tag stringclasses 9
values | question_body stringlengths 61 12.9k | accepted_answer stringlengths 38 36.4k | second_answer stringlengths 63 33k |
|---|---|---|---|
linear-algebra | <p>A matrix is diagonalizable iff it has a basis of eigenvectors. Now, why is this satisfied in case of a real symmetric matrix ? </p>
| <p>Suppose the ground field is $\mathbb C$. It is immediate then that every square matrix can be triangulated. Now, symmetry certainly implies normality ($A$ is normal if $AA^t=A^tA$ in the real case, and $AA^*=A^*A$ in the complex case). Since normality is preserved by similarity, it follows that if $A$ is symmetric, ... | <p>This question is about the spectral theorem for (finite dimensional) real Euclidean spaces, which says that in such a space any self-adjoint operator is diagonalisable (over the real numbers) with mutually orthogonal eigenspaces (so that orthonormal bases of eigenvectors exist). This is of course a classic result th... |
geometry | <p>I'm thinking about a circle rolling along a parabola. Would this be a parametric representation?</p>
<p>$(t + A\sin (Bt) , Ct^2 + A\cos (Bt) )$</p>
<p>A gives us the radius of the circle, B changes the frequency of the rotations, C, of course, varies the parabola. Now, if I want the circle to "match up" with the... | <p>(I had been meaning to blog about roulettes a while back, but since this question came up, I'll write about this topic here.)</p>
<p>I'll use the parametric representation</p>
<p>$$\begin{pmatrix}2at\\at^2\end{pmatrix}$$</p>
<p>for a parabola opening upwards, where $a$ is the focal length, or the length of the se... | <p>If I understand the question correctly:</p>
<p>Your parabola is $p(t)=(t,Ct^2)$. Its speed is $(1,2Ct)$, after normalization it is $v(t)=(1,2Ct)//\sqrt{1+(2Ct)^2)}$, hence the unit normal vector is $n(t)=(-2Ct,1)/\sqrt{1+(2Ct)^2)}$. The center of the circle is at $p(t)+An(t)$. The arc length of the parabola is $\i... |
linear-algebra | <blockquote>
<p>Let $ A, B $ be two square matrices of order $n$. Do $ AB $ and $ BA $ have same minimal and characteristic polynomials?</p>
</blockquote>
<p>I have a proof only if $ A$ or $ B $ is invertible. Is it true for all cases?</p>
| <p>Before proving $AB$ and $BA$ have the same characteristic polynomials show that if $A_{m\times n}$ and $B_{n\times m} $ then characteristic polynomials of $AB$ and $BA$ satisfy following statement: $$x^n|xI_m-AB|=x^m|xI_n-BA|$$ therefore easily conclude if $m=n$ then $AB$ and $BA$ have the same characteristic poly... | <p>If $A$ is invertible then $A^{-1}(AB)A= BA$, so $AB$ and $BA$ are similar, which implies (but is stronger than) $AB$ and $BA$ have the same minimal polynomial and the same characteristic polynomial.
The same goes if $B$ is invertible.</p>
<p>In general, from the above observation, it is not too difficult to show th... |
geometry | <p>The volume of a $d$ dimensional hypersphere of radius $r$ is given by:</p>
<p>$$V(r,d)=\frac{(\pi r^2)^{d/2}}{\Gamma\left(\frac{d}{2}+1\right)}$$</p>
<p>What intrigues me about this, is that $V\to 0$ as $d\to\infty$ for any fixed $r$. How can this be? For fixed $r$, I would have thought adding a dimension would ... | <p>I suppose you could say that adding a dimension "makes the volume bigger" for the hypersphere, but it does so even more for the unit you measure the volume with, namely the unit <em>cube</em>. So the numerical value of the volume does go towards zero.</p>
<p>Really, of course, it is apples to oranges because volume... | <p>The reason is because the length of the diagonal cube goes to infinity.</p>
<p>The cube in some sense does exactly what we expect. If it's side lengths are $1$, it will have the same volume in any dimension. So lets take a cube centered at the origin with side lengths $r$. Then what is the smallest sphere which ... |
geometry | <p>This is an idea I have had in my head for years and years and I would like to know the answer, and also I would like to know if it's somehow relevant to anything or useless.
I describe my thoughts with the following image:<br>
<img src="https://i.sstatic.net/UnAyt.png" alt="enter image description here"><br>
What wo... | <p>I found this problem interesting enough to make a little animation along the line of @Blue's diagram (but I didn't want to edit their answer without permission):</p>
<p><img src="https://i.sstatic.net/5le9i.gif" alt="enter image description here"></p>
<p><em>Mathematica</em> syntax for those who are interested:</p... | <p><img src="https://i.sstatic.net/0Z1P6.jpg" alt=""></p>
<p>Let $O$ be the center of the square, and let $\ell(\theta)$ be the line through $O$ that makes an angle $\theta$ with the horizontal line.
