Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Representing the statement using Quantifiers I want to represent the statement "Some numbers are not real " using quantifiers. I have been told by my teacher that the correct way to represent this is
num(x) : x is a number
real(x) : x is real
∃x (num(x) ^ ¬real(x))
This made sense, i can translate this statement in... | Your version is wrong because "A and B" is not the same as "if A, then B".
For instance, there exists a horse H such that if H is 50 feet tall, then I win the lottery.
It is sadly not true that there exists a horse H such that H is 50 feet tall and I win the lottery.
More pointedly, (A implies B) is true when A is fals... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/369546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Does every non-trivial element of $\mathbb{Z}_p$ generate the group? This just popped up in my head and I just wanted to make sure if I'm right.
Every element (except the identity element $0$) of the group $\mathbb{Z}_p$ (under addition and $p$ is prime) is a generator for the group. For example, $\mathbb{Z}_5 = \langl... | You can show that $\mathbb{Z}_p$ is a field, so if $H= \langle h \rangle$ with $h \neq 0$, $1 \in H$ since $h$ is invertible; you deduce that $H= \mathbb{Z}_p$, ie. $h$ is a generator.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/369628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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What are zigzag theories, and why are they called that? I've encountered the term zigzag theory while randomly clicking my way through the internet. It is given here. I haven't been able to find a clear explanation of what constitutes a zigzag theory. Here, it is said that they have to do with non-Cantorian sets, which... | As a footnote to Arthur Fischer, here's an additional quote from Michael Potter's Set Theory and its Philosophy:
In 1906, Russell canvassed three forms a solution to the paradoxes might take:
the no-class theory, limitation of size, and the zigzag theory. It is striking that a century later all of the theories that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/369689",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Does a symmetric matrix with main diagonal zero is classified into a separate type of its own? And does it have a particular name? I have a symmetric matrix as shown below
$$\begin{pmatrix} 0&2&1&4&3 \\ 2&0&1&2&1 \\ 1&1&0&3&2 \\4&2&3&0&1 \\ 3&1&2&1&0\end{pmatrix}$$
Does this matrix belong to a particular type?
I am ... | This is a hollow matrix. You can say that the sum of its eigenvalues equals zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/369735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
What is the probability that a random $n\times n$ bipartite graph has an isolated vertex? By a random $n\times n$ bipartite graph, I mean a random bipartite graph on two vertex classes of size $n$, with the edges added independently, each with probability $p$.
I want to find the probability that such a graph contains... | This can be done using inclusion/exclusion. We have $n+n$ conditions for the individual vertices being isolated. There are $\binom nk\binom nl$ combinations of these conditions that require $k$ particular vertices in $X$ and $l$ particular vertices in $Y$ to be isolated, and the probability for this is $q^{kn+ln-kl}$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/369830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Prove By Mathematical Induction (factorial-to-the-fourth vs power of two) Prove $(n!)^{4}\le2^{n(n+1)}$ for $n = 0, 1, 2, 3,...$
Base Step: $(0!)^{4} = 1 \le 2^{0(0+1)} = 1$
IH: Assume that $(k!)^{4} \le 2^{k(k+1)}$ for some $k\in\mathbb N$.
Induction Step: Show $(k+1!)^{4} \le 2^{k+1((k+1)+1)}$
Proof: $(k+1!)^{4} = (... | You are doing well up to $(k+1)^4*(k!)^4 \le (k+1)^4*2^{k(k+1)}$ That is the proper use of the induction hypothesis. Now you need to argue $(k+1)^4 \le \frac {2^{(k+1)(k+2)}}{2^{k(k+1)}}=2^{(k+1)(k+2)-k(k+1)}=2^{2(k+1)}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/369882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Relation between Galois representation and rational $p$-torsion Let $E$ be an elliptic curve over $\mathbb{Q}$. Does the image of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ under the mod $p$ Galois representation tell us whether or not $E$ has rational $p$-torsion or not?
| Yes, it does, and in a rather straightforward way. The $p$-torsion in $E(\mathbb{Q})$ is precisely the fixed vectors under the Galois action. In particular, $E$ has full rational $p$-torsion if and only if the mod $p$ representation is trivial.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/369955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Lagrangian subspaces Let $\Lambda_{n}$ be the set of all Lagrangian subspaces of $C^{n}$, and $P\in \Lambda_{n}$. Put $U_{P} = \{Q\in \Lambda_{n} : Q\cap (iP)=0\}$. There is an assertion that the set $U_{P}$ is homeomorphic to the real vector space of all symmetric endomorphisms of $P$. And then in the proof of it ther... | Remember these are Lagrangians and thus half-dimensional. It's easiest to see what is going on if you take $P = \mathbb{R}^n$. This simplifies notation and also is somewhat easier to understand, imo.
We are given a Lagrangian subspace $Q$, transverse to $i \mathbb{R}^n$. Then, consider the linear map $Q \to \mathbb{R}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/370012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why isn't $\lim \limits_{x\to\infty}\left(1+\frac{1}{x}\right)^{x}$ equal to $1$? Given $\lim \limits_{x\to\infty}(1+\frac{1}{x})^{x}$, why can't you reduce it to $\lim \limits_{x\to\infty}(1+0)^{x}$, making the result "$1$"? Obviously, it's wrong, as the true value is $e$. Is it because the $\frac{1}{x}$ is still so... | In the expression
$$\left(1+\frac{1}{x}\right)^x,$$
the $1+1/x$ is always bigger than one. Furthermore, the exponent is going to $\infty$ and (I suppose) that any number larger than one raised to infinity should be infinity. Thus, you could just as easily ask, why isn't the limit infinity?
Of course, the reality is t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/370125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
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Minimal value of a polynomial I do not know the following statement is true or not:
Given $1<x_0<2$, there exists $\delta>0$ such that for any n, define $A=\{ f(x)=\sum\limits_{i=0}^{n}a_ix^i\}$ where $a_i\in\{0\,,1\}$, then for any $f\,,g \in A$ and their degrees are the same, we have $\delta\leq|f(x_0)-g(x_0))|$ or ... | Let $x_0$ be a real root of the polynomial $1-X^2-X^3+X^4$ in the interval $(1,2)$; it exists by the intermediate value theorem because this polynomial has value $0$ and derivative $-1$ at $X=1$, and value $5$ at $X=2$. Then with $f(x)=1+x^4$ and $g(x)=x^2+x^3$ one has $f(x_0)=g(x_0)$, so no positive $\delta\leq|f(x_0)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show $\det \left[T\right]_\beta=-1$, for any basis $\beta$ when $Tx=x-2(x,u)u$, $u$ unit vector Let $u$ be a unit vector in an $n$ dimensional inner product space $V$.
Define the orthogonal operator
$$
Tx= x - 2 (x,u)u,
$$
where $x \in V$. Show
$$
\det A = -1,
$$
whenever $A$ is a matrix representation of $T$.
I can sh... | To complement rschwieb's answer, he is giving you way to determine a specific basis to perform your calculation in easily. Then recall that if $A$ and $A'$ are two matrices that represent the same linear transformation in two different bases, then $A' = P^{-1} A P$ where $P$ is a change of basis matrix. In particular... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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In how many ways can five letters be posted in 4 boxes? Question : In how many ways can 5 letters be posted in 4 boxes?
Answer 1: We take a letter. It can be posted in any of the 4 boxes. Similarly, next letter can be posted in 4 ways, and so on. Total number = $4^5$.
Answer 2: Among the total number of positions of 5 ... | The way to understand this problem (and decide which answer is correct) is to ask what you are counting exactly. If all the letters are distinct then the first answer sounds better because it counts arrangements that only differ by where each particular letter goes.
The second answer appears to make the letters and t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/370439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 1
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Finding the Laplace Transform of sin(t)/t I'm in a Differential Equations class, and I'm having trouble solving a Laplace Transformation problem.
This is the problem:
Consider the function
$$f(t) = \{\begin{align}&\frac{\sin(t)}{t} \;\;\;\;\;\;\;\; t \neq 0\\& 1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; t = 0\end{align}$$
a) ... | Just an small hint:
Theorem: If $\mathcal{L}\{f(t)\}=F(s)$ and $\frac{f(t)}{t}$ has a laplace transform, then $$\mathcal{L}\left(\frac{f(t)}{t}\right)=\int_s^{\infty}F(u)du$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/370491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Characteristic Polynomial of a Linear Map I am hoping for some help with this question from a practice exam I am doing before a linear algebra final.
