INSTRUCTION stringlengths 61 6.63k | RESPONSE stringlengths 1 11k |
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Factorising certain polynomials During lesson we were given a starter activity which was to try and factorise polynomials and see what happened.
The polynomials were
$x^3-8$,
$x^3-3x^2+ x -3$
and
$x^4 - 16$.
I could not work out what happened to them, and it's bugging me. If anyone can explain then it'd be helpful! Th... | a^3 - b^3 = (a - b)(a^2 + ab + b^2). So x^3 - 8 = (x - 2)(x^2 + 2x + 4).
For x^3 - 3x^2 + x - 3, factor by grouping the first two terms together and taking out the GCF and grouping the second two and doing the same. This should give you
x^2(x - 3) + 1(x - 3). This gives you a greatest common factor of (x - 3) that you ... |
Is restriction of scalars a pullback? I am reading some handwritten notes, and scribbled next to a restriction of scalars functor, are the words "a pullback".
I don't understand why this might be the case.
In particular, consider a field $k$ with a map $\varphi:k\rightarrow A$ for $A$ an associative unital $k$-algebra.... | The sense in which "pullback" is being used here is the same sense it's being used in this Wikipedia article; that is, it's being used to refer to the process of "pulling back" a morphism $A \to \text{End}(R)$ (an $A$-module) along the morphism $k \to A$ to get a morphism $k \to \text{End}(R)$ (a $k$-module) by precomp... |
The characteristic and minimal polynomial of a companion matrix The companion matrix of a monic polynomial $f \in \mathbb F\left[x\right]$ in $1$ variable $x$ over a field $\mathbb F$ plays an important role in understanding the structure of finite dimensional $\mathbb F[x]$-modules.
It is an important fact that the ch... | Suppose your matrix is over a field $\mathbb{F}$. Look at $G = \mathbb F[x]/f$, where $f$ is your polynomial of degree $n$. Then $G$ is a vector space over $\mathbb{F}$, and $C(f)$ is the matrix (with respect to the basis $1,x,x^2,\ldots,x^{n-1}$) corresponding to the linear operator $g \mapsto x \cdot g$.
Since $f = 0... |
Does the series $\sum\limits_{n=1}^{\infty}\frac{\sin(n-\sqrt{n^2+n})}{n}$ converge? I'm just reviewing for my exam tomorow looking at old exams, unfortunately I don't have solutions. Here is a question I found : determine if the series converges or diverges. If it converges find it's limit.
$$\displaystyle \sum\li... | The key here is that $n - \sqrt{n^2 + n}$ converges to $-{1 \over 2}$ as $n$ goes to infinity:
$$n - \sqrt{n^2 + n}= (n - \sqrt{n^2 + n}) \times {n + \sqrt{n^2 + n} \over n + \sqrt{n^2 + n}}$$
$$= {n^2 - (n^2 + n) \over n + \sqrt{n^2 + n}} = -{n \over n + \sqrt{n^2 + n}}$$
$$= -{1 \over 1 + \sqrt{1 + {1 \over n}}}... |
0.246810121416...: Is it a algebraic number? Is it algebraic the number 0.2468101214 ...? (After point, the natural numbers are juxtaposed pairs).
| No, this number is transcendental. The proof by Mahler mentioned in a comment shows this.
A good reference to learn about basic transcendental number theory is the book "Making transcendence transparent: an intuitive approach to classical transcendental number theory", by Edward Burger and Robert Tubbs, Springer-Verla... |
What will be the remainder? I'm stuck with this problem I'm trying to solve from about an hour. Here's the question.
What is the remainder when (3^202)+137 is divided by 101?
There are 4 options -> 36, 45, 56, 11
I want to know the answer of the question with proper and possibly easiest method to solve the problem.
Tha... | HINT $\ 101\ $ is prime so a little Fermat $\rm\ \Rightarrow\ \ 3^{101}\ \equiv\ 3\ \ \Rightarrow\ \ 3^{202}\ \equiv\ \ldots\ (mod\ 101)$
Since your comment reveals you are not familiar with modular arithmetic, here is an alternative.
By Fermat's little theorem $101$ divides $\: 3^{101}-3\: $ so it also divides $\r... |
Solving $2x - \sin 2x = \pi/2$ for $0 < x < \pi/2$ What is $x$ in closed form if $2x-\sin2x=\pi/2$, $x$ in the first quadrant?
| The solution is given by $$\displaystyle x = \pi/4 + D/2$$
where $\displaystyle D$ is the root of $\cos y = y$
The root of $\displaystyle \cos y = y$ is nowadays known as the Dottie Number and apparently has no known "closed form" solution. If you consider this number to be part of your constants, then the above can... |
Why steenrod commute with transgression I'm reading Hatcher's notes on spectral sequences and he mentions that steenrod squares commute with the coboundary operator for pairs (X,A) which would then explain why these operations commute with the transgression. It says it's because
that coboundary operator can be defined... | I realized that your question wasn't exactly about the Steenrod axioms themselves, but about the definition of the coboundary operator involving suspension. In reduced homology, the boundary operator $\partial$ for the pair $(X,A)$ (where the inclusion $i:A\rightarrow X$ is a cofibration) can be defined to come from t... |
When does a subbase of a base generate the same topology? Suppose that $\mathcal{B}$ is a base for a topology on a space $X$. Is there a nice way of thinking about how we can modify $\mathcal{B}$ (for instance, to simplify computations) without changing the topology it generates? It seems non-trivial to compute the top... | Since a topology generated by a base consists of open sets that are union of basic open sets, you may drop, from a given base, any open set that is a union of open sets in the same base and get a smaller base.
|
How to compute a 2x2 Homography out of 3 corresponding points? In 1D projective geometry,
I want to compute the 2x2 Homography matrix $H$ (in homogeneous coordinates), given 3 pairs of corresponding points.
i.e. I want to find H such that:
$$\left(\begin{array}{cc}
h_{11} & h_{12}\\
h_{21} & h_{22}\end{array}\right)... | Your answer is mathematically correct, however I figured out another way to solve this equation, which leads to a simpler result.
I applied the technique for 2D projective geometry, which is described here to the 1D case and it works out fine.
|
What type of triangle satisfies: $8R^2 = a^2 + b^2 + c^2 $? In a $\displaystyle\bigtriangleup$ ABC,R is circumradius and $\displaystyle 8R^2 = a^2 + b^2 + c^2 $ , then $\displaystyle\bigtriangleup$ ABC is of which type ?
| $$\sin^2A+\sin^2B+\sin^2C$$
$$=1-(\cos^2A-\sin^2B)+1-\cos^2C$$
$$=2-\cos(A+B)\cos(A-B)-\cos C\cdot\cos C$$
$$=2-\cos(\pi-C)\cos(A-B)-\cos\{\pi-(A+B)\}\cdot\cos C$$
$$=2+\cos C\cos(A-B)+\cos(A+B)\cdot\cos C\text{ as }\cos(\pi-x)=-\cos x$$
$$=2+\cos C\{\cos(A-B)+\cos(A+B)\}$$
$$=2+2\cos A\cos B\cos C$$
$(1)$ If $2+2\cos ... |
Moments and non-negative random variables? I want to prove that for non-negative random variables with distribution F:
$$E(X^{n}) = \int_0^\infty n x^{n-1} P(\{X≥x\}) dx$$
Is the following proof correct?
$$R.H.S = \int_0^\infty n x^{n-1} P(\{X≥x\}) dx = \int_0^\infty n x^{n-1} (1-F(x)) dx$$
using integration by parts... | Here's another way. (As the others point out, the statement is true if $E[X^n]$ actually exists.)
