1-2x)^n e^x dx \right\rvert \leq \left( \frac{1}{8} \right)^n(e-1)
\end{align}
therefore, as... | The case is trivial when $n$ is even ($I_n$ always $>0$), so we just assume $n$ is odd. Write
$$I_n=\underbrace{\int_0^{1/2} x^n(1-2x)^n e^x~\mathrm dx}_{J_1}+\underbrace{\int_{1/2}^1 t^n(1-2t)^n e^t~\mathrm dt}_{J_2}.$$
By considering the area under the curve, $J_1>0$ while $J_2<0$.
Claim: $-J_2> J_1$ when $n$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2989409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\mathbb{R}^2\setminus E$ is path-connected
Let $E$ be the set of all points in $\mathbb{R}^2$ having both coordinates rational. Prove that the space $\mathbb{R}^2\setminus E$ is path-connected.
Path-connected definition: A topological space $(X,\tau)$ is said to be path-connected if given $a,b\in X$, ther... | Let $C=\{c_1,c_2,\dots\}\subset \mathbb R^2$ be any countable set. Then $\mathbb R^2\setminus C$ is path connected.
Proof: Suppose $p,q$ are distinct points in $\mathbb R^2\setminus C.$ Consider the set of rays emanating from $p$ that contain a point of $C;$ the set of such rays is countable. Same thing for for $q.$ Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2989879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Growth of Digamma function
For $1\le \sigma \le 2$ and $t\ge 2$, $s=\sigma+it$ prove that $\displaystyle \frac{\Gamma'(s)}{\Gamma(s)}=O(\log t)$.
From Stirling's formula we have, $\displaystyle \Gamma(s)\approx \sqrt{2\pi}\exp\{s\log s-s-\frac 12 \log s\}$.
Then, $\displaystyle \frac{\Gamma'(s)}{\Gamma(s)}\approx\log... | Let's use the following formula, from Abramowitz and Stegun, valid everywhere in the strip of interest:
$$
\psi(z)=-\gamma+\sum_{n=1}^{\infty}\frac{z-1}{n(n+z-1)}.
$$
Letting $a=\sigma-1\in[0,1]$,
$$
\psi(\sigma+it)+\gamma=\sum_{n=1}^{\infty}\frac{a+it}{n(n+a+it)}=\sum_{n=1}^{\infty}\frac{\left(a+it\right)(n+a-it)}{n\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2990150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Show that the space of increasing, bounded function is not totally bounded w.r.t. $\sup$-norm Here is the exercise that I got.
Verify that the class $\mathcal{G} = \left\{ g: \mathbb{R} \to [0,1], g \text{ is increasing}\right\}$ is not totally bounded for the supremum norm on $\mathbb{R}$.
I am trying to prove this ... | The below will work for functions into $[-\pi/2,\pi/2]$. You would just need to shift and stretch a little for your case. Recall that $\arctan(x)$ is strictly increasing and ranges in $(-\pi/2,\pi/2)$. Replacing $x$ by $\alpha x$ for $\alpha>1$ does a horizontal compression. So for large $\alpha$, $\arctan(\alpha x)$ w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2990351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Proof for $\lor$ Elim: rule in Soundness Theorem So far I have been told to assume the line is invalid and then arrive at a contradiction.
Suppose the first invalid step derives the sentence $C$ by an application of $\lor$ Elim to the sentences $A\lor B$ and $A$ and $B$ appearing earlier in the proof. Let $P_1,\ldots,P... | There's no compelling reason to use proof by contradiction. The rule consists of a valid one in intuitionstic positive logic, so like anything else in intuitiionistic positive logic it can get proved without the use of proof by contradiction.