Let $T_1, T_2$ be the linear maps from $C^{\infty}(\mathbb{R})$ to $C^{\infty}(\mathbb{R})$ given by
$$T_1(f)=f'' - 3f' + 2f$$
$$T_2(f)=f''-f'-2f$$
(a) Write out the cha... | Here's how to solve the second boxed problem.
First, for every $v\in\beta$, write $T(v)$ in the basis $\beta$:
$$
\begin{align}
T(1) &= 2\cdot 1+0\cdot x+0\cdot x^2 \\
T(x) &= 3\cdot 1+3\cdot x+0\cdot x^2 \\
T(x^2) &= 0\cdot 1+6\cdot x+3\cdot x^2
\end{align}
$$
Now, the scalars appearing in these equations become the c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/370573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Integration function spherical coordinates, convolution How can I calculate the following integral explicitly:
$$\int_{R^3}\frac{f(x)}{|x-y|}dx$$
where $f$ is a function with spherical symmetry that is $f(x)=f(|x|)$?
I tried to use polar coordinates at $x=0$ but it didn't help. Any idea on how to do this? Do you think ... | This is a singular integral and so you can expect some weird behavior but if $f$ has spherical symmetry, then I would change to spherical coordinates. Then you'll have $f(x) = f(r)$. $dx$ will become $r^2\sin(\theta)drd\theta d\phi$. The tricky part is then what becomes of $|x-y|$. Recall that $|x-y| = \sqrt{(x-y)\cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/370648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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introductory reference for Hopf Fibrations I am looking for a good introductory treatment of Hopf Fibrations and I am wondering whether there is a popular, well regarded, accessible book. ( I should probably say that I am just starting to learn about vector bundles. )
If anyone with more experience could point me in th... | These notes might also add some motivation to the topic coming in from Physics:
http://www.itp.uni-hannover.de/~giulini/papers/DiffGeom/Urbantke_HopfFib_JGP46_2003.pdf
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/370718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
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Free modules have no infinitely divisible elements Let $F$ be a free $\mathbb Z$-module. How can we show that $F$ has no non-zero infinitely divisible element? (An element $v$ in $F$ is called infinitely divisible if the equation $nx = v$ has solutions $x$ in $F$ for infinitely many integers $n$.)
| By definition, $F$ has a basis $(b_i)_{i \in I}$. Suppose
$$
v = a_1 b_1 + \dots + a_k b_k,
$$
for $a_i \in \Bbb{Z}$, is divisible by infinitely many $n$. Choose $n$ positive, larger than all the $\lvert a_i \rvert$, so that $v$ is divisible by $n$. If $v = n x$ for
$$
x = x_1 b_1 + \dots + x_k b_k
$$
then $n x_i = a_i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/370790",
"timestamp": "2023-03-29T00:00:00",
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Prove that a sequence diverges Let $b > 1$. Prove that the sequence $\frac{b^n}{n}$ diverges to $\infty$
I know that I need to show that $\dfrac{b^n}{n} \geq M $, possibly by solving for $n$, but I am not sure how.
If I multiply both sides by $n$, you get $b^n \geq Mn$, but I don't know if that is helpful.
| You could use L'Hospital's rule:
$$
\lim_{n\rightarrow\infty}\frac{b^n}{n}=\lim_{n\rightarrow\infty}\frac{b^n\log{b}}{1}=\infty
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/370876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding the limit of function - irrational function How can I find the following limit:
$$ \lim_{x \rightarrow -1 }\left(\frac{1+\sqrt[5]{x}}{1+\sqrt[7]{x}}\right)$$
| Let $x=t^{35}$. As $x \to -1$, we have $t \to-1$. Hence,
$$\lim_{x \to -1} \dfrac{1+\sqrt[5]{x}}{1+\sqrt[7]{x}} = \lim_{t \to -1} \dfrac{1+t^7}{1+t^5} = \lim_{t \to -1} \dfrac{(1+t)(1-t+t^2-t^3+t^4-t^5+t^6)}{(1+t)(1-t+t^2-t^3+t^4)}$$
I am sure you can take it from here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/371106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is it always true that $\lim_{x\to\infty} [f(x)+c_1]/[g(x)+c_2]= \lim_{x\to\infty}f(x)/g(x)$? Is it true that $$\lim\limits_{x\to\infty} \frac{f(x)+c_1}{g(x)+c_2}= \lim\limits_{x\to\infty} \frac{f(x)}{g(x)}?$$ If so, can you prove it? Thanks!
| Think of it this way: the equality is true only when $f(x), g(x)$ completely 'wash out' the additive constants at infinity. To be more precise, suppose $f(x), g(x) \rightarrow \infty$. Then
$$
\frac{f(x) + c_1}{g(x) + c_2} = \frac{f(x)}{g(x)} \frac{1 + c_1/f(x)}{1 + c_2 / g(x)}
$$
In the limit as $x \rightarrow \infty$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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elementary ring proof with composition of functions and addition of functions as operations
Consider the set $\mathcal{F}=\lbrace f\mid \mathcal{f}:\Bbb{R}\to\Bbb{R}\rbrace$, in which an additive group is defined by addition of functions and a second operation defined as composition of functions. The question asks to ... | Taking on the suggestion of Sammy Black, consider $g = h = \mathbf{1} = \text{the identity function}$, and $f$ any function.
Suppose $f \circ (\mathbf{1} + \mathbf{1}) = f \circ \mathbf{1} + f \circ \mathbf{1} = f + f = 2 f$.
So for all $x \in \Bbb{R}$ you should have $f( 2 x) = 2 f(x)$. Now think of a function $f$ whi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/371223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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An infinite union of closed sets is a closed set? Question: {$B_n$}$ \in \Bbb R$ is a family of closed sets.
Prove that $\cup _{n=1}^\infty B_n$ is not necessarily a closed set.
What I thought: Using a counterexample: If I say that each $B_i$ is a set of all numbers in range $[i,i+1]$ then I can pick a sequence $a_n \i... | But since $\infty\notin\Bbb R$ we cannot use it as a counterexample. To see that is indeed the case note that your union is the set $[1,\infty)$, which is closed. Why is it closed? Recall that $A\subseteq\Bbb R$ is closed if and only if every convergent sequence $a_n$ whose elements are from $A$, has a limit inside $A$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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General solution of $yy′′+2y'''=0$ How do you derive the general solution of $yy''+2y'''= 0$?
please help me to derive solution thanks a lot
| NB: This is a general way to reduce the order of the equation. This doesn't solve your question as is, but rather gives you a starting point.
In this equation, the variable $t$ does not appear, hence one can substitute: $$y'=p(y), \hspace{7pt}y''=p'(y)\cdot y'=p'p,\hspace{7pt}y'''=p''\cdot y'\cdot p+p'\cdot p'\cdot y'=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/371369",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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General solution of a differential equation $x''+{a^2}x+b^2x^2=0$ How do you derive the general solution of this equation:$$x''+{a^2}x+b^2x^2=0$$where a and b are constants.
Please help me to derive solution thanks a lot.
| First make the substitution:
$x=-\,{\frac {6y}{{b}^{2}}}-\,{\frac {{a}^{2}}{{2b}^{2}}}$
This will give you the differential equation:
$y^{''} =6y^{2}-\frac{a^{4}}{24}$ which is to be compared with the second order differential equation for the Weierstrass elliptic function ${\wp}(t-\tau_{0},g_2,g_3)$:
${\wp}^{''} =6{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/371452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that a tensor field is of type (1,2) Let $J\in\operatorname{end}(TM)=\Gamma(TM\otimes T^*M)$ with $J^2=-\operatorname{id}$ and for $X,Y\in TM$, let
$$N(X,Y):=[JX,JY]-J\big([JX,Y]-[X,JY]\big)-[X,Y].$$
Prove that $N$ is a tensor field of type (1,2).
Since I heard $N$ is the Nijenhuis tensor and saw its component f... | Being a tensor field means that you have to show $C^{\infty}(M)$-Linearity. (1,2)-Tensor just means, that it eats two vector fields at spits out another vector field which is obvious from the definition.
So look at $N(X,fY)$ for $f \in C^{\infty}(M)$ and use the properties of the Lie-bracket, to show $N(X,fY)=fN(X,Y)$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/371522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If f is integrable on $[a,b]$, prove $f^{q}$ is integrable on $[a,b]$ Let $q $ be a rational number. Suppose that $a < b,\ 0 < c < d $, and that $f : [a,b] $->$ [c,d] $. If $f$ is integrable on $[a,b]$, then prove that $f^{q}$ is integrable on $[a,b]$.
