instruction stringlengths 12 30k |
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**First:** You **don't** choose $\epsilon$!
In an $\epsilon$-$\delta$ proof, you begin by taking an *arbitrary* $\epsilon$; you then need to show that it is possible to choose a $\delta$ that "works" for *that* $\epsilon$. So you are going at things *exactly* backwards, by first trying to work with $\delta$.
**S... |
Qiaochu is right: categories are much more general. Let me provide a concrete example which, while in a sense trivial, may illustrate just how much more you could get from a category than you can from a reflexive and *transitive* relation.
Consider the positive integers $\mathbb P$. The usual total order $\leqslant$... |
When we consider the Taylor Series expansion of $f(x)=b^x$ for some $b \in \mathbb{R}$, we see that $$b^x = 1 + \sum_{n=1}^{\infty}\frac{(\log(b))^n}{n!}x^n.$$ We can substitute $x$ for $b^x$ to find that $$ b^{b^{x}} = 1 + \sum_{n=1}^{\infty}\frac{(\log(b))^n}{n!}b^{xn}.$$
Now, let's say that we want to find a fun... |
Could real iterates of the Taylor Series expansion of $b^x$ help to find a way to define tetration? |
Qiaochu is right: categories are much more general. Let me provide a concrete example which, while in a sense trivial, may illustrate just how much more you could get from a category than you can from a reflexive and *transitive* relation.
Consider the positive integers $\mathbb P$. The usual total order $\leqslant$... |
Qiaochu is right: categories are much more general. Let me provide a concrete example which, while in a sense trivial, may illustrate just how much more you could get from a category than you can from a reflexive and *transitive* relation.
Consider the positive integers $\mathbb P$. The usual total order $\leqslant$... |
When we consider the Taylor Series expansion of $f(x)=b^x$ for some $b \in \mathbb{R}$, we see that $$b^x = 1 + \sum_{n=1}^{\infty}\frac{(\log(b))^n}{n!}x^n.$$ We can substitute $x$ for $b^x$ to find that $$ b^{b^{x}} = 1 + \sum_{n=1}^{\infty}\frac{(\log(b))^n}{n!}b^{xn}.$$
Now, let's say that we want to find a fun... |
Suppose that $f:U\subset\mathbb{C}\to\mathbb{C}$, where $U$ is a region in the complex plane, is a holomorphic function.
If $c\in\mathbb{R}$ is a regular value for $\text{Re}(f(z))$ then it follows from implicit function theorem that
$\text{Re}(f(z))^{-1}(c)$ is at least locally a differentiable curve in the plan... |
What Re(f(z))=c is if f is a holomorphic function ? |
Many years ago a lecturer of mine had a photocopy of a page from a book containing a math problem ( I think it was a simple quadradic equation ) that was stated/solved in Cuneiform, Arabic, Latin scripts and Finally in modern math notation.
I have contacted my lecturer but he has no idea where it was from, nor I hav... |
At university, we are currently introduced in various methods of proving that a series converges. For example the ComparisonTest, the RatioTest or the RootTest. However we aren't told of how to calculate the point of convergence. The only series of which we know the point of convergence is the harmonic series.
I be... |
At university, we are currently introduced in various methods of proving that a series converges. For example the ComparisonTest, the RatioTest or the RootTest. However we aren't told of how to calculate the point of convergence. The only series of which we know the point of convergence is the geometric series.
I b... |
There's a brief note in [this book][1] on how Khayyam bumped into having to solve a cubic.
I'll only make the note that you should remember the context of the time: there was no concept of negative, much less complex, solutions. Corresponding to our current Cartesian system, Khayyam only looked at intersections in t... |
"there have to be at least some methods to calculate the point of convergence for simple series, haven't there?"
If you are sure a given sequence of numbers converges, you have to just keep iterating everything manually, unless you can think of a clever trick. I suggest you study how Euler proved that $\zeta(2)=\fr... |
There's a brief note in [this book][1] on how Khayyam bumped into having to solve a cubic.