The line $\ell(\theta)$ intersects with the lower side of the square at a point $M_\theta$, with
$OM_\theta=\dfrac{a}{2... |
linear-algebra | <blockquote>
<p>Show that the determinant of a matrix $A$ is equal to the product of its eigenvalues $\lambda_i$.</p>
</blockquote>
<p>So I'm having a tough time figuring this one out. I know that I have to work with the characteristic polynomial of the matrix $\det(A-\lambda I)$. But, when considering an $n \times ... | <p>Suppose that <span class="math-container">$\lambda_1, \ldots, \lambda_n$</span> are the eigenvalues of <span class="math-container">$A$</span>. Then the <span class="math-container">$\lambda$</span>s are also the roots of the characteristic polynomial, i.e.</p>
<p><span class="math-container">$$\begin{array}{rcl} \d... | <p>I am a beginning Linear Algebra learner and this is just my humble opinion. </p>
<p>One idea presented above is that </p>
<p>Suppose that $\lambda_1,\ldots \lambda_2$ are eigenvalues of $A$. </p>
<p>Then the $\lambda$s are also the roots of the characteristic polynomial, i.e.</p>
<p>$$\det(A−\lambda I)=(\lambda_... |
logic | <p>Most of the systems mathematicians are interested in are consistent, which means, by Gödel's incompleteness theorems, that there must be unprovable statements.</p>
<p>I've seen a simple natural language statement here and elsewhere that's supposed to illustrate this: "I am not a provable statement." which leads to ... | <p>Here's a nice example that I think is easier to understand than the usual examples of Goodstein's theorem, Paris-Harrington, etc. Take a countably infinite paint box; this means that it has one color of paint for each positive integer; we can therefore call the colors <span class="math-container">$C_1, C_2, $</span... | <p>Any statement which is not logically valid (read: always true) is unprovable. The statement $\exists x\exists y(x>y)$ is not provable from the theory of linear orders, since it is false in the singleton order. On the other hand, it is not disprovable since any other order type would satisfy it.</p>
<p>The statem... |
probability | <p>Could you kindly list here all the criteria you know which guarantee that a <em>continuous local martingale</em> is in fact a true martingale? Which of these are valid for a general local martingale (non necessarily continuous)? Possible references to the listed results would be appreciated.</p>
| <p>Here you are :</p>
<p>From Protter's book "Stochastic Integration and Differential Equations" Second Edition (page 73 and 74)</p>
<p>First :
Let $M$ be a local martingale. Then $M$ is a martingale with
$E(M_t^2) < \infty, \forall t > 0$, if and only if $E([M,M]_t) < \infty, \forall t > 0$. If $E([M,M]... | <p>I found by myself other criteria that I think it is worth adding to this list.</p>
<p>5) $M$ is a local martingale of class DL iff $M$ is a martingale</p>
<p>6) If $M$ is a bounded local martingale, then it is a martingale.</p>
<p>7) If $M$ is a local martingale and $E(\sup_{s \in [0,t]} |M_s|) < \infty \, \fo... |
geometry | <p>In the textbook I am reading, it says a dimension is the number of independent parameters needed to specify a point. In order to make a circle, you need two points to specify the $x$ and $y$ position of a circle, but apparently a circle can be described with only the $x$-coordinate? How is this possible without the ... | <p>Suppose we're talking about a unit circle. We could specify any point on it as:
$$(\sin(\theta),\cos(\theta))$$
which uses only one parameter. We could also notice that there are only $2$ points with a given $x$ coordinate:
$$(x,\pm\sqrt{1-x^2})$$
and we would generally not consider having to specify a sign as being... | <p>Continuing ploosu2, the circle can be parameterized with one parameter (even for those who have not studied trig functions)...