Suppose that (A $\lor$ B) is true, (A$\rightarrow$C) is true, and (B$\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2990466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Evaluate $\lim_{n\to \infty}(0.9999+\frac{1}{n})^n$ $\lim_{n\to \infty}(0.9999+\frac{1}{n})^n$
Using Binomial theorem:
$(0.9999+\frac{1}{n})^n={n \choose 0}*0.9999^n+{n \choose 1}*0.9999^{n-1}*\frac{1}{n}+{n \choose 2}*0.9999^{n-2}*(\frac{1}{n})^2+...+{n \choose n-1}*0.9999*(\frac{1}{n})^{n-1}+{n \choose n}*(\frac{1}{n... | Hint Look at $n\gt 10000$
For $n\gt 10000$, we have $0.9999 + \frac{1}{n} \leq 0.9999+ \frac{1}{10001}\lt1$, so $\lim_{n\rightarrow\infty}(0.9999 + \frac{1}{n})^n \leq \lim_{n\rightarrow\infty} (0.9999 + \frac{1}{10001})^n = 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2990642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 2
} |
If $\lim_{n\rightarrow\infty} S_{n} = \infty$, prove that $\lim_{n\rightarrow\infty}\sqrt{S_{n}} = \infty$ I have no idea how to go about this and I have been stuck on it for 3 days.
The formula we have for infinite limits is:
For every $M>0$, there exists an $N \in \mathbb{N}$ such that for $n > N \implies S_{n} > M$.... | $\fbox{For every $M>0$, there exists an $N \in \mathbb{N}$ such that for $n > N \implies S_{n} > M$.}$
Let $M$ be arbitrary and choose $N \in \mathbb{N}$ such that for $n>N$, $S_n > M^2$. Then for $n>N$, $\sqrt{S_n} > M$
If you want to be really verbose, you could say:
Let $M$ be arbitrary. By supposition, there exis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2990745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Find a basis and dimension of V When finding a basis for a vector space defined by $V = \{p(x) \in P_3(\Bbb R) \mid p(3)=0\text{ and }p(2)=0\}$, I started by showing $p(x) = (x-2)(x-3)q(x)$ where $q(x)\in P_1(\Bbb{R})$.
So, $p(2) = 8a+4b+2c+d = 0 $
And, $p(3) = 27a+9b+3c+d = 0$
I am wondering do I set these equal and s... | Go for $$\{(x-2)(x-3), x(x-2)(x-3)\}$$ for your basis.
These two polynomials will span your space and they are linearly independent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2990882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find a general solution for $\int_{0}^{\infty} \sin\left(x^n\right)\:dx$ So, I was recently working on the Sine Fresnal integral and was curious whether we could generalise for any Real Number, i.e.
$$I = \int_{0}^{\infty} \sin\left(x^n\right)\:dx$$
I have formed a solution that I'm uncomfortable with and was hoping f... | Here is an alternative approach that avoids complex numbers and series altogether. To get round these two obstacles I will use a property of the Laplace transform.
Let
$$I = \int_0^\infty \sin (x^n) \, dx, \qquad n > 1.$$
We begin by enforcing a substitution of $x \mapsto x^{1/n}$. This gives
$$I = \frac{1}{n} \int_0^\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2991201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 3
} |
solving an inseparable differential equation involving exponentials Solve $2xe^y$ + $e^x$ + ($x^2$ + 1)$e^y$$\frac{dy}{dx}$ = 0 with $y$ = 0 when $x$ = 0.
So this is clearly an inseparable differential equation so I thought the standard way to approach this was with a substitution but I cannot think of anything that wo... | Hint:
$\frac{\partial}{\partial y}(2xe^y+e^x) = \frac{\partial}{\partial x} (x^2+1)e^y$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2991342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Three space property I want to show that Finite dimensionality is a three space property. Let $X$ be a normed linear space and let $Y$ be a closed subspace of $X$. If $Y$ and $X/Y$ are finite dimensional spaces, then I want to show $X$ to be finite dimensional.