I think that the proof involves the binomial theorem. My book has ... | Hints:
*
*$f$ is Riemann integrable and $g$ is continuous then the composition $g \circ f$ -- when this makes sense --is Riemann integrable (Theorem 8.18 of these notes).
*If $f: [a,b] \rightarrow [c,d]$ is continuous and monotone, then the inverse function $f^{-1}$ exists and is continuous (Theorem 5.39 of loc. ci... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Confusion about Banach Matchbox problem While trying to solve Banach matchbox problem, I am getting a wrong answer. I dont understand what mistake I made. Please help me understand.
The problem statement is presented below (Source:Here)
Suppose a mathematician carries two matchboxes at all times: one in his left pocke... | Apart from doubling $p$ at the end, your answer is correct: your denominator is actually equal to $1$. It can be rewritten as
$$\frac1{2^{2n}}\sum_{i=0}^n\binom{2n-i}n2^i=\frac1{2^{2n}}\sum_{m=n}^{2n}\binom{m}n2^{2n-m}=\frac1{2^{2n}}\sum_{i=0}^n\binom{n+i}n2^{n-i}\;,$$
and
$$\begin{align*}
\sum_{i=0}^n\binom{n+i}n2^{n-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/371728",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Does this $3\times 3$ matrix exist? Does a real $3\times 3$ matrix $A$ that satisfies conditions $\operatorname{tr}(A)=0$ and $A^2+A^T=I$ ($I$ is an identity matrix) exist?
Thank you for your help.
| [Many thanks to user1551 for this contribution.] First, let us show that the eigenvalues of $A$ must be real. The equation $A^2+A^T=I$ implies that
$$A^T=I−A^2 \quad\text{and}\quad (A^T)^2+A=I.$$ Substituting the first equation into the second yields
$$I−2A^2+A^4+A=I \quad\Longrightarrow\quad A^4−2A^2+A=A(A−I)(A^2+A−I)... | {
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 1
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Optimizing the area of a rectangle A rectangular field is bounded on one side by a river and on the other three sides by a fence. Additional fencing is used to divide the field into three smaller rectangles, each of equal area. 1080 feet of fencing is required. I want to find the dimensions of the large rectangle that ... | @Gorg, you are on the right track. You can either solve this problem using small y or large Y. Your equations are set up correctly with small y, and the answer I get if you want to compare with what you get is $$x=135 \text{ and}\ y=180$$. Good job :)
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What should be the intuition when working with compactness? I have a question that may be regarded by many as duplicate since there's a similar one at MathOverflow.
*
*In $\mathbb{R}^n$ the compact sets are those that are closed and bounded, however the guy who answered this question and had his answer accepted says ... | You may read various descriptions and consequences of compactness here. But be aware that compactness is a very subtle finiteness concept. The definitive codification of this concept is a fundamental achievement of $20^{\,\rm th}$ century mathematics.
On the intuitive level, a space is a large set $X$ where some notion... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "133",
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Show inequality of integrals (cauchy-schwarz??) $f:[0,1]\to\mathbb{C}$ continuous and differentiable and $f(0)=f(1)=0$.
Show that
$$
\left |\int_{0}^{1}f(x)dx \right |^2\leq\frac{1}{12}\int_{0}^{1} \left |f'(x)\right|^2dx
$$
Well I know that
$$
\left |\int f(x)\cdot g(x)\ dx \right|^2\leq \int \left |f(x) \right|^2dx\... | Integrate by parts to get
$$
\int_0^1 f(x)\,dx = -\int_0^1 (x-1/2)f'(x)\,dx,
$$
and then use Cauchy-Schwarz:
$$
\begin{align*}
\left|\int_0^1 (x-1/2)f'(x)\,dx\right|^2 & \leq \int_0^1(x-1/2)^2\,dx\int_0^1|f'(x)|^2\,dx \\
& = \frac{1}{12}\int_0^1|f'(x)|^2\,dx
\end{align*}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/371965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
boundary map in the (M-V) sequence Let $K\subset S^3$ be a knot, $N(K)$ be a tubular neighborhood of $K$ in $S^3$, $M_K$ to be the exterior of $K$ in $S^3$, i.e., $M_K=S^3-\text{interior of }{N(K)}$.
Now, it is clear that $\partial M_K=\partial N(K)=T^2$, the two dimensional torus, and when using the (M-V) sequence to ... | The generator of $H_3(S_3)$ can be given by taking the closures of $M_K$ and $N(K)$ and triangulating them so the triangulation agrees on the boundary, then taking the union of all the simplices as your cycle. The boundary map takes this cycle and sends it to the common boundary of its two chunks, which is exactly the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
disconnected/connected graphs Determine whether the statements below are true or false. If the statement is true, then
prove it; and if it is false, give a counterexample.
(a) Every disconnected graph has a vertex of degree 0.
(b) A graph is connected if and only if some vertex is connected to all other vertices.
Plea... | So that this doesn't remain unanswered: Yes, all of your reasoning is correct.
| {
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"source": "stackexchange",
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Application of quadratic functions to measurement and graphing thanks for any help!
Q1. Find the equation of the surface area function of a cylindrical grain silo. The input variable is the radius (r). (the equation is to be graphed using a graphics calculator in the following question)
Height (h) = 5 meters
Radius (r)... | (SA = Surface Area)
*
*SA (silo) = SA (cylinder) + $\frac{1}{2}$ SA (sphere)
*SA (cylinder) = $2\pi r h $
*SA (sphere) = $4\pi r^2$
So we have,
SA (silo) = SA (cylinder) + $\frac{1}{2}$ SA (sphere) = $2\pi r h + \frac{1}{2}4\pi r^2 = 2\pi r h + 2 \pi r^2 = 2 ~\pi~ r(h + r) = 2 ~\pi~ r(5 + r)$
Plot:
| {
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"source": "stackexchange",
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Why does the theta function decay exponentially as $x \rightarrow \infty$? I'm trying to understand the proof of the functional equation for the L-series of primitive, even Dirichlet characters.
For even, primitive characters we have $$\theta_\chi(x):=\sum_{n\in \mathbb{Z}} \chi(n)\exp\left(\frac{-\pi}{q}n^2x^2\right).... | So, using Sanchez's hint, I think I have
$$\begin{array}{rll}\theta_\chi(x) & \leq & 2\exp\left(\frac{-\pi}{q}x^2\right)+\sum_{|n|\geq 2}\exp\left( \frac{-\pi}{q}n^2x^2\right) \\
& \leq & 2\exp\left(\frac{-\pi}{q}x^2\right)+\sum_{|n|\geq 4}\exp\left( \frac{-\pi}{q}n x^2\right) \\
\text{(expanding the sum} \\ \text{ as... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
inequality involving complex exponential Is it true that
$$|e^{ix}-e^{iy}|\leq |x-y|$$ for $x,y\in\mathbb{R}$? I can't figure it out. I tried looking at the series for exponential but it did not help.
Could someone offer a hint?
| One way is to use
$$
|e^{ix} - e^{iy}| = \left|\int_x^ye^{it}\,dt\right|\leq \int_x^y\,dt = y-x,
$$
assuming $y > x$.
| {
"language": "en",
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"source": "stackexchange",
"question_score": "21",
"answer_count": 6,
"answer_id": 1
} |
Integrate $e^{f(x)}$ Just wondering how I can integrate $\displaystyle xe^{ \large {-x^2/(2\sigma^2)}}$
Tried using substitution where $U(x) = x^2$ but I kept getting a $x^2$ at the denominator which is incorrect.
I understand that $\displaystyle \int e^{f(x)} = e^{\large \frac{f(x)}{f'(x)}}$ if $f(x)$ is linear, howev... | A close relative of your substitution, namely $u=-\dfrac{x^2}{2\sigma^2}$, works.
In "differential" notation, we get $du=-\frac{1}{\sigma^2} x\,dx$, so $x\,dx=-\sigma^2\,du$.
Remark: Or more informally, let's guess that the answer is $e^{-x^2/(2\sigma^2)}$. Differentiate, using the Chain Rule. We get $-\frac{x}{\sig... | {
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What will be the units digit of $7777^{8888}$? What will be the units digit of $7777$ raised to the power of $8888$ ?
