instruction stringlengths 12 30k |
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How do I solve for $x$ from this equation?
$$-1/x^2 + 9/(4-x-y)^2 = 0$$
I need to get this into $x=$"blah"?
Also. How do you guys get that funky formatting? |
For a university project, I am implementing an automated regression analysis tool.
However, I have very little background in statistics.
So what books / articles / material would you suggest I could use to brush up on this topic, based on your experiences?
Thanks |
Recommended Reading on Regression Analysis? |
Question:
Let a be a quadratic residue to a prime modulus p. Prove a^((p-1)/2) = 1 mod p
My attempt at a solution:
a is a quadratic residue
=> a is a residue class of p which has even index c relative to a primitive root g
=> a = g^c mod p
=> a = g^(2k) mod p where 2k=c
=> g^(2kv) = g^(c) mod p fo... |
My algebra is sooo rusty so this should be an easy one!
Area of a rectangle is
A = W x H
If you're given A (say 150) and a ratio of W:H (say 3:2) how can you calculate W and H? |
I learned of the [plate trick](http://en.wikipedia.org/wiki/Plate_trick) via Wikipedia, which states that this is a demonstration of the fact that SU(2) double-covers SO(3). It also offers a link to an animation of the "belt trick" which is apparently equivalent to the plate trick (I do not yet possess enough reputatio... |
Can somebody explain the plate trick to me? |
I learned of the [plate trick](http://en.wikipedia.org/wiki/Plate_trick) via Wikipedia, which states that this is a demonstration of the fact that SU(2) double-covers SO(3). It also offers a [link][1] to an animation of the "belt trick" which is apparently equivalent to the plate trick (I do not yet possess enough repu... |
I remember a homework question in elementary school. Something like this: Billy and Jane's house is x blocks east and y blocks north of school. Billy walks home by walking east for x blocks and then north for y blocks. Jane decides to take a short cut: she walks alternately a block north and a block east. (There is a p... |
Let $a$ be a natural number.
Suppose
$a^4 \neq 1 \mod 29$
$a^7 \neq 1 \mod 29$
$a^{14} \neq 1 \mod 29$
Calculate the order of $a$ modulo 29.
Using Euler's Criterion, I have $a^{14} = a^{(29-1)/2} \neq 1 \mod 29$.
How would I go about solving this problem? I have not seen this kind of problem befor... |
What do you know about the multiplicative group modulo 29? What are the possible orders of nonzero elements mod 29? (It's not hard to answer if you know the structure of the group; it's also possible to compute directly, since there are only 28 elements to check.)
Once you know the structure, you should be able to... |
Presumably, $\gcd(a,29)=1$; else, $a\equiv 0\pmod{29}$, and all your incongruences hold, and there is no order as Sivaram points out.
From Fermat's Little Theorem, you know that $a^{28}\equiv 1\pmod{29}$.
Now, what is the *smallest* positive integer $j$ such that $a^j\equiv 1\pmod{29}$? It is at most $28$. If we ... |
Question:
Let $a$ be a quadratic residue to a prime modulus $p$. Prove $a^{(p-1)/2} \equiv 1 \pmod{p}$.
My attempt at a solution:
\begin{align*}
&a\text{ is a quadratic residue}\\\
&\Longrightarrow a\text{ is a residue class of $p$ which has even index $c$ relative to a primitive root $g$}\\\
&\Longrigh... |
Bill has succintly told you how to prove the result. But you were also asking for comments on your proposed argument. I will address that.
In line 5, where did the $v$ come from, and what is its role? Notice that you can take $v=1$ and what you write is true. So how is this giving you any information?
In line 9, ... |
This is motivated by looking at the calculus exams of some of my undergraduate students. A recurring mistake is assuming that the derivative of the product of functions is a product of derivatives and the derivative of the quotient of two functions is the quotient of their derivatives.
This might be an ill formed q... |
This is motivated by looking at the calculus exams of some of my undergraduate students. A recurring mistake is assuming that the derivative of the product of functions is a product of derivatives and the derivative of the quotient of two functions is the quotient of their derivatives.