$$
x = \frac{2t}{1+t^2},\qquad y=\frac{1-t^2}{1+t^2}
$$</p>
|
probability | <p>Whats the difference between <em>probability density function</em> and <em>probability distribution function</em>? </p>
| <p><strong>Distribution Function</strong></p>
<ol>
<li>The probability distribution function / probability function has ambiguous definition. They may be referred to:
<ul>
<li>Probability density function (PDF) </li>
<li>Cumulative distribution function (CDF)</li>
<li>or probability mass function (PMF) (statement ... | <p>The relation between the probability density funtion <span class="math-container">$f$</span> and the cumulative distribution function <span class="math-container">$F$</span> is...</p>
<ul>
<li><p>if <span class="math-container">$f$</span> is discrete:
<span class="math-container">$$
F(k) = \sum_{i \le k} f(i)
$$</s... |
logic | <p>For some reason, be it some bad habit or something else, I can not understand why the statement "p only if q" would translate into p implies q. For instance, I have the statement "Samir will attend the party only if Kanti will be there." The way I interpret this is, "It is true that Samir will attend the party only ... | <p>Think about it: "$p$ only if $q$" means that $q$ is a <strong>necessary condition</strong> for $p$. It means that $p$ can occur <strong>only when</strong> $q$ has occurred. This means that whenever we have $p$, it must also be that we have $q$, as $p$ can happen only if we have $q$: that is to say, that $p$ <strong>... | <p>I don't think there's really anything to <em>understand</em> here. One simply has to learn as a fact that in mathematics jargon the words "only if" invariably encode that particular meaning. It is not really forced by the everyday meanings of "only" and "if" in isolation; it's just how it is.</p>
<p>By this I mean ... |
linear-algebra | <p>I happened to stumble upon the following matrix:
$$ A = \begin{bmatrix}
a & 1 \\
0 & a
\end{bmatrix}
$$</p>
<p>And after trying a bunch of different examples, I noticed the following remarkable pattern. If $P$ is a polynomial, then:
$$ P(A)=\begin{bmatrix}
P(a) & P'(a) \... | <p>If $$ A = \begin{bmatrix}
a & 1 \\
0 & a
\end{bmatrix}
$$
then by induction you can prove that
$$ A^n = \begin{bmatrix}
a^n & n a^{n-1} \\
0 & a^n
\end{bmatrix} \tag 1
$$
for $n \ge 1 $. If $f$ can be developed into a power series
$$
f(z) = \sum_{n=0}^\in... | <p>It's a general statement if <span class="math-container">$J_{k}$</span> is a Jordan block and <span class="math-container">$f$</span> a function matrix then
<span class="math-container">\begin{equation}
f(J)=\left(\begin{array}{ccccc}
f(\lambda_{0}) & \frac{f'(\lambda_{0})}{1!} & \frac{f''(\lambda_{0})}{2!} ... |
linear-algebra | <p>I'm in the process of writing an application which identifies the closest matrix from a set of square matrices $M$ to a given square matrix $A$. The closest can be defined as the most similar.</p>
<p>I think finding the distance between two given matrices is a fair approach since the smallest Euclidean distance is ... | <p>Some suggestions. Too long for a comment:</p>
<p>As I said, there are many ways to measure the "distance" between two matrices. If the matrices are $\mathbf{A} = (a_{ij})$ and $\mathbf{B} = (b_{ij})$, then some examples are:
$$
d_1(\mathbf{A}, \mathbf{B}) = \sum_{i=1}^n \sum_{j=1}^n |a_{ij} - b_{ij}|
$$
$$
d_2(\mat... | <p>If we have two matrices $A,B$.
Distance between $A$ and $B$ can be calculated using Singular values or $2$ norms.</p>
<p>You may use Distance $= \vert(\text{fnorm}(A)-\text{fnorm}(B))\vert$
where fnorm = sq root of sum of squares of all singular values. </p>
|
linear-algebra | <p>The largest eigenvalue of a <a href="https://en.wikipedia.org/wiki/Stochastic_matrix" rel="noreferrer">stochastic matrix</a> (i.e. a matrix whose entries are positive and whose rows add up to $1$) is $1$.</p>
<p>Wikipedia marks this as a special case of the <a href="https://en.wikipedia.org/wiki/Perron%E2%80%93Frob... | <p>Here's a really elementary proof (which is a slight modification of <a href="https://math.stackexchange.com/questions/8695/no-solutions-to-a-matrix-inequality/8702#8702">Fanfan's answer to a question of mine</a>). As Calle shows, it is easy to see that the eigenvalue $1$ is obtained. Now, suppose $Ax = \lambda x$ f... | <p>Say <span class="math-container">$A$</span> is a <span class="math-container">$n \times n$</span> row stochastic matrix. Now:
<span class="math-container">$$A \begin{pmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{pmatrix} =
\begin{pmatrix}
\sum_{i=1}^n a_{1i} \\ \sum_{i=1}^n a_{2i} \\ \vdots \\ \sum_{i=1}^n a_{ni}
\end{pmatri... |
combinatorics | <p>I'm having a hard time finding the pattern. Let's say we have a set</p>
<p>$$S = \{1, 2, 3\}$$</p>
<p>The subsets are:</p>
<p>$$P = \{ \{\}, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\} \}$$</p>
<p>And the value I'm looking for, is the sum of the cardinalities of all of these subsets. That is, ... | <p>Here is a bijective argument. Fix a finite set $S$. Let us count the number of pairs $(X,x)$ where $X$ is a subset of $S$ and $x \in X$. We have two ways of doing this, depending which coordinate we fix first.</p>