Can someone do the math with explaining the fact "units digit of $7777$ raised to the power of $8888$"?
| the units digit of a number is the same as the number mod 10
so we just need to compute $7777^{8888} \pmod {10}$
first $$7777^{8888} \equiv 7^{8888} \pmod {10}$$ and secondly by Eulers totient theorem $$7^{8888} \equiv 7^{0} \equiv 1 \pmod {10}$$ by Eulers totient theorem (since $\varphi(10)=4$ and $4 \mid 8888$)
so th... | {
"language": "en",
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"source": "stackexchange",
"question_score": "3",
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Mean Value Property of Harmonic Function on a Square A friend of mine presented me the following problem a couple days ago:
Let $S$ in $\mathbb{R}^2$ be a square and $u$ a continuous harmonic function on the closure of $S$. Show that the average of $u$ over the perimeter of $S$ is equal to the average of $u$ over the ... | Consider the isosceles right triangles formed from two sides of the square and a diagonal.
Let's consider the first such triangle. Call the sides $L_1, L_2$ and $H_1$ for legs and hypotenuse; for sake of convenience, our square is the unit square, so we give the triangle $T_1$, legs $L_1$ from $(0,0)$ to $(1,0)$ and $... | {
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"source": "stackexchange",
"question_score": "10",
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Continuity of one partial derivative implies differentiability Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function such that the partial derivatives with respect to $x$ and $y$ exist and one of them is continuous. Prove that $f$ is differentiable.
| In short: the problem reduces to the easy case when $f$ depends solely on one variable. See the greyish box below for the formula that does the reduction.
It suffices to show that $f$ is differentiable at $(0,0)$ with the additional assumption that $\frac{\partial f}{\partial x}(0,0)=\frac{\partial f}{\partial y}(0,0)=... | {
"language": "en",
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"source": "stackexchange",
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Modular Exponentiation Give numbers $x,y,z$ such that $y \equiv z \pmod{5}$ but $x^y \not\equiv x^z \pmod{5}$
I'm just learning modular arithmetic and this questions has me puzzled. Any help with explanation would be great!
| Take $x=2$, $y=3$, $z=8$. Then $x^y \bmod 5 = 3$ but $x^z \bmod 5 = 1$.
| {
"language": "en",
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"question_score": "3",
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If $x,y$ are positive, then $\frac1x+\frac1y\ge \frac4{x+y}$ For $x$, $y$ $\in R^+$, prove that $$\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}$$
Could someone please help me with this inequality problem? I have tried to use the AM-GM inequality but I must be doing something wrong. I think it can be solved with the AM-GM bu... | Here is a solution with AM-GM:
$$\frac{1}{x}+\frac{1}{y} \geq \frac{2}{\sqrt{xy}}$$
$$x+y \geq 2 \sqrt{xy} \Rightarrow \frac{1}{\sqrt{xy}} \geq \frac{2}{x+y}\Rightarrow \frac{2}{\sqrt{xy}} \geq \frac{4}{x+y}$$
Also you can note that
$$(x+y)(\frac{1}{x}+\frac{1}{y}) \geq 4$$
is just Cauchy-Schwarz.
| {
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"source": "stackexchange",
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Find a lower bound Let $M$ be an $N\times N$ symmetric real matrix, and let $J$ be a permutation of the integers from 1 to $N$, with the following properties:
*
*$J:\{1,...,N\}\rightarrow\{1,...,N\}$ is one-to-one.
*$J$ is its own inverse: $J(J(i))=i$.
*$J$ has at most one fixed point, that is, there's at most one... | Not really an answer, rather a reformulation and an observation.
UPD: The answer is in the addendum
Any such permutation $S$ can be considered as an $N\times N$ matrix with one $1$ and $N-1$ zeros in each raw and each column. Note that $S^{-1}=S^T$. The number of fixed points coincides with $\mathrm{Tr}\,S$. You are al... | {
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"source": "stackexchange",
"question_score": "4",
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Photograph of Marjorie Rice I'm giving a presentation this weekend about Marjorie Rice's work on tilings. The only photograph I have of her (from her website) is small and pixelated, and I haven't been able to make contact with her to ask her for a better one. I'd be most grateful if you could point me to a better phot... | See Marjorie Rice, page 2 from a newsletter published by Key Curriculum Press, on Tesselations. There's a photo of Marjorie Rice (on the left), at the lower left of page 2 of the newsletter. The link will take you to the pdf.
I don't know if this is an improvement over your current picture, but I thought I'd post thi... | {
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Is mathematical induction necessary in this situation? I was reading "Number Theory" by George E. Andrews.
On P.17, where he proves that for each pair of positive integers a,b, gcd(a,b) uniquely exists, I came up with a question.
The approach he used is probably the most common one, that is, to make use of Euclidean A... | Here are the spots where induction is required:
"Since $b>r_0>r_1>....≥0$, there exists $k$ such that $r_k=0$." Not true for real numbers, right?
"I think this does not need Principle of Mathematical Induction because k is a finite number." But how do we know it's finite? You could descend forever in some rings.
Person... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to find solutions for linear recurrences using eigenvalues Use eigenvalues to solve the system of linear recurrences
$$y_{n+1} = 2y_n + 10z_n\\
z_{n+1} = 2y_n + 3z_n$$
where $y_0 = 0$ and $z_0 = 1$.
I have absolutely no idea where to begin. I understand linear recurrences, but I'm struggling with eigenvalues.
| Set $x_n=[y_n z_n]^T$, and your system becomes $x_{n+1}=\left[\begin{smallmatrix}2&10\\2&3\end{smallmatrix}\right]x_n$. Iteration becomes matrix exponentiation. If your eigenvalues are less than 1 in absolute value, the matrix approaches 0. If an eigenvalue is bigger than 1 in absolute value, you get divergence. It... | {
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Show that the equation $\cos(x) = \ln(x)$ has at least one solution on real number I have question
Q
Show that the equation $\cos (x) = \ln (x)$ has at least one solution on real number.
to solve this question by using intermediate value theorem
we let $f(x)=\cos (x)-\ln (x)$
we want to find $a$ and $b$
but what i... | Hint: $\cos$ is bounded whereas $\ln$ is increasing with $\lim\limits_{x\to 0^+} \ln(x) =- \infty$ and $\lim\limits_{x \to + \infty} \ln(x)=+ \infty$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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4 dimensional numbers I've tought using split complex and complex numbers toghether for building a 3 dimensional space (related to my previous question). I then found out using both together, we can have trouble on the product $ij$. So by adding another dimension, I've defined $$k=\begin{pmatrix}
1 & 0\\
0 & -1
\end{p... | You have discovered split-quaternions. You can compare the multiplication table there and in your question.
This algebra is not commutative and has zero divisors. So, it combines the "negative" traits of both quaternions and tessarines. On the other hand it is notable to be isimorphic to the $2\times2$ matrices. Due to... | {
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"source": "stackexchange",
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Domain of the function $f(z) = \sqrt{z^2 -1}$ What will be the domain of the function $f(z) = \sqrt{z^2 -1}$?
My answers are: $(-\infty, -1] \cup [1, \infty)$ OR $\mathbb{R} - \lbrace1>x\rbrace$ OR $\mathbb {R}$, such that $z \nless 1$.
| The first part of your answer (before the "or") is correct:
The domain of your function, in $\mathbb R$ is indeed $(-\infty, -1]\cup [1, \infty).$ That is, the function is defined for all real numbers $z$ such that $z \leq -1$ or $z \geq 1$.
Did you have any particular reason you included: this as your answer, along w... | {
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Generalization of metric that can induce ordering I was wondering if there is some generalization of the concept metric to take positive and negative and zero values, such that it can induce an order on the metric space? If there already exists such a concept, what is its name?
For example on $\forall x,y \in \mathbb ... | You will have to lose some of the other axioms of a metric space as well since the requirement that $d(x,y)\ge 0$ in a metric space is actually a consequence of the other axioms: $0=d(x,x)\le d(x,y)+d(y,x)=2\cdot d(x,y)$, thus $d(x,y)\ge 0$. This proof uses the requirements that $d(x,x)=0$, the triangle inequality, and... | {
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Conformally Map Region between tangent circles to Disk Suppose we are given two circles, one inside the other, that are tangent at a point $z_0$. I'm trying to map the region between these circles to the unit disc, and my thought process is the following:
I feel like we can map $z_0$ to $\infty$, but I'm not really sur... | I just recently solved this myself.
