Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Why $\mathbb Z_2\otimes_{\mathbb Z}\mathbb Z_3\cong \{0\}$? I just stated tensor product, and I have problem to see how it works. So, why $$\mathbb Z_2\otimes_{\mathbb Z}\mathbb Z_3\cong \{0\}\ \ ?$$
| Let $a\otimes b\in \mathbb Z_2\otimes_{\mathbb Z} \mathbb Z_3$. Then
$$a\otimes b= (3a)\otimes b=a\otimes (3b)=a\otimes 0=0\otimes 0=0_{\mathbb Z_2\otimes_{\mathbb Z}\mathbb Z_3}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2141161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve integral $\int_1^2x\,d[x^2]$ with floor differential.
How can I evaluate the integral
$$\int_1^2x\,d[x^2]$$
I think this meaningless, because the area below separate points is not defined.
Note: $[.]$ is the floor function.
| Compute this as Riemann-Stieltjes integral: $$\int\limits_a^b f(x)\text{d}g(x)=\int\limits_a^b f(x)g'(x)\text{d}x$$ under suitable regularity asssumptions (which are in this case satisfied).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2141251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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$\tan^{-1}(x^2-\frac{x^4}{2}+\frac{x^6}{4}-\cdots)+ \cot^{-1}(x^4-\frac{x^8}{2}+\frac{x^{12}}{4}-\cdots)=\pi/ 2$
Find $x$ with $0<|x|<2$ such that
$$\tan^{-1}(x^2-\frac{x^4}{2}+\frac{x^6}{4}-\cdots)+ \cot^{-1}(x^4-\frac{x^8}{2}+\frac{x^{12}}{4}-\cdots)=\pi/ 2$$
My try:
$(x^2-\frac{x^4}{2}+\frac{x^6}{4}-\cdots)=\alp... | What I can come up with is as below
$\tan^{-1}(\alpha) =A \implies \tan(A)=\alpha$
$\cot^{-1}(\beta) =B \implies \cot(B)=\beta$
Also, we have
$A+B=\frac{\pi}{2}$
Add $\frac{\pi}{2}$ to both sides
$A+B+\frac{\pi}{2}=\pi$
Take $\tan$ from both sides and use the famous formula of $\tan(x+y)=\frac{\tan(x)+\tan(y)}{1-\tan(x... | {
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"timestamp": "2023-03-29T00:00:00",
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The principal value of an integral Prove the following
$$\int^\infty_0 \frac{\tan(x)}{x}=\frac{\pi}{2} $$
This question was posted on some forum, but i think it should be rewritten as
$$PV\int^\infty_0 \frac{\tan(x)}{x}=\frac{\pi}{2} $$
Because if the discontinuoities of the zeros of $\cos(x)$.
My attempt
Consider the... | $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newc... | {
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"timestamp": "2023-03-29T00:00:00",
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$X_1$ and $X_2$ stochastically larger than $Y_1$ and $Y_2$ implies $X_1+X_2$ stochastically larger than $Y_1+Y_2$ The random variables $X_1$, $X_2$, $Y_1$ and $Y_2$ are all mutually independent and I think one can show this in two different ways. In one paper I just came to a point where this may need to be shown. But ... | \begin{align}
G_{Y_1+Y_2}(z)&=\int_{-\infty}^{+\infty}G_{Y_1}(z-x)dG_{Y_2}(x)
\ge \int_{-\infty}^{+\infty}F_{X_1}(z-x)dG_{Y_2}(x)\\
&=\iint_{x+y\le z}dF_{X_1}(x)dG_{Y_2}(y)
= \int_{-\infty}^{+\infty}G_{Y_2}(z-y)dF_{X_1}(y)\\
&\ge \int_{-\infty}^{+\infty}F_{X_2}(z-y)dF_{X_1}(y)=F_{X_1+X_2}(z).
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2141590",
"timestamp": "2023-03-29T00:00:00",
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Find the average value of $2x^2 + 5x + 2$ on the interval where $x \in [1,3]$. Find the average value of the following function:
$p(x) = 2x^2 + 5x + 2$
on the interval $1 \le x \le 3$.
I know that I need to find $u$, $du$, $v$, and $dv$ and set it up into an definite integral but I don't know what to make them t... | You find the average of a function via applying the following formula:
$\frac {1}{b-a} \int_{a}^{b} f(x) dx$
$\frac {1}{2} \int_{1}^{3} 2x^2+5x+2 dx$
Wolfram says that the integral returns $124/3$, so
$\frac {1}{2}* \frac {124}{3} = 20.667$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2141730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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If an element is not in the set difference $A\setminus B \: \:?$ Given two sets$$A=\{1,3,5\}\quad,\quad B=\{1,3,8\}.$$ Then I compute the $A\setminus B=\{5\}.$
But my book$^\dagger$said: ... . Also, observe that $x\notin A\setminus B$ does not mean that $x\notin A\lor x\in B$. Why?
I don't know how to explain the Why,... | You could also use the identity $A\setminus B =A \cap B^c$, where $B^c$ is the complement of $B$ with respect to the universe, say $U$ Then $$ x \notin A \setminus B \iff x \notin A \cap B^c.$$ Applying DeMorgan’s laws (in the “set form” one gets $$ x \notin A \cap B^c \iff x \in (A \cup B)^c \iff x \in A^c \cap B,$$ t... | {
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"timestamp": "2023-03-29T00:00:00",
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Every linear mapping satisfies $f(0) = 0$ In my textbook is written that every linear mapping $ f: V \to V'$ satisfies $f(0)=0$. But what about the mappings between the polynomials like $p(x) = ax^2 + bx + c$, where $c$ is nonzero? Elementary probably, but fundamental.
| You are getting confused between $f$ and $p$.
$f$ is a function that applies to the coefficients of the polynomial and returns new coefficients.
For clarity, we can write
$$f(a,b,c)=(a',b',c')$$
and a necessary condition for linearity is
$$f(0,0,0)=(0,0,0).$$
For instance,
$$f(a,b,c)=(2a,b+c,c-a),$$ also understood as... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2142156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What kind of manifold is the quotient of $ \mathbb C^n $by a (non-mirror) crystallographic group A lattice of rank $k $ on $\mathbb C ^n$ is a discrete subgroup of $(\mathbb C ^n,+)$. It has the form
$$\Gamma_k = \mathbb Z e_1 + \mathbb Z e_2+ \cdots + \mathbb Z e_k$$
where $\{e_i\}_{i=1,\cdots,k}$ is an $\mathbb R$ in... | I would call them "Kahler Bieberbach maniflolds" or "Kahler flat manifolds". You can find some discussion and references here.
| {
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Show the isomorphism $X/\{0\} \cong X$ $X/\{0\}$ is the set of singletons of $X$ so we define a mapping $T(x) = x$. Since $\{0\} \subseteq \ker T$ we have a linear mapping $\hat{T}:X/\{0\}\to X$.
But $\{0\}$ is exactly equal to $\ker T$ so $X/\ker T \cong ran T = X$ since the mapping is surjective.
Does this make sen... | So yes, your reasoning seems to be correct. Although it took me a while to fully understand what you are doing here. So let's make it rigorous (for example by giving explicitely domains and codomains of each map).
Let $X$ be a group/ring/module/linear space (or anything that has an analogue of the first isomorphism the... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Count how many unique eulerian path are in a graph? I know how to check if graph have an eulerian path. But, I wonder, is there any general solution to count, how many unique eulerian path exists in a graph?
| For a case of directed graph there is a polynomial algorithm, bases on BEST theorem about relation between the number of Eulerian circuits and the number of spanning arborescenes, that can be computed as cofactor of Laplacian matrix of graph.
Undirected case is intractable unless $P \ne \#P$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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random walk with hitting probability If I am at $0$, I can go up by $1$ with probability $q$, and go down by $1$ with probability $(1-q)$, what is the probability I hit $10$ at some time?
Let the probability I hit $i$ at some time be $p_i$, then,
*
*$p_2=p_1^2$ since I hit $1$ and then hit another $1$
*$p_1=q+(1-q)... | You can think of this as a grid where we go up with probability $q$. We will hit 10 if there are 10 more ups than downs. That is, P(hit 10) $= \sum_{n \geq 10} \sum_{k=10}^{n} {n \choose k} q^{k} (1-q)^{k-10}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2142659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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reqired help solving $\cos(x)=-\cos(y)$ If I have $\cos(x)=-\cos(x+\alpha)$,
can I solve it by doing
$x=-(x+\alpha+2\pi)$ and $x=-(-(x+\alpha+2\pi))$?
It's probably a stupid question but I'm really confused.
| HINT: $$\cos(x)+\cos(y)=2\cos\left(\frac{x-y}{2}\right)\cos\left(\frac{x+y}{2}\right)$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Odd / Even integrals My textbook doesn't really have an explanation for this so could someone explain this too me.