Let $B$ be a finite basis of $Y$. Then it can be extended... | Let $x_1,\ldots,x_n\in X$ be such that $(x_1+Y,\ldots,x_n+Y)$ is a basis of $X/Y$ and let $(y_1,\ldots,y_m)$ be a basis of $Y$. If $x\in X$, then there are scalars $\alpha_1,\ldots,\alpha_n$ such that$$x+Y=\alpha_1(x_1+Y)+\cdots+\alpha_n(x_n+Y).$$So, $x-\sum_{k=1}^n\alpha_kx_k\in Y$ and therefore there are scalars $\be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2991446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
If $\sin x+\sin^2x+\sin^3x=1$, then find $\cos^6x-4\cos^4x+8\cos^2x$
If $\sin x+\sin^2x+\sin^3x=1$, then find $$\cos^6x-4\cos^4x+8\cos^2x$$
My Attempt
\begin{align}
\cos^2x&=\sin x+\sin^3x=\sin x\cdot\big(1+\sin^2x\big)\\
\text{ANS}&=\sin^3x\cdot\big(1+\sin^2x\big)^3-4\sin^2x\cdot\big(1+\sin^2x\big)^2+8\sin x\cdot\bi... | Let $t=\sin x$ and solve the cubic $$t^3+t^2+t=1$$
Wolfram Alpha gives the real solution as
$$t=(1/3) (-1 - 2/(17 + 3 \sqrt {33})^{1/3} + (17 + 3 \sqrt{33})^{1/3})$$
Plug the real solution of the above to get $$(1-t^2)^3 -4(1-t^2)^2+8(1-t^2) =4$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2991604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 4
} |
Integral Representation of $\frac{\mathrm{sign}(x)}{|x|^{s}}$ In this paper, on page 433 (4.12), the authors used an integral formula of the function $\frac{\mathrm{sign}(x)}{|x|^{s}}$, which is
$$|x|^{-s}\mathrm{sign}(x)=\frac{2}{\Gamma(\frac{s+1}{2})}\int_{0}^{\infty}dyy^{s}xe^{-x^{2}y^{2}}$$
for any real non-zero $... | It's a Laplace transform. Roughly, powers of $y$ transform to powers of $x^{-1}$ because $x,y$ have opposite dimension (as seen from the $x^2y^2$ term in the exponential). Changing variable $u=y^2$ would tidy it up, and then it would be essentially the definition of the Gamma function, which is the Laplace transform of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2991695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Prove that If $p$ is prime s.t. $0Prove that If $p$ is prime s.t. $0<n\leq p$ , then $p|[{p! \over {(p-n)!(n)!}}]$.
I know that if $p|q$ , then $q=kp$, for some integer number $k$.
But I don’t know how to prove that $p$ divided like above.
Is it working to use proof by induction?
| $(p-n)!$ and $n!$ do not contain $p$ as a factor, so neither of them can divide $p$. However $\binom{p}{n}$ is an integer, containing $p$ as a factor. So $p$ divides $\binom{p}{n}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2991996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How to compute the limit,$\lim_{x\rightarrow 0}\frac{3x^2-3x\sin x}{x^2+x\cos1/x}$ How to compute the limit,
$$\lim_{x\rightarrow 0}\frac{3x^2-3x\sin x}{x^2+x\cos\frac{1}{x}}$$
| As noticed the limit doesn't exist indeed we can consider the sequence as $x_n \to 0$ such that
$$\cos\frac{1}{x_n}=2x_n \implies \frac{3x^2-3x\sin x}{x^2+x\cos\frac{1}{x}}=\frac{3x_n^2-3x_n\sin x_n}{3x_n^2}=1-\frac{\sin x_n}{x_n} \to 1-1=0$$
and the sequence $x_n \to 0$ such that
$$\cos\frac{1}{x_n}=-x_n+x_n^3 \im... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2992131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
How many $S\subseteq\mathcal{P}(A)$ contain each element of $A$ an even number of times? Let $A=\{1,2,...,n\}$. Let the powerset of $A$ be $\mathcal{P}(A)$.
We call $S\subseteq \mathcal{P}(A)$ a paired family of subsets if $\forall a\in A$, the number of elements of $S$ that contain $a$ is even.
For $|A|=n$, how many... | Answer is $2^{2^n-n}$.
I'll give two (morally the same) proofs.
Proof Using Linear Algebra
Identifying subsets with indicator functions, we can view every $\mathcal{S}\subseteq\mathcal{P}(A)$ as an element of the $\mathbb{F}_2$-vectorspace $(\mathbb{F}_2)^{\mathcal{P}(A)}$. For each $a\in A$, define an $\mathbb{F}_2$-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2992265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find a remainder when dividing some number $n\in \mathbb N$ with 30
A number n when you divide with 6 give a remainder 4, when you divide with 15 the remainder is 7. How much is remainder when you divide number $n$ with $30$?