This might be an ill formed q... |
Suppose you want to find $E(X)$ for a non-negative random variable $X$. So you can use $\int_{0}^{\infty} [1-F(x)] \ dx$. To get $1-F(x)$, all you have to do is integrate as follows: $\int_{x}^{\infty} f(t) \ dt$? In other words, instead of integrating from $t=0$ to $t= x$, we can just integrate from $t = x$ to $t = \i... |
What's the name of this stochastic process? |
Is there a simple algorithm for efficient decomposition on this special kind of graph? |
I'm looking for an algorithm to solve this problem:
Given set of n points on 2D euclidean space, create a net of rectilinear edges, so that:
1. Every two points are connected with shortest edge.
2. Sum of all edges is minimal (minimized).
That's clearly a NP-hard problem and it's similar to Steiner tree problem b... |
Is it known whether every finite abelian group is the ideal class group of the ring of integers in some number field? If so, is it still true if we consider only imaginary quadratic fields? |
Is it known whether every finite abelian group is isomorphic to the ideal class group of the ring of integers in some number field? If so, is it still true if we consider only imaginary quadratic fields? |
I want to motivate the theory of $C_0$-semigroups to someone, and the following question was asked:
> What is an example of a non-separable linear PDE?
Preferably a simple one. |
I want to motivate the theory of $C_0$-semigroups to someone, and the following question was asked:
> What is an example of a non-separable linear PDE?
Preferably a simple homogeneous one. |
Suppose $0.6$ of students submit assigned homework on time. In a typical class of $40$ students a professor has assigned homework. If a student submits homework on time, he gets $5$ points. Otherwise he gets $0$ points. Find the variance of the total number of points by all students.
So this is a binomial distribut... |
In Chapter 1 of *Polynomials* by Victor Prasolov, Springer, 2001, the following theorem is proved. (p.3)
> Theorem 1.1.4 (Ostrovsky). Let
> $f(x)=x^{n}-b_{1}x^{n-1}-\cdots -b_{n}$,
> where all the numbers $b_{i}$ are non-negative and at least one of them
> is nonzero. If the greatest common
> divisor of the in... |
i have a question to special cases of Sasaki-manifolds. Let $(M, g, \xi, \eta, \Phi)$ a Sasaki-manifold.
In special case maybe $M=S^{2m+1} \cong \mathbb{C}^{m+1}$. This is a Sasaki manifold. But what is $\Phi$?
Analogous: Let $M=\mathbb{R}^{2m+1}$, it is also Sasaki, but what is $\xi$ or $\Phi$?
Thanks and bes... |
$M$, $N$ are sets, $f: M \rightarrow N$ is a function, $R \subseteq M\times M$ is an equivalence relation on $M$, $\pi: M \rightarrow M/R$ is the canonical projection (i.e. $\pi(x)=[x]_R$), and $\ker f \subseteq M\times M$ is defined as $\{(x,y) \in M \times M: f(x)=f(y)\}$.
Show that $R \subseteq \ker f \Leftrighta... |
I was wondering if there are any general results to this problem. Explicitly, given an $n$-cube and $n$ distinct colors, in how many ways can we paint the faces of the cube such that no two faces with the same color intersect. Here we consider two colorings the same if they can be obtained from one another by rotations... |
$M$, $N$ are sets, $f: M \rightarrow N$ is a function, $R \subseteq M\times M$ is an equivalence relation on $M$, $\pi: M \rightarrow M/R$ is the canonical projection (i.e. $\pi(x)=[x]_R$), and $\ker f \subseteq M\times M$ is defined as $\{(x,y) \in M \times M: f(x)=f(y)\}$.
Show that $R \subseteq \ker f \Leftrighta... |
Your "$\Leftarrow$" proof is a muddled. First, $R$ is *not* necessarily equal to $\mathrm{ker} f$, but that is what you are asserting is happening when you say "$R$ can be defined as...". Second, I don't see you showing that every element of $R$ is in $\mathrm{ker} f$, which is what you need to do.