Let $z_1$ be the center of the inner tangent circle such that $|z_1 - z_0| =r$ and let $z_2$ be the center of the larger circle with $|z_2 - z_0| = R$ with $R>r$. Rotate and translate your circles so that $z_0$ lies on the real axis (as so does $z_1$) and $z_2$ is 0. You can map this... | {
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"timestamp": "2023-03-29T00:00:00",
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Does one always use augmented matrices to solve systems of linear equations? The homework tag is to express that I am a student with no working knowledge of math.
I know how to use elimination to solve systems of linear equations. I set up the matrix, perform row operations until I can get the resulting matrix into row... | I certainly wouldn't use an augmented matrix to solve the following:
$x = 3$
$y - x = 6$
When you solve a system of equations, if doing so correctly, one does indeed perform "elementary row operations", and in any case, when working with any equations, to solve for $y$ above, for example, I would add $x$ to each side o... | {
"language": "en",
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Finding intersection of 2 planes without cartesian equations?
The planes $\pi_1$ and $\pi_2$ have vector equations:
$$\pi_1: r=\lambda_1(i+j-k)+\mu_1(2i-j+k)$$
$$\pi_2: r=\lambda_2(i+2j+k)+\mu_2(3i+j-k)$$
$i.$ The line $l$ passes through the point with position vector $4i+5j+6k$ and is parallel to both $\pi_1$ a... | 1) One method you can use to find line $L_1$ is to make equations in $x, y, z$ for each plane and solve them, instead of finding the cross product:
$$x=\lambda_1+2\mu_1, y=\lambda_1-\mu_1, z=-\lambda_1+\mu_1$$
$$x=\lambda_2+3\mu_2, y=\lambda_2+\mu_2, z=\lambda_2-\mu_2$$
What we'll do is find the line of intersection; t... | {
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"timestamp": "2023-03-29T00:00:00",
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Finding the closed form for a sequence My teacher isn't great with explaining his work and the book we have doesn't cover anything like this. He wants us to find a closed form for the sequence defined by:
$P_{0} = 0$
$P_{1} = 1$
$\vdots$
$P_{n} = -2P_{n-1} + 15P_{n-2}$
I'm not asking for a straight up solution, I just ... | A related problem. Here is a start. Just assume your solution $P_n=r^n$ and plug in back in the eq. to find $r$
$$ P_{n} = -2P_{n-1} + 15P_{n-2} \implies r^n+2r^{n-1}-15r^{n-2}=0 $$
$$ \implies r^{n-2}(r^2+2r-15)=0 \implies r^2+2r-15=0 $$
Find the roots of the above polynomial $r_1, r_2$ and construct the general sol... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Derivatives using the Limit Definition How do I find the derivative of $\sqrt{x^2+3}$? I plugged everything into the formula but now I'm having trouble simplifying.
$$\frac{\sqrt{(x+h)^2+3}-\sqrt{x^2+3}}{h}$$
| Keaton's comment is very useful. If you multiply the top and bottom of your expression by $\sqrt{(x+h)^2+3}+\sqrt{x^2+3}$, the numerator should simplify to $2xh+h^2$. See if you can finish the problem after that.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Clarification on expected number of coin tosses for specific outcomes. As seen in this question, André Nicolas provides a solution for 5 heads in a row.
Basically, for any sort of problem that relies on determining this sort of probability, if the chance of each event is 50/50, then no matter what the composition of va... | No, it is not the same. For this pattern, the argument holds well (if you keep flipping heads) until four tosses. But on the fifth toss, if you flip heads you are done, but if you flip tails you are not back to the start-you have potentially flipped the first of your winning series. You will need to consider states ... | {
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Simplex on Linear Program with equations My linear program instead of inequations also contains one equation. I do not understand how to handle this in every tutorial I searched the procedure is to add slack variables to convert the inequations to equations. My lp is the following:
Minimize x4
Subject to:
3x1+7x2+8x3<... | I have to be honest, my simplex is rusty. But perhaps you could split the equation into two inequalities:
$$x_1+x_2+x_3\leq 1$$
$$-x_1-x_2-x_3\leq -1$$
This is exactly what some solvers do that can't handle mixtures of inequalities and equations.
| {
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"timestamp": "2023-03-29T00:00:00",
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} |
Analytically continue a function with Euler product I would like to estimate the main term of the integral
$$\frac{1}{2\pi i} \int_{(c)} L(s) \frac{x^s}{s} ds$$
where $c > 0$, $\displaystyle L(s) = \prod_p \left(1 + \frac{2}{p(p^s-1)}\right)$.
Question: How to estimate the integral? In other words, is there any way to... | Let $\rho(d)$ count the number of solutions $x$ in $\frac{Z}{dZ}$, to $x^2\equiv \text{-1 mod d}$, then we have
$$\sum_{n\leq x}d(n^2+1)=2x\sum_{n\leq x}\frac{\rho(n)}{n}+O(\sum_{n\leq x}\rho(n))$$
By multiplicative properties of $\rho(n)$ we have,
$$\rho(n)=\chi(n)*|\mu(n)|$$
Where $\chi(n)$ is the non principal char... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/374088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Is the ideal $(X^2-3)$ proper in $\mathbb{F}[[X]]$?
Let $\mathbb{F}$ be a field and $R=\mathbb{F}[[X]]$ be the ring of formal power series over $\mathbb{F}$. Is the ideal $(X^2-3)$ proper in $R$? Does the answer depend upon $\mathbb{F}$?
Clearly $X^2-3=(X+\sqrt3)(X-\sqrt3)$ and hence $X^2-3$ is not zero.
I have no id... | The element $\sum _{i \geq 0} a_i X^i \in R$ is invertible in $R$ if and only if $a_0\neq 0$.
The key to the proof of that relatively easy result is the identity $(1-X)^{-1}=\sum_{i \geq 0} 1. X^i \in R $
In your question $a_0=-3$ so that the element $X^2-3$ is invertible in $R=F[[X]]$ (which is equivalent to the ide... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Addition table for a 4 elements field Why is this addition table good,
\begin{matrix}
\boldsymbol{\textbf{}+} & \mathbf{0} & \boldsymbol{\textbf{}1} & \textbf{a} &\textbf{ b}\\
\boldsymbol{\textbf{}0} & 0 & 1 & a & b\\
\boldsymbol{\textbf{}1} & 1 & 0 & b & a\\
\boldsymbol{\textbf{} a} & a & b & 0 & 1\\
\boldsymbol{... | If you have only one operation, it is difficult to speak about field. But, it is well-known that:
1) there exists exactly two groups (up to isomorphism) with 4 elements: one is ${\mathbb Z}/2{\mathbb Z}\times{\mathbb Z}/2{\mathbb Z}$ (the first table) and the other one is ${\mathbb Z}/4{\mathbb Z}$ (the second table)
2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/374233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
understanding $\mathbb{R}$/$\mathbb{Z}$ I am having trouble understanding the factor group, $\mathbb{R}$/$\mathbb{Z}$, or maybe i'm not. Here's what I am thinking.
Okay, so i have a group $G=(\mathbb{R},+)$, and I have a subgroup $N=(\mathbb{Z},+)$. Then I form $G/N$. So this thing identifies any real number $x$ wit... | You can also use the following nice facts. I hope you are inspired by them.
$$\mathbb R/\mathbb Z\cong T\cong\prod_p\mathbb Z(p^{\infty})\cong\mathbb R\oplus(\mathbb Q/\mathbb Z)\cong\mathbb C^{\times}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/374380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
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Formulate optimization problem My research area has "nothing to do with mathematics" but I still find it full of optimization problems. Therefore, I would like to learn to formulate and solve such problems, even though I am not encouraged to do it (at least at the moment; maybe the situation will change after I have pr... | The notation in the question looks fine. So, you have a function $F$ of four real variables $p_{11},\dots,p_{22}$, defined by
$$F(p_{11},p_{12},p_{21},p_{22}) = f(p_{11}x+p_{12}y,\ p_{21}x+p_{22}y) \tag2$$
If $f$ is differentiable, then so is $F$. Therefore, the gradient descent can be used; how successful it will be ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/374439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Moment of inertia of a circle A wire has the shape of the circle $x^2+y^2=a^2$. Determine the moment of inertia about a diameter if the density at $(x,y)$ is $|x|+|y|$
Thank you
| Consider a small segment of the wire, going from $\theta$ to $\theta +d\theta$. The length of the small segment is $a \,d\theta$. The density varies, but is approximately $a|\cos\theta|+a|\sin \theta|$.