Let $Y = X^n$. $Y$ is non-negative if $X$ is.
We know
$$E[Y] = \int_0^{\infty} P(Y \geq t) dt,$$
so
$$E[X^n] = \int_0^{\infty} P(X^n \geq t) dt.$$
Then, perform the change of variables $t = x^n$. This immediately yie... |
Partitioning a graph (clustering of point sets in 2 dimensions) I am given $n$ points in 2D.(Each of say approximately equal weight). I want to partition it into $m$ clusters ($m$ can be anything and it is input by the user) in such a way that the center of mass of each cluster is "far" from center of mass of all other... | The keyword is "clustering" as mentioned in Moron's answer. Any problem of this type will be NP-complete. In practice, K-means is not bad in its runtime or (depending very much on the application) its results. Like the simplex algorithm for linear programming, it can take exponential time in the worst case, but its ... |
Let $a$ be a quadratic residue modulo $p$. Prove $a^{(p-1)/2} \equiv 1 \bmod p$.
Question:
Let $a$ be a quadratic residue to a prime modulus $p$. Prove $a^{(p-1)/2} \equiv 1 \pmod{p}$.
My attempt at a solution:
\begin{align*}
&a\text{ is a quadratic residue}\\\
&\Longrightarrow a\text{ is a residue class of $p$ w... | Bill has succintly told you how to prove the result. But you were also asking for comments on your proposed argument. I will address that.
In line 5, where did the $v$ come from, and what is its role? Notice that you can take $v=1$ and what you write is true. So how is this giving you any information?
In line 9, you ar... |
Integral with Tanh: $\int_{0}^{b} \tanh(x)/x \mathrm{d} x$ . What would be the solution when 'b' does not tends to infinity though a large one? two integrals that got my attention because I really don't know how to solve them. They are a solution to the CDW equation below critical temperature of a 1D strongly correlate... | For $x$ large, $\tanh x$ is very close to $1$. Therefore for large $b$, $$\int_0^b \frac{\tanh x}{x} \, \mathrm{d}x \approx C + \int^b \frac{\mathrm{d}x}{x} = C' + \log b.$$ You can prove it rigorously and obtain a nice error bound if you wish. Your post indicates a specific value of $C'$, but for large $b$, any two "c... |
Convergence of integrals in $L^p$ Stuck with this problem from Zgymund's book.
Suppose that $f_{n} \rightarrow f$ almost everywhere and that $f_{n}, f \in L^{p}$ where $1<p<\infty$. Assume that $\|f_{n}\|_{p} \leq M < \infty$. Prove that:
$\int f_{n}g \rightarrow \int fg$ as $n \rightarrow \infty$ for all $g \in L^{q}... | Here is a proof which is not based on Egoroff's theorem. As Jonas T points out in an other answer, Fatou's Lemma imply
$$\int|f|^p=\int\lim |f_n|^p =\int\liminf |f_n|^p \le \liminf\int |f_n|^p \le M^p$$
and hence $f\in L^p$.
At this point we may assume $f=0$ (consider the problem with $f_n$ replaced by $f_n-f$), and... |
Calculus, find the limit, Exp vs Power? $\lim_{x\to\infty} \frac{e^x}{x^n}$
n is any natural number.
Using L'hopital doesn't make much sense to me. I did find this in the book:
"In a struggle between a power and an exp, the exp wins."
Can I refer that line as an answer? If the fraction would have been flipped, then ... | HINT: One way of looking at this would be: $$\frac{1}{x^{n}} \biggl[ \biggl(1 + \frac{x}{1!} + \frac{x^{2}}{2!} + \cdots + \frac{x^{n}}{n!}\biggr) + \frac{x^{n+1}}{(n+1)!} + \cdots \biggr]$$
I hope you understand why i put the brackets inside those terms.
|
Yet another inequality: $|a+b|^p<2^p(|a|^p+|b|^p)$ Let $a$ and $b$ be real numbers and $p>0$. What is the best way to prove that $|a+b|^p<2^p(|a|^p+|b|^p)$?
| Well,
\begin{align*}
|a + b|^p \leq (|a| + |b|)^p &\leq 2^p \text{max}\{|a|^p, |b|^p\}\\
&\leq 2^{p - 1} (|a|^p + |b|^p - |a^p - b^p|)\\
&\leq 2^{p - 1} (|a|^p + |b|^p)
\end{align*}
|
Zero to the zero power – is $0^0=1$? Could someone provide me with a good explanation of why $0^0=1$?
My train of thought:
$x>0$
$0^x=0^{x-0}=0^x/0^0$, so
$0^0=0^x/0^x=\,?$
Possible answers:
*
*$0^0\cdot0^x=1\cdot0^0$, so $0^0=1$
*$0^0=0^x/0^x=0/0$, which is undefined
PS. I've read the explanation on mathforum.org,... | A clear and intuitive answer can be provided by ZFC Set-Theory. As described in Enderton's 'Elements of Set Theory (available free for viewing here; see pdf-page 151): http://sistemas.fciencias.unam.mx/~lokylog/images/stories/Alexandria/Teoria%20de%20Conjuntos%20Basicos/Enderton%20H.B_Elements%20of%20Set%20Theory.pdf, ... |
Solving the recurrence relation that contains summation of nth term $$T(n)=1+2\sum_{i=1}^{n-1}T(i) , \quad n > 1$$
$$T(1)=1$$
any hint or how to solve?
| Using a spreadsheet, I note that $T(n)=3^{(n-1)}$ This is easily verified by induction.
$T(1)=1=3^0$.
Then if it is true up to $n$, $$T(n+1)=1+2\sum_{i=0}^{n-1}3^i=1+2\frac{3^n-1}{3-1}$$
|
density of 3D Gaussian distribution For a 2D Gaussian distribution with
$$ \mu = \begin{pmatrix} \mu_x \\ \mu_y \end{pmatrix}, \quad \Sigma = \begin{pmatrix} \sigma_x^2 & \rho \sigma_x \sigma_y \\ \rho \sigma_x \sigma_y & \sigma_y^2 \end{pmatrix},
$$
its probability density function is
$$
f(x,y) = \frac{1}{2 \pi... | There is a standard, general formula for the density of the joint normal (or multivariate normal) distrubution of dimension $n$, provided that the ($n \times n$) covariance matrix $\Sigma$ is non-singular (see, e.g., this or this). In particular, you can apply for $n=3$. When the covariance matrix is singular, the dist... |
Curve of a fixed point of a conic compelled to pass through 2 points Suppose that in the plane a given conic curve is compelled to pass through two fixed points of that plane.
What are the curves covered by a fixed point of the conic, its center (for an ellipse), its focus, etc. ?
(I apologize for the bad English ...) ... | First, a few animations:
These were generated by a parabola with focal length $a=1$ and distance between two points $c=5$. The first one has the focus of the parabola as the tracing point, while the second one has the vertex as tracing point.
Now, for the mathematics: using whuber's and Blue's comments as a possible... |
Why would I want to multiply two polynomials? I'm hoping that this isn't such a basic question that it gets completely laughed off the site, but why would I want to multiply two polynomials together?
I flipped through some algebra books and have googled around a bit, and whenever they introduce polynomial multiplicati... | When you take calculus, you will need to factor a polynomial p as a product of two polynomials a and b. If you know how polynomial multiplication works, then finding factorizations is easier. Learn how to multiply now so that you can factor easily later. :)
|
When functions commute under composition Today I was thinking about composition of functions. It has nice properties, its always associative, there is an identity, and if we restrict to bijective functions then we have an inverse.