that mean $n=6k_1+4$, $n=15k_2+7$, and $n=30k_3+x$, so I need to find $x$. And $30=6*5$ or ... | Alternatively:
$$n\equiv 4 \pmod{6} \Rightarrow 5n\equiv 20 \pmod{30};\\
n\equiv 7 \pmod{15} \Rightarrow 2n\equiv 14 \pmod{30}.$$
Add the two:
$$7n \equiv 34\equiv 4 \pmod{30} \Rightarrow \\
91n\equiv n\equiv 52 \equiv 22 \pmod{30}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2992393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Prove that a connected graph with $n$ vertices and $n+2$ edges is planar. What I have tried is that a tree has $n-1$ number of edges. So if there are $n$ edges there would be a cycle. This cycle would contribute to one face of the graph. Similarly for the other 3 edges. I think this would add one face each to the graph... | Let $G$ be a connected graph on $n$ vertices with $n+2$ edges. Perform the following reductions on $G$.
If a vertex $v$ of $G$ has degree $1$, then $G$ is planar iff $G-v$ is plannar. Thus, we may remove all vertices of degree $1$ from $G$ successively, so that at the end, $G$ has no vertex of degree $1$.
If $v$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2992490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Correct notation for rounding off numbers e.g. $201.7 \times 0.028 = 5.6476 = 5.6$ What would be the correct notation to rounding off numbers:
Option 1
$$201.7 \times 0.028 = 5.6476 = 5.6$$
Option 2
$$201.7 \times 0.028 = 5.6476 \approx 5.6$$
PS: I am not sure I inserted the most appropriate tag for this question.
Than... | The second way
$$201.7 \times 0.028 = 5.6476 \approx 5.6$$
is to prefer unless we don't specify in some way that we are rounding the result to the first decimal digit.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2992632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Proving every metrizable space is normal space
A topological space $X,\tau$ is said to be normal space if for each pair of disjoint closed sets $A$ and $B$, there exists open sets $U$ and $V$ such that $A\subseteq U$,$B\subseteq V$ and $U\cap V=\emptyset$. Prove that every metrizable space is normal space.
If $X,\tau... | Let $X$ be a metrizable and let $d: X^{2} \rightarrow \mathbb{R} $ be a metric which defines the topology of $X$. We want to show that $X$ is Normal space. Let $ F_1$ and $F_1$ be disjoint closed subsets of $X$. Let
$$ (X-F_2) = \left\{ x\in X: d(x,F_1) < d(x,F_2) \right\}$$
and
$$ (X-F_1) = \left\{ x\in X: d(x,F_1) > ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2992758",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
On an expected value inequality. Given $X$ a random variable that takes values on all of $\mathbb{R}$ with associated probability density function $f$ is it true that for all $r > 0$
$$E \left[ \int_{X-r}^{X+r} f(x) dx \right] \ge E \left[ \int_{X-r}^{X+r} g(x) dx \right]$$
for any other probability density function $g... | Unfortunately, your intuitive conjecture is INCORRECT.
Let $f(x)$ be the PDF of the random variable $X$ and $F(x)$ be its cumulative PDF, so that $F'(x)=f(x)$, or
$$F(x)=\int_{-\infty}^x f(t)dt$$
Similarly, let $g(x)$ be another PDF with cumulative PDF $G(x)$. Then the expected value of the integral
$$\int_{X-r}^{X+r} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2992859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
PDE Laplace equation. Integral representation form and Green function Let $\Omega$ be a domain in $\mathbb{R}^{d}$ and assume that for any $y \in \Omega$ there is a function $h_{y} \in C^{2}(\overline{\Omega})$ such that
\begin{equation}
\label{eq8.1}
\begin{cases}
\Delta h_{y}(x) = 0 \text{ in } \Omega \\
h_... | It follows from the divergence theorem applied to the vector field $h_y\nabla u - \nabla h_y u$. Then, the we get the terms term $\partial_v u h_y = \partial_vE$ and $\partial_v h_y u$ which cancels the term from $\partial_v G u$, so we recover the original integral representation for $u$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2993067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Are Christoffel symbols structure coefficients? For a chart $(U,x^1,...,x^n)$ in an $n$ dimensional manifold, the the christoffel symbols for a covariant derivative are defined by $\nabla_{\partial_j}\partial_i=\Gamma_{ij}^k\partial_k$. For a general algebra of dimension $n$ with multiplication $\beta$, the structure c... | The answer to the first question is not, because the algebra is an infinite-diennsional algebra, while you have only $n^3$ Christoffel symbols. And regarding the second question, if the connection is the Levi-Civita connection then the differences you say are zero, because the Levi-Civita connection is torsion-free.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2993200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Counting ways to divide $10$ kids into $2$ teams of $5$. Why divide by $2!\,$? Here's the question:
In order to play a game of basketball, $10$ kids at a playground divide themselves into two teams of $5$ each. How many different divisions are possible?