If f(x) is even, then what can we say about: $$\int_{-2}^{2} f(x)dx$$
If f(x) is odd, then what can we say about $$\int_{-2}^{2} f(x)dx$$
I guessed they both are zero? For the first one if its even would... | If $f(x)$ is even then $f(-x) = f(x)$. So $$\int_{-2}^2 f(x) \, \mathrm{d}x = \int_{-2}^0 f(x)\, \mathrm{d}x + \int_0^2 f(x) \, \mathrm{d}x = \int_0^2 f(-x) \, \mathrm{d}x + \int_0^2 f(x) \, \mathrm{d}x$$
But then $f(-x) = f(x)$ so that simplifies to $2\int_0^2 f$.
Similarly, if $f$ is odd - that is: $f(-x) = -f(x)$ w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2142870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Vector that is orthogonal to one vector in a plane, automatically the normal? I'm trying to understand why if a vector is orthogonal to one vector in a plane, why it wouldn't be orthogonal to all vectors in that plane?
Sketches/diagrams would be helpful.
| Think of the $xy$-plane in $\mathbb{R}^3$ and the vector $\hat{\text{i}}$. $\hat{\text{j}}$ is orthogonal to $\hat{\text{i}}$ but it's not orthogonal/normal to the plane.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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How to normalize this exponentially distributed data? I have this histogram of data...what is the most proper way to prepare it for consumption in a neural network? I know how to normalize/standardize other types of data, but I'm wondering what to do with this kind of distribution.
| When normalising inputs to a Neural net, you want the numbers to be in a similar range across different inputs, so inputs which tend to have much larger absolute values don't dwarf the contributions from smaller ones. You need to preserve the Y values (frequencies) but can change the X scale with various transforms (po... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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A problem from a math contest about powers of two Recently I have participated in a math contest. One of the tasks really attracted me, but I still can't find the solution of it. Maybe you can help me?
Problem:
Given a natural number A which consists of 20 digits. Someone wrote the number AA...A(101 times) and removed... | First of all, note that instead of removing the last $11$ digits, we might as well remove the first $11$ digits — this change just corresponds to permuting the digits of $A$. For example, if $A=11223344556677889900$, the entire number will look like
$$
\underbrace{11223344556677889900}_{\text{repeated 100 times}} \,112... | {
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"url": "https://math.stackexchange.com/questions/2143192",
"timestamp": "2023-03-29T00:00:00",
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Upper bound on number of bipartite subgraphs Given a graph with $n$ vertices and $m$ edges, and fix positive integers $a+b\leq n$. What are some upper bounds on the number of (ordered) pairs of subsets of vertices $(A,B)$ such that $|A|=a,|B|=b$, and any vertex in $A$ has an edge to any vertex in $B$?
If we only use th... | All right, you already have some finite upper bound :-) I'll give exact bounds for some special cases. Let $k = k(n, m, a, b)$ be maximum number of desired pairs of subsets.
If $m = \binom{n}{2}$ then as far as you know $k = \frac{n!}{a!b!(n - a - b)!}$. If $m < ab$, then $k = 0$. If $m = ab$, then $k = 1 + [a = b]$.
I... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Angles in pyramid
We have to find the value of $\cos\theta$.
I tried it alot. But could not able to do it.
Solution given in the book
| HINTS:
Shall be brief, hope you fill in details.
Let $AB= 4 b$ then side of inner square on base $ 2 \sqrt 2 b = 2d $ say
Let $OA = h. \, $ Projections on midface and base are $ h \cos \beta, h \sin \beta \,$ where $ 2\beta = AOB$
If $H$ is side of dihedral angle $ 2 \gamma$ between slant faces
$$ \dfrac{1}{H^2}=\dfr... | {
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"url": "https://math.stackexchange.com/questions/2143534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Compute $\lim\limits_{x \to 0} \frac{ \sqrt[3]{1+\sin x} - \sqrt[3]{1- \sin x }}{x}$
Compute $\lim\limits_{x \to 0} \dfrac{ \sqrt[3]{1+\sin x} - \sqrt[3]{1- \sin x }}{x}$
Original question was to solve $\lim\limits_{x \to 0} \dfrac{ \sqrt[3]{1+ x } - \sqrt[3]{1-x }}{x}$ and it was solved by adding and subtracting 1 i... | $$\lim _{ x\to 0 }{ \frac { { (1+\sin x) }^{ \frac { 1 }{ 3 } }-{ (1-\sin x) }^{ \frac { 1 }{ 3 } } }{ x } } \frac { \left( { (1+\sin x) }^{ \frac { 2 }{ 3 } }+{ (1+\sin x) }^{ \frac { 1 }{ 3 } }{ (1-\sin x) }^{ \frac { 1 }{ 3 } }+{ (1-\sin x) }^{ \frac { 2 }{ 3 } } \right) }{ \left( { (1+\sin x) }^{ \fr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 3
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How to make N equal instalments of any amount? I am working on a financial software, in it I would like to offer an option of paying for an item in 3,6 or 12 instalments. I would be deducting each instalment amount via there Credit Card every month.
for example if I have an item that
costs $700,
and I want to make 12... | The easiest ways are sometimes the best ones...
You can add the resting to the first installment.
r=((700.00*100)%12)/100; //=00.04
installment[0]+=r; //=58.37
I dont think that would be a big problem - the resting for 12 months would be at most $0.11\$$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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concurrence of three lines in a quadrilateral Prove that the lines joining the midpoints of opposite sides of a quadrilateral and the line joining the midpoints of the diagonals of the quadrilateral are concurrent.
| Let $E=\frac{A+B}{2}$ and $F=\frac{C+D}{2}$ be the midpoints of two opposite sides.
Let $G=\frac{A+D}{2}$ and $H=\frac{C+B}{2}$ be the midpoints of another two opposite sides.
Let $J=\frac{A+C}{2}$ and $K=\frac{B+D}{2}$ be the midpoints of diagonals.
The midpoint of section $EF$ is a point $\frac{E+F}{2}=\frac{A+B+C+D}... | {
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If $\gcd(x,y)=1$, and $x^2 + y^2$ is a perfect sixth power, then $xy$ is a multiple of $11$ This is a problem that I don't know how to solve:
Let $x, y, z$ integer numbers such that $x$ and $y$ are relatively primes and $x^2+y^2=z^6$ . Show that $x\cdot y$ is a multiple of $11$.
| It's not complete answer
By computer search, it seems to be valid for co-prime $x$ and $y$.
Now, I'm trying to give the general solution of $(x,y,z)$.
\begin{align*}
(a+bi)^3 &= a(a^2-3b^2)+b(3a^2-b^2)i \\[7pt]
(a^2+b^2)^3 &=
\underbrace{a^2(a^2-3b^2)^2}_{\Large{m^2}}+
\underbrace{b^2(3a^2-b^2)^2}_{\Large{n^2}}... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 2
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Continuity of $t\mapsto f(t,h(t))$ Let $f:[0,T]\times X \to Y$ be a map where $X$ and $Y$ are Hilbert spaces. We have that $t \mapsto f(t,x)$ is continuous, and so is $x \mapsto f(t,x)$.
Let $h:[0,T] \to X$ be continuous. Does it follow that
$$t \mapsto f(t, h(t))\quad\text{ is continuous}?$$
$f$ is better than a Cara... | Look at this question and then create $h$ such that it moves to and fro the two paths used in the answer to provide a counter example:
| {
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Sum of a series using squeeze theorem How do I find using the Squeeze theorem
$$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{\sqrt{n^2+k}} \;,$$
using the fact that
$$ \lim_{n\to \infty}\frac{n}{\sqrt{n^2+n}}=1.$$
Thank you very much for your help,
C.G
| Note that
$$\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\sqrt{n^2+n}}\leq \frac{n}{\sqrt{n^2+1}}$$
because $\sqrt{n^2+1}\leq\sqrt{n^2+k}$ for $k\geq1$.
On the other hand, by a similar reasoning, using that $\sqrt{n^2+n}\geq\sqrt{n^2+k}$ for $k\geq1$:
$$\frac{n}{\sqrt{n^2+n}}\leq\frac{1}{\sqrt{n^2+1}... | {
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"timestamp": "2023-03-29T00:00:00",
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Why do mathematicians need to define a zero exponent? I think a zero exponent can be defined logically from various premises;
*
*We can define an exponent as the number of times the base "appears" in a multiplication process.
Thus: $2^3 = 2*2*2 $, $2^1 = 2 $, $2^0 = ( )$ doesn't appear at all.
Here's the thi... | One part of the answer is "calculus". You want to have some theory of continuous functions, limits, derivatives, integration, ODE and so on; in 18. and 19. century, it has been remarkably successful to describe phenomena from physics. Exponential functions are one of the cornerstone of all of this; even the simplest eq... | {
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"source": "stackexchange",
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How to show a sequence is monotonically decreasing and a null sequence? For example Let the sequence be $a_n=\frac{n+1}{n^2}$. I proved that $a_n$ is a null sequence by factoring out the $n^2$ .My question is how do i prove that it is monotonically decreasing? . Do i find the limit of the ratio of $\frac{a_{n+1}}{a_n}$... | we compute $$a_{n+1}-a_n=\frac{n+2}{(n+1)^2}-\frac{n+1}{n^2}=\frac{n^2(n+2)-(n+1)^3}{n^2(n+1)^2}=-\frac{n^2+3n+1}{n^2(n+1)^2}<0$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Is an infinite Cartesian product of well ordered sets well ordered? If $A_1, \ldots, A_n$ are well ordered sets, then so is the Cartesian product $\prod_{i=1}^n A_i$ under the dictionary order. Am I right?