But then I thought about commutativity. My first intuition was that bijective self maps... | This question may also be related to how certain functions behave under functions of their variables. In this context, the property of commuting with binary operators, such as addition and multiplication, can be used to define classes of functions:
*
*additive commutation: if $g(x, y) = x + y$, then $f\big(g(x, y)\b... |
If there are $200$ students in the library, how many ways are there for them to be split among the floors of the library if there are $6$ floors? Need help studying for an exam.
Practice Question:
If there are $200$ students in the library, how many ways are there for them to be split among the floors of the library if... | Note that if they are distinguishable then the number of ways is given by $6^{200}$ since each of the 200 students have $6$ choices of floors.
However, we are given that the students are indistinguishable.
Hence, we are essentially interested in solving $a_1 + a_2 + a_3 + a_4 + a_5 + a_6 = 200$, where $a_i$ denotes the... |
Find formula from values Is there any "algorithm" or steps to follow to get a formula from a table of values.
Example:
Using this values:
X Result
1 3
2 5
3 7
4 9
I'd like to obtain:
Result = 2X+1
Edit
Maybe using excel?
Edit 2
Additional info... | (This is way too complicated to use it here, one can always expect a desired polynomial that fits all the points.)
One of the possible algorithm is Langrange Interpolating Polynomial.
For a polynomial $P(n)$ of degree $(n-1)$ passes through $n$ points:
$$(x_1,y_1=f(x_1)),\ldots,(x_n,y_n=f(x_n))$$
We have
$$P(x)=\sum_{... |
conversion of a powerseries $-3x+4x^2-5x^3+\ldots $ into $ -2+\frac 1 x - 0 - \frac 1 {x^3} + \ldots $ This is initially a funny question, because I've found this on old notes but I do not find/recover my own derivation... But then the question is more general.
Q1:
I considered the function
$ f(x) = - \frac {2x^2+... | Divide the numerator and denominator of $f(x)$ by $x^2$ and set $y=1/x$ then expand for $y$ and you have your expansion at infinity.
|
Algebraic Identity $a^{n}-b^{n} = (a-b) \sum\limits_{k=0}^{n-1} a^{k}b^{n-1-k}$ Prove the following: $\displaystyle a^{n}-b^{n} = (a-b) \sum\limits_{k=0}^{n-1} a^{k}b^{n-1-k}$.
So one could use induction on $n$? Could one also use trichotomy or some type of combinatorial argument?
| You can apply Ruffini's rule. Here is a copy from my Algebra text book (Compêndio de Álgebra, VI, by Sebastião e Silva and Silva Paulo) where the following formula is obtained:
$x^n-a^n\equiv (x-a)(x^{n-1}+ax^{n-2}+a^2x^{n-3}+\cdots +a^{n-2}x+a^{n-1}).$
Translation: The Ruffini's rule can be used to find the quotient ... |
Rational Numbers and Uniqueness Let $x$ be a positive rational number of the form $\displaystyle x = \sum\limits_{k=1}^{n} \frac{a_k}{k!}$ where each $a_k$ is a nonnegative integer with $a_k \leq k-1$ for $k \geq 2$ and $a_n >0$. Prove that $a_1 = [x]$, $a_k = [k!x]-k[(k-1)!x]$ for $k = 2, \dots, n$, and that $n$ is t... | Since $a_k \le k-1$ for $k \ge 2$ we have
$$ \lfloor x \rfloor = \left\lfloor \sum_{k=1}^n \frac{a_k}{k!} \right\rfloor \le a_1
+ \left\lfloor \sum_{k=2}^n \frac{k-1}{k!} \right\rfloor = a_1 $$
as the latter term is $0$ since $ \sum_{k=2}^n \frac{k-1}{k!} = \sum_{k=2}^n \lbrace \frac{1}{(k-1)!} - \frac{1}{k!} \rbrace... |
Do all manifolds have a densely defined chart? Let $M$ be a smooth connected manifold. Is it always possible to find a connected dense open subset $U$ of $M$ which is diffeomorphic to an open subset of R$^n$?
If we don't require $U$ to be connected, the answer is yes: it is enough to construct a countable collection o... | Depending on what you consider a manifold, the long line may be a counterexample.
And for a non-connected manifold, surely the answer is no? Take your favorite smooth manifold, and take a disjoint union of more than $\mathfrak{c}$ copies of it. Again, unless your definition of "manifold" rules this out (by assuming s... |
Proof by induction $\frac1{1 \cdot 2} + \frac1{2 \cdot 3} + \frac1{3 \cdot 4} + \cdots + \frac1{n \cdot (n+1)} = \frac{n}{n+1}$ Need some help on following induction problem:
$$\dfrac1{1 \cdot 2} + \dfrac1{2 \cdot 3} + \dfrac1{3 \cdot 4} + \cdots + \dfrac1{n \cdot (n+1)} = \dfrac{n}{n+1}$$
| Every question of the form: prove by induction that
$$\sum_{k=1}^n f(k)=g(n)$$
can be done by verifying two facts about the functions
$f$ and $g$:
*
*$f(1)=g(1)$
and
*$g(n+1)-g(n)=f(n+1)$.
|
$\epsilon$-$\delta$ limit proof, $\lim_{x \to 2} \frac{x^{2}-2x+9}{x+1}$ Prove that $\lim\limits_{x \to 2} \frac{x^{2}-2x+9}{x+1}$ using an epsilon delta proof.
So I have most of the work done. I choose $\delta = min{\frac{1}{2}, y}$,
$f(x)$ factors out to $\frac{|x-3||x-2|}{|x+1|}$
But $|x-3| \lt \frac{3}{2}$ for $... | I'm going to go out on a limb and guess that you're trying to show the limit is 3 and that $f(x) = {x^2 - 2x + 9 \over x + 1} - 3$. I suggest trying to translate what you've done into the fact that $|{x^2 - 2x + 9 \over x + 1} - 3| < {3 \over 5}|x - 2|$ whenever $|x - 2| < {1 \over 2}$.
This means that if you choose ... |
Elementary Row Operations - Interchange a Matrix's rows Let's consider a $2\times 2$ linear system:
$$
A\bf{u} = b
$$
The solution will still be the same even after we interchange the rows in $A$ and $B$. I know this to be true because algebraically, we will get the same set of equations before and after the row inter... | Let us consider a $2 \times 2$ example. We will then extend this higher dimensions.
Let $$A = \begin{bmatrix}A_{11} & A_{12}\\A_{21} & A_{22} \end{bmatrix}$$
$$b = \begin{bmatrix}b_1 \\b_2 \end{bmatrix}$$
So you now want to solve $Ax_1 = b$.
$x_1$ is given by $A^{-1}b$.
Now you swap the two rows of $A$ and $b$. Call ... |
Guidance on a Complex Analysis question My homework question: Show that all zeros of $$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... | Hint: This kind of questions are usually handled using Rouche's 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.