Take a particular diameter, say with rectangular equation $y=(\tan\phi) x$, or better, $x\sin \phi -y\cos\phi=0$. Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/374527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Weather station brain teaser I am living in a world where tomorrow will either rain or not rain. There are two independent weather stations (A,B) that can predict the chance of raining tomorrow with equal probability 3/5. They both say it will rain, what is the probability of it actually rain tomorrow?
My intuition is ... | After a couple months of thinking, a friend of mine have pointed out that the question lacks one piece of information: the unconditional probability distribution of rain $P(rain)$. The logic is that, if one is to live in an area that is certain to rain everyday, the probability of raining is always 1. Then Kaz's analys... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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which texts on number theory do you recommend? my close friend intend to study number theory
and he asked me if i know a good text on it , so i thought that you guys can help me to help him !
he look for a text for the beginners and for a first course
he will study it as a self study ..
so what texts do you recommend... | My two pennyworth: for introductory books try ...
*
*John Stillwell, Elements of Number Theory (Springer 2002). This is by a masterly expositor, and is particularly approachable.
*G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers (OUP 1938, and still going strong with a 6th edition in 2008). Also... | {
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can a ring homomorphism map an integral domain to a non integral domain? i understand that if two rings are isomorphic, and one ring is an integral domain, so must the other be.
however, consider two rings, both commutative rings with unity. is it possible that one ring contains zero divisors and one does not while th... | Given any unital ring $R$ (with multiplicative identity $1_R$, say), there is a unique ring homomorphism $\Bbb Z\to R$ (take $1\mapsto 1_R$ and "fill in the blanks" from there).
This may be an injective map, even if $R$ has zero divisors. For example, take $$R=\Bbb Z[\epsilon]:=\Bbb Z[x]/\langle x^2\rangle.$$ Surjectiv... | {
"language": "en",
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"source": "stackexchange",
"question_score": "7",
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Does $\frac{x}{x}=1$ when $x=\infty$? This may be a dumb question:
Does $\frac{x}{x}=1$ when $x=\infty$?
I understand why $\frac{x}{x}$ is undefined when $x=0$: This can cause errors if an equation is divided by $x$ without restrictions.
Also, $\frac{\infty}{\infty}$ is undefined. So when I use $\frac{x}{x}=1$ to ... | You cannot really say $x = \infty$ because $\infty \not \in \mathbb{R}$
What you do is, you take the limes. Limes means not, that $x=a$, but that $x$ is getting closeser and closer to $a$. For example:
$$\lim_{x\mapsto 0}\frac{1}{x}=\infty$$ because the divisor gets smaller and smaller
$$\frac{1}{2}=0.5 \\\frac{1}{1}=... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
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"answer_id": 4
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A Criterion for Surjectivity of Morphisms of Sheaves? Suppose that $f: \mathcal{F} \rightarrow \mathcal{G}$ is a morphism of sheaves on a topological space $X$. Consider the following statements.
1) $f$ is surjective, i.e. $\text{Im } f = \mathcal{G}$.
2) $f_{p}: \mathcal{F}_p \rightarrow \mathcal{G}_p$ is surjective f... | This can be found in every complete introduction to sheaves or algebraic geometry and comes down to the fact that the functor $F \mapsto (F_x)_{x \in X}$ is faithful and exact.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/374854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Use L'Hopital's rule to evaluate $\lim_{x \to 0} \frac{9x(\cos4x-1)}{\sin8x-8x}$ $$\lim_{x \to 0} \frac{9x(\cos4x-1)}{\sin8x-8x}$$
I have done this problem a couple of times and could not get the correct answer. Here is the work I have done so far http://imgur.com/GDZjX26 . The correct answer was $\frac{27}{32}$, did I... | You have to use L'Hopitals 3 times we have $$\begin{align} \lim_{x\to 0}\frac{9x(\text{cos}(4x)-1}{\text{sin}(8x)-8x}&=\lim_{x\to 0}\frac{(9 (\text{cos}(4 x)-1)-36 x \text{sin}(4 x))}{(8 \text{cos}(8 x)-8)}\\
&=\lim_{x\to 0}\frac{-1}{64}\frac{(-72 \text{sin}(4 x)-144 x \text{cos}(4 x))}{\text{sin}(8x)}\\&=\lim_{x\to 0}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/374907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Calculating a complex derivative of a polynomial What are the rules for derivatives with respect to $z$ and $\bar{z}$ in polynomials?
For instance, is it justified to calculate the partial derivatives of $f(z,\bar{z})=z^3-2z+\bar{z}-(\overline{z-3i})^4$ as if $z$ and $\bar{z}$ were independent? i.e. $f_z=3z^2-2$ and $f... | I would first write $$ f(z,\bar z)=z^3−2z+\bar z−(\bar z+3i)^4 $$
and then treat $z$ and $\bar z$ as independent parameters.
Then I have
$$f_z=3z^2−2$$ $$f_{\bar z}=1−4(\bar z+3i)^3$$
Am I right?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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number of terms of a sum required to get a given accuracy How do I find the number of terms of a sum required to get a given accuracy. For example a text says that to get the sum $\zeta(2)=\sum_{n=1}^{\infty}{\frac{1}{n^2}}$ to 6 d.p. of accuracy, I need to add 1000 terms. How do in find it for a general series?
| If you have a sum $S=\sum_{n=1}^{\infty} a(n)$ that you want to estimate with a partial sum, denote by $R$ the residual error
$$
R(N) = S-\sum_{n=1}^N a(n) = \sum_{n=N+1}^\infty a(n)
$$
If all $a(n)$ are nonnegative then $R(N)\ge a(N+1)$, so to estimate within a given accuracy $\epsilon$ you need $N$ at least large eno... | {
"language": "en",
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Proof of an Integral inequality Let $f\in C^0(\mathbb R_+,\mathbb R)$ and $a\in\mathbb R_+$, $f^*(x)=\dfrac1x\displaystyle\int_0^xf(t)\,dt$ when $x>0$, and $f^*(0)=f(0)$.
Show that
$$
\int_0^a(f^*)^2(t)\,dt\le4\int_0^af^2(t)\,dt$$
I tried integration by part without success, and Cauchy-Schwarz is not helping here.
Tha... | Assume without loss of generality that $f\geqslant0$. Writing $(f^*)^2(t)$ as
$$
(f^*)^2(t)=\frac1{t^2}\int_0^tf(y)\left(\int_0^tf(x)\mathrm dx\right)\mathrm dy=\frac2{t^2}\int_0^tf(y)\left(\int_0^yf(x)\mathrm dx\right)\mathrm dy,
$$
and using Fubini, one sees that
$$
A=\int_0^a(f^*)^2(t)\mathrm dt=2\int_0^af(y)\int_0^... | {
"language": "en",
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Automorphism of Graph $G^n$ I try to define the automorphism of $G^n$ where $G$ is a graph and $G^n = G \Box \ldots \Box G$,( $n$ times, $\Box$ is the graph product).
I think that : $\text{Aut}(G^n)$ is $\text{Aut}(G) \wr S_n$ where $S_n$ is the symmetric group of $\{1,\ldots,n\}$ but I have no idea how to prove it b... | You are right, provided you assume that $G$ is prime relative to the Cartesian product, the automorphism group of the $n$-th Cartesian power of $G$ is the wreath product as you stated.
The standard reference for this is Hammack, Richard; Imrich, Wilfried; Klavžar, Sandi: Handbook of product graphs. (There is an older v... | {
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Evaluation of a limit with integral Is this limit
$$\lim_{\varepsilon\to 0}\,\,\varepsilon\int_{\mathbb{R}^3}\frac{e^{-\varepsilon|x|}}{|x|^2(1+|x|^2)^s}$$
with $s>\frac{1}{2}$, zero?.