I'll only make the note that you should remember the context of the time: there was no concept of negative, much less complex, solutions. Corresponding to our current Cartesian system, Khayyam only looked at intersections in t... |
In a previous post, http://math.stackexchange.com/questions/11849/w1w2-wna-w1a-wna-for-n-integer-2-w0-and-aconstant-real-0 the following inequality has been proven
${\left( {\sum\limits_{i = 1}^n {{W_i}} } \right)^a} \le \sum\limits_{i = 1}^n {{W_i}^a} $ where ${W_i}$>0, 0>a>1.
I guess it is more correct to say tha... |
f(sum(Xi)) <= sum ( f(Xi) ) where Xi>0, For what functions is this true ? |
In <a href="http://math.stackexchange.com/questions/11849/w1w2-wna-w1a-wna-for-n-integer-2-w0-and-aconstant-real-0">a previous post</a>, the following inequality has been proven
$${\left( {\sum\limits_{i = 1}^n {{W_i}} } \right)^a} \le \sum\limits_{i = 1}^n {{W_i}^a}$$
where $W_i\gt 0$, $0\lt a\lt 1$.
I guess it is ... |
$f\left(\sum X_i\right) \leq \sum f(X_i)$, where $X_i\gt 0$; for what functions is this true ? |
I am working through an economics [paper][1] and I need to take the derivative of the following function:
$h\left(\overline{\omega}\right) = \int^{\infty}_{\overline{\omega}} \omega \Phi \left(d\omega\right)$
Even though I don't understand it well, I can do the derivative for the case
$g\left(\overline{\omega}... |
What does this expression mean?
$$f\in C^2[a,b]$$
More specifically, I don't know what $C$ means. |
I play a game known as Bejeweled (I'm sure some of you have heard of it) which has the following properties:
> Given a planar graph with every vertex having degree 4, any path of 3 (or more) contiguous colors is considered "complete." Additionally, any two adjacent vertices may be "swapped" to produce said path. I... |
What is the minimal coloring of a planar graph with swaps? |
Will every $n^{th}$ root of $2$ be an irrational number? If yes, how can I prove that?
|
Is $n^{th}$ root of 2 an irrational number? |
Given $N$ random iid variables, $X_1, \ldots, X_N$, with a uniform probability distribution on $[0, 1)$ what is the distribution of $\displaystyle \max_{i = 1 \ldots N}(X_i)$? |
There's a brief note in [this book][1] on how Khayyam bumped into having to solve a cubic.
I'll only make the note that you should remember the context of the time: there was no concept of negative, much less complex, solutions. Corresponding to our current Cartesian system, Khayyam only looked at intersections in t... |
There's a brief note in [this book][1] on how Khayyam bumped into having to solve a cubic.
I'll only make the note that you should remember the context of the time: there was no concept of negative, much less complex, solutions. Corresponding to our current Cartesian system, Khayyam only looked at intersections in t... |
Let $$p_n(x)e^{-x^2}$$ be the $n$th derivative of $$e^{-x^2}.$$
Find a formula for $p_n(x)$. We have $p_1(x)=-2x, p_2(x)=4x^2-2$, etc. But what is the general formula for $p_n$? |
I am reminded of this question that appeared in a regional Physics Olympiad I had appeared. I was wondering if there is a "mathematical" way of doing it.
If you start from a point $A$ at midnight along a specified path down a mountain and reach a specified point $B$ exactly 24 hours later. At this point you reverse ... |
There's a brief note in [this book][1] on how Khayyam bumped into having to solve a cubic.