Is this finite product also well ordered in the anti-dictionary order?
Now suppose that $J$ is an infinite set ... | The result does not hold for infinite products.
Here's a counterexample: take each $A_i$ to be $\{0, 1\}$ ordered the usual way, $i\in\mathbb{N}$. Then let $e_i$ be the string with a $1$ in the $i$th place and a $0$ everywhere else; what can you say about $e_i$ versus $e_j$ if $i<j$?
Also, it's worth noting that the d... | {
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"url": "https://math.stackexchange.com/questions/2144604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Infinite dimensional spaces and norm equivalence Let $X$ be an infinite dimensional space over a field $F$ and $B$ a basis of $X$.
We define two norms in $X$
$||x||_1=\sum_{i=1}^n|k_i|$ and $||x||_2=max\{|k_1|...|k_n|\}$ $\forall x \in X$,where $x=\sum_{i=1}^nk_ib_i$ and $b_1...b_n \in B$
Prove that these two norms ar... | Since $X$ is infinite dimensional, $B$ is infinite. Take a sequence $b_1,b_2,\ldots$ of independent vectors in $B$ and define $x_n=\sum_{i=1}^n b_i$. What are the norms of $x_n$ in both case? What happens when $n\to \infty$?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Let $\alpha = \omega + \omega^2 + \omega^4$ and $\beta = \omega^3 + \omega^5 + \omega^6$. Let $\omega$ be a complex number such that $\omega^7 = 1$ and $\omega \neq 1$. Let $\alpha = \omega + \omega^2 + \omega^4$ and $\beta = \omega^3 + \omega^5 + \omega^6$. Then $\alpha$ and $\beta$ are roots of the quadratic
[x^2 + p... | method is due to Gauss. This book 1875. $x^2 + x + 2.$
gp-pari to check; note a^8 = a
? x = a + a^2 + a^4
%1 = a^4 + a^2 + a
? q = x^2 + x + 2
%2 = a^8 + 2*a^6 + 2*a^5 + 2*a^4 + 2*a^3 + 2*a^2 + a + 2
?
If we switched to one of the real numbers $$ t = \omega + \omega^6, $$ we would have a root of
$$ t^3 + t^2 - 2 t -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2144834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that the subsequence of a convergent series with nonnegative terms converges Let $\sum_{k=1}^\infty a_k$ be a convergent series with nonnegative terms and let $a_{n_k}$ be a subsequence of {$a_k$}. Prove that the series $\sum_{k=1}^\infty a_{n_k}$ converges.
I am having trouble figuring out how to prove this and ... | HINT Remember that the limit of the series is defined as the limit of the sequence of partial sums. In the case where the series only has non-negative terms, the sequence of partial sums is monotonically increasing. If you take away terms, you can only decrease the value of the sum and it's still monotonic. Now remembe... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let {$t_n$} = $\sum_{k=n}^{\infty}a_k$. Prove that the sequence {$t_n$} converges to 0. Suppose that $\sum_{k=1}^{\infty}a_k$ is a convergent series.
a) Prove that the series $\sum_{k=n}^{\infty}a_k$ converges for each positive integer n.
b)Let {$t_n$} = $\sum_{k=n}^{\infty}a_k$. Prove that the sequence {$t_n$} conver... | Since $\sum a_k$ is convergent the sequence $s_n=\sum_{k=1}^na_k$ of partial sums is convergent and hence Cauchy. This means that:
$$\forall \epsilon>0,\,\exists n_0 \in \mathbb{N},\,\forall m,n\geq n_0,\,|s_m-s_n|\leq \epsilon$$
Notice that if $m>n$, then $s_m-s_n =\sum_{k=n+1}^ma_n$. Letting $m\to\infty$, we concude ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Probability for Type I and Type II error
P$\bigg(20 < \bar{x} < 35 | \bar{x} \sim N(32, \frac{25^2}{30})\bigg)$
I know the answer is $18.7\%$ (according to my notes) I am not sure how to get to this value.
Also, how will the method change if I had
P$\bigg(\bar{x} <20 \bigcup \bar{x} > 35 | \bar{x} \sim N(30, \frac{... | I am getting different number for your first expression. But this does not matter. (Also, note that your notation $\mathbb{P}[\bar{x}<20|\bar{x}\sim\mathcal{N}(\mu,\sigma)]$ is somewhat nonstandard since it is used for 'conditional on' but you are conditioning on $\bar{x}$ having some distribution, which is not an even... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What's meta-arithmetic? I am not able to find a definition of meta-arithmetic. Thus I am asking here this
Question. Do you know what's means meta-arithmetic? If do you know it, can you explain me in easy words what is the feasible/possible meaning of such definition for meta-arithmetic? Many thanks.
Feel free to add... | In general, see Metamathematics:
"the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories."
So, unless specified otherwise, it is the "logico-mathematical" study of arithmetic theory.
Some sources :
*
*Sthepen... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Derivative of $f: \mathcal M_{n\times n}(\mathbb R)\rightarrow \mathcal M_{n\times n}(\mathbb R)$ given by $f(X)=X^2$ Let $\mathcal M_{n\times n}(\mathbb R)$ be the set of $n\times n$ matrices with real entries and consider $f: \mathcal M_{n\times n}(\mathbb R)\rightarrow \mathcal M_{n\times n}(\mathbb R)$ given by $f(... | $$B:\Bbb R^{n\times n}\times \Bbb R^{n\times n}\to \Bbb R^{n\times n}, (A,B)\mapsto A\cdot B$$
Then $B\circ (\mathrm{id},\mathrm{id}):\Bbb R^{n\times n}\to\Bbb R^{n\times n}, X\mapsto X^2$ is differentiable at $X$ by your lemma and has differential
$$B[\mathrm{id},\mathrm{id}]_X'(A)=B[\mathrm{id}'_X (A),\mathrm{id}(X)]... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Openness and closedness in R Decide the closedness and openness of {x:0<|x|<1 and (1/x)∉N}. I know the answer that the set is open but not closed. But I am not getting why?
| Your set $A$ is open because it is equal to union of intervals $$ ( \frac{1}{n} ; \frac{1}{n + 1} )$$
and $A$ is not closed because $$ 0 \in \mathbb{R} - A $$ but every interval $$ ( -\varepsilon ; \varepsilon)$$ have nonempty intersection with $A$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A doubt on a proof of $\lim \frac{\sin x}{x}$ as $x\to 0$ provided in Simmons's Calculus with Analytic Geometry I'm having difficulty understanding a proof of $\lim_{x\to0}\frac{\sin{x}}{x}=1$ provided in Simmons's Calculus With Analytic Geometry, pg. 72. The proof goes as follows:
Let $P$ and $Q$ be two nearby points ... | One can interpret the proof in a more abstract, intuitive way. As arc length decreases, its "curviness" disappears (i.e. it becomes a straight line). We can say this as $\sin(\theta) \to \theta $ as $ \theta \to 0$. Taking this into account:
$$\lim_{\theta \to 0}\frac{\sin\theta}{\theta} = \lim_{\theta \to 0} \frac{\th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2145758",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Do we need to use limits while finding definite integral of a piecewise function? Let $f:[a,b]\to \mathbb{R}$ be a piecewise function such that
$$f(x) = \left\{
\begin{array}{c}
g(x) \hspace{1cm} a\le x<\alpha \\
h(x) \hspace{1cm} \alpha \le x \le b
\end{array}
\right. $$
where $\alpha \in (a,b)$. If we need to fin... | From the comments I understood that your main concern is that $g(\alpha)$ is not defined and hence we can't talk of the symbol $$\int_{a}^{\alpha}g(x)\,dx$$ You are right here. But we have to talk of $\int_{a}^{\alpha}f(x)\,dx$ and not about integral of $g$ here.
The problem is handled by introducing another function $... | {
"language": "en",
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"source": "stackexchange",
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Discrete Math: Seating at a circular table - Possible Problem and My thinking:
Imagine a circular table, and you want to sit 7 people around it. The total arrangements would be 7!/7 or 6!. So, the order of left or right does not matter because we can sit these people anywhere and in any direction we want.
However, If t... | Also I realized something:
If two people, assuming A and B individuals insist on sitting next to each other in a circular table of 7 people. Then now we should keep A and B together while we still can switch the places of C D E F G people. In another words the arrangements would be 2*(5!) because A to the right of B (B... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove there are $x_1,x_2\in(a,b)$ such that $\frac{f(x_2)-f(x_1)}{x_2-x_1}=f'(\xi)$ Let $f(x) \in C^2$ in $(a,b)$ and $f''(\xi)\not=0, \xi \in (a,b)$. Prove there are $x_1,x_2\in(a,b)$ such that $$\frac{f(x_2)-f(x_1)}{x_2-x_1}=f'(\xi) \tag{1}$$ So I don't really understand what's the dufficulty of that task. Why can't ... | Hint: Special case: $f'(\xi) = 0$ and $f''(\xi) >0.$ Because $f$ is $C^2,$ we have $f''>0$ in some neighborhood of $\xi.$ This is enough to give a strict local minimum of $f$ at $\xi.$ Now stare at the picture to see there are $x_1<\xi < x_2$ such that $f(x_1) = f(x_2);$ of course you need to prove this. This gives the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2146225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to prove $|\int^{b}_{a}f(x)dx| \leq \int^{b}_{a} |f(x)|dx$ using riemann sum. $$|\int^{b}_{a}f(x)dx| \leq \int^{b}_{a} |f(x)|dx$$
We know for a fact that $a \leq |a|$.