The key is choosing wisely another function $f(z)$ with which to compare in the inequality in Rouche's theorem and such that you ... |
What are the conditions for existence of the Fourier series expansion of a function $f\colon\mathbb{R}\to\mathbb{R}$ What are the conditions for existence of the Fourier series expansion of a function $f\colon\mathbb{R}\to\mathbb{R}$?
| In addition to Carleson's theorem (stated by AD above), which gives a sufficient condition for pointwise convergence almost everywhere, one might also consider the following theorem about uniform convergence:
Suppose $f$ is periodic. Then, if $f$ is $\mathcal{C}^0$ and piecewise $\mathcal{C}^1$, $S_N(f)$ converges uni... |
Given a function $f(x)$ where $x$ is uniformly distributed between $a$ and $b$, how do I find the probability density function of $f$? For example, if $f(x) = \sin x$ and $x$ is uniformly distributed on $[0, \pi]$, how is the equation found that satisfies the probability distribution function of $f(x)$? I imagine the d... | Note that $\sin(x)$ increases from $x = 0$ to $x = {\pi \over 2}$, then decreases from ${\pi \over 2}$ to $\pi$, in a way symmetric about ${\pi \over 2}$. So for a given $0 \leq \alpha \leq 1$, the $x \in [0,\pi]$ for which $\sin(x) \leq \alpha$ consists of two segments, $[0,\beta]$ and $[\pi - \beta, \pi]$, where $\be... |
Is $\lim\limits_{n \to \infty}\frac{1}{n}\left( \cos{\frac{\pi}{n}} + \cos{\frac{2\pi}{n}} + \ldots + \cos{\frac{n\pi}{n}} \right)$ a Riemann sum? This is probably simple, but I'm solving a practice problem:
$\lim_{n \to \infty}\frac{1}{n}\left( \cos{\frac{\pi}{n}} + \cos{\frac{2\pi}{n}} + \ldots +\cos{\frac{n\pi}{n}} ... | The key to this last assertion is the simple fact that $$\cos(\pi - x) = -\cos(x).$$ Said symmetry can be observed directly from the definition of the cosine function via the unit circle.
|
Roots of Legendre Polynomial I was wondering if the following properties of the Legendre polynomials are true in general. They hold for the first ten or fifteen polynomials.
*
*Are the roots always simple (i.e., multiplicity $1$)?
*Except for low-degree cases, the roots can't be calculated exactly, only approximate... | The Abramowitz–Stegun Handbook of Mathematical Functions claims on page 787 that all the roots are simple: http://convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&Page=787
|
Can we reduce the number of states of a Turing Machine? My friend claims that one could reduce the number of states of a given turning machine by somehow blowing up the tape alphabet. He does not have any algorithm though. He only has the intuition.
But I say it's not possible. Else one could arbitrarily keep decreasi... | I think this in the same vein as creating a compression algorithm that will compress any given file, i.e. than we can compress the output again and again, until we reach a single bit that will represent all possible files. Yet, compression algorithms do exist, and they do compress some files.
So, even it the number of ... |
Euler's formula for connected planar graphs Euler's formula for connected planar graphs (i.e. a single connected component) states that $v-e+f=2$. State the generalization of Euler's formula for planar graphs with $k$ connected components (where $k\geq1$).
The correct answer is $v-e+f=1+k$, but I'm not understanding t... | Consider 2 components...
Both are similar components now for first excluding face f4 three faces for each component is considered so for both components
V - E + (F-1) = 1
since, V = 10, E = 12
So, for adding both we get 2V - 2E + 2F-2 = 2
Now we will consider face F4 which will be unbounded face for whole graph and ... |
Help solving a differential equation my Calculus II class is nearing the end of the quarter and we've just started differential equations to get ready for Calculus III. In my homework, I came upon these problems.
One of the problems was:
Find the general solution to the differential equation
$$\frac{dy}{dt} = t^3 + 2t... | In the first question, you are given the derivative in terms of the variable. But in the second question, you are given an expression for the derivative that involves the function. For instance, it would be one thing if you were told $\frac{dy}{dx} = x$ (which would mean that $y = \frac{1}{2}x^2 + C$), and a completely... |
Embedding torus in Euclidean space For $n > 2$, is it possible to embed $\underbrace{S^1 \times \cdots \times S^1}_{n\text{ times}}$ into $\mathbb R^{n+1}$?
| As Aaron Mazel-Gee's comment indicates, this follows from induction. Although you only asked about $n > 2$, it actually holds for $n \geq 1$.
The base case is $n = 1$, i.e. $S^1$ embeds in $\mathbb{R}^2$, which is clear
For the inductive step, suppose that $T^{k-1}$ embeds in $\mathbb{R}^k$. Then $T^k = T^{k-1}\times S... |
Is a regular sequence ordered? A regular sequence is an $n$-fold collection $\{r_1, \cdots, r_n\} \subset R$ of elements of a ring $R$ such that for any $2 \leq i \leq n$, $r_i$ is not a zero divisor of the quotient ring
$$ \frac R {\langle r_1, r_2, \cdots, r_{i-1} \rangle}.$$
Does the order of the $r_i$'s matter? T... | Here is a general result for when any permutations of elements of a regular sequence forms a regular sequence:
Let $A$ be a Noetherian ring and $M$ a finitely generated $A$-module. If $x_1,...,x_n$ be an $M$-sequence s.t. $x_i \in J(A)$ for $1 \leq i \leq n$, where $J(A)$ is the Jacobson radical of $A$, then any permut... |
Typical applications of Fubini's theorem and Radon-Nikodym Can someone please share references (websites or books) where I can find problems related with Fubini's theorem and applications of Radon-Nikodym theorem? I have googled yes and don't find many problems. What are the "typical" problems (if there are any) relate... | One basic example of what you can do with Fubini's theorem is to represent an integral of a function of a function in terms of its distribution function. For instance, there is the formula (for reasonable $\phi$, $f$ basically arbitrary but nonnegative)
$$ \int \phi \circ f d\mu = \int_t \mu(\{ f> t\}) \phi'(t) dt$$ wh... |
For a Planar Graph, Find the Algorithm that Constructs A Cycle Basis, with each Edge Shared by At Most 2 Cycles In a planar graph $G$, one can easily find all the cycle basis by first finding the spanning tree ( any spanning tree would do), and then use the remaining edge to complete cycles. Given Vertex $V$, edge $E$,... | I agree with all the commenters who say that you should just find a planar embedding. However, I happened to stumble across a description that might make you happy:
Let $G$ be a three-connected
planar graph and let $C$ be a cycle.
Let $G/C$ be the graph formed by
contracting $C$ down to a point. Then
$C$ is a ... |
Can $n!$ be a perfect square when $n$ is an integer greater than $1$?
Can $n!$ be a perfect square when $n$ is an integer greater than $1$?
Clearly, when $n$ is prime, $n!$ is not a perfect square because the exponent of $n$ in $n!$ is $1$. The same goes when $n-1$ is prime, by considering the exponent of $n-1$.
Wha... | √n ≤ n/2 for n ≥ 4. Thus if p is a prime such that n/2 < p ≤ n, we have
√n < p → n < p² so p² cannot be a factor of n if n ≥ 4.
|
How many ways are there for 8 men and 5 women to stand in a line so that no two women stand next to each other? I have a homework problem in my textbook that has stumped me so far. There is a similar one to it that has not been assigned and has an answer in the back of the textbook. It reads:
How many ways are there f... | *
*M * M * M * M * M * M * M * M * .
there are 9 "*" .
8 "M" .
now select 5 * from 9 * then == c(9,5)
|
Partitioning an infinite set Can you partition an infinite set, into an infinite number of infinite sets?
| Note: I wrote this up because it is interesting, and 'rounds out' the theory, that we can partition $X$ into an infinite number of blocks with each on being countably infinite.