The solution given is:
$$\frac{10!}{5!5!2!}$$
My question is
... | Think about it this way:
Let's line up all ten people, and let's say that the first five go on team 1, and the next five on team 2. Now, there are $10!$ different ways to line up the $10$ people. However, note that you get the same two teams if:
*
*you shuffle the first five people. Having, say, persons 2 and 4 chan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2993316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to simplify a multiplication of several summations? The formula is $$-\sin(i)\sum_{n=0}^\infty (\frac{w}{2i})^n\sum_{n=0}^\infty (\frac{w}{i})^n\sum_{n=0}^\infty \frac{(-1)^n}{(2n)!} (w)^{2n-1}.$$
I only want to get the coefficient of the $w^{-1}$ term, and the coefficients of other terms are negligible, so it look... | By the definition of Laurent series multiplication, the only terms that will contribute to $w^{-1}$ in the product are the lowest-indexed terms (said another way, the first coefficient in the product series is the product of the first coefficients.)
Those are $\left(\frac{w}{2i}\right)^0$, $\left(\frac{w}{i}\right)^0$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2993471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Logical equivalence of ¬p→q Just wondering what other ways $\neg p \to q$ can be expressed.
I know that $p\to q$ is logically equivalent to $\neg p\lor q$, hence I think that $\neg p\to q$ has the same logical equivalence as $p\lor q$.
| You have pretty much given the answer yourself already. Logical equivalence of $p$ and $q$ is given if $p$ is true if and only if $q$ is true (and hence $p$ is false iff $q$ is false). Since $p$ and $q$ can only be true or false, you can use truth tables and check whether logical equivalence is given. From there you ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2993594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Error in calculation of number of ways to put $20$ identical balls in $4$ labelled boxes if each box contains at most $18$ balls The number of ways to put $20$ identical balls in $4$ labelled boxes in such a way that each box contains at most $18$ balls is,
$$(a)~~\binom{24}{4}-16~~(b)~~\binom{24}{4}-10~~(c)~~\binom{... | Okay so using we know that the number of ways without restrictions is simply $$\binom{n+r-1}{r-1}$$ And using $n=20,r=4$ we get $$\binom{23}{3}$$Now if we put $19$ balls in any one box then we have $\binom{4}1$ ways of selecting the one box and then we also have to select the next box to hold the last ball which can be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2993718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Solve for x negative exponential I have the following equation:
$y = 14857x^{-1.092}$
I know $y = 43$, how do I rewrite this to solve for $x$.
e.g. $43 = 14857x^{-1.092}$
| you have $$43 = 14857x^{-1.092}$$
write it as $$x^{-1.092}=\frac{1}{x^{1.092}}=\frac{43}{14857}$$
You get $$x^{1.092}=\frac{14857}{43}$$
Divide for the power $1.092$: $$x=(\frac{14857}{43})^{\frac{1}{1.092}}=211.154$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2993846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$\max\{a_1,a_2,\dots,a_n\}$ converges for a convergent sequence $a_n$ I am tackling the following question and want to be sure that my reasoning is fine.
Let $a_n$ be a convergent sequence s.t $\displaystyle \lim_{n\to\infty}a_n=a$. Let $$b_n\triangleq\max\{a_1,a_2,\dots,a_n\}$$
Prove that $b_n$ converges. Also, is ... | What you did is fine and if you missed something is that an even simpler example than yours can be found. Just take$$a_n=\begin{cases}1&\text{ if }n=1\\0&\text{ otherwise.}\end{cases}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2994000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Type-I vs. type-II error in statistical hypotheses testing Let us consider standard statistical hypotheses testing:
$$\alpha=P\{\text{type}-I \text{ error}\}=P\{\text{Rejecting } H_0 \text{ when }H_0\text{ is true}\}$$
and
$$\beta=P\{\text{type}-II \text{ error}\}=P\{\text{Accepting } H_0 \text{ when }H_1\text{ is ... | Suppose you have two boxes of dice, one is a box of fair dice in which all faces are equally likely. The other has loaded dice for which the probability of getting a six is is 1/3. The labels are missing so you will roll a sample of 50 dice from each box to try to identify which box has the loaded dice.