I'll only make the note that you should remember the context of the time: there was no concept of negative, much less complex, solutions. Corresponding to our current Cartesian system, Khayyam only looked at intersections in t... |
The first thing you have to understand is that notes are not uniquely defined. Everything depends on what tuning you use. I'll assume we're talking about <a href="http://en.wikipedia.org/wiki/Equal_temperament">equal temperament</a> here. In equal temperament, a half-step is the same as a frequency ratio of $\sqrt[1... |
Can anyone recommend a good self-contained reference for completion of rationals to get reals using Cauchy sequences? |
I made the following observation following a suggestion from a friend. This question is about the Microsoft calculator program implemented on Windows. Since I don't have access to any other calculator at the moment, I cannot confirm whether this is a general feature.
If you enter an integer in the calculator progra... |
Let f: A -> B. The compliments indicated below are taken within A or B. \begin{equation}
(eg. \overline{S} = A - S )
\end{equation}
Prove that NOT F(S) is a subset of F(NOT S) for all S that is a subset of A if and only if f is surjective.
\begin{equation}
\overline{F(S)} \subset f(\overline{S})
\end{equation... |
**First:** You **don't** choose $\epsilon$!
In an $\epsilon$-$\delta$ proof, you begin by taking an *arbitrary* $\epsilon$; you then need to show that it is possible to choose a $\delta$ that "works" for *that* $\epsilon$. So you are going at things *exactly* backwards, by first trying to work with $\delta$.
**S... |
Let $f\colon A\to B$. The complements indicated below are taken within $A$ or $B$.
Prove that $f(S)^c\subset f(S^c)$ for all $S\subset A$ if and only if $f$ is surjective.
------------
So I need to prove $f(A) = B$ right?
How can I prove this?
|
Let $f:A \rightarrow B$.
The compliments indicated below are taken within $A$ or $B$ i.e.
\begin{equation}
\overline{S} = A - S
\end{equation}
$\overline{f(S)} \subseteq f(\overline{S})$, $\forall S \subseteq A$, if and only if $f$ is surjective.
So i need to prove $f(A) = B$ right?
How can I prove t... |
Let $f:A \rightarrow B$.
The compliments indicated below are taken within $A$ or $B$ i.e.
\begin{equation}
\overline{S} = A - S
\end{equation}
$\overline{f(S)} \subseteq f(\overline{S})$, $\forall S \subseteq A$, if and only if $f$ is surjective.
So I need to prove $f(A) = B$ right?
How can I prove t... |
What does "$f\in C^2[a,b]$" mean? |
Let $f:A \rightarrow B$.
The complements indicated below are taken within $A$ or $B$.
I need to prove that
$f(S)^c \subseteq f(S^c)$, $\forall S \subseteq A$, if and only if $f$ is surjective.
So I need to prove $f(A) = B$ right?
How can I prove this?
|
> Mathematically, is there any difference between white and black notes, or do we make the distinction just for historical reasons?
There is no mathematical difference between the white and black notes. Adjacent notes on modern piano keyboards are typically tuned 1/12 of an octave apart. Quiaochu explains this most ... |
J.M., since when is 24 decimal digits accuracy considered a bad approximation?
Original poster, they could show sqrt(5)**2 as
5.000000000000000000000001232
but since the display probably has fewer display digits than this, they chop it to "5" as that is the closest representation in the allowed number of digit... |
Pages [142][1]-154 of Stoll's *Set theory and logic* looks like a good self-contained treatment. The chapter starts on [page 130][2] and is called "The extension of the natural numbers to the real numbers."
[1]: http://books.google.com/books?id=3-nrPB7BQKMC&lpg=PP1&pg=PA142#v=onepage&q&f=false
[2]: http://bo... |
Let Q= {q_1,...,q_n} be a set of permutations of some finite set X. Let
Q' be {q in Q^* | q=1 in the symmetric group on X}. Is Q' a regular language? |
Construct a graph $\Gamma$ having as vertices the elements of the symmetric group $S(X)$, and for each $g\in S(X)$ and each $i\in\{1,\dots,n\}$ an arrow from the vertex $g$ to the vertex $q_ig$ labeled $q_i$. Upgrade $\Gamma$ into a deterministic automatic so that the initial state is the identity element $e$ of $S(X)$... |
Let's consider a 2x2 linear system:
A **u** = **b**
The solution will still be the same even after we interchange the rows in A and B. I know this to be true because algebraically, we will get the same set of equations before and after the row interchange.