The limit of a product is the product of limit, so I evaluate
$$\lim_{\varepsilon\to 0}\,\,\int_{\mathbb{R}^3}\frac{e^{-\varepsilon|x|}... | What you did is correct. The only thing you have to take care is that in general, dominated convergence theorem applies for sequences. Here there is no problem since the convergence is monotonic.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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In what sense is the derivative the "best" linear approximation? I am familiar with the definition of the Frechet derivative and it's uniqueness if it exists. I would however like to know, how the derivative is the "best" linear approximation. What does this mean formally? The "best" on the entire domain is surely wron... | Say the graph of $L$ is a straight line and at one point $a$ we have $L(a)=f(a)$. And suppose $L$ is the tangent line to the graph of $f$ at $a$. Let $L_1$ be another function passing through $(a,f(a))$ whose graph is a straight line. Then there is some open interval $(a-\varepsilon,a+\varepsilon)$ such that for eve... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
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how to prove: $A=B$ iff $A\bigtriangleup B \subseteq C$ I am given this: $A=B$ iff $A\bigtriangleup B \subseteq C$. And $A\bigtriangleup B :=(A\setminus B)\cup(B\setminus A)$.
I dont know how to prove this and I dont know where to start.
please give me guidance
| Hint: For an arbitrary set $C$, what is the one and only set that is the subset of every set?
So given $\,A\triangle\,B \subseteq C$, where $C$ is any arbitrary set, what does this tell you about the set $A\triangle B$?
And what does that tell you about the relationship between $A$ and $B$?
| {
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"timestamp": "2023-03-29T00:00:00",
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How can I able to show that $(S ^{\perp})^{\perp}$ is a finite dimensional vector space. Let $H$ be a Hilbert space and $S\subseteq H$ be a finite subset. How can I able to show that $(S ^{\perp})^{\perp}$
is a finite dimensional vector space.
| What you want to prove is that, for any $S\subset H$,
$$
S^{\perp\perp}=\overline{\mbox{span}\,S}
$$
One inclusion is easy if you notice that $S^{\perp\perp}$ is a closed subspace that contains $S$. The other inclusion follows from
$$
H=\overline{\mbox{span}\,S}\oplus S^\perp
$$
and the uniqueness of the orthogonal c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Contour Integral of $\int \frac{a^z}{z^2}\,dz$. My task is to show $$\int_{c-i\infty}^{c+i\infty}\frac{a^z}{z^2}\,dz=\begin{cases}\log a &:a\geq1\\ 0 &: 0<a<1\end{cases},\qquad c>0.$$So, I formed the contour consisting of a semi-circle of radius $R$ and center $c$ with a vertical line passing through $c$. I am having t... | For $a>1$, consider the contour $$(c-iT \to c+iT) \cup (c+iT \to -R +iT) \cup (-R+iT \to -R - iT) \cup (-R-iT \to c-iT),$$where $R>0$.
For $a<1$, consider the contour $$(c-iT \to c+iT) \cup (c+iT \to R +iT) \cup (R+iT \to R - iT) \cup (R-iT \to c-iT),$$where $R>0$.
Then let $R \to \infty$ and then $T \to \infty$.
The m... | {
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"source": "stackexchange",
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Primes in $\mathbb{Z}[i]$ I need a bit of help with this problem
Let $x \in \mathbb{Z}[i]$ and suppose $x$ is prime, therefore $x$ is not a unit and cannot be written as a product of elements of smaller norm. Prove that $N(x)$ is either prime in $\mathbb{Z}$ or else $N(x) = p^2$ for some prime $p \in \mathbb{Z}$.
thank... | Hint $\ $ Prime $\rm\:w\mid ww' = p_1^{k_1}\!\cdots p_n^{k_n}\:\Rightarrow\:w\mid p_i\:\Rightarrow\:w'\mid p_i' = p_i\:\Rightarrow\:N(w) = ww'\mid p_i^2$
Here $'$ denotes the complex conjugation.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Characterization of short exact sequences The following is the first part of Proposition 2.9 in "Introduction to Commutative Algebra" by Atiyah & Macdonald.
Let $A$ be a commutative ring with $1$. Let $$M'
\overset{u}{\longrightarrow}M\overset{v}{\longrightarrow}M''\longrightarrow
0\tag{1} $$ be sequence of $A$-modul... | I think I have found the solution using Zach L's hint.
Let $N=\operatorname{coker}(v)=M''/\operatorname{Im}(v)$, and let $p\in\operatorname{Hom}(M'', N)$ be the canonical map $p: M''\to M''/\operatorname{Im(v)}=N$. We observe for every $x\in M$, we have
$$p(v(x))=v(x)+\operatorname{Im}(v)=0+\operatorname{Im}(v)=0_{M''... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/375763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Analytical Solution to a simple l1 norm problem Can we solve this simple optimization problem analytically?
$ \min_{w}\dfrac{1}{2}\left(w-c\right)^{2}+\lambda\left|w\right| $
where c is a scalar and w is the scalar optimization variable.
| Set $f(w)=\frac{1}{2}(w-c)^2+\lambda |w|$, equal to $\frac{1}{2}(w-c)^2\pm\lambda w$. We find $f'(w)=w-c\pm \lambda$. Setting this to zero gives $c\pm \lambda$ as the only critical values of $f$. As $w$ gets large, $f(w)$ grows without bound, so the minimum is going to be at one of the two critical values. At those... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/375838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Let $X$ and $Y$ be 2 disjoint connected subsets of $\mathbb{R}^{2}$, can $X \cup Y =\mathbb{R}^{2}$? Let $X$ and $Y$ be 2 disjoint connected subsets of $\mathbb{R}^{2}$. Then can $$X \cup Y =\mathbb{R}^{2}$$
I think this cannot be true, but I don't know of a formal proof. Any help would be nice.
| consider $X:=\{(x,0) :x>0\}$ and $Y:=\mathbb{R^2}-X$ ,Both are connected and disjoint but $X\cup Y=\mathbb{R^2}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/375920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Rank of the difference of matrices Let $A$ and $B$ be to $n \times n$ matrices. My question is:
Is $\operatorname{rank}(A-B) \geq \operatorname{rank}(A) - \operatorname{rank}(B)$ true in general? Or maybe under certain assumptions?
| Set $X=A-B$ and $Y=B$. You are asking whether $\operatorname{rank}(X) + \operatorname{rank}(Y) \ge \operatorname{rank}(X+Y)$. This is true in general. Let $W=\operatorname{Im}(X)\cap\operatorname{Im}(Y)$. Let $U$ be a complementary subspace of $W$ in $\operatorname{Im}(X)$ and $V$ be a complementary subspace of $W$ in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/375982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Error estimate, asymptotic error and the Peano kernel error formula Find the error estimate by approximating $f(x)$ and derive a numerical integration formula for $\int_0^l f(x) \,dx$ based on approximating $f(x)$ by the straight line joining $(x_0, f(x_0))$ and $(x_1, f(x_1))$, where the two points $x_0$ and $x_1 = h ... | For this I think you can use trapezoidal rule. You can approximate $f(x)$ by the straight line joining $(a,f(a))$ and $(b,f(b))$ Then by integrating the formula for this straight line, we get the approximation $$I_1(f)=\frac{(b-a)}{2}[f(a)+f(b)].$$ To get the error formula we get $$f(x)-\frac{b-x)f(a)+(x-a)f(b)}{b-a}=(... | {
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Showing that if $\lim\limits_{x \to a} f'(x)=A$, then $f'(a)$ exists and equals $A$
Let $f : [a; b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable in $(a, b)$. Show that if $\lim\limits_{x \to a} f'(x)=A$, then $f'(a)$ exists and equals $A$.
I am completely stuck on it. Can somebody help me please? Than... | Let $\epsilon>0$. We want to find a $\delta>0$ such that if $0\lt x-a\lt\delta$ then $\left|\dfrac{f(x)-f(a)}{x-a}-A\right|\lt\epsilon$. If $x\gt a$ then MVT tell us that $\dfrac{f(x)-f(a)}{x-a}=f'(c)$ for some $c\in[a,x]$.
Now use that $\displaystyle\lim_{c\to a^+}f'(c)=A$ to find that $\delta$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/376130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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How to find a power series representation for a divergent product? Euler used the identity $$ \frac{ \sin(x) }{x} = \prod_{n=1}^{\infty} \left(1 - \frac{x^2}{n^2 \pi^2 } \right) = \sum_{n=0}^{\infty} \frac{ (-1)^n }{(2n + 1)! } x^{2n} $$ to solve the Basel problem. The product is obtained by noting that the sine functi... | I never gave the full answer :)
$$\prod_{n=1}^{\infty} (1-x/n)$$
When analysing a product it's often easiest to consider the form $\prod_{n=1}^{\infty} (1+f(n))$ given that $\sum_{n=1}^{\infty}f(n)^m=G(m)$; $f(n)=-x/n$ Then $G(m)=(-x)^m\zeta(m)$.
$$\prod_{n=1}^{\infty} (1-x/n)=e^{\sum_{m=1}^{\infty}\frac{(-1)^{m+1}x^m\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/376245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Definition of minimal and characteristic polynomials I have defined the characteristic and minimal polynomial as follows, but have been told this is not strictly correct since det$(-I)$ is not necessarily 1, so my formulae don't match for $A=0$, how can I correct this?