In that directio... |
<a href="http://mathworld.wolfram.com/CatalansConstant.html">Quite</a> a <a href="http://mathworld.wolfram.com/MertensConstant.html">few</a> mathematical constants are known to arise in several branches of mathematics (more <a href="http://en.wikipedia.org/wiki/Euler_product">here</a>). I have no doubt that they are u... |
Not many people know that WA is also able of giving you all kinds of possible closed forms concerning fundamental constants, e.g.:
http://www.wolframalpha.com/input/?i=2.9299372410244
You can also click "more"... |
When do the Freshman's dream product and quotient rules for differentiation hold? |
I learned of the [plate trick](http://en.wikipedia.org/wiki/Plate_trick) via Wikipedia, which states that this is a demonstration of the fact that SU(2) double-covers SO(3). It also offers a [link][1] to an animation of the "belt trick" which is apparently equivalent to the plate trick. Since I've thought most about th... |
Let $A$ be a commutative ring with identity. If $A$ has finite number of prime ideals $p_1,...p_n$ and moreover $\prod_{i=1}^n p_i^{k_i} = 0$ for some $k_i$. Are the prime ideals necessarily maximal? |
Ring with finitely many prime ideals with an extra condition. Are they maximal? |
I was wondering if there are any general results to this problem. Explicitly, given an $n$-cube and $n$ distinct colors, in how many ways can we paint the faces of the cube. Here we consider two colorings the same if they can be obtained from one another by rotations.
For $n=3$ this number is $57$ by Burnside's lem... |
The rank of a finitely-generated abelian group is the size of the maximal $\mathbb{Z}$-linear subset; loosely this means the number of distinct copies of $\mathbb{Z}$ contained as direct summands. I have come across a situation where a notion somewhat like rank may be useful for finite abelian groups (which obviously a... |
Analogue of rank for finite abelian groups? |
The rank of a finitely-generated abelian group is the size of the maximal $\mathbb{Z}$-linearly-independent subset; loosely this means the number of distinct copies of $\mathbb{Z}$ contained as direct summands. I have come across a situation where a notion somewhat like rank may be useful for finite abelian groups (whi... |
Can I draw a net for a steinmetz solid from two cylinders with a compass?
That is, can we flatten the net? I often make a model using paper and a compass-- it looks about right... is it really a valid method of construction? |
Can I draw a net for a steinmetz solid with a compass? |
No, but the counterexample is trivial. Take any integral domain with finitely many prime ideals which is not a field. For example, the localization $\mathbb{Z}_{(p)}$ of the integers at a prime p. The zero ideal is non-maximal and prime so, trivially, $\prod_{i=1}^np_i=0$. Maybe this isn't exactly what you were meaning... |
When I was trying to find closed-form representations for odd zeta-values, I used $$ \Gamma(z) = \frac{e^{-\gamma \cdot z}}{z} \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{-1} e^{\frac{z}{n}} $$ and rearranged it to $$ \frac{\Gamma(z)}{e^{-\gamma \cdot z}} = \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{-1} e^... |
Uses of Divergent Series and their summation-values in mathematics ? |
When I was trying to find closed-form representations for odd zeta-values, I used $$ \Gamma(z) = \frac{e^{-\gamma \cdot z}}{z} \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{-1} e^{\frac{z}{n}} $$ and rearranged it to $$ \frac{\Gamma(z)}{e^{-\gamma \cdot z}} = \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{-1} e^... |
Alright, I'm trying to figure out how to calculate a critical value using t-distribution in Microsoft Excel... ex. a one-tailed area of 0.05 with 39 degrees of freedom: t=1.685
I know the answer, but how do I get this? I've tried TDIST() TINV() and TTEST() but they all give me different answers. This web calculator... |
Suppose $x \geq 0$, $y \geq 0$ and $0<p<1$. Why is the following inequality true?
$|x^{p}-y^{p}| \leq |x-y|^p$
|
When I was trying to find closed-form representations for odd zeta-values, I used
$$ \Gamma(z) = \frac{e^{-\gamma \cdot z}}{z} \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{-1} e^{\frac{z}{n}} $$
and rearranged it to
$$ \frac{\Gamma(z)}{e^{-\gamma \cdot z}} = \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{... |
A vector $v \in \mathbb{Z}^n$ is primitive if there does not exist some vector $v' \in \mathbb{Z}^n$ and some $k \in \mathbb{Z}$ such that $v = k v'$ and $k \geq 2$.