Let $H_0: \text... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2994150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Conditional Probability Drawing Candy I figured out the first one and need help with the second, with the second one could you provide solutions for both with and without replacement?
I have two bowls of candy. One is supposed to be filled with candy-covered chocolates, the other filled with fruit-flavored candies with... | With replacement:
$$P(A|CFF) = \frac{\frac{2}{3}\cdot (\frac{1}{3})^2}{\frac{2}{3}\cdot (\frac{1}{3})^2+\frac{1}{4}\cdot (\frac{3}{4})^2} = \frac{128}{371}$$
There is an inconsistency here. That is, if both bowls are large enough not to effect the proportion ratio then solutions with and without replacement are theoret... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2994224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Hints about the limit $\lim_{x \to \infty} ((1+x^2)/(x+x^2))^{2x}$ without l'Hôpital's rule? I've tried to evaluate $\lim_{x \to \infty} \left(\frac {1+x^2}{x+x^2}\right)^{2x}$ as $$\lim_{x \to \infty} \left(\left(\frac {1+ \frac{1}{x^2}}{1+ \frac{1}{x}}\right)^{x}\right)^{2}$$
So the denominator goes to $e^2$, but I d... | HINT
We have
$$\left(\frac {1+x^2}{x+x^2}\right)^{2x}=\left(\frac {x+x^2+1-x}{x+x^2}\right)^{2x}=\left[\left(1+\frac {1-x}{x+x^2}\right)^{\frac {x+x^2}{1-x}}\right]^{\frac {2x(1-x)}{x+x^2}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2994404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Find the smallest positive odd integer n such that φ(n)/n = 7680/12121 In a previous problem, I was able to deduce that if you have φ(n)/n = a/b where gcd(a,b) = 1 then the largest prime factor of b must also be the largest prime factor of n.
I found the prime factorization of both integers:
7680 = 2^9*3*5
12121 = 17*... | From "previous problem", you know that $31$ is the largest prime factor of $n$. If $n=31^km$ with $31\nmid m$ and $k\ge 1$, then $\phi(n)/n=\frac{30}{31}\phi(m)/m$ does not depend on $k$, so we take $k=1$ in order to minimize. This leaves us with the new (simpler) problem to find the minimal $m$ with
$$ \frac{\phi(m)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2994476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
A Nerve functor into any $\infty$-comos $\mathcal{N}: Cat \to \mathcal{K}$ I believe there is a notion of a nerve functor into any $\infty$-cosmos $\mathcal{K}$. My inclination is that it would be defined as the colimit of the constant functor that sends all objects to the terminal object in $\mathcal{K}$, which exist... | Your formula would, interpreted literally, produce a coproduct of terminal objects indexed by the connected components of your category. There is an interpretation of your proposal as a lax colimit which works when these exist, but they are not assumed in a general cosmos. It's probably clearer to define $\mathcal N(J)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2994646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Random Variable with Characteristic function $\frac{1}{2-\phi(t)}$ I am given that $X$ has c.f. $\phi(t)$, I need to find the random variable whose c.f. is equal to $\frac{1}{2-\phi(t)}$ in terms of $X$.
My idea is that express $\frac{1}{2-\phi(t)}$ as a series, since $|\phi(t)| \leq 1$ so we have
$$
\frac{1}{2-\phi(t)... | You already got the answer, you are just not completely writing that out. In probability context we usually write
$$\displaystyle Z = \sum_{i=1}^A Y_i$$
as a random sum, with the convention that $Z = 0$ when $A = 0$. We assumed $A$ is independent of $Y_i$ also, and they are defined as what you written. Then by law of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2994784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How to prove $\sum_{d\mid q}\frac{\mu(d)\log d}{d}=-\frac{\phi(q)}{q}\sum_{p\mid q}\frac{\log p}{p-1}$? Prove that $$\sum_{d\mid q}\frac{\mu(d)\log d}{d}=-\frac{\phi(q)}{q}\sum_{p\mid q}\frac{\log p}{p-1},$$
where $\mu$ is Möbius function, $\phi$ is Euler's totient function, and $q$ is a positive integer.