<p><strong>First way</strong>: For each set $X$, there are $|X|$ elements $x \in X$, so the count is $... | <p>Each time an element appears in a set, it contributes $1$ to the value you are looking for. For a given element, it appears in exactly half of the subsets, i.e. $2^{n-1}$ sets. As there are $n$ total elements, you have $$n2^{n-1}$$ as others have pointed out.</p>
|
linear-algebra | <p>I'm starting a very long quest to learn about math, so that I can program games. I'm mostly a corporate developer, and it's somewhat boring and non exciting. When I began my career, I chose it because I wanted to create games.</p>
<p>I'm told that Linear Algebra is the best place to start. Where should I go?</p>
| <p>You are right: Linear Algebra is not just the "best" place to start. It's THE place to start.</p>
<p>Among all the books cited in <a href="http://en.wikipedia.org/wiki/Linear_algebra" rel="nofollow noreferrer">Wikipedia - Linear Algebra</a>, I would recommend:</p>
<ul>
<li>Strang, Gilbert, Linear Algebra a... | <p>I'm very surprised no one's yet listed Sheldon Axler's <a href="https://books.google.com/books?id=5qYxBQAAQBAJ&source=gbs_similarbooks" rel="nofollow noreferrer">Linear Algebra Done Right</a> - unlike Strang and Lang, which are really great books, Linear Algebra Done Right has a lot of "common sense", and ... |
linear-algebra | <p>In which cases is the inverse of a matrix equal to its transpose, that is, when do we have <span class="math-container">$A^{-1} = A^{T}$</span>? Is it when <span class="math-container">$A$</span> is orthogonal? </p>
| <p>If $A^{-1}=A^T$, then $A^TA=I$. This means that each column has unit length and is perpendicular to every other column. That means it is an orthonormal matrix.</p>
| <p>You're right. This is the definition of orthogonal matrix.</p>
|
matrices | <p>Is there an intuitive meaning for the <a href="http://mathworld.wolfram.com/SpectralNorm.html">spectral norm</a> of a matrix? Why would an algorithm calculate the relative recovery in spectral norm between two images (i.e. one before the algorithm and the other after)? Thanks</p>
| <p>The spectral norm (also know as Induced 2-norm) is the maximum singular value of a matrix. Intuitively, you can think of it as the maximum 'scale', by which the matrix can 'stretch' a vector.</p>
<p>The maximum singular value is the square root of the maximum eigenvalue or the maximum eigenvalue if the matrix is sy... | <p>Let us consider the singular value decomposition (SVD) of a matrix <span class="math-container">$X = U S V^T$</span>, where <span class="math-container">$U$</span> and <span class="math-container">$V$</span> are matrices containing the left and right singular vectors of <span class="math-container">$X$</span> in the... |
number-theory | <p>The question is written like this:</p>
<blockquote>
<p>Is it possible to find an infinite set of points in the plane, not all on the same straight line, such that the distance between <strong>EVERY</strong> pair of points is rational?</p>
</blockquote>
<p>This would be so easy if these points could be on the sam... | <p>You can even find infinitely many such points on the unit circle: Let $\mathscr S$ be the set of all points on the unit circle such that $\tan \left(\frac {\theta}4\right)\in \mathbb Q$. If $(\cos(\alpha),\sin(\alpha))$ and $(\cos(\beta),\sin(\beta))$ are two points on the circle then a little geometry tells us th... | <p>Yes, it's possible. For instance, you could start with $(0,1)$ and $(0,0)$, and then put points along the $x$-axis, noting that there are infinitely many different right triangles with rational sides and one leg equal to $1$. For instance, $(3/4,0)$ will have distance $5/4$ to $(0,1)$.</p>
<p>This means that <em>mo... |
probability | <p>I give you a hat which has <span class="math-container">$10$</span> coins inside of it. <span class="math-container">$1$</span> out of the <span class="math-container">$10$</span> have two heads on it, and the rest of them are fair. You draw a coin at random from the jar and flip it <span class="math-container">$5$<... | <p>To convince your friend that he is wrong, you could modify the question:</p>
<blockquote>
<p>A hat contains ten 6-sided dice. Nine dice have scores 1, 2, 3, 4, 5, 6, and the other dice has 6 on every face. Randomly choose one dice, toss it <span class="math-container">$1000$</span> times, and write down the results.... | <p>The main idea behind this problem is a topic known as <em>predictive posterior probability</em>.</p>
<p>Let <span class="math-container">$P$</span> denote the probability of the coin you randomly selected landing on heads.</p>
<p>Then <span class="math-container">$P$</span> is a random variable supported on <span cl... |
geometry | <p>What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane? </p>
<p>One approach is to find the length of each side from the coordinates given and then apply <a href="https://en.wikipedia.org/wiki/Heron's_formula" rel="noreferrer"><em>Heron's ... | <p>What you are looking for is called the <a href="http://en.wikipedia.org/wiki/Shoelace_formula" rel="nofollow noreferrer">shoelace formula</a>:</p>
<p><span class="math-container">\begin{align*}
\text{Area}
&= \frac12 \big| (x_A - x_C) (y_B - y_A) - (x_A - x_B) (y_C - y_A) \big|\\