For a paper I'm writing right now, I'd like to know that a "random" vector in $\mathbb{Z}^n$ is primitive. Let me make this precise.
Let $\|\cdot\|... |
In 'An Introduction to Probability Theory and Applications' by W. Feller I encountered this apparently innocuous problem.
> A throw of twelve dice can result in
> $6^{12}$ different outcomes, to all of
> which we attribute equal
> probabilities. The event that each
> face appears twice can occur in as
> many wa... |
When I was trying to find closed-form representations for odd zeta-values, I used
$$ \Gamma(z) = \frac{e^{-\gamma \cdot z}}{z} \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{-1} e^{\frac{z}{n}} $$
and rearranged it to
$$ \frac{\Gamma(z)}{e^{-\gamma \cdot z}} = \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{... |
Can we determine if a binomial coefficient C(n,p) is even or odd, without calculating its value? (p<n,p and n are positive integers) |
C(n,p): even or odd? |
When I was trying to find closed-form representations for odd zeta-values, I used
$$ \Gamma(z) = \frac{e^{-\gamma \cdot z}}{z} \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{-1} e^{\frac{z}{n}} $$
and rearranged it to
$$ \frac{\Gamma(z)}{e^{-\gamma \cdot z}} = \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{... |
Can we determine if a binomial coefficient $C(n,p)$ is even or odd, without calculating its value?
($p<n$,$p$ and $n$ are positive integers) |
$C(n,p)$: even or odd? |
The case of four dice is not so hard. You require XXYY in some order, where X != Y. So let the first throw be X. Then there are three possibilities: XXYY, XYXY, and XYYX. Each has probability 1/6 x 5/6 x 1/6 = 5/216. So we get 5/72, which is about 0.0694. This agrees very well with your simulation!
The question is, ... |
Can we determine if a binomial coefficient $C(n,p)$ is even or odd, without calculating its value?
($p\lt n$, $p$ and $n$ are positive integers) |
Which of the following mappings are linear?
(a)$\varphi : \mathbb{R} \rightarrow \mathbb{R}^{2}, \varphi(x) = (1 - x, (1 - x)^{2}),$
(b)$\varphi : \mathcal{F}(\mathbb{R},\mathbb{R}) \rightarrow \mathbb{R}, \varphi(f) = f(1),$
(c)$\varphi : \mathbb{R}[t] \rightarrow \mathbb{R}[t], \varphi(f) = t \cdot f', $
... |
which of the following are linear mappings? |
In 'An Introduction to Probability Theory and Applications' by W. Feller I encountered this apparently innocuous problem.
> A throw of twelve dice can result in
> $6^{12}$ different outcomes, to all of
> which we attribute equal
> probabilities. The event that each
> face appears twice can occur in as
> many wa... |
Assume $M$ is a (finitely generated) $A$-module such that $\wedge^n M$ is free of rank $1$ for some $n \geq 1$. Does it follow that $M$ is free of rank $n$? Or at least locally free of rank $n$?
In general, is there a way of characterizing local free modules via exterior powers and tensor products?
|
When I was trying to find closed-form representations for odd zeta-values, I used
$$ \Gamma(z) = \frac{e^{-\gamma \cdot z}}{z} \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{-1} e^{\frac{z}{n}} $$
and rearranged it to
$$ \frac{\Gamma(z)}{e^{-\gamma \cdot z}} = \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{... |
When I was trying to find closed-form representations for odd zeta-values, I used
$$ \Gamma(z) = \frac{e^{-\gamma \cdot z}}{z} \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{-1} e^{\frac{z}{n}} $$
and rearranged it to
$$ \frac{\Gamma(z)}{e^{-\gamma \cdot z}} = \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{... |
Presumably, $\gcd(a,29)=1$; else, $a\equiv 0\pmod{29}$, and all your incongruences hold, and there is no order as Sivaram points out.
From Fermat's Little Theorem, you know that $a^{28}\equiv 1\pmod{29}$.