Also, prove using riemann sum.
$$\lim_{n \to \infty} \sum^{n}_{i=1}f(x^*)\Delta x = \int^{b}_{a}f(x)dx$$
Thus,
$$|\lim_{n \to \infty} \sum^{n}_{i=1}... | By the triangle inequality we have,
$$|b_1+b_2| \leq |b_1|+|b_2|$$
This implies
$$|b_1+b_2+b_3| \leq |b_1+b_2|+|b_3|$$
$$\leq |b_1|+|b_2|+|b_3|$$
Etc.
It follows that,
$$\sum_{i=1}^{n} |b_i| \geq |\sum_{i=1}^{n} b_i|$$
If you want to more rigorously show this use induction.
Now choose $b_i=f(x^{*}) \Delta x$. Assuming ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Why is $-\ \frac{1}{2}\ln(\frac{1}{9})$ equal to $\frac{\ln(9)}{2}$? I solved this problem in my textbook but noticed their solution was different than mine.
$1. \ 9e^{-2x}=1$
$2. \ e^{-2x}=\frac{1}{9}$
$3. -2x=\ln(\frac{1}{9})$
$4. \ x=-\ \frac{1}{2}\ln(\frac{1}{9})$
However, the answer that my textbook gives is $\f... | Note that $$\ln x +\ln y =\ln xy, \; \ln 1=\ln e^{0}=0$$
If $x, y$ are positive reals, as seen here. From this, $$\ln x +\ln \frac{1}{x}=0 \iff \ln x =-\ln \frac{1}{x}$$
So $$\ln \frac{1}{9}=-\ln 9$$
So $-\frac{1}{2}\ln(\frac{1}{9})=\frac{\ln(9)}{2}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2146571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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If $X_1, \ldots , X_n$ are independent RVs from Gamma$(1,\beta)$ and $S = \sum_i X_i$ find the $P(X_1 > 1 | S = s)$
If $X_1, \ldots , X_n$ are independent RVs from Gamma$(1,\beta)$ and $S = \sum_i^n X_i$ find the $P(X_1 > 1 | S = s)$.
Attempt: What I know so far is that $S\sim$ Gamma$(n,\beta)$ which is continue. By ... | You have correctly calculated the conditional density:
$$(n-1)\frac{(s-x_1)^{n-2}}{s^{n-1}}=(n-1)\frac1s \left(1-\frac{x_1}{s}\right)^{n-2} \text{ for } 0<x_1<s$$
is a PDF of minimum of $n-1$ independent r.v. with Uniform distribution on $[0,s]$.
The CDF for $0<x_1<s$ is
$$P(X_1\leq x_1|S=s)=\int\limits_0^{x_1} (n-1)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Probability of getting a certain number of heads in coin toss problem with differently biased coins with different numbers There are $K$ bags of coins, each bag contains $m_k$ ($k=1,\dots,K$) coins, and thus we have $M:=\sum_{k=1}^K m_k$ coins in total.
Coins in the $k$-th bag satisfies $P(\text{H})=p_k$, $P(\text{T})=... | Your thoughts are along the right lines, particularly this:
For the $k$-th bag, the probability of getting $n_k$ heads is ${m_k \choose n_k} p_k^{n_k}(1-p_k)^{m_k-n_k}$.
The only remaining thing is: what can the values of $n_k$ be, so that the total number of heads is $N$? Well, the only constraint is that $n_1 + n_2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proving a relation between area of triangle and square of its rational sides Let d be a positive integer.The question is to prove that there exists a right angled triangle with rational sides and area equal to $d$ if and only if there exists an Arithmetic Progression $x^2,y^2,z^2$ of squares of rational numbers whose c... | Hint:
If $y^2- d = x^2, y^2, y^2+d = z^2$, then: $z-x $ and $z+x $ are the legs of the right-angled triangle, area $= \frac {z^2-x^2}{2} = d $ and rational hypotenuse $2y $ because, $2 (x^2+z^2)=4y^2$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Condition of two hyperbolas do not intersect Given two hyperbolas h1(with foci and center) and h2(with foci and center), In what condition these hyperbolas will not intersect to each other?. I can get the condition when h1 and h2 are standard hyperbola (parallel to axis and the center is the origin). I want to find the... | Let us consider the reference equilateral hyperbola with equation $xy=1$ (see blue curve on graphics below). Any rectangular hyperbola can be obtained from the reference with a rotation (angle $\theta$) followed by a translation (vector $\binom{a}{b}$). One should obtain the following result :
*
*if $\theta\neq0$, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Factor $9(a-1)^2 +3(a-1) - 2$ I got the equation $9(a-1)^2 +3(a-1) - 2$ on my homework sheet. I tried to factor it by making $(a-1)=a$ and then factoring as a messy trinomial. But even so, I couldn't seem to get the correct answer; they all seemed incorrect.
Any help would be greatly appreciated.
Thank you so much in... | $x=a-1\\
9x^2+3x-2=0\\
\Delta=9+72=81\\
\sqrt{\Delta}=9\\
x=\frac{-3 \pm 9}{18} = \pm \frac{1}{2}-\frac{1}{6}\\
a=x+1=\pm \frac{1}{2}-\frac{1}{6}+1 = \pm \frac{1}{2}+\frac{5}{6}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why does this method for finding cube roots work? While on the Internet I came across a formula for cube root using recursion. The formula was:
$$ use \space x_1 = \frac{a}{3} \space (a \space is \space the \space number \space we \space want \space to \space find \space cube \space root \space of)$$
Then use the form... | If you want to solve for $x$, consider $$f(x)=x^3-a\implies f'(x)=3x^2$$ Now, using Newton method $$x_{n+1}=x_ n-\frac{f(x_n)}{f'(x_n)}=x_n-\frac{x_n^3-a}{3x_n^2}=\frac{2x_n^3+a}{3x_n^2}=\frac 23x_ n+\frac a {3x_n^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2147244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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A matrix is given in echelon form This is my first time posting a question and I'm really stuck on this one.
Help would be appreciated.
I'm given this matrix
$$\left[
\begin{array}{ccc|c}
2&3&0&9\\0&1&\lambda+6&4\\
0&0&\lambda^2-5\lambda+6&9-3\lambda
\end{array}\right],
$$
and I need to find for which λ ∈ ℝ this mat... | Use the determinant formula for the first three columns of the matrix (that compose an upper triangular matrix, so you just need to get the product of the diagonal elements).
There is one solution if and only if the determinant you got is not equal to zero (Determinant theorem).
Then, check the values for which the det... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Adjoint Norms in Banach Space If $T:X\to Y$ is a bounded linear transformation of Banach spaces $X$ and $Y$, then there is an adjoint transformation $Y^*\to X^*$ that satifies $<Tx,y^*> =<x,T^{*}y*>$ for all $x\in X$ and $y^*\in Y^*.$
One standard result is that $\left \| T \right \|=\left \| T^{*} \right \|.$ I have s... | If we have a double-indexed family $\{ u_{\alpha\beta} : \alpha \in A, \beta \in B\}$, then we have
$$\sup \: \{ u_{\alpha\beta} : \alpha \in A, \beta \in B\} = \sup_{\beta\in B} \sup\: \{u_{\alpha\beta} : \alpha \in A\} = \sup_{\alpha\in A} \sup\: \{ u_{\alpha\beta} : \beta \in B\}.$$
Clearly, since $C\subset D \impli... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can we show that $\sum_{n=0}^{\infty}{2n^2-n+1\over 4n^2-1}\cdot{1\over n!}=0?$ Consider
$$\sum_{n=0}^{\infty}{2n^2-n+1\over 4n^2-1}\cdot{1\over n!}=S\tag1$$
How does one show that $S=\color{red}0?$
An attempt:
$${2n^2-n+1\over 4n^2-1}={1\over 2}+{3-2n\over 2(4n^2-1)}={1\over 2}+{1\over 2(2n-1)}-{1\over (2n+1)}... | Hint:
\begin{eqnarray}
&&\sum_{n=0}^{\infty}{2n^2-n+1\over 4n^2-1}\cdot{1\over n!}\\
&=&\sum_{n=0}^{\infty}{(2n^2+n)-(2n-1)\over 4n^2-1}\cdot{1\over n!}\\
&=&\sum_{n=0}^{\infty}{2n^2+n\over 4n^2-1}\cdot{1\over n!}-\sum_{n=0}^{\infty}{2n-1\over 4n^2-1}\cdot{1\over n!}\\
&=&\sum_{n=1}^{\infty}{1\over 2n-1}\cdot{1\over (n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2147519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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About solvable groups Let $L$ be a finite Galois group that's solvable. So by definition there exists a chain of normal subgroups of $L$ such that $1 = G_{0} \trianglelefteq G_{1} \trianglelefteq \dots \trianglelefteq G_{n}=L $ where all the $G_{i+1}/G_{i}$ are abelian.