However, the vectors in columns of A and B are d... |
I was assisting a TA for an introductory calculus class with the following limit,
$$\lim_{x \rightarrow \infty} \left(\frac{20x}{20x+4}\right)^{8x}$$
and I came to simple solution which involved evaluating the "reciprocal" limit
$$\lim_{z \rightarrow 0} \left(\frac{1}{1+\frac{z}{5}}\right)^{8/z}$$
by using ... |
Where does this sequence converge?
sqrt(7),sqrt(7+sqrt(7)),sqrt(7+sqrt(7+sqrt(7))),...
|
I have the following constraint functions:
$$g_{i_{min}} \leq y_{i+1}-y_{i} \leq g_{i_{max}}$$
$$y_{i_{max}}-y_{i} \geq h_{i}$$
$$v_{i_{min}} \leq \Biggl[\frac{(y_{i+1}-y_{i})^{1/2} (h_{i}+a)^{2/3}}{(h_{i}+b)^{2/3}}\Biggr] \leq v_{i_{max}}$$
$$Q_{i_{min}} \leq \Biggl[\frac{(y_{i+1}-y_{i})^{1/2} (h_{i}+a)^{5... |
I have the following constraint functions:
$$g_{i_{min}} \leq y_{i+1}-y_{i} \leq g_{i_{max}}$$
$$y_{i_{max}}-y_{i} \geq h_{i}$$
$$v_{i_{min}} \leq \Biggl[\frac{(y_{i+1}-y_{i})^{1/2} (h_{i}+a)^{2/3}}{(h_{i}+b)^{2/3}}\Biggr] \leq v_{i_{max}}$$
$$Q_{i_{min}} \leq \Biggl[\frac{(y_{i+1}-y_{i})^{1/2} (h_{i}+a)^{5... |
Lets imagine that we have some experiments. Each experiment may result in one of the outcomes: A, B, C. So we have probabilities distribution for each experiment $P_A, P_B, P_C$ which is context-dependent.
E.g.:
1. $Context_1 \Rightarrow \{P_A^1, P_B^1, P_C^1\},$ experimental outcome is A
2. $Context_2 \Righ... |
What is the standard method to estimate probabilities of events based on observations? |
The [maximum likelihood estimate][1] is a typical method for estimating the probability distribution given some outcomes. To describe it simply, you assume that your probability distribution you are sampling from is actually one from some set $\Omega$, and you are trying to find which one fits best the data (i.e. whic... |
I hope this question is not off-topic, but I'm currently discovering the existence of mixing processes, which attempt to express the dependence between random variables using miscellaneous coefficients ($\alpha$, $\beta$, $\phi$ and $\rho$ are the ones I've met so far).
Are there any references you would advise that... |
This is my first post here. Apologies if there are any misformats or mis-tagging.
Also if the post doesn't belong here, sorry, and I will delete it.
I derived this formula from a probability calculation. How would I go about finding the maximum value for the following?
$\frac{T}{S}\frac{S!}{(S-D)!}\frac{(T-D)!... |
This is my first post here. Apologies if there are any misformats or mis-tagging.
Also if the post doesn't belong here, sorry, and I will delete it.
I derived this formula from a probability calculation. How would I go about finding the maximum value for the following?
$\frac{T}{S}\frac{S!}{(S-D)!}\frac{(T-D)!... |
> **Can a finite group G have a cyclic Sylow 3-subgroup of order 9, such that the intersection of the Sylow 3-subgroups of G has order exactly 3?**
In classifying the finite groups with cyclic Sylow 3-subgroups of order 9, it seems reasonable to split into cases based on the size of the 3-core (the intersection of t... |
I am learning group theory, and while learning automorphisms, I came across conjugation as an example in many textbooks. Though the definition itself, (and when considering the case of abelian groups), it seems pretty innocent, I have to admit that I have no intuition about whats happening, or why even such a map is im... |
Scott Aaronson once basically did this for a bunch of theorems in computer science <a href="http://scottaaronson.com/blog/?p=152">here</a> and <a href="http://scottaaronson.com/blog/?p=153">here</a>. I particularly like this one:
> Suppose a baby is given some random examples of grammatical and ungrammatical senten... |
One way to think about conjugation is as a generalization of changing coordinates when rewriting matrices or, from a physical point of view, as the process of "seeing a group from a different perspective."