Now, what is the *smallest* positive integer $j$ such that $a^j\equiv 1\pmod{29}$? It is at most $28$. If we ... |
When I was trying to find closed-form representations for odd zeta-values, I used
$$ \Gamma(z) = \frac{e^{-\gamma \cdot z}}{z} \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{-1} e^{\frac{z}{n}} $$
and rearranged it to
$$ \frac{\Gamma(z)}{e^{-\gamma \cdot z}} = \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{... |
I know it is unknown if there are infinitely many primes of the form $n^2+1$. Is it known if there is a positive integer $k$ such that $|\{n\in\mathbb{Z}:n^2+1 \text{ has at most k prime factors}\}|=\infty$? |
If $f \in L(0,1)$ show that $x^{n} f(x) \in L(0,1)$ for $n \in \mathbb{N}$ and that $\displaystyle \int_{0}^{1} x^{n} f(x) dx \rightarrow 0$ as $n \infty$.
Is the following attempt correct?
Since $0<x<1$ then $0<x^{n}<1$ also $x^{n}$ is measurable (being cts) and $f$ is measurable because it is in $L(0,1)$ so $... |
If $f \in L(0,1)$ show that $x^{n} f(x) \in L(0,1)$ for $n \in \mathbb{N}$ and that $\displaystyle \int_{0}^{1} x^{n} f(x) dx \rightarrow 0$ as $n \to \infty$.
Is the following attempt correct?
Since $0<x<1$ then $0<x^{n}<1$ also $x^{n}$ is measurable (being cts) and $f$ is measurable because it is in $L(0,1)$ so... |
In Chapter 1 of *[Polynomials][1]* by Victor Prasolov, Springer, 2001, the following theorem is proved. (p.3)
> Theorem 1.1.4 (Ostrovsky). Let
> $f(x)=x^{n}-b_{1}x^{n-1}-\cdots -b_{n}$,
> where all the numbers $b_{i}$ are non-negative and at least one of them
> is nonzero. If the greatest common
> divisor of t... |
It is a well-known fact that
$f\colon a \mapsto \left(\begin{array}{cc}
1 & 1\\
0 & 1\\
\end{array}\right), ~ b \mapsto \left(\begin{array}{cc}
1 & 0\\
1 & 1\\
\end{array}\right)$
is a monomorphism from the free monoid generated by $a$ and $b$ to the matrix monoid $\mathbb{Z}^{2\times 2}$.
1. Is there ... |
Stuck with this problem from Zgymund's book.
Suppose that $f_{n} \rightarrow f$ almost everywhere and that $f_{n}, f \in L^{p}$ where $1<p<\infty$. Assume that $||f_{n}||_{p} < \infty$. Prove that:
$\int f_{n}g \rightarrow \int fg$ as $n \rightarrow \infty$ for all $g \in L^{q}$ such that $\frac{1}{p} + \frac{1}... |
Suppose $x \geq 0$, $y \geq 0$ and $0<p<1$. Why is the following inequality true?
$|x^{p}-y^{p}| \leq |x-y|^p$
|
Stuck with this problem from Zgymund's book.
Suppose that $f_{n} \rightarrow f$ almost everywhere and that $f_{n}, f \in L^{p}$ where $1<p<\infty$. Assume that $||f_{n}||_{p} \leq M < \infty$. Prove that:
$\int f_{n}g \rightarrow \int fg$ as $n \rightarrow \infty$ for all $g \in L^{q}$ such that $\frac{1}{p} + \... |
Can I solve this with induction? |
I have taken a look at the following topics:
http://math.stackexchange.com/questions/3069/number-of-infinite-sets-with-different-cardinalities
http://math.stackexchange.com/questions/1467/cardinality-of-all-cardinalities
http://math.stackexchange.com/questions/8013/are-there-uncountably-infinite-orders-of-infi... |
I have taken a look at the following topics:
http://math.stackexchange.com/questions/3069/number-of-infinite-sets-with-different-cardinalities
http://math.stackexchange.com/questions/1467/cardinality-of-all-cardinalities
http://math.stackexchange.com/questions/8013/are-there-uncountably-infinite-orders-of-infi... |
What is the number of Latin rectangles given the number of reduced Latin rectangles? |
How many committees of 5 can be chosen from 10 people if each committee includes Alice and
excludes Bob?