Now, it's said that one can assume the $G_{i+1}/G... | Take a finite group $G$ and a normal subgroup $H_0\subseteq G$. Then there is a one-to-one correspondence between normal subgroups of $G/H_0$ and normal subgroups of $G$ that contain $H_0$.
If $G/H_0$ is abelian, then by the structure theorem of finite abelian groups, we have that $G/H_0$ is isomorphic to a finite prod... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Let $\mathcal{D}$ be an additive category, and $f$ a morphism that is both monic and epic. Must $f$ be an isomorphism? Basic theory of Abelian categories tells us that this is true if $\mathcal{D}$ is abelian. However, is this still true if we don't necessarily have kernels or cokernels?
Tag 05R4 in the stacks project... | Nope. Topological abelian groups form an additive category, as can be seen directly or as a property of the category of abelian group objects in any category with finite products-a biproduct of $x,y$ in any category enriched over abelian groups is given by an object $z$ with morphisms $i_1,i_2:x,y\to z,p_1,p_2:z\to x,y... | {
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Prove using mathematical induction: for $n \ge 1, 5^{2n} - 4^{2n}$ is divisible by $9$ I have to prove the following statement using mathematical induction.
For all integers, $n \ge 1, 5^{2n} - 4^{2n}$ is divisible by 9.
I got the base case which is if $n = 1$ and when you plug it in to the equation above you get 9 an... | You're very close. Now add and subtract $4^{2k}$ in the first term to obtain
$$ 5^{2k}\cdot 25-4^{2k}\cdot 16=25\cdot (5^{2k}-4^{2k})+(25-16)\cdot 4^{2k}=25\cdot (5^{2k}-4^{2k})+9\cdot 4^{2k} $$
The first term is divisible by $9$ by the induction hypothesis, hence the whole expression is divisible by $9$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2147877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Derivative of a large product I need help computing
$$
\frac{d}{dx}\prod_{n=1}^{2014}\left(x+\frac{1}{n}\right)\biggr\rvert_{x=0}
$$
The answer provided is $\frac{2015}{2\cdot 2013!}$, however, I do not know how to arrive at this answer. Does anyone have any suggestions?
| This big product is hard to work with, but we can turn it into an easier-to-work-with summation by taking advantage of logarithms. Let $\displaystyle f(x) = \prod_{n=1}^{2014} \left(x + \frac{1}{n} \right)$.
Chain rule gives $\displaystyle \Big(\ln(f(x)) \Big)' = \frac{f'(x)}{f(x)}$, which means $f'(x) = f(x) \Big( ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2147977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 8,
"answer_id": 3
} |
Why is the set $K$ recursively enumerable? I am trying to understand why the set $K=\{i \mid M_i(i) \;\mathsf{halts}\}$ is recursively enumerable, where $M_i(i)$ is a Turing machine that is given its own index (in the standard enumeration of Turing machines) as input ("the halting problem").
If we cannot determine if $... | Just because you have a machine that outputs the elements of $K$ doesn't mean they "are all known".
The problem is that $M_K$ does not necessarily output the elements in order, so just observing the run of that machine will never allow you to conclude that some number is not in $K$. You can fun the machine for a year a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2148115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Distribution of the Maximum of a (infinite) Random Walk Let $S_0 = 0$ and define $S_n = \sum^n_{i = 1} X_i$ such that
\begin{align*}
\mathbb P(X_i = 1) &= p \\
\mathbb P(X_i = -1) &= 1 - p = q
\end{align*}
for $p < \frac{1}{2}$. Find the distribution of $Y = \max \{S_0, S_1, S_2, ...\}$.
My attempt at a solution: One k... | Define a function $f(n):= \big( \frac{1-p}{p} \big)^n$. Then $f$ is a harmonic function for this random walk. In other words, $f(n) = pf(n+1)+(1-p)f(n-1)$. Therefore $Y_n:=f(S_n)$ is a martingale.
For the rest of the problem, fix an integer $N \geq 0$. We will use the martingale property to compute the probability $P(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2148246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Integral of product of CDF and PDF of a random variable For a continuous real random variable $X$ with CDF $F_X(x)$ and PDF $f_X(x)$ I want to prove the following
$$\int\limits_{-\infty}^{\infty}x(2F_x(x)-1)f_x(x)dx\geq 0$$
I was thinking of integration by parts but $x$ complicates it. Any hints?
| Let $X$ and $Y$ denote i.i.d. random variables whose cdf are given by $F$.
Then $P(\max{\{X,Y\}}\leq x) = P(X\leq x, Y \leq x) = P(X \leq x)P(Y \leq x) = F(x)^2$.
Therefore the cdf of $\max\{X,Y\}$ is given by $\frac{d}{dx}F(x)^2 = 2F(x)f(x)$.
We clearly have that $X \leq \max\{X,Y\}$, so that $E[X] \leq E[\max\{X,Y\}]... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2148424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
compute genus of sphere and torus I am learning differential geometry by programming them and seeing their shapes. But topology is absolutely mysterious to me. For example, a sphere
$$
x^2+y^2+z^2=r^2
$$
has genus 0 (no holes).
and a torus
$$
\left(R- \sqrt{x^2+y^2} \right)^2+z^2=r^2
$$
has genus 1 (with one hole).
B... | Any compact Riemann surface $R$ is homeomorphic to a sphere with handles. The number $g$ of handles is called the genus of $R$. With this standard definition we see that the first example, the sphere without handles, has genus zero, whereas the torus can be deformed (the hole becomes a handle) to a sphere with $1$ han... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2148541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Inequality of arithmetic and geometric mean I did a proof for inequality below, anyone has a other proof?
Let $a$ and $b$ be positive real numbers, and $t$ the parameter. Prove that:
$$a+b\geq 2\sqrt{1-t^2}\sqrt{ab}+(a-b)t$$
| Let $f (t)=Rhs $ of inequality . So differentiating and setting it equal to $0$ gives $\frac {2t}{\sqrt {1-t^2}}=\frac {a-b}{\sqrt {ab}} $ squaring both sides and solving we have $t=\frac {a-b}{a+b} $ thus putting the value of $t $ in original equation and simplifying we get it as $a^2+b^2$ now both $a,b $ are positive... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2148700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove $|\det(J_{\phi}(a))|=k$ where $v(\phi (A))=kv(A)$ and $A$ is a Jordan Measurable set.
$(i)$ Let $\phi: \mathbb R^n \to \mathbb R^n$ be continuously differentiable homeomorphism such that $J_{\phi}(a)$ is invertible matrix for every $a \in \mathbb R^n$.
Suppose there is a constant $k>0$ such that for every Jordan... | Surely you know:
Lemma: If $f,g$ are integrable and $\int_{A}f=\int_{A}g$ for every measurable set $A$, then $f=g$ almost everywere.
Then,
by the change of variables theorem:
$$ \int_{A}|det(J_{\phi}(x)|dx =\int_{\phi(A)}dx=v(\phi(A))=kv(A)=\int_{A}kdx$$
for every (Jordan)measurable set $A$. Then, $|det(J_{\phi}(x)|=k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2148814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Is it logical to attempt differentiation of y=1? Is y=1 different from y=x^0? I am wary of getting in over my head, after initial searching it is apparent to me that I don't know how to properly phrase this question for the answer I want. I am only working at high school level but in class we learnt differentiation and... | The other answers are correct, but they miss the bigger error you made: you write
since $y=1^2$, we have ${dy\over dx}=2\cdot 1^{2-1}$.
But this is misapplying the power rule! Remember that this rule says $${d\over dx}(x^n)=nx^{n-1},$$ but - in the highlighted step above - you've conflated $x$ and $1$! (Note that the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2148881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
How to perform this manipulation? (1) $ z^2y+xy^2+x^2z-(x^2y+xz^2+y^2z) $
(2) $ (x-y)(y-z)(z-x) $
How to go from STEP (1) to STEP (2). Nothing I do seems to work. I tried combining terms but that doesn't help.
I do not want to go from step 2 to step 1. I arrived at step 1 in some question and I need to g... | Another method :
Rewrite (1) as
$$-yx^2 + x^2 z + x y^2 - x z^2 - y^2 z + y z^2 + \underline{ xyz - xyz}$$
Group the terms as follows :
$$(\underline{xyz - xz^2 - y^2z + yz^2} ) - ( \underline{ x^2y-x^2z-xy^2+xyz})$$
Factor out $z$ from first underlined expression and $x$ from the second :
$$ = z(xy - xz -y^2 + yz) -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2148970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Computing $7^{13} \mod 40$ I wanted to compute $7^{13} \mod 40$. I showed that
$$7^{13} \equiv 2^{13} \equiv 2 \mod 5$$
and
$$7^{13} \equiv (-1)^{13} \equiv -1 \mod 8$$.