I can get
\be... | Let me write $n$ instead of $q$.
We have
\begin{align}
\sum_{d|n}\frac{\mu(d)\log(d)}d
&=\sum_{d|n}\frac{\mu(d)}d\sum_{p|d}\log(p)\\
&=\sum_{p|n}\log(p)\sum_{p|d|n}\frac{\mu(d)}d\\
&=\frac 1n\sum_{p|n}\log(p)\sum_{p|d|n}\mu(d)\frac nd
\end{align}
Write $n=p^em$ with $p\nmid m$.
Then $\varphi(n)=p^{e-1}(p-1)\varphi(m)$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2994900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Nullity of the same matrix over a finite and infinite field Is there a relation between the nullity of the same matrix over a finite field and a infinite field?
Or is there a prime power q so there the differents of nullity of a matrix over, let's say the rational numbers and the field $\Bbb{F_q}$ is small?
| Your question makes most sense if you ask it about a matrix of integers, say $A$. It is slightly easier to talk about the rank, so I will do that: the Rank-Nullity Theorem lets one translate to nullities.
Given such a matrix matrix $A$ we can find (by the Smith Normal Form algorithm) invertible matrices of integers $P,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2995107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Egorov’s Theorem (?) Let $(X, \mathbb A, m)$ be a measurable space and let $\{f_n : X \to \mathbb R\}_{n \in \mathbb N}$ be a sequence of Borel measurable functions. If such sequence converges $m$-almost everywhere to some Borel measurable $f: X \to \mathbb R$, I have to prove that for any $\epsilon > 0$ there exists $... | Yes, it is and additionally you need a finite measure space. It is wrong for not-finite measures: For example take $f_n = 1_{[n,n+1]}$, then $f_n \rightarrow 0$ pointwise, but if $\lambda(A) < \varepsilon < 1$, then we must have $\lambda([n,n+1] \setminus A ) >0$. Thus
$$\sup_{x \in \mathbb{R} \setminus A} |f(x) - 0| =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2995213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
contradiction proof on divides Suppose a,b ∈ Z. If 4 | $(a^2 + b^2)$ then a and b are not both odd.
So, assuming that 4 | $(a^2 + b^2)$ and $a$ and $b$ are odd
this gives $4k=(2l+1)^2+(2u+1)^2$ for some $k,l,u\in z$
eventually leading to $4k=4(l^2+l+u)+2(u^2+1)$
The RHS is not a multiple of 4 when $u=2$ contradiction.... | It's not yet valid, because you haven't shown why the RHS cannot be a multiple of $4$. You cannot simply set $u=2$, because the $u$ you have is already determined by $b$, since $b=2u+1$.
To correct your proof, re-think how you got from
$$4k=(2l+1)^2 + (2u+1)^2$$
to
$$4k = 4(l^2+l+u) + 2(u^2+1)$$
because I think you w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2995327",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Short mathematical proofs for teaching I am looking for short proofs in order to illustrate undergraduate notions. Most of the time students struggle with technical exercises without having the time, before going on with the following semester, to realize some great applications or insights about the objects introduced... | For a class on abstract algebra, there is the proof that the complex numbers are an algebraically closed field, following Artin. This is very short and can be presented in $10$ to $30$ minutes on the blackboard. One obtains it as a "great application" of Galois theory. For details, see also here:
Is there a purely alge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2995474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Integrate squared trigonometric function I'm trying to integrate $\int_a^b \left( \frac{1}{1+x^2} \right)^2 dx$
I know that $\frac{d}{dx} \arctan(x) = \frac{1}{1+x^2}$, but how can I integrate with the squared part?