Let's make this precise. Let $G$ be a group and let $X$ be a set on which $G$ acts faithfully. For example, i... |
One way to think about conjugation is as a generalization of changing coordinates when rewriting matrices or, from a physical point of view, as the process of "seeing a group from a different perspective."
Let's make this precise. Let $G$ be a group and let $X$ be a set on which $G$ acts faithfully. For example, i... |
> **Can a finite group G have a cyclic Sylow 3-subgroup of order 9, such that the intersection of the Sylow 3-subgroups of G has order exactly 3, without having non-identity normal subgroups of order coprime to 3?**
In classifying the finite groups with cyclic Sylow 3-subgroups of order 9, it seems reasonable to spl... |
In other words, if X = Y in distribution, is it true that EX = EY?
I think this must be true, but I've tried to prove it a few times and I always get stuck.
Thanks in advance for any hints or reference. |
Does equality in distribution imply equality of expected value? |
How would one go about showing how many solutions the following congruence has?
x^2 + y^2 = 23 mod 93 |
> **Can a finite group G have a cyclic Sylow 3-subgroup of order 9, such that the intersection of the Sylow 3-subgroups of G has order exactly 3, without having non-identity normal subgroups of order coprime to 3?**
In classifying the finite groups with cyclic Sylow 3-subgroups of order 9, it seems reasonable to spl... |
Where does this sequence converge?
$\sqrt{7},\sqrt{7+\sqrt{7}},\sqrt{7+\sqrt{7+\sqrt{7}}}$,...
|
This is a start:
You would like to know for which y it holds that 23 - y^2 is a square modulo 93. As 93 is 31*3, you can find these y by finding the squares of the form 23 - y^2 modulo 3 and modulo 31. Can you go on from there? |
How would one go about showing how many solutions the following congruence has?
$$x^2 + y^2 \equiv 23 \pmod{93}.$$ |
My homework question: Show that all zeros of p(z)=z^4 + 6z + 3 lie in the circle of radius 2 centered at the origin.
I know p(z) has a zero-count of 4 by using the Fund. Th. of Alg. Then
using the Local Representation Th; the ∫ n/z+a = 4(2i∏). I am assuming a=0 since we are centered at the origin. I apologize fo... |
If you are working modulo a prime $p$, then you can obtain the result using the Legendre symbol or by simple trial and error if it is small. For instance, working modulo $3$, as CJost suggests, you are trying to solve $x^2 + y^2 \equiv 23 \equiv 2 \pmod{3}$. then you want to solve $x^2 \equiv 2-y^2\pmod{3}$. Since $y^2... |
> **Possible Duplicate:**
> [$a^{1/2}$ is either an integer or an irrational number.](http://math.stackexchange.com/questions/4467/a1-2-is-either-an-integer-or-an-irrational-number)
<!-- End of automatically inserted text -->
Will every $n^{th}$ root of $2$ be an irrational number? If yes, how can I prove that?
|
**Hint:** This kind of questions are usually handled using [Rouche's Theorem](http://en.wikipedia.org/wiki/Rouch%C3%A9%27s_theorem). I suggest you look it up in the wikipedia article, where you can see an example of its usage. Also here's an [example](http://books.google.co.cr/books?id=wDsrQTPUaSsC&pg=PA77&dq=rouche%27... |
With help from Maple, I got
$$\left(\frac{ax+by+cz}{x-y}\right)^2+\left(\frac{ay+bz+cx}{y-z}\right)^2+\left(\frac{az+bx+cy}{z-x}\right)^2-(c-a)^2-(c-b)^2$$
equal to
$$\frac{(c(x^3+y^3+z^3)+(a-c)(x^2y+y^2z+z^2x)+(b-c)(x^2z+y^2x+z^2y)-3(a+b-c)xyz)^2}{(x-y)^2(y-z)^2(x-z)^2}$$ which of course is $\ge 0$.