Therefore, I have that $7^{13} - 2$ is a multiple of $5$, whereas $7^{13} +1$ is a multiple of $8$. I wanted to make both equal, so I solved $-2 + 5k... | You don't necessarily have to use the fact $40=8\times 5$ (but if you do, look up "Chinese remainder theorem".)
Otherwise, you know that
$$7^2\equiv 49\equiv 9\pmod{40}.$$
So
$$7^3\equiv 63\equiv 23\pmod{40},$$
so
$$7^4\equiv 161\equiv 1\pmod{40}.$$
Then
$$7^{13}\equiv 7^{4\times 3}\times 7\equiv 1\times 7\equiv 7\pm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2149062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 1
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How does one show that $\sum_{k=1}^{n}\sin\left({\pi\over 2}\cdot{4k-1\over 2n+1}\right)={1\over 2\sin\left({\pi\over 2}\cdot{1\over 2n+1}\right)}?$ Consider
$$\sum_{k=1}^{n}\sin\left({\pi\over 2}\cdot{4k-1\over 2n+1}\right)=S\tag1$$
How does one show that $$S={1\over 2\sin\left({\pi\over 2}\cdot{1\over 2n+1}\right... | Following the idea suggested to pass the problem to the complex:
$$\sum_{k=1}^{n}\sin\left({\pi\over 2}\cdot{4k-1\over 2n+1}\right)=\Im\left(\sum_{k=1}^{n}\exp\left(i{\pi\over 2}\cdot{4k-1\over 2n+1}\right)\right)$$
Now, let us manipulate the exponentials directly.
$$\exp\left(i{\pi\over 2}\cdot{4k-1\over 2n+1}\right)=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2149149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How to find the maximum and minimum value of $\left|(z_1-z_2)^2 + (z_2-z_3)^2 + (z_3-z_1)^2\right|$ (where $|z_1|=|z_2|=|z_3|=1$)? How to find the maximum and minimum value of $\left|(z_1-z_2)^2 + (z_2-z_3)^2 + (z_3-z_1)^2\right|$ (where $|z_1|=|z_2|=|z_3|=1$ are complex numbers.) ?
My try:
$$\begin{align}\left|(z_1-z_... | Given
$$(z_1-z_2)+(z_2-z_3)+(z_3-z_1)=0$$
and
$$\left|(z_1-z_2)^2+(z_2-z_3)^2+(z_3-z_1)^2\right|=\\
\left|z_1^2-2z_1z_2+z_2^2+z_2^2-2z_2z_3+z_3^2+z_3^2-2z_3z_1+z_1^2\right|=\\
2\left|z_1^2+z_2^2+z_3^2-z_1z_2-z_2z_3-z_3z_1\right|=\\
2\left|z_1(z_1-z_2)+z_2(z_2-z_3)+z_3(z_3-z_1)\right|=\\
2\left|z_1(z_1-z_2)+z_2(z_2-z_3)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2149243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 1
} |
Can a double series of $\ln(r)^m r^k$ converge to $r^\alpha$? Consider a series like
$$\sum_{k=0}^\infty\sum_{m=0}^\infty c_{k,m}\ln(r)^m r^k.$$
Can there be such a non-integer $\alpha$ and a set of $c_{k,m}$ that this series would converge to $r^\alpha$ in some neighborhood of $r=0$, and that partial sums would approx... | Both $\ln(r)$ and $r^\alpha$ are multi-valued functions in the complex plane, but I'm assuming we're taking compatible branches so that
$r^\alpha = \exp(\alpha \ln(r))$ for $r \ne 0$. $r=0$ is a problem: see below.
If you didn't want to include $r=0$, just take $c_{km} = 0$ for $k > 0$ and
$c_{0,m} = \alpha^m/m!$. T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2149344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Other Idea to show an inequality $\dfrac{1}{\sqrt 1}+\dfrac{1}{\sqrt 2}+\dfrac{1}{\sqrt 3}+\cdots+\dfrac{1}{\sqrt n}\geq \sqrt n$ $$\dfrac{1}{\sqrt 1}+\dfrac{1}{\sqrt 2}+\dfrac{1}{\sqrt 3}+\cdots+\dfrac{1}{\sqrt n}\geq \sqrt n$$
I want to prove this by Induction
$$n=1 \checkmark\\
n=k \to \dfrac{1}{\sqrt 1}+\dfrac{1}{... | Integrals:
$$\sum_{k=1}^n\frac1{\sqrt n}\ge\int_1^{n+1}\frac1{\sqrt x}\ dx=2\sqrt{n+1}-2$$
And it's very easy to check that
$$2\sqrt{n+1}-2\ge\sqrt n$$
for $n\ge2$.
A visuallization of this argument:
From the red lines down, that area represents a sum. From the blue line down, that represents an integral. Clearly, t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2149448",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 4
} |
Does an eigenvalue that does NOT have multiplicity usually have a one-dimensional corresponding eigenspace? I'm trying to understand this statement in my book:
In general, the multiplicity of an eigenvalue is greater than or equal to the dimension of its eigenspace.
So then, if an eigenvalue does NOT occur as a multi... | The eigenspace of a particular eigenvalue is guaranteed to have dimension at least $1$. This is because in finding the eigenvalues of a matrix $A$, we require $\det(A-\lambda I)=0$, which guarantees that the system $(A-\lambda I)\mathbf{x}=\mathbf{0}$ will have a nontrivial solution. So there will be at least one eigen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2149568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Avoiding Matrix Inversion I have as input values the matrices $A,B\in\mathbb{R}^{n\times n}$, where $B$ is invertible, vector $\vec{b}\in\mathbb{R}^n$, and $\alpha\in\mathbb{R}$. Denoting the identity matrix by $I_d$, I am computing the value of
$$(\alpha A+I_d)(A+B^{-1})\vec{b}.$$
However, as computing the value of $... | If you want to avoid computing $B^{-1}$, you can just compute $B^{-1}\vec b$ separately:
*
*Compute $B^{-1}\vec b$ by solving system of linear equations given by matrix $B$ with RHS $\vec b$.
*Compute $A\vec b$.
*Add 1.+2. You get $(A+B^{-1})\vec b$.
*Multiply 3. by $(\alpha A+I_d)$. You get $(\alpha A+I_d)(A+B^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2149656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Rational Inequalities Error I'm slowly working my way through Serge Lang's Basic Mathematics and I'm having difficulty with this solution to a basic inequality:
$\qquad(1)$
$${-2x+5\over x+3}<1
$$
I started by making two assumptions about the denominator, which must be either greater than or less than zero. Hopefully o... | You went from $$-2x+5 > x+3$$ to $$-3x > -3$$ which is an arithmetic error.
Let's suppose for the moment that $3-5 = -3$ and examine what you did. You showed the following two facts:
*
*if $x < -3$ and $\frac{-2x+5}{x+3} < 1$ then $x<1$
*if $x > -3$ and $\frac{-2x+5}{x+3} < 1$ then $x > 1$.
So you have shown tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2149767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Regular expression - write language in MSO
$$(ab^*a)^*$$ (a) Define this language in MSO
(b) Decide if this language is definable in first order logic
(b) It seems to be simple. We know that we can't distinguish in $n$ rounds (wit $n$ quantifiers) two linear order with length $2^{n}+1$ and $2^{n}+2$. These two orde... | Hint. Guess a subset of the $\mathtt a$s that you pretend are $\mathtt c$s instead, and recognize $(\mathtt a \mathtt b^* \mathtt c)^*$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2149933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Is all normed space also inner product space? 1) I know that all inner product space is also a normed space with the norm induce by the scalar product, but is the reciprocal true ? I mean, is all normed space also a inner product space ?
2) I know that all normed space is a metric space with the metric induced by the n... | A norm is induced b yan inner product iff it satisfies the paralellogram law
$$2||u||^2 + 2||v||^2 = ||u -v||^2 + ||u+v||^2$$
And e.g. the supremum norm on $\mathbb{R}^2$ already fails this.
Even if we have a metric topological vector space with a translation invariant metric $(V,d)$ which is moreover complete (a so-ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2150044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
How do I find the derivative of the function $f(u)=5\sqrt{u}$ Find the derivative of $f(u)=5\sqrt{u}$.
This just drives me crazy. I am able to solve this problem with some hand-waving, in this case - standard methods like power rules and so on. Piece of cake. Problem is, I want to solve this equation with non-standard ... | I'm not sure about how to prove this using non-standard analysis but I think I know a good way to start, with a standard trick to deal with the difference of square roots (leaving out the $5$):
$$
\begin{align}
\Delta y & = (\sqrt{u + \Delta u} - \sqrt{u}) \times
\frac{ \sqrt{u + \Delta u} + \sqrt{u} }{ \sqrt{u +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2150142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to do this step quickly in Chinese remainder theorem I have
$ \begin{cases} x \equiv 2 \pmod 3 \\ x \equiv 4 \pmod 7 \\ x \equiv 5 \pmod8 \end{cases} $
and I don't know how to do this quickly in this step:
$56x_1 \equiv 1 \pmod 3 $ implies $x_1 = 2$
The question is, how to find $x_1, x_2, x_3$ fast? In case $x_1$:... | One thing to speed things up would be that it seems like you aren't taking advantage of the fact that you can reduce the number $56$ modulo $3$ without affecting anything.
$$56x\equiv1\bmod 3 \quad\leadsto\quad 2x\equiv 1\bmod 3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2150248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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How to show $(a + b)^n \leq a^n + b^n$, where $a, b \geq 0$ and $n \in (0, 1]$? Does anyone happen to know a nice way to show that $(a+b)^n \le a^n+b^n$, where $a,b\geq 0$ and $n \in (0,1]$? I figured integrating might help, but I've been unable to pull my argument full circle. Any suggestions are appreciated :)
| Assume that
$a \ge b$.