&= \frac12 \big| x_A y_B + x... | <p>You know that <strong>AB × AC</strong> is a vector perpendicular to the plane ABC such that |<strong>AB × AC</strong>|= Area of the parallelogram ABA’C. Thus this area is equal to ½ |AB × AC|.</p>
<p><a href="https://i.sstatic.net/3oDbh.png" rel="noreferrer"><img src="https://i.sstatic.net/3oDbh.png" alt="enter imag... |
linear-algebra | <p>I am looking for an intuitive reason for a projection matrix of an orthogonal projection to be symmetric. The algebraic proof is straightforward yet somewhat unsatisfactory.</p>
<p>Take for example another property: $P=P^2$. It's clear that applying the projection one more time shouldn't change anything and hence t... | <p>In general, if $P = P^2$, then $P$ is the projection onto $\operatorname{im}(P)$ along $\operatorname{ker}(P)$, so that $$\mathbb{R}^n = \operatorname{im}(P) \oplus \operatorname{ker}(P),$$ but $\operatorname{im}(P)$ and $\operatorname{ker}(P)$ need not be orthogonal subspaces. Given that $P = P^2$, you can check th... | <p>There are some nice and succinct answers already. If you'd like even more intuition with as little math and higher level linear algebra concepts as possible, consider two arbitrary vectors <span class="math-container">$v$</span> and <span class="math-container">$w$</span>.</p>
<h2>Simplest Answer</h2>
<p>Take the do... |
differentiation | <p>Are continuous functions always differentiable? Are there any examples in dimension <span class="math-container">$n > 1$</span>?</p>
| <p>No. <a href="http://en.wikipedia.org/wiki/Karl_Weierstrass" rel="noreferrer">Weierstraß</a> gave in 1872 the first published example of a <a href="http://en.wikipedia.org/wiki/Weierstrass_function" rel="noreferrer">continuous function that's nowhere differentiable</a>.</p>
| <p>No, consider the example of $f(x) = |x|$. This function is continuous but not differentiable at $x = 0$.</p>
<p>There are even more bizare functions that are not differentiable everywhere, yet still continuous. This class of functions lead to the development of the study of fractals.</p>
|
linear-algebra | <p>Here's a cute problem that was frequently given by the late Herbert Wilf during his talks. </p>
<p><strong>Problem:</strong> Let $A$ be an $n \times n$ matrix with entries from $\{0,1\}$ having all positive eigenvalues. Prove that all of the eigenvalues of $A$ are $1$.</p>
<p><strong>Proof:</strong></p>
<blockquo... | <p>If one wants to use the AM-GM inequality, you could proceed as follows:
Since $A$ has all $1$'s or $0$'s on the diagonal, it follows that $tr(A)\leq n$.
Now calculating the determinant by expanding along any row/column, one can easily see that the determinant is an integer, since it is a sum of products of matrix en... | <p>Suppose that A has a column with only zero entries, then we must have zero as an eigenvalue. (e.g. expanding det(A-rI) using that column). So it must be true that in satisfying the OP's requirements we must have each column containing a 1. The same holds true for the rows by the same argument. Now suppose that we ha... |
matrices | <p>If the matrix is positive definite, then all its eigenvalues are strictly positive. </p>
<p>Is the converse also true?<br>
That is, if the eigenvalues are strictly positive, then matrix is positive definite?<br>
Can you give example of $2 \times 2$ matrix with $2$ positive eigenvalues but is not positive definite?... | <p>I think this is false. Let <span class="math-container">$A = \begin{pmatrix} 1 & -3 \\ 0 & 1 \end{pmatrix}$</span> be a 2x2 matrix, in the canonical basis of <span class="math-container">$\mathbb R^2$</span>. Then A has a double eigenvalue b=1. If <span class="math-container">$v=\begin{pmatrix}1\\1\end{pmatr... | <p>This question does a great job of illustrating the problem with thinking about these things in terms of coordinates. The thing that is positive-definite is not a matrix $M$ but the <em>quadratic form</em> $x \mapsto x^T M x$, which is a very different beast from the linear transformation $x \mapsto M x$. For one t... |
differentiation | <p>I am a Software Engineering student and this year I learned about how CPUs work, it turns out that electronic engineers and I also see it a lot in my field, we do use derivatives with discontinuous functions. For instance in order to calculate the optimal amount of ripple adders so as to minimise the execution time ... | <p>In general, computing the extrema of a continuous function and rounding them to integers does <em>not</em> yield the extrema of the restriction of that function to the integers. It is not hard to construct examples.</p>
<p>However, your particular function is <em>convex</em> on the domain <span class="math-container... | <p>The main question here seems to be "why can we differentiate a function only defined on integers?". The proper answer, as divined by the OP, is that we can't--there is no unique way to define such a derivative, because we can interpolate the function in many different ways. However, in the cases that you... |
geometry | <p>I need to find the volume of the region defined by
$$\begin{align*}
a^2+b^2+c^2+d^2&\leq1,\\
a^2+b^2+c^2+e^2&\leq1,\\
a^2+b^2+d^2+e^2&\leq1,\\
a^2+c^2+d^2+e^2&\leq1 &\text{ and }\\
b^2+c^2+d^2+e^2&\leq1.