I am marking this as community wiki. If several people downvote this and/or post comments that they disagree with my sentiments, I'll delete th... |
Correlation between out of phase signals Say I have a numeric sequence A and a set of sequences B that vary with time.
I suspect that there is a relationship between one or more of the B sequences and sequence A, that changes in Bn are largely or wholly caused by changes in sequence A. However there is an unknown tim... | Take a look at dynamic time warping. I think it's just the solution you need. I've used the R package 'dtw' which is described here. http://cran.r-project.org/web/packages/dtw/dtw.pdf
|
My Daughter's 4th grade math question got me thinking Given a number of 3in squares and 2in squares, how many of each are needed to get a total area of 35 in^2?
Through quick trial and error (the method they wanted I believe) you find that you need 3 3in squares and 2 2in squares, but I got to thinking on how to solve ... | There is an algorithmic way to solve this which works when you have two types of squares.
if $\displaystyle \text{gcd}(a,b) = 1$, then for any integer $c$ the linear diophantine equation $\displaystyle ax + by = c$ has an infinite number of solution, with integer $\displaystyle x,y$.
In fact if $\displaystyle x_0, y_0$... |
Example of a non-commutative rings with identity that do not contain non-trival ideals and are not division rings I'm looking for an example of a non-commutative ring, $R$, with identity s.t $R$ does not contain a non-trival 2 sided ideal and $R$ is not a division ring
| If you mean two-sided ideals, you are looking for simple rings (that are not division rings):
http://en.wikipedia.org/wiki/Simple_ring
E.g. as in the wikipedia article, is the ring of matrices (of a certain size) over a field. Clearly this is not a division ring since not every matrix is invertible (matrices with zero ... |
The staircase paradox, or why $\pi\ne4$ What is wrong with this proof?
Is $\pi=4?$
| (non rigorous) If you repeat the process a million times it "seems" (visually) that the perimeter approaches in length to the circumference, but if you magnify the picture of a single "tooth" to full screen, you will notice a big difference from the orthogonal segments and the arc of the circumference. No matter how ma... |
subgroups of finitely generated groups with a finite index Let $G$ be a finitely generated group and $H$ a subgroup of $G$. If the index of $H$ in $G$ is finite, show that $H$ is also finitely generated.
| Hint: Suppose $G$ has generators $g_1, \ldots, g_n$. We can assume that the inverse of each generator is a generator. Now let $Ht_1, \ldots, Ht_m$ be all right cosets, with $t_1 = 1$. For all $i,j$, there is $h_{ij} \in H$ with $t_i g_j = h_{ij} t_{{k}_{ij}}$, for some $t_{{k}_{ij}}$. It's not hard to prove that $H$ is... |
What is the maximum number of primes generated consecutively generated by a polynomial of degree $a$? Let $p(n)$ be a polynomial of degree $a$. Start of with plunging in arguments from zero and go up one integer at the time. Go on until you have come at an integer argument $n$ of which $p(n)$'s value is not prime and c... | Here is result by Rabinowitsch for quadratic polynomials.
$n^2+n+A$ is prime for $n=0,1,2,...,A-2$ if and only if $d=1-4A$ is squarefree and the class number of $\mathbb{Q}[\sqrt{d}]$ is $1$.
See this article for details.
http://matwbn.icm.edu.pl/ksiazki/aa/aa89/aa8911.pdf
Also here is a list of imaginary quadratic f... |
Charpit's Method Find the complete integral of partial differential equation
$$\displaystyle z^2 = pqxy $$
I have solved this equation till auxiliary equation:
$$\displaystyle \frac{dp}{-pqy+2pz}=\frac{dq}{-pqx+2qz}=\frac{dz}{2pqxy}=\frac{dx}{qxy}=\frac{dy}{pxy} $$
But I have unable to find value of p and q.
EDIT:
p... | A much easier solution can be obtained by introducing new dependent/independent variables U=log u, X=log x, Y=log y. Then, with P,Q denoting the first partial derivatives of U with respect to X,Y, respectively, the PDE becomes
PQ=1,
which can be solved very easily by Charpit's method.
|
How to get a reflection vector? I'm doing a raytracing exercise. I have a vector representing the normal of a surface at an intersection point, and a vector of the ray to the surface. How can I determine what the reflection will be?
In the below image, I have d and n. How can I get r?
Thanks.
| $$r = d - 2 (d \cdot n) n$$
where $d \cdot n$ is the dot product, and
$n$ must be normalized.
|
A stereographic projection related question This might be an easy question, but I haven't been able to up come up with a solution.
The image of the map $$f : \mathbb{R} \to \mathbb{R}^2, a \mapsto (\frac{2a}{a^2+1}, \frac{a^2-1}{a^2+1})$$
is the unit circle take away the north pole. $f$ extends to a function $$g: \mat... | Note that although $a$ is complex, is valid :
$$\left(\frac{2a}{a^2+1}\right)^2+\left(\frac{a^2-1}{a^2+1}\right)^2= \frac{4a^2}{(a^2+1)^2}+\frac{(a^2-1)^2}{(a^2+1)}=$$
$$\frac{4a^2+a^4-2a^2+1}{(a^2+1)^2}=\frac{(a^4+2a^2+1)}{(a^2+1)^2}=\frac{(a^2+1)^2}{(a^2+1)^2}=1$$
Thus is also an circle
EDIT
Is say, the points of t... |
Intersection of neighborhoods of 0. Subgroup? Repeating for my exam in commutative algebra.
Let G be a topological abelian group, i.e. such that the mappings $+:G\times G \to G$ and $-:G\to G$ are continuous. Then we have the following Lemma:
Let H be the intersection of all neighborhoods of $0$ in $G$. Then $H$ is a s... | If $U$ is a neighbourhood of $0$ then so is $-U=\{-x:x\in U\}$.
This shows that if $x\in H$ then $-x\in H$.
To show that $H$ is closed under addition, use the fact that
if $U$ is a neighbourhood of $0$ then there is another
neighbourhood $V$ of $0$ with $V+V\subseteq U$. The existence
of $V$ follows from the continuity... |
Mapping Irregular Quadrilateral to a Rectangle I have a camera looking at a computer monitor from varying angles. Since the camera is a grid of pixels, I can define the bounds of the monitor in the camera image as:
I hope that makes sense. What I want to do is come up with an algorithm to translate points within this ... | HINT
$A,B,C,D$ are not in the same plane.
A very approximate rectangular projection ratio ... by area projections with extended boundary length), may be obtained considering boundary vector lenghts.
$$\frac{\frac12(|u \times v|+|w \times a|)}{(|u|+|v|+|w|+|a|)^2}$$
The rectangle can be now re-sized.
|
how do you solve $y''+2y'-3y=0$? I want to solve this equation:
$y''+2y'-3y=0$
I did this:
$y' = z$
$y'' = z\dfrac{dz}{dy}$
$z\dfrac{dz}{dy}+2z-3y=0$
$zdz+2zdy-3ydy=0$
$zdz=(3y-2z)dy$
$z=3y-2z$
$z=y$
$y=y'=y''$
???
now, I'm pretty sure I did something wrong. could you please correct.
| You can also write it in matrix form:
$u=(y',y)$, $u'=\big(\matrix{-2 & 3 \\ \hphantom- 1 & 0}\big) u$. Find the eigenvalues and eigenvectors, turn it into a diagonal system whose solution is simple. Go back to the original coordinates.
|
Does $R[x] \cong S[x]$ imply $R \cong S$? This is a very simple question but I believe it's nontrivial.