I've tried substitution with no success.
| Full Work
Perform a substitution:
$$x = tan(u)$$
$$dx = sec^2(u)du$$
Then:
$$\int\nolimits\frac{sec^2(u)}{(1+tan^2(u))^2} du$$
Use the trig identity:
$$1+tan^2(x) = sec^2(x)$$
Then:
$$\int\nolimits\frac{sec^2(u)}{sec^4(u)} du$$
Simplify:
$$\int\nolimits\frac{1}{sec^2(u)} du$$
$$\int\nolimits{cos^2(u)} du$$
Use the ide... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2995619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 5
} |
Longest element of Weyl Group for $G_2$ Let $\mathfrak{g}$ be a semisimple Lie Algebra, $\mathfrak{t}$ a Cartan Subalgebra, $\Phi$ the corresponding set of roots, $\Delta \subset \Phi$ a root basis and $W$ the Weyl Group with respect to $\Delta$
I am having trouble finding the longest element of the Weyl Group $w_0$ ... | While Travis' answer gives a nice hands-on calculation, I like to point out two answers to related questions which put things in perspective:
Anton Geraschenko's answer here states, among other things, that the longest element in most simple types (actually, all except $A_{n \ge 2}, D_{2n+1}$ and $E_6$) is just $-id$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2995727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Proving Equivalence Relations, Constructing and Defining Operations on Equivalence Classes I think I have an intuitive sense of how ordered pairs can function to specify equivalence classes when used in the construction of integers and rationals, for example. I put the cart before the horse, however, and am less well v... | Equivalence relations can be defined on any set $X$.
For a relation $R$ on a set $X$, $R$ is an equivalence relation if $R$ is reflexive, symmetric, and transitive. So to check/prove that a relation $R$ on $X$ is an equivalence relation, you need to check that $R$ satisfies those three properties.
Now given an equivale... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2995887",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Testing whether the following integral converges
Determine whether the following integral converges or diverges:
$$I=\int_0^\infty\frac{x^{80}+\sin(x)}{\exp(x)}\,dx.$$
Since
$$0\overset{?}{<}\frac{x^{80}+\sin(x)}{e^x}\leq\frac{x^{80}+1}{e^x}\sim\frac{x^{80}}{e^x}$$
and if we let $f(x)=x^{80}/e^x$ and $g(x)= \sqrt{... |
Is there a more straightforward way to show that the above integral converges?
We can actually evaluate the integral of interest. Note that we have
$$\begin{align}
\int_0^\infty (x^{80}+\sin(x))e^{-x}\,dx&=\int_0^\infty x^{80}e^{-x}\,dx+ \int_0^\infty \sin(x)e^{-x}\,dx\\\\
&\Gamma(81)+\frac12\\\\
&=80!+0.5
\end{alig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2995984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Understanding abstract algebra proof of Fermat's Little Theorem The following proof of Fermat's Little Theorem is taken verbatim from Fraleigh's A First Course in Abstract Algebra:
For any field, the nonzero elements form a group under the field multiplication. In particular, for $\mathbb Z_p$, the elements $$1,2,3,\l... | Due to the hint from Bill Dubuque and Doug M, I think I have resolved my own problem. The bolded claim in the proof which is integral to the rest of the proof is only applicable when $\mathbb Z_p$ is indeed a field, which requires that every element in it has a multiplicative inverse, so each of them is coprime with th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2996246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find the sum of $1-\frac17+\frac19-\frac1{15}+\frac1{17}-\frac1{23}+\frac1{25}-\dots$
Find the sum of $$1-\frac17+\frac19-\frac1{15}+\frac1{17}-\frac1{23}+\frac1{25}-\dots$$
a) $\dfrac{\pi}8(\sqrt2-1)$
b) $\dfrac{\pi}4(\sqrt2-1)$
c) $\dfrac{\pi}8(\sqrt2+1)$
d) $\dfrac{\pi}4(\sqrt2+1)$
I have tried a lot.. But i can'... | We are looking for
$$1+\sum_{k=1}^\infty \left(\frac1{8k+1}-\frac1{8k-1}\right)=1-\sum_{k=1}^\infty \frac{2}{64k^2-1}\approx 1-\frac1{32}\zeta(2)\approx \frac{\pi}{8}(\sqrt2+1)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2996420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
} |
How are the known digits of $\pi$ guaranteed? When discussing with my son a few of the many methods to calculate the digits of $\pi$ (15 yo school level), I realized that the methods I know more or less (geometric approximation, Monte Carlo and basic series) are all convergent but none of them explicitly states that th... | The simplest method to explain to a child is probably the polygon method, which states that the circumference of a circle is bounded from below by the circumference of an inscribed regular $n$-polygon and from above by the circumference of a circumscribed polygon.
Once you have a bound from below and above, you can gua... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2996541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "40",
"answer_count": 11,
"answer_id": 1
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.