\end{align*}$$
I don't necessarily need a full solution but any starting points ... | <p>It turns out that this is much easier to do in <a href="http://en.wikipedia.org/wiki/Hyperspherical_coordinates#Hyperspherical_coordinates">hyperspherical coordinates</a>. I'll deviate somewhat from convention by swapping the sines and cosines of the angles in order to get a more pleasant integration region, so the ... | <p>There's reflection symmetry in each of the coordinates, so the volume is $2^5$ times the volume for positive coordinates. There's also permutation symmetry among the coordinates, so the volume is $5!$ times the volume with the additional constraint $a\le b\le c\le d\le e$. Then it remains to find the integration bou... |
geometry | <p>Source: <a href="http://www.math.uci.edu/%7Ekrubin/oldcourses/12.194/ps1.pdf" rel="noreferrer">German Mathematical Olympiad</a></p>
<h3>Problem:</h3>
<blockquote>
<p>On an arbitrarily large chessboard, a generalized knight moves by jumping p squares in one direction and q squares in a perpendicular direction, p, q &... | <p>Case I: If $p+q$ is odd, then the knight's square changes colour after each move, so we are done.</p>
<p>Case II: If $p$ and $q$ are both odd, then the $x$-coordinate changes by an odd number after every move, so it is odd after an odd number of moves. So the $x$-coordinate can be zero only after an even number of ... | <p>This uses complex numbers.</p>
<p>Define $z=p+qi$. Say that the knight starts at $0$ on the complex plane. Note that, in one move, the knight may add or subtract $z$, $iz$, $\bar z$, $i\bar z$ to his position.</p>
<p>Thus, at any point, the knight is at a point of the form:
$$(a+bi)z+(c+di)\bar z$$
where $a$ and $... |
linear-algebra | <p>When someone wants to solve a system of linear equations like</p>
<p>$$\begin{cases} 2x+y=0 \\ 3x+y=4 \end{cases}\,,$$</p>
<p>they might use this logic: </p>
<p>$$\begin{align}
\begin{cases} 2x+y=0 \\ 3x+y=4 \end{cases}
\iff &\begin{cases} -2x-y=0 \\ 3x+y=4 \end{cases}
\\
\color{maroon}{\implies} &\beg... | <p>You wrote this step as an implication: </p>
<blockquote>
<p>$$\begin{cases} -2x-y=0 \\ 3x+y=4 \end{cases} \implies \begin{cases} -2x-y=0\\ x=4 \end{cases}$$</p>
</blockquote>
<p>But it is in fact an equivalence:</p>
<p>$$\begin{cases} -2x-y=0 \\ 3x+y=4 \end{cases} \iff \begin{cases} -2x-y=0\\ x=4 \end{cases}$$<... | <p>The key is that in solving this system of equations (or with row-reduction in general), every step is <em>reversible</em>. Following the steps forward, we see that <em>if</em> $x$ and $y$ satisfy the equations, then $x = 4$ and $y = -8$. That is, we conclude that $(4,-8)$ is the only possible solution, assuming a ... |
probability | <p>Given the rapid rise of the <a href="http://en.wikipedia.org/wiki/Mega_Millions">Mega Millions</a> jackpot in the US (now advertised at \$640 million and equivalent to a "cash" prize of about \$448 million), I was wondering if there was ever a point at which the lottery became positive expected value (EV), and, if s... | <p>I did a fairly <a href="http://www.circlemud.org/~jelson/megamillions">extensive analysis of this question</a> last year. The short answer is that by modeling the relationship of past jackpots to ticket sales we find that ticket sales grow super-linearly with jackpot size. Eventually, the positive expectation of a... | <p>An interesting thought experiment is whether it would be a good investment for a rich person to buy every possible number for \$175,711,536. This person is then guaranteed to win! Then you consider the resulting size of the pot (now a bit larger), the probability of splitting it with other winners, and the fact th... |
game-theory | <p>The <a href="https://en.wikipedia.org/wiki/Monty_Hall_problem" rel="nofollow">Monty Hall problem or paradox</a> is famous and well-studied. But what confused me about the description was an unstated assumption.</p>
<blockquote>
<p>Suppose you're on a game show, and you're given the choice of three
doors: behind... | <p>The car probably doesn't come out of the host's salary, so he probably doesn't really want to minimize the payoff, he wants to maximize the show's ratings. But OK, let's
suppose he did want to minimize the payoff, making this a zero-sum game.