I would like to know if the following is true:
If $R$ and $S$ are rings and $R[x]$ and $S[x]$ are isomorphic as rings, then $R$ and $S$ are isomorphic.
Thanks!
If there isn't a proof (or disproof) of the general ... | Here is a counterexample.
Let $R=\dfrac{\mathbb{C}[x,y,z]}{\big(xy - (1 - z^2)\big)}$, $S=\dfrac{\mathbb{C}[x,y,z]}{\big(x^2y - (1 - z^2)\big)}$. Then, $R$ is not isomorphic to $S$ but, $R[T]\cong S[T]$.
In many variables, this is called the Zariski problem or cancellation of indeterminates and is largely open. Here is... |
Show that a continuous function has a fixed point Question: Let $a, b \in \mathbb{R}$ with $a < b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$.
I suppose this has to do with the basic definition of continuity. The definition I am using i... | Consider $x-f(x)$ and use Intermediate Value Theorem.
|
Applications for Homology The Question: Are there any ways that "applied" mathematicians can use Homology theory? Have you seen any good applications of it to the "real world" either directly or indirectly?
Why do I care? Topology has appealed to me since beginning it in undergrad where my university was more into pu... | There are definite real world applications. I would look at the website/work of Gunnar Carlsson (http://comptop.stanford.edu/) and Robert Ghrist (http://www.math.upenn.edu/~ghrist/). Both are excellent mathematicians.
The following could be completely wrong: Carlsson is one of the main proponents of Persistent Homolog... |
Probability Problems Problem:
The probability that a man who is 85 years. old will die before attaining the age of 90 is $\frac13$. A,B,C,D are four person who are 85 years old. what is the probability that A will die before attaining the age of 90 and will be the first to die ?
| The probability of dying by 90 is the same for all four. The probability that A dies first is simply 1/4 since we are give no more information. Since the two events are independent, the probability of their conjunction, i.e., that A dies and is the first to die, is simply the product of the two probabilities, or 1/12... |
How to solve the following system? I need to find the function c(k), knowing that
$$\sum_{k=0}^{\infty} \frac{c(k)}{k!}=1$$
$$\sum_{k=0}^{\infty} \frac{c(2k)}{(2k)!}=0$$
$$\sum_{k=0}^{\infty} \frac{c(2k+1)}{(2k+1)!}=1$$
$$\sum_{k=0}^{\infty} \frac{(-1)^k c(2k+1)}{(2k+1)!}=-1$$
$$\sum_{k=0}^{\infty} \frac{(-1)^k c(2... | You are looking for a function $\displaystyle f(z) = \sum_{k \ge 0} \frac{c(k)}{k!} z^k$ satisfying
$$f(1) = 1$$
$$f(-1) = -1$$
$$f(i) = -i.$$
Infinitely many functions have this property. There is a unique quadratic polynomial $p(z)$ with this property (for example by Lagrange interpolation), and for any entire fu... |
What is the value of $1^x$? I am trying to understand why $1^{x}=1$ for any $x\in\mathbb{R}$
Is it OK to write $1^{x}$? As the base 1 should not equal 1 for $1^{x}$ to be an exponential function?
Is $1^{x}=1$ just because it is defined to be so?
If possible please refer me to a book or article that discusses this top... | I think you will at least agree that $1^x$=$1$ if $x$ is any natural number (since this is just 1 times itself x times). We can extend this to all integer values of $x$ by using the facts $a^0$=$1$ for all non-zero $a$ and $x^{-c}$=$1\over{x^c}$. Then we have $1^x$=1 for all rational numbers $x$ by the fact $1^{b\ove... |
Infinitely differentiable How can one find if a function $f$ is infinitely differentiable?
| By differentiating it an infinite number of times?
But seriously, that's what you do. Just that usually you can infer what the higher order derivatives will be, so you don't have to compute it one by one.
Example To see that $\sin(x)$ is infinitely differentiable, you realize the following: $\frac{d}{dx}\sin(x) = \co... |
Are these transformations of the $\beta^\prime$ distribution from $\beta$ and to $F$ correct? Motivation
I have a prior on a random variable $X\sim \beta(\alpha,\beta)$ but I need to transform the variable to $Y=\frac{X}{1-X}$, for use in an analysis and I would like to know the distribution of $Y$.
Wikipedia states:
... | 2 and 3) Both of these transformations are correct; you can prove them with the cdf (cumulative distribution function) technique. I don't see any limitations on using them.
Here is the derivation for the first transformation using the cdf technique. The derivation for the other will be similar. Let $X \sim \beta(\a... |
In every power of 3 the tens digit is an even number How to prove that in every power of $3$, with natural exponent, the tens digit
is an even number?
For example, $3 ^ 5 = 243$ and $4$ is even.
| It's actually interesting.
If you do a table of multiples of 1, 3, 7, 9 modulo 20, you will find a closed set, ie you can't derive an 11, 13, 17 or 19 from these numbers.
What this means is that any number comprised entirely of primes that have an even tens-digit will itself have an even tens-digt. Such primes are 3... |
How to sum up this series? How to sum up this series :
$$2C_o + \frac{2^2}{2}C_1 + \frac{2^3}{3}C_2 + \cdots + \frac{2^{n+1}}{n+1}C_n$$
Any hint that will lead me to the correct solution will be highly appreciated.
EDIT: Here $C_i = ^nC_i $
| Let's assume $C_i=\binom ni$. I'll give a solution that is not precalculus level. Consider first the equality
$$ (1+x)^n=C_0+xC_1+x^2C_2+\dots+x^nC_n. $$
This is the binomial theorem.
Integrate from 0 to t. On the left hand side we get $\frac{(1+t)^{n+1}-1}{n+1}$ and on the right hand side $\sum \frac1{i+1}t^{i+1}C_i$... |
Is possible to simplify $P = N^{ CN + 1}$ in terms of $N$? Having: $P = N^{CN + 1}$;
How can I simplify this equation to $N = \cdots$?
I tried using logarithms but I'm stucked...
Any ideas?
| The equation $P=N^N$ doesn't have an elementary solution, but taking logarithms you can find approximate solutions (or even solutions in the form of transseries). In your case, take logarithms to find $(cn+1)\log n = p$. Assuming $p$ and so $n$ are large, $n\log n \approx p/c$. Therefore $n \approx p/c$, and so $$n \ap... |
Computer Programs for Pure Mathematicians Question: Which computer programs are useful for a pure mathematician to familiarize themselves with?
Less Briefly: I was once told that, now-a-days, any new mathematician worth his beans knows how to TeX up work; when I began my graduate work one of the fourth year students to... | mpmath, which is a part of sage has great special functions support.
nickle is good for quick things, and has C like syntax.
|
Generalizing Cauchy-Riemann Equations to Arbitrary Algebraic Fields Can it be done?
For an arbitrary quadratic field $Q[\sqrt{d}]$, it's easy to show the equations are simply $ f_x = -\sqrt{d} f_y $, where $ f : Q[\sqrt{d}] \to Q[\sqrt{d}]$. I'm working on the case of $Q[\theta]$, when $\theta$ is a root of $\theta^3 -... | I don't know if this will help, but I thought about something like this when I was an undergrad. I was thinking about the Jugendtraum: the fact that abelian extensions of imaginary quadratic fields can be described by values of analytic functions on $\mathbb{C}$. My thought was the following: Let $K=\mathbb{Q}(\sqrt{-D... |
Development of a specific hardware architecture for a particular algorithm How does a technical and theoretical study of a project of implementing an algorithm on a specific hardware architecture?