But with n... |
My homework question: Show that all zeros of $p(z)=z^4 + 6z + 3$ lie in the circle of radius $2$ centered at the origin.
I know $p(z)$ has a zero-count of $4$ by using the Fundamental Theorem of Algebra. Then using the Local Representation Theorem the $\int \frac{n}{z+a} = 4(2 \pi i)$. I am assuming $a=0$ since we... |
Are [order][1] and [binary relation][2] the same thing, or order is just some special binary relation? If it is the latter, what kinds of properties distinguish an order from a general binary relation?
Thanks and regards!
[1]: http://en.wikipedia.org/wiki/Order_theory
[2]: http://en.wikipedia.org/wiki/Bina... |
Are [order][1] and [binary relation][2] the same thing, or order is just some special binary relation? If it is the latter, what kinds of properties distinguish an order from a general binary relation?
Thanks and regards!
[1]: http://en.wikipedia.org/wiki/Order_theory
[2]: http://en.wikipedia.org/wiki/Bina... |
Are [order][1] and [binary relation][2] the same thing, or order is just some special binary relation? If it is the latter, what kinds of properties distinguish an order from a general binary relation?
Thanks and regards!
*****************
Sorry for the confusion already caused. What I thought was that a par... |
If you are working modulo a prime $p$, then you can obtain the result using the Legendre symbol or by simple trial and error if it is small. For instance, working modulo $3$, as CJost suggests, you are trying to solve $x^2 + y^2 \equiv 23 \equiv 2 \pmod{3}$. then you want to solve $x^2 \equiv 2-y^2\pmod{3}$. Since $y^2... |
Are [order][1] and [binary relation][2] the same thing, or order is just some special binary relation? If it is the latter, what kinds of properties distinguish an order from a general binary relation?
Thanks and regards!
*****************
Sorry for the confusion already caused. What I thought was that a par... |
Equations it's on Latex Format, I'm still trying to understand how MathJax works.
Given the following integral:
\int_0^{+\infty } \frac{1}{x \sqrt{x}} \, dx
I'm pretty sure that does not converge.
And if the integral was a indefinite one, also I'm almost sure the result is ln(sqrt(x)) using substitution.
If ... |
Equations it's on Latex Format, I'm still trying to understand how MathJax works.
Given the following integral:
$\int_0^{+\infty } \frac{1}{x \sqrt{x}} \, dx$
I'm pretty sure that does not converge.
And if the integral was a indefinite one, also I'm almost sure the result is ln(sqrt(x)) using substitution.
I... |
Equations it's on Latex Format, I'm still trying to understand how MathJax works.
Given the following integral:
$\int_0^{+\infty } \frac{1}{x \sqrt{x}} \, dx$
I'm pretty sure that does not converge.
And if the integral was a indefinite one, also I'm almost sure the result is $\ln(\sqrt x)$ using substitution... |
This is probably a bit of a boring and ridiculously noob question, so apologies in advance, but:
Is everything in maths essentially built on adding, subtracting, multiplying and dividing? |
I remember reading in RPF's biographical book, "Surely you're joking Mr. Feynman" that he used to have timed contest about writing down the biggest number using standard symbols.
Instead of the time restriction, I was wondering what if we restrict the number of characters, to say $N=10$. What is the largest number ... |
I'm trying to understand the proof for Lemma 4.8 in Lyndon & Schupp Combinatorial Group Theory, page 26, proposition 4.8.