Then the optimal value of the game (in terms of the probability of winni... | <p>Thanks for your answer, Robert. If the optimal value is 1/3 as you showed, then I suppose there must be infinitely many mixed strategies that the host could employ that would be in equilibrium. If, as I mentioned in the question, the host offers the switch to 2/3 of correct guessers and 1/3 of incorrect guessers, 1/... |
probability | <p>This is a really natural question for which I know a stunning solution. So I admit I have a solution, however I would like to see if anybody will come up with something different. The question is</p>
<blockquote>
<p>What is the probability that two numbers randomly chosen are coprime?</p>
</blockquote>
<p>More f... | <p><strong>Here is a fairly easy approach.</strong>
<strong>Let us start with a basic observation:</strong></p>
<p><span class="math-container">$\bullet$</span> Every integer has the probability "1" to be divisible by 1.</p>
<p><span class="math-container">$\bullet$</span> A given integer is either even or od... | <p>Let's look at the function <span class="math-container">$$S(x)=\sum_{\begin{array}{c}
m,n\leq x\\
\gcd(m,n)=1\end{array}}1.$$</span> </p>
<p>Then notice that <span class="math-container">$$S(x)=\sum_{m,n\leq x}\sum_{d|\gcd(m,n)}\mu(d)=\sum_{d\leq x}\sum_{r,s\leq\frac{x}{d}}\mu(d)= \sum_{d\leq x}\mu(d)\left[\frac{... |
logic | <p>I enjoy reading about formal logic as an occasional hobby. However, one thing keeps tripping me up: I seem unable to understand what's being referred to when the word "type" (as in type theory) is mentioned.</p>
<p>Now, I understand what types are in programming, and sometimes I get the impression that types in log... | <p><strong>tl;dr</strong> Types only have meaning within type systems. There is no stand-alone definition of "type" except vague statements like "types classify terms". The notion of type in programming languages and type theory are basically the same, but different type systems correspond to different type theories. O... | <blockquote>
<p>I've found they don't really help me to understand the underlying concept, partly because they are necessarily tied to the specifics of a particular type theory. If I can understand the motivation better it should make it easier to follow the definitions.</p>
</blockquote>
<p>The basic idea: In ZFC s... |
differentiation | <p>I never understand what the trigonometric function sine is..</p>
<p>We had a table that has values of sine for different angles, we by hearted it and applied to some problems and there ends the matter. Till then, sine function is related to triangles, angles.</p>
<p>Then comes the graph. We have been told that the... | <p>The sine function doesn't actually operate on angles, it's a function from the real numbers to the interval [-1, 1] (or from the complex numbers to the complex numbers).</p>
<p>However, it just so happens that it's a very useful function when the input you give it relates to angles. In particular, if you express an... | <p>Imagine the unit circle in the usual Cartesian plane: the set of pairs $(x, y)$ where $x$ and $y$ are real numbers. The unit circle is the set of all such pairs a distance of exactly $1$ from the origin.</p>
<p>Imagine a point moving around the circle. As it travels around the circle, it makes an angle of $t$ <em>r... |
matrices | <p>More precisely, does the set of non-diagonalizable (over $\mathbb C$) matrices have Lebesgue measure zero in $\mathbb R^{n\times n}$ or $\mathbb C^{n\times n}$? </p>
<p>Intuitively, I would think yes, since in order for a matrix to be non-diagonalizable its characteristic polynomial would have to have a multiple ro... | <p>Yes. Here is a proof over $\mathbb{C} $.</p>
<ul>
<li>Matrices with repeated eigenvalues are cut out as the zero locus of the discriminant of the characteristic polynomial, thus are algebraic sets. </li>
<li>Some matrices have unique eigenvalues, so this algebraic set is proper.</li>
<li>Proper closed algebraic set... | <p>Let $A$ be a real matrix with a non-real eigenvalue. It's rather easy to see that if you perturb $A$ a little bit $A$ still will have a non-real eigenvalue. For instance if $A$ is a rotation matrix (as in Georges answer), applying a perturbed version of $A$ will still come close to rotating the vectors by a fixed an... |
combinatorics | <blockquote>
<p>All numbers <span class="math-container">$1$</span> to <span class="math-container">$155$</span> are written on a blackboard, one time each. We randomly choose two numbers and delete them, by replacing one of them with their product plus their sum. We repeat the process until there is only one number ... | <p>Claim: if <span class="math-container">$a_1,...,a_n$</span> are the <span class="math-container">$n$</span> numbers on the board then after n steps we shall be left with <span class="math-container">$(1+a_1)...(1+a_n)-1$</span>.</p>
<p>Proof: <em>induct on <span class="math-container">$n$</span></em>. Case <span cl... | <p>Another way to think of Sorin's observation, without appealing to induction explicitly:</p>
<p>Suppose your original numbers (both the original 155 numbers and later results) are written in <em>white</em> chalk. Now above each <em>white</em> number write that number plus one, in <em>red</em> chalk. Write new red co... |
differentiation | <p>Which derivatives are eventually periodic?</p>
<p>I have noticed that is $a_{n}=f^{(n)}(x)$, the sequence $a_{n}$ becomes eventually periodic for a multitude of $f(x)$. </p>
<p>If $f(x)$ was a polynomial, and $\operatorname{deg}(f(x))=n$, note that $f^{(n)}(x)=C$ if $C$ is a constant. This implies that $f^{(n+i)}... | <p>The sequence of derivatives being globally periodic (not eventually periodic) with period $m$ is equivalent to the differential equation </p>
<p>$$f(x)=f^{(m)}(x).$$</p>
<p>All solutions to this equation are of the form $\sum_{k=1}^m c_k e^{\lambda_k x}$ where $\lambda_k$ are solutions to the equation $\lambda^m-1... | <p>Let's also look at it upside down. You can define analytical (infinitely differentiable) functions with their Taylor series $\sum \frac{a_n}{n!}x^n$. Taylor series are simply all finite and infinite polynomials with coefficient sequences $(a_n)$ that satisfy the series convergence criteria ($a_n$ are the derivatives... |
differentiation | <p>As referred <a href="https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule" rel="noreferrer">in Wikipedia</a> (see the specified criteria there), L'Hôpital's rule says,</p>
<p><span class="math-container">$$
\lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{f'(x)}{g'(x)}
$$</span></p>
<p>As</p>
<p><span class="mat... | <p>With L'Hôpital's rule your limit must be of the form <span class="math-container">$\dfrac 00$</span>, so your antiderivatives must take the value <span class="math-container">$0$</span> at <span class="math-container">$c$</span>. In this case you have <span class="math-container">$$\lim_{x \to c} \frac{ \int_c^x f(t... | <p>I recently came across a situation where it was useful to go through exactly this process, so (although I'm certainly late to the party) here's an application of L'Hôpital's rule in reverse:</p>
<p>We have a list of distinct real numbers $\{x_0,\dots, x_n\}$.