Given an $n$-by-$n$ matrix $A$, the characteristic... | There are two (nearly identical) ways to define the characteristic polynomial of a square $n\times n$ matrix $A$. One can use either
*
*$\det(A-I x)$ or
*$\det(Ix-A)$
The two are equal when $n$ is even, and differ by a sign when $n$ is odd, so in all cases, they have the same roots. The roots are the most impor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/376305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Does limit means replacing $x$ for a number? I don't understand limit so much. For example I see $\lim_{x \to -3}$. And I always just put $-3$ everywhere I see $x$. I feel like I'm doing something wrong, but it seems correct all the time.
| Substitution "works" many times; it works but not always: $$\lim_{x\to a} f(x) = f(a)\quad \text{${\bf only}$ when $f(x)$ is defined and continuous at $a$}$$
and this is why understanding the limit of a function as the limiting value (or lack of one) when $x$ is approaching a particular value: getting very very near th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/376383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Why is $PGL_2(5)\cong S_5$? Why is $PGL_2(5)\cong S_5$? And is there a set of 5 elements on which $PGL_2(5)$ acts?
| As David Speyer explains,
there are 15 involutions of $P^1(\mathbb F_5)$ without fixed points (one might call them «synthemes»). Of these 15 involutions 10 («skew crosses») lie in $PGL_2(\mathbb F_5)$ and 5 («true crosses») don't. The action of $PGL_2(\mathbb F_5)$ on the latter ones gives the isomorphism $PGL_2(\mathb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/376464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Monotonic Lattice Paths and Catalan numbers Can someone give me a cleaner and better explained proof that the number of monotonic paths in an $n\times n$ lattice is given by ${2n\choose n} - {2n\choose n+1}$ than Wikipedia
I do not understand the how they get ${2n\choose n+1}$ and I do not see how this is the number of... | There are $\binom{2n}{n+1}$ monotonic paths from $\langle 0,0\rangle$ to $\langle n-1,n+1\rangle$: such a path must contain exactly $(n-1)+(n+1)=2n$ steps, any $n+1$ of those steps can be vertical, and the path is completely determined once you know which $n+1$ of the $2n$ steps are vertical. Every monotonic path from ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/376522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to find the order of these groups? I don't know why but I just cannot see how to find the orders of these groups:
$YXY^{-1}=X^2$
$YXY^{-1}=X^4$
$YXY^{-1}=X^3$
With the property that $X^5 = 1$ and $Y^4 =1$
How would I go about finding the order? The questions asks me to find which of these groups are isomorphic.
Th... | Hint: You should treat those relations as a rule on how to commute $Y$ past $X$, for example the first can be written:
$$YX = X^2Y$$
Then you know that every element can be written in the form $X^nY^m$ for some $n$ and $m$. Use the orders of $X$ and $Y$ to figure out how many elements there are of this form.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/376593",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Path independence of an integral? I'm studying for a test (that's why I've been asking so much today,) and one of the questions is about saying if an integral is path independent and then solving for it.
I was reading online about path independence and it's all about vector fields, and I'm very, very lost.
This is the... | Seeing other answers, the follםwing perhaps doesn't grab the OP's intention, but here it is anyway.
Putting $\,z=x+iy\implies\,z^2=x^2-y^2+2xyi\,$ , so along the $\,y$-axis from zero to $\,i\,$ we get:
$$x=0\;,\;\;0\le y\le 1\implies \frac1{1-z^2}=\frac1{1+y^2}\;,\;\;dx=0 \;,\;\;dz=i\,dy\;,\;\ \;\text{so}$$
$$\int\limi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/376654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
surface area of a sphere above a cylinder I need to find the surface area of the sphere $x^2+y^2+z^2=4$ above the cone $z = \sqrt{x^2+y^2}$, but I'm not sure how. I know that the surface area of a surface can be calculated with the equation $A=\int{\int_D{\sqrt{f_x^2+f_y^2+1}}}dA$, but I'm not sure how to take into acc... | Hint: Use spherical coordinates. $dA = r^2\sin\theta d\theta d\phi$ with $0<\theta<\pi$. The surface area becomes $\iint_D dA$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/376740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Find the derivative of y with respect to x,t,or theta, as appropriate $$y=\int_{\sqrt{x}}^{\sqrt{4x}}\ln(t^2)\,dt$$
I'm having trouble getting started with this, thanks for any help.
| First Step
First, we need to recognize to which variables you are supposed to differentiate with respect. The important thing to realize here is that if you perform a definite integration with respect to one variable, that variable "goes away" after the computation. Symbolically:
$$\frac{d}{dt}\int_a^b f(t)\,dt = 0$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/376810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How do I show that $6(4^n-1)$ is a multiple of $9$ for all $n\in \mathbb{N}$? How do I show that $6(4^n-1)$ is a multiple of $9$ for all $n\in \mathbb{N}$? I'm not so keen on divisibility tricks. Any help is appreciated.
| You want it to be a multiple of $9$, it suffices to show you can extract a pair of 3's from this. The $6$ has one of the 3's, and $4^n-1$ is 0 mod 3 so you're done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/376861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Probability of Multiple Choice first attempt and second attempt A multiple choice question has 5 available options, only 1 of which is correct. Students are allowed 2 attempts at the answer. A student who does not know the answer decides to guess at random, as follows:
On the first attempt, he guesses at random among t... | First try to find the sample space $S$ for the question. There are five equally likely choices, so $S=\{c_1,\cdots, c_5\}$ and the event $E \subset S$ is choosing the correct answer, and there is only one correct answer i.e. $|E|=1.$ Therefore the probability is $\frac{|E|}{|S|}=\frac 15.$
Do the same to determine the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/376934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What does it mean for a set to exist? Is there a precise meaning of the word 'exist', what does it mean for a set to exist?
And what does it mean for a set to 'not exist' ?
And what is a set, what is the precise definition of a set?
| In mathematics, you do not simply say, for example, that set $S$ exists. You would add some qualifier, e.g. there exists a set $S$ with some property $P$ common to all its elements.
Likewise, for the non-existence of a set. You wouldn't simply say that set $S$ does not exist. You would also add a qualifier here, e.g. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/376989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 5,
"answer_id": 1
} |
Identity Law - Set Theory I'm trying to wrap my head around the Identity Law, but I'm having some trouble.
My lecture slides say:
$$
A \cup \varnothing = A
$$
I can understand this one. $A$ union nothing is still $A$. In the same way that $1 + 0$ is still $1$.
However, it goes on to say:
$$
A \cup U = U
$$
I don't see ... | Well $U$ is "the universe of discourse" -- it contains everything we'd like to talk about. In particular, all elements of $A$ are also in $U$.
In the "circles" representation, you can think of $U$ as the paper on which we draw circles to indicate sets like $A$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/377056",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Dissecting a proof of the $\Delta$-system lemma (part II) This is part II of this question I asked yesterday. In the link you can find a proof of the $\Delta$-system lemma. In case 1 it uses the axiom of choice (correct me if I'm wrong). Now one can also prove the $\Delta$-system lemma differently, for example as follo... | You need some AC to prove the statement just for a family of pairs of $\omega_1$.
If $\omega_1$ is the union of a countable family $\{B_n:n \in \omega \}$ of countable sets (which is consistent with ZF!), then the family $\{\{n, \beta\}: n<\omega, \beta \in B_n-\omega \}$ does not contain an uncountable $\Delta$-system... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/377113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Pascal's triangle and combinatorial proofs This recent question got me thinking, if a textbook (or an exam) tells a student to give a combinatorial proof of something involving (sums of) binomial coefficients, would it be enough to show that by Pascal's triangle these things do add up, or would you fail an answer like ... | I would argue that a combinatorial proof is something more substantial than pointing out a pattern in a picture! If we are at the level of "combinatorics" then we are also at the level of proofs and as such, the phrase "combinatorial proof" asks for a proof but in the combinatorial (or counting) sense.
A proof by exam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/377190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
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