In the very end of this proof, we have: $N \unlhd F$ where $F$ is a finitely generated free group. $[F:N]=p$, $p$ prime. $K \unlhd N$, $K \unlhd F $. $u,v \in N$ such that their images in $N/K$ a... |
Simplified Version: $N \unlhd F$, $a,b \in N$. $[F:N]=p$, prime. $a \sim b$ in $F$ (conjugates). Are $a \sim b$ in $N$? (more inside) |
I'll do one better: I'll write you one (mostly because I actually have already written one). Recall that a first course in analysis typically asserts that there exists a field with an order and the least upper bound property, and you then develop the theory of sequences and series from there. Thus our goal should be to... |
I have the following constraint functions:
$$g_{i_{min}} \leq y_{i+1}-y_{i} \leq g_{i_{max}}$$
$$y_{i_{max}}-y_{i} \geq h_{i}$$
$$v_{i_{min}} \leq \Biggl[\frac{(y_{i+1}-y_{i})^{3} (h_{i}+a)^{4}}{(h_{i}+b)^{4}}\Biggr] \leq v_{i_{max}}$$
$$Q_{i_{min}} \leq \Biggl[\frac{(y_{i+1}-y_{i})^{3} (h_{i}+a)^{10}}{(h_{... |
I'll do one better: I'll write you one (mostly because I actually have already written one). Recall that a first course in analysis typically asserts that there exists a field with an order and the least upper bound property, and you then develop the theory of sequences and series from there. Thus our goal should be to... |
I'm surprised nobody has mentioned **Mathematics and the Imagination** by Edward Kasner and James Newman; originally published in 1940, still <a href="http://www.amazon.com/Mathematics-Imagination-Edward-Kasner/dp/0486417034/ref=sr_1_1?ie=UTF8&qid=1290829129&sr=8-1">available</a> in an edition from Dover. Among other t... |
What does it mean for a polynomial with integer coefficients to have a root in p-adic integers? |
The group $S_n \times S_n \times S_n$ acts on the set of [Latin squares][1] $L$ of order $n$, with $\theta:=(\alpha,\beta,\gamma)$, starting from $L$, permuting the rows by $\alpha$, permuting the columns by $\beta$ and permuting the symbols by $\gamma$. Each $\theta \in S_n \times S_n \times S_n$ is called an _isotop... |
Why is there no Latin square of order 7 with an autotopism group of order 7? |
another way to get this result is to first recall that $S := X_1+X_2$ and $R := X_2/X_1$ are independent. [this is well-known and can be proved by the usual method of derived distributions for two random variables. the mapping $(X_1,X_2) \to (R,S)$ is 1-1 and easily inverted. this is a standard example or exercise in m... |
I'll do one better: I'll write you one (mostly because I actually have already written one). Recall that a first course in analysis typically asserts that there exists a field with an order and the least upper bound property, and you then develop the theory of sequences and series from there. Thus our goal should be to... |
another way to get this result is to first recall that $S := X_1+X_2$ and $R := X_2/X_1$ are independent. [this is well-known and can be proved by the usual method of derived distributions for two random variables. the mapping $(X_1,X_2) \to (R,S)$ is 1-1 and easily inverted. this is a standard example or exercise in m... |
Well, just for fun I thought I'd try to answer the question with the operations restricted to:
- nullary operations $\{0,1,2,\ldots,9\}$ and
- binary operation $*$ (i.e. multiplication).
It turns out that
9999999999
is the greatest in this case. The closest runner-up is:
99999*9999 = 99989000... |
The problem is "Calculating $\oint_{L} \frac{xdy - ydx}{x^2 + y^2}$, where L is a smooth curve that does not pass throught the orignal point".
Here is my solution:
Let $$P(x,y) = \frac{-y}{x^2 + y^2}, Q(x,y) = \frac{x}{x^2 + y^2}$$
Get $$\frac{\partial{P}}{\partial{y}} = \frac{y^2 - x^2}{(x^2 + y^2)^2} = \frac{\... |
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