If $a = b = 0$,
the result is immediate.
If $a > 0$,
divide
$(a+b)^n \le a^n+b^n
$
by
$a^n$
to get
$(1+b/a)^n \le 1+(b/a)^n
$.
Since
$b \le a$,
$0 \le b/a \le 1$,
so this becomes
$(1+x)^n \le 1+x^n
$
where
$x = b/a$.
Let $f(x)
=1+x^n-(1+x)^n
$.
$f(0) = 0$
and
$f(1)
=2-2^n
\ge 0
$
since
$0 < n \le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2150372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
Prove that $\lim_{x\to 0}\frac{e^x-1}{x}=1$. I need help understanding this proof:
Prove that $$\lim_{x\to 0}\frac{e^x-1}{x}=1.$$
For $x>0$ and $n\in\Bbb N$: $$1\leq\frac{(1+\frac {x}{n})^n -1}{x}=\frac{1}{n}[(1+\frac{x}{n})^{n-1}+...+1]\leq(1+\frac{x}{n})^{n-1}$$
For $n\rightarrow \infty$ we have
$$1\leq\frac{f(x)-1}... | HINT: $$\lim_{n \to 0} \left(1 + \frac{x}{n}\right)^n = e^x$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2150480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Recursive sequence convergence.: $s_{n+1}=\frac{1}{2} (s_n+s_{n-1})$ for $n\geq 2$, where $s_1>s_2>0$ The problem is the following: suppose $s_1>s_2>0$ and let $s_{n+1}=\frac{1}{2} (s_n+s_{n-1})$ for $n\geq 2$. Show that ($s_n$) converges.
Now, here is what I figured out:
*
*$s_2<s_4$: Base Case for induction that $... | Consider the characteristic equation of your recurrence:
$X^2-\frac1{2}X-\frac1{2}=0$
Which has solutions $1$ and $-\frac1{2}$.
Therefore the general expression for $s_n$ is:
$s_n=a\cdot (1)^n+b\cdot(-\frac1{2})^n=a+b\cdot(-\frac1{2})^n$
You can determine the value of the constants $a$ and $b$ using
$s_1$ and $s_2$.
Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2150607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Sum of powers divisible by 53
Show that $1^{26} + 2^{26} + 3^{26} + \cdots + 26^{26} $ is divisible by $53$.
Inspired by another more basic question asked here... I'm interested to see what elegant proofs you come up with (and mine is also given below in an answer).
| Let $ S $ be the given sum, and fix $ \alpha \in \mathbb F_{53}^{\times} $ such that $ \alpha $ is not a root of $ X^{26} - 1 $. Then, (the following equalities are in $ \mathbb F_{53} $)
$$ 2S = \sum_{k=1}^{52} k^{26} = \sum_{k=1}^{52} (\alpha k)^{26} = \alpha^{26} \cdot 2S $$
(Note that $ x \to \alpha x $ is a permu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2150699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Can I use a binomial distribution for this exercise? A work team is made up of five Computer Engineers and nine Computer Technologists. If five team members are randomly selected and assigned a project, what is the probability that the project team will include exactly three Technologists?
| Not binomial. No, you cannot use a binomial distribution because sampling is without
replacement. (No team member can be chosen to fill two places doing the project.)
If a Technologist is chosen first (probability 9/14), then the probability of choosing another
Technologist on the next draw is different (probability 8/... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2150791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
proof $|\sinh(x)|\leq|3x|$ for $ -\frac{1}{2}I am supposed to prove that $|\sinh(x)|\leq3|x|$ for $|x|<\frac{1}{2}$.
I know I am supposed to use $|e^x-1|\leq3|x|$ for $|x|<\frac{1}{2}$.
I am completely stuck, and I don't know how to approach this, so any help is greatly appreciated!
| In THIS ANSWER, I showed using only the limit definition of the exponential function and Bernoulli's Inequality that the exponential function satisfies the inequalities
$$\bbox[5px,border:2px solid #C0A000]{1+x\le e^x\le\frac{1}{1-x}} \tag 1$$
for $x<1$.
Let $f(x) = 6x-e^x+e^{-x}$. Then, applying $(1)$ to $f(x)$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2150905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find the coefficient of $x^{29}$ in the given polynomial. The polynomial is :
$$
\left(x-\frac{1}{1\cdot3}\right) \left(x-\frac{2}{1\cdot3\cdot5}\right) \left(x-\frac{3}{1\cdot3\cdot5\cdot7}\right) \cdots \left(x-\frac{30}{1\cdot3\cdot5\cdots61}\right)
$$
What I've done so far : The given polynomial is an expression of... | Let
$p_m(x)
=\prod_{k=1}^{m} (x-\dfrac{k}{\prod_{j=1}^{k+1}(2j-1)})
$
Since,
in the usual way,
$\begin{array}\\
\prod_{j=1}^{k+1}(2j-1)
&=\dfrac{\prod_{j=1}^{2k+1}j}{\prod_{j=1}^k(2j)}\\
&=\dfrac{(2k+1)!}{2^kk!}\\
\end{array}
$
$p_m(x)
=\prod_{k=1}^{m} (x-\dfrac{k2^kk!}{(2k+1)!})
$
The coefficient of $x^{m-1}$
is $(-1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2151030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
How can I prove if $R$ is a domain where every submodule is a summand, then it is a field? Suppose $R$ is a domain with the property that, for $R$-modules, every
submodule is a summand.
I would like to show $R$ is a field.
Stating the definitions I know that for any submodule $A$ there exsists a summand $B$ where $A \... | Let $I$ be an ideal of $R$. It is sufficient to show that either $I = 0$ or $I = R$.
Let us assume $I \neq 0$ and try to get $I = R$. From the assumption submodule (ideal) $I$ of $R$ has a complement $J$:
$$ I + J = R, \quad I \cap J = 0.$$
If $J \neq 0$ then, since $R$ is a domain, we have
$$ 0 \neq IJ \subseteq I \c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2151162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Calculate Inverse Laplace Transform I need to get the Inverse Laplace Transform badly for the following function
$$\frac{1}{\beta + \sqrt{p}} e^{-\alpha \sqrt{p + \gamma}},$$
$\alpha,\, \beta,\, \gamma$ being some parameters.
I have looked through Erdelyi's book of tables and found only the expression for
$$\mathcal{L}... | Using the convolution theorem:
$$\text{f}\left(t\right)=\mathscr{L}_\text{s}^{-1}\left[\frac{\exp\left(-\alpha\cdot\sqrt{\gamma+\text{s}}\right)}{\beta+\sqrt{\text{s}}}\right]_{\left(t\right)}=\mathscr{L}_\text{s}^{-1}\left[\frac{1}{\beta+\sqrt{\text{s}}}\right]_{\left(t\right)}\space*\space\mathscr{L}_\text{s}^{-1}\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2151280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
A formula for $\sin(\pi/2^n)$ May be this a duplicate, but I did not find any question related.
I found the following formula, but there was no proof of it:
$$2\sin\left(\frac{\pi}{2^{n+1}}\right)=\sqrt{2_1-\sqrt{2_2+\sqrt{2_3+\sqrt{2_4+\cdots\sqrt{2_n}}}}}$$
where
$$2_k=\underbrace{222\cdots222}_{k\text { times}}.$$
(... | Here's my repeated half-angle approach (I know, this is definitely not a great way to deal with it, but still am posting it here. This is my first answer, here in this website, so please bear with me..):
We know
$2\cos^2 \theta =1+\cos 2\theta\implies \cos \theta =\sqrt{\frac{1+\cos 2\theta}{2}}.$
Taking positive sign ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2151405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
show that $\cosh x \geq1$ Can someone help me show that $\cosh \geq1$.
I know that $\cosh x = \frac{1}{2}(e^x+s^{-x})$
I think I'm supposed to use the identity: $\cosh^2x - \sinh^2x=1$
| Note $$\cosh^2 x =1+\sinh^2 x \ge 1$$
So $\cosh^2 x \ge 1$. Since $\cosh x$ is positive, we have $$\cosh x \ge 1$$
Done! You could also say $$\cosh x =\frac{e^x+e^{-x}}{2} \ge \sqrt{e^{x} \times e^{-x}}=1$$
From $\text{AM-GM}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2151551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What's the relationship between geometric multiplicity of eigenvalues and dim ker(T)? Let $T: \mathbb{R}^7 \rightarrow \mathbb{R}^7$ be a diagonalizable linear operator with characteristic polynomial give by $p(t) = t(t-1)^2(t+2)^3(t -3)$.