Function example:
%Expresion 1:
y1 = exp (- (const1 + x) ^ 2 / ((conts2 ^ 2)),
y2 = y1 * const3
Where x is the input varia... | You have several issues to consider. First, your output format allows very many fewer output values than the double standard. If your exponent is 2-based, your output will be of the form $\pm m*2^e$ where m has 2048 values available and e has 32 values (maybe they range from -15 to +16). The restricted range of e me... |
Best Strategy for a die game You are allowed to roll a die up to six times. Anytime you stop, you get the dollar amount of the face value of your last roll.
Question: What is the best strategy?
According to my calculation, for the strategy 6,5,5,4,4, the expected value is $142/27\approx 5.26$, which I consider quite hi... | Just work backwards. At each stage, you accept a roll that is >= the expected gain from the later stages:
Expected gain from 6th roll: 7/2
Therefore strategy for 5th roll is: accept if >= 4
Expected gain from 5th roll: (6 + 5 + 4)/6 + (7/2)(3/6) = 17/4
Therefore strategy for 4th roll is: accept if >= 5
Expected gain fr... |
Projection of a lattice onto a subspace Let $G$ be a $n \times n$ matrix with real entries and let $\Lambda = \{x^n \colon \exists i^n \in \mathbb{Z}^n \text{ such that } x^n = G \cdot i^n\}$ define a lattice. I am interested in projecting the lattice points onto a $k$-dimensional subspace $U$ with $k < n$. Let $A$ be ... | The technique you suggest is sometimes used to construct quasi-crystaline structures or in mathematical parlance, aperiodic tilings. If you take a high dimensional lattice and you project on an appropriate subspace U, the result can be the lattice structure of a quasi-crystal.
|
Homeomorphism of the unit disk onto itself which does not extend to the boundary It is well known that any conformal mapping of the unit disk onto itself extends to the unit cirle.
However, is there an homeomorphism of the unit disk onto itself which does not extend to a continuous function on the closed unit disk?
If... | This following gives an example, probably:
Let $H=\{(x,y)\in\mathbb R^2:y>0\}$ be the upper half plane in $\mathbb R^2$ and let $\bar H$ be its closure. Let me give you an homeo $f:H\to H$ which does not extend to an homeo $\bar f:\bar H\to\bar H$, and which fixes the point of infinity on the $x$-axis. Then you can con... |
Poincare Duality Reference In Hatcher's "Algebraic Topology" in the Poincaré Duality section he introduces the subject by doing orientable surfaces. He shows that there is a dual cell structure to each cell structure and it's easy to see that the first structure gives the cellular chain complex, while the other gives ... | See also my 2011 Bochum lectures The Poincare duality theorem and its converse I., II.
|
System of linear equations Common form of system of linear equations is A*X = B, X is unknown. But how to find A, if X and B are known?
A is MxN matrix, X is column vector(N), B is column vector(M)
| Do it row by row. Row $k$ in $A$ multiplied by the column vector $X$ equals the $k$th entry in the vector $B$. This is a single equation for the $N$ entries in that row of $A$ (so unless $X$ is zero, you get an $(N-1)$-parameter set of solutions for each row).
|
Please explain how Conditionally Convergent can be valid? I understand the basic idea of Conditionally Convergent (some infinitely long series can be made to converge to any value by reordering the series). I just do not understand how this could possibly be true. I think it defies common sense and seems like a clear v... | It deserves to be better known that there are simple cases where one can give closed forms for some rearrangements of alternating series. Here are a couple of interesting examples based on results of Schlömilch in 1873. Many further results can be found in classical textbooks on infinite series, e.g. those by Bromwich ... |
Projective closure Is the projective closure of an infinite affine variety (over an algebraically closed field, I only care about the classical case right now) always strictly larger than the affine variety? I know it is an open dense subset of its projective closure, but I don't think it can actually be its own projec... | For your own example: the projectivization of your curve is given by $X^2 + Y^2 - Z^2= 0$. There is the point corresponding to the projective equivalence class of $(1,i,0)$, which does not belong to the affine curve.
For more general varieties given by a single equation, you can always similarly find a projective poin... |
Can a polynomial size CFG over large alphabet describe any of these languages: Can a polynomial size CFG over large alphabet describe any of these languages:
*
*Each terminal appears $0$ or $2$ times
*Word repetition $\{www^* \mid w \in \Sigma^*\}$ (word repetition of an arbitrary word $w$)
"Polynomial size" Cont... | The language described in (2) is not context free. Take the word $a^nba^nba^nb$ for $n$ large enough, and use the pumping lemma to get a word not in the language.
As for (1), consider a grammar in CNF (Chomsky Normal Form) generating that language. Let $n = |\Sigma|$. Consider all $S=(2n)!/2^n$ saturated words containi... |
The logic behind the rule of three on this calculation First,
to understand my question, checkout this one:
Calculating percentages for taxes
Second,
consider that I'm a layman in math.
So, after trying to understand the logic used to get the final result. I was wondering:
Why multiply $20,000 by 100 and then divide b... | The "rule of three" is an ancient ad-hoc mindless rote rule of inference that is best ignored. Instead, you should strive to learn the general principles behind it - namely, the laws of fraction arithmetic. Let's consider the example at hand. You seek the number of dollars $\rm\:X\:$ such that when decremented by $\:1... |
Fiction "Division by Zero" By Ted Chiang Fiction "Division by Zero" By Ted Chiang
I read the fiction story "Division by Zero" By Ted Chiang
My interpretation is the character finds a proof that arithmetic is inconsistent.
Is there a formal proof the fiction can't come true? (I don't suggest the fiction can come true).
... | Is there a formal proof the fiction can't come true?
No, by Gödel's second incompleteness theorem, formal systems can prove their own consistency if and only if they are inconsistent. So given that arithmetic is consistent, we'll never be able to prove that it is. (EDIT: Actually not quite true; see Alon's clarificatio... |
an example of a continuous function whose Fourier series diverges at a dense set of points Please give me a link to a reference for an example of a continuous function whose Fourier series diverges at a dense set of points. (given by Du Bois-Reymond). I couldn't find this in Wikipedia.
| Actually, such an almost-everywhere divergent Fourier series was constructed by Kolmogorov.
For an explicit example, you can consider a Riesz product of the form:
$$ \prod_{k=1}^\infty \left( 1+ i \frac{\cos 10^k x}{k}\right)$$
which is divergent. For more examples, see here and here.
Edit: (response to comment). Yes... |
Probability that a random permutation has no fixed point among the first $k$ elements Is it true that $\frac1{n!} \int_0^\infty x^{n-k} (x-1)^k e^{-x}\,dx \approx e^{-k/n}$ when $k$ and $n$ are large integers with $k \le n$?
This quantity is the probability that a random permutation of $n$ elements does not fix any of ... | Update: This argument only holds for some cases. See italicized additions below.
Let $S_{n,k}$ denote the number of permutations in which the first $k$ elements are not fixed. I published an expository paper on these numbers earlier this year. See "Deranged Exams," (College Mathematics Journal, 41 (3): 197-202, 2010... |
Bi invariant metrics on $SL_n(\mathbb{R})$ Does there exist a bi-Invariant metric on $SL_n(\mathbb{R})$. I tried to google a bit but I didn't find anything helpful.
| If $d$ were a bi-invariant metric on $\operatorname{SL}_n(\mathbb{R})$, we would be able to restrict it to a bi-invariant metric on $\operatorname{SL}_2(\mathbb{R})$ as in Jason's answer.