How many committees of 5 can be chosen from 12 people if each committee
(i) includes Alice;
(ii) excludes Bob;
(iii) includes Alice and excludes Bob;
(iv) includes at least one of either Alice or Bob
Plea... |
I was reading Martin Gardner's _Mathematical Games_ column on the Ulam spiral which appeared in the March 1964 issue of _Scientific American_. (The spiral actually featured on the cover of that issue.) Gardner makes the following statement:
>The grid on the cover suggests that throughout the entire number series ... |
Ulam spiral: Is there an "unusual amount of clumping" in prime-rich quadratic polynomials? |
How to find the domain of the function $\sqrt{ \log_{\frac{1}{2}} x}$ ? |
This isn't a complete answer, but just an idea about your
first question. Let $P$ be a rank one projective module
over the commutative ring $A$, and $P^*=\mathrm{Hom}_A(P,A)$
be its dual. Then for $M=P\oplus P^*$, $\bigwedge^2 M\cong
P\otimes_A P^*\cong A$ is free. There must be $P$ for which $M$
isn't free, but I... |
I've got a Python script that grinds out Amicable Pairs by trying out each integer: i, i+1, i+2, ... . I've now found all APs with 1 <= n <= 65 million. At this point I'm finding that it takes 24 minutes per each additional million. To get into much larger n I'll need to find a way to filter out substantial numbers of ... |
Just today I had a bet with my friend over the following problem:
How many winning configurations can you have in a nxn Tic-Tac-Toe game where players win if a they get n/2 in either a row or column, consecutively.
n is even.
For example, in a 4x4 game, players win if a they get 2 of their symbols in either a ... |
I've been having some rather morbid thoughts lately, so, naturally, I decided to share:
Suppose I have a deadly disease, which has a chance of killing me every day that it is left uncured. Also, suppose that there's this operation I can go through, but it is not without its risks: If it is successful, I won't have t... |
To operate or not to operarate? |
Let $S^1$ the unit circle in $\mathbb{R}^2$ and
$$d: S^1\times S^1\to\mathbb{R}$$
$$d(\theta_1,\theta_2) = \left\{
\begin{array}{ll}
|\theta_1-\theta_2| & \mbox{if } |\theta_1-\theta_2|\le \pi \\
2\pi-|\theta_1-\theta_2| & \mbox{else}
\end{array}
\right.$$
I'm trying to prove that... |
It is nature to understand the need for scalar(numbers), but why do we invent vectors? Who invent it and for what? |
Why do we need vectors and who invent it? |
Why do we need vectors and who invented it? |
A case study of a particular application:
Vectors are absolutely necessary for game development, for instance. How is a 3D model, which may be positioned at any point in space and rotated in any direction and is viewed by a camera pointing in an arbitrary direction, actually rendered?
First, the model is stored i... |
It is nature to understand the need for scalar(numbers), but why do we invent vectors? Who invent it and for what?
EDIT: As George Lowther pointed out the problem is too broad, I added the following questions as concrete supplements.
1. It's easy for our human to understand the law of addition of scalar number, b... |
$\lim_{x\to\infty} \frac{e^x}{x^n}$
n is any natural number.
Using L'hopital doesn't make much sense to me. I did find this in the book:
"In a struggle between a power and an exp, the exp wins."
Can I refer that line as an answer? If the fraction would have been flipped, then the limit would be zero. But ... |
Calculus, find the limit, Exp vs Power? |
Willie covered a lot of what I wanted to say; however I'd like to make a little historical digression: before we ever had the concept of a vector, there was the quaternion, William Rowan Hamilton's generalization of the usual complex numbers. They proved very convenient for physical applications, and thus the use of qu... |
Why isn't L'Hôpital a good solution? Just use induction. |
I greatly enjoyed the Lights Out game described here http://en.wikipedia.org/w/index.php?title=Lights_Out_(game)&oldid=395986075 (I am sorry I had to link to an older page because some wikidiot keeps deleting most of the page).
Its mathematical analysis is here: http://www.jaapsch.net/puzzles/lomath.htm (it's just ... |
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