Calculate $\dim(\ker(T - Id), \dim(Im(T + 2Id)), \dim(Im(T))$
I'm thinking whats... | Since $T$ is diagonalizable, it has an eigenbasis (in other words, you can find a linearly independent collection of eigenvectors which span $\mathbb{R}^7$). Let $p(t)$ be the characteristic polynomial of $T$ (or the matrix associated to $T$). Then,
*
*$\ker(T-\lambda I)$ is a linear space of dimension equal to t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2151676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Find a function $f(x)$ so that $f(x)=f'(x)$ I know the answer is $e^x$, but supposing I have no idea, what would be the steps required to get to that answer? Note that starting to test different functions is not a valid answer here as there are infinitely many.
I started by doing
$$
f(x)=\frac{d}{dx}f(x)
$$
so
$$
\in... | You can solve this using differential equations.
You have:
$$\frac{df}{dx}=f$$
This is a separable ODE, so if $f\neq 0$:
$$\int \frac{1}{f}~df=\int dx$$
Integrate both sides, and you should get the set of solutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2151763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Distributing pebbles The rules to this "game" are simple, but after checking 120 starting positions, i can still not find a single pattern that consistantly holds. I am grateful for the smallest of suggestions.
Rules:
You have two bowls with pebbles in each of them.
Pick up the pebbles from one bowl and distribute the... | By the result of Simon, $T^k(n,n)=(1,2n−1)$ for some $k∈N$ if and only if $(1,2n−1)$ is in the orbit of $(n,n)$.
$(1,2n−1)$ is in the orbit of $(n,n)$ if and only if $(n,n)$ is in the orbit of $(1,2n−1)$.
Indeed, $$T^2(1,2n-1)=T(2n,0)=(n,n)$$ and it is obvious that there exist $a$ such that $T^a(1,2n-1)=(1,2n-1)$. $T^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2151868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 1
} |
Prove that $\tan^{-1}{\frac{1}{5}} \approx \frac{\pi}{16}$ using complex number method
Prove that $\tan^{-1}{\frac{1}{5}} \approx \frac{\pi}{16}$ using
complex number method.
Hint: Take $z=5+i$
What is meant by complex number method in this problem? I couldn't think of do the above proof using complex numbers. Any ... | $(5+i)^4 = 476 + 480i $ Since $476$ is approximately $480$, the angle of the line from $0$ to $476 + 480i$ made with the x-axis is about $45$ degrees or $\pi/4$ radians.
So $\arctan (1/5) $ is about $\pi/4 \over 4$ or $\pi/16$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2152022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Ring which satisfies that if $S$ is subring of $R$ then $R^\times\cap S\ne S^\times$ Let $R$ be a unital ring and $S\le R$ be a unital subring with $1_S = 1_R$.
Then it can be shown that $S^\times\subset R^\times\cap S$, where $R^\times$ and $S^\times$ are the sets of units of respective rings. (Statement *)
There can... | For (1), consider the (unital) ring $\mathbb{R}^2$ where addition and multiplication are calculated coordinate-wise, i.e.
$$(a,b)+(c,d) = (a+c,b+d)$$
$$(a,b)\cdot (c,d) = (a\cdot c,b\cdot d)$$
Then the subset $\mathbb{R}\times \{0\} = \{(a,0) \mid a\in\mathbb{R}\}$. In this case, $\mathbb{R}\times \{0\}$ is a subring ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2152273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
A good example of set A such that $(A')\cap (A')'=\emptyset$ in $\mathbb{R}^2$ I tried to show that $A=\{1/n:n\in\mathbb{N}\}$ is such that $A'=\{0\}$ but $(A')'=\emptyset$ but I can't imagine a set in $\mathbb{R}^2$ with the same property and obviusly non trivial, like $A=\{(0,1/n):n\in\mathbb{N}\}$ or similary, Can y... | The only way this is possible is if $A^{\prime\prime}$ is empty, since $A^{\prime\prime} \subseteq A^\prime$ for any $A \subseteq \mathbb R^2$. Below let $d$ denote the usual Euclidean metric on $\mathbb R^2$.
*
*Suppose that $x \in A^{\prime\prime}$, and let $\varepsilon > 0$. Then there is a $y \in A^\prime$ with ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2152381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Show there exists $v\in V$ such that $v\not\in V_{i}$ for any $1\leq i\leq k$
Show that if $V$ is a nonzero vector space over $\mathbb{R}$, and $V_{1}, V_{2},\ldots V_{k}$ are proper subspaces of $V$ then there exists $v\in V$ such that $v\not\in V_{i}$ for any $1\leq i\leq k$
I can prove the case for $k=2$ but I can... | A proof using some topology in the finite dimensional case (in the general case and using only linear algebra, see learnmore's answer) :
If each of the $V_i$ is proper then they're all closed sets with empty interior, which, according to the Baire Category theorem implies that their union also has empty interior : the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2152464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
What is the probability that a string of length $L$ occurs at least twice? Suppose, $N$ random digits $$x_1,x_2,\cdots ,x_N$$ with $0\le x_j\le 9$ for $j=1,2,\cdots , N$ and a positibe integer $L$ is given.
What is the probability that a block of $L$ consecutive digits appears at least twice in the above sequence ?
T... | Given a string of length $N$, and a positive integer $L$, there can be $N-L+1$ blocks possible. Also, given $L$, the fraction of strings of length $L$ which have consecutive numbers equals $p = \frac{\begin{pmatrix}10 \\ 1\end{pmatrix}}{10^L}$ (choosing the first digit and then fixing next $L-1$ consecutive numbers). T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2152651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What does "open set" mean in the concept of a topology? Given the following definition of topology, I am confused about the concept of "open sets".
2.2 Topological Space. We use some of the properties of open sets in the case of metric spaces in order to define what is meant in general by a class of open sets and by a... | In an abstract topological space, "open set" has no definition!
You simply decide (as part of making your topological space) which sets you want to call open -- those are the sets you put into $\mathcal T$. Whatever you decide to call open will be called open, as long as your decision meets the condition "$\emptyset, X... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2152735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 8,
"answer_id": 3
} |
3 plate dinner problem Consider n people dining in a circular table. Each of them is ordering one of three plates. What is the probability that no two people sitting next to one another will order the same plate?
I intuitively think that every person except the first one has 2 choices as he cannot order the same as the... | You are considering problem when people are sitting in a line. This is a good idea, but it is not enough to compute probability that it is possible to make a circle from this line. When is it possible? If and only if the last of $n$ people has chosen plate different with previous one and the first one. So we need to co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2152866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
A group of order 30 has at most 7 subgroups of order 5
Show that a group of order 30 has at most 7 subgroups of order 5.
This should be a basic question (from an introductory algebra course), but I got no clue... Please help!
| It's going to have fewer than $7$, but proving that requires some tools it doesn't sound like you have.
Two distinct subgroups of order $5$ intersect only in the identity (by Lagrange's theorem). This means no two will share any nonidentity elements. Thus if there are $m$ subgroups of order $5$, you can calculate exac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2152948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Relation between implication and subset relation If $a \implies b$ can we then generally say that $a \subseteq b$ ?
For example:
if $a: x > 15$ and $b: x >10$ then clearly $a \implies b$ and if we look at the sets represented by a $\lbrace16,17,18..\rbrace$ and b $\lbrace 11,12,13... \rbrace$ it is also obvious that ... | No but you can say:
Let P be (x ∈ A)
Let Q be (x ∈ B)
A ⊆ B ≡ ∀x, P → Q
⊆ is for sets
→ is for propositions
Note: You cannot say the same for A ⊂ B
Why?
Let A be {1}
Let B be {1}
Let P be (x ∈ A)
Let Q be (x ∈ B)
A ⊂ B is false.
∀x, P → Q is true.
Therefore A ⊂ B cannot be equivalent to ∀x, P → Q
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2153046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Find the area of an infinitesimal elliptical ring. I have an ellipse given by,
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=c$$
and another ellipse that is infinitesimally bigger than the previous ellipse, i.e.
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=c+dc$$
I want to find the area enclosed by the ring from $x$ to $x+dx$ but I don't kno... | As the ellipse can be derived from a circle by means of a dilation along $y$ of ratio $b/a$,
we can find the desired area by considering a circular ring with radii $a\sqrt c$ and $a\sqrt{c+dc}$ and then multiplying the result by $b/a$. A $y$ section of such a circular ring at $x$ has a width
$$
dy=\sqrt{a^2(c+dc)-x^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2153179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Show that $|f(x)|$ is continuous at $a$
Suppose that $f(x)$ is continuous at $a$. Show that the function $|f(x)|$ is continuous at $a$.
Proof:
Since $f(x)$ is continuous at $a$ then, $$\lim_{x\to a}f(x)=f(a)$$
Show that $|f(x)|$ is continouos at $a$ $$\lim_{x\to a}|f(x)|=...$$
From here I can not figure a way to fini... | First, show that the function $g(x)=|x|$ is continuous on reals, then use the fact that the composition of two continuous functions is also continuous ($|f(x)|= g \circ f (x)$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2153334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
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