And then we would have $$ d\left(\left[{\begin{array}{cc}
t & 0 \\
0 & 1/t \\
\end{array}}\right]\left[{\begin{array}{cc}
... |
Convolution of multiple probability density functions I have a series of tasks where when one task finishes the next task runs, until all of the tasks are done. I need to find the probability that everything will be finished at different points in time. How should I approach this? Is there a way to find this in polynom... | If the durations of the different tasks are independent, then the PDF of the overall duration is indeed given by the convolution of the PDFs of the individual task durations.
For efficient numerical computation of the convolutions, you probably want apply something like a Fourier transform to them first. If the PDFs a... |
An Eisenstein-like irreducibility criterion I could use some help with proving the following irreducibility criterion. (It came up in class and got me interested.)
Let p be a prime. For an integer $n = p^k n_0$, where p doesn't divide $n_0$, set: $e_p(n) = k$. Let $f(x) = a_n x^n + \cdots + a_1 x + a_0$ be a polynomial... | Apply Eisenstein's criterion to ${1 \over p^{n-1}}x^nf({p \over x})$.
|
How to find $10 + 15 + 20 + 25 + \dots + 1500$ During a test I was given the following question:
What is $$10+15+20+25+...+1490+1495+1500=?$$
After process of elimination and guessing, $ came up with an answer. Can anyone tell me a simple way to calculate this problem without having to try to actually add the numbers?... | Hint 3 (Gauss): $10 + 1500 = 15 + 1495 = 20 + 1490 = \cdots = 1500 + 10$.
|
Find the Angle ( as Measured in Counter Clock Wise Direction) Between Two Edges This is a similar question to this one, but slightly different.
The question is given two edges ($e_1$ and $e_2$, with the vertex coordinates known), how to find the angles from $e_1$ to $e_2$, with the angles measured in anti clock wise d... | The way to get the smaller angle spanned by $\mathbf e_1=(x_1,y_1)$ and $\mathbf e_2=(x_2,y_2)$ is through the expression
$\min(|\arctan(x_1,y_1)-\arctan(x_2,y_2)|,2\pi-|\arctan(x_1,y_1)-\arctan(x_2,y_2)|)$
where $\arctan(x,y)$ is the two-argument arctangent.
|
Good Book On Combinatorics What is your recommendation for an in-depth introductory combinatoric book? A book that doesn't just tell you about the multiplication principle, but rather shows the whole logic behind the questions with full proofs. The book should be for a first-year-student in college. Do you know a good ... | Alan Tucker's book is rather unreadable. I'd avoid it. Nick Loehr's Bijective Combinatorics text is much more thorough, and it reads like someone is explaining mathematics to you. It mixes rigor and approachability quite well.
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Finding the change point in data from a piecewise linear function Greetings,
I'm performing research that will help determine the size of observed space and the time elapsed since the big bang. Hopefully you can help!
I have data conforming to a piecewise linear function on which I want to perform two linear regression... | (This was supposed to be a comment, but it got too long.)
The problem of piecewise linear regression has been looked into many times before; I do not currently have access to these papers (and thus cannot say more about them), but you might want to look into these:
This paper (published in a physiology journal, of all ... |
famous space curves in geometry history? For an university assignment I have to visualize some curves in 3 dimensional space.
Until now I've implemented Bézier, helix and conical spiral.
Could you give me some advice about some famous curves in geometry history?
| Though is it is not 3D, the Clothoid or Cornu Spiral is an amazing curve. It surely can be made 3D by adding a simple extra parameter $z(t)=t$. It has infinite length but converges to two points in the plane. It has several applications in optics and road engineering, for example. An it looks quite nice:
I found a 3D ... |
integral test to show that infinite product $ \prod \limits_{n=1}^\infty\left(1+\frac{2}{n}\right)$ diverges This is part of an assignment that I need to get a good mark for - I'd appreciate it if you guys could look over it and give some pointers where I've gone wrong.
(apologies for the italics)
$$\prod_{n=1}^\infty... | For $a_n \ge 0 $ the infinite product $\prod_{n=1}^\infty (1+ a_n)$ converges precisely when the infinite sum $\sum_{n=1}^\infty a_n $ converges, since
$$1+ \sum_{n=1}^N a_n \le \prod_{n=1}^N (1+ a_n) \le \exp \left( \sum_{n=1}^N a_n \right) . $$
So you only need consider $ \sum_{n=1}^\infty 2/n $ and you can use your ... |
Using the second principle of finite induction to prove $a^n -1 = (a-1)(a^{n-1} + a^{n-2} + ... + a + 1)$ for all $n \geq 1$ The hint for this problem is $a^{n+1} - 1 = (a + 1)(a^n - 1) - a(a^{n-1} - 1)$
I see that the problem is true because if you distribute the $a$ and the $-1$ the terms cancel out to equal the left... | HINT $\ $ Put $\rm\ f(n) = a^n+a^{n-1}+\:\cdots\:+1\:.\ \ $ Then
$\rm\ a^{n+1}-1\ = \ (a+1)\ (a^n-1) - a\ (a^{n-1}-1)$
$\rm\phantom{\ a^{n+1}-1\ } =\ (a+1)\ ((a-1)\ f(n-1) - a\ (a-1)\ f(n-2))\quad $ by strong induction
$\rm\phantom{\ a^{n+1}-1\ } =\ (a-1)\ ((a+1)\ f(n-1)- a\ f(n-2))\quad$
$\rm\phantom{\ a^{n+1}-1\ }... |
How to make a sphere-ish shape with triangle faces? I want to make an origami of a sphere, so I planned to print some net of a pentakis icosahedron, but I have a image of another sphere with more polygons:
I would like to find the net of such model (I know it will be very fun to cut).
Do you know if it has a name ?
| Here is a net of a buckyball, from GoldenNumber.net:
It should be possible to turn this into the kind of net you're looking for by replacing the pentagons and hexagons with 5 and 6 isosceles triangles (the heights of the triangles determine the "elevation" of the center vertex from the original pentagonal/hexagonal fa... |
Making Change for a Dollar (and other number partitioning problems) I was trying to solve a problem similar to the "how many ways are there to make change for a dollar" problem. I ran across a site that said I could use a generating function similar to the one quoted below:
The answer to our problem ($293$) is the
co... | You should be able to compute it using a Partial Fraction representation (involving complex numbers). For instance see this previous answer: Minimum multi-subset sum to a target
Note, this partial fraction expansion needs to be calculated only one time. Once you have that, you can compute the way to make change for an ... |
Equation for a circle I'm reading a book about Calculus on my own and am stuck at a problem, the problem is
There are two circles of radius $2$ that have centers on the line $x = 1$ and pass through the origin. Find their equations.
The equation for circle is $(x-h)^2 + (y-k)^2 = r^2$
Any hints will be really appreci... | HINT: Find out what points $(x, y)$ with $x = 1$ also have distance $2$ to the origin. What would these points represent?
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Why is the Möbius strip not orientable? I am trying to understand the notion of an orientable manifold.
Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ such that $\textrm{det}(J(\phi_{\alpha} \circ \phi_{\beta}^{-1}))> 0$ (where define... | If you had an orientation, you'd be able to define at each point $p$ a unit vector $n_p$ normal to the strip at $p$, in a way that the map $p\mapsto n_p$ is continuous. Moreover, this map is completely determined once you fix the value of $n_p$ for some specific $p$. (You have two possibilities, this uses a tangent pla... |
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