We define the $(n+1)$th <em>nodal polynomial</em> as
$$... |
linear-algebra | <p>Let $A$ be an $n \times n$ matrix. Then the solution of the initial value problem
\begin{align*}
\dot{x}(t) = A x(t), \quad x(0) = x_0
\end{align*}
is given by $x(t) = \mathrm{e}^{At} x_0$.</p>
<p>I am interested in the following matrix
\begin{align*}
\int_{0}^T \mathrm{e}^{At}\, dt
\end{align*}
for some $T>... | <p><strong>Case I.</strong> If <span class="math-container">$A$</span> is nonsingular, then
<span class="math-container">$$
\int_0^T\mathrm{e}^{tA}\,dt=\big(\mathrm{e}^{TA}-I\big)A^{-1},
$$</span>
where <span class="math-container">$I$</span> is the identity matrix.</p>
<p><strong>Case II.</strong> If <span class="math... | <p>The general formula is the power series</p>
<p>$$ \int_0^T e^{At} dt = T \left( I + \frac{AT}{2!} + \frac{(AT)^2}{3!} + \dots + \frac{(AT)^{n-1}}{n!} + \dots \right) $$</p>
<p>Note that also</p>
<p>$$ \left(\int_0^T e^{At} dt \right) A + I = e^{AT} $$</p>
<p>is always satisfied.</p>
<p>A sufficient condition fo... |
logic | <p>There are many classic textbooks in <strong>set</strong> and <strong>category theory</strong> (as possible foundations of mathematics), among many others Jech's, Kunen's, and Awodey's.</p>
<blockquote>
<p>Are there comparable classic textbooks in <strong>type theory</strong>, introducing and motivating their matt... | <p>Although not as comprehensive a textbook as, say, Jech's classic book on set theory, Jean-Yves Girard's <a href="http://www.paultaylor.eu/stable/Proofs+Types"><em>Proofs and Types</em></a> is an excellent starting point for reading about type theory. It's freely available from translator Paul Taylor's website as a P... | <p>There are two main settings in which I see type theory as a foundational system.</p>
<p>The first is intuitionistic type theory, particularly the system developed by Martin-Löf. The book <em>Intuitionistic Type Theory</em> (1980) seems to be floating around the internet. </p>
<p>The other setting is second-order (... |
logic | <p>What are some good online/free resources (tutorials, guides, exercises, and the like) for learning Lambda Calculus?</p>
<p>Specifically, I am interested in the following areas:</p>
<ul>
<li>Untyped lambda calculus</li>
<li>Simply-typed lambda calculus</li>
<li>Other typed lambda calculi</li>
<li>Church's Theory of... | <p><img src="https://i.sstatic.net/8E8Sp.png" alt="alligators"></p>
<p><a href="http://worrydream.com/AlligatorEggs/" rel="noreferrer"><strong>Alligator Eggs</strong></a> is a cool way to learn lambda calculus.</p>
<p>Also learning functional programming languages like Scheme, Haskell etc. will be added fun.</p>
| <p>Recommendations:</p>
<ol>
<li>Barendregt & Barendsen, 1998, <a href="https://www.academia.edu/18746611/Introduction_to_lambda_calculus" rel="noreferrer">Introduction to lambda-calculus</a>;</li>
<li>Girard, Lafont & Taylor, 1987, <a href="http://www.paultaylor.eu/stable/Proofs+Types.html" rel="noreferrer">Pr... |
differentiation | <p>I've got this task I'm not able to solve. So i need to find the 100-th derivative of $$f(x)=e^{x}\cos(x)$$ where $x=\pi$.</p>
<p>I've tried using Leibniz's formula but it got me nowhere, induction doesn't seem to help either, so if you could just give me a hint, I'd be very grateful.</p>
<p>Many thanks!</p>
| <p>HINT:</p>
<p>$e^x\cos x$ is the real part of $y=e^{(1+i)x}$</p>
<p>As $1+i=\sqrt2e^{i\pi/4}$</p>
<p>$y_n=(1+i)^ne^{(1+i)x}=2^{n/2}e^x\cdot e^{i(n\pi/4+x)}$</p>
<p>Can you take it from here?</p>
| <p>Find fewer order derivatives:</p>
<p>\begin{align}
f'(x)&=&e^x (\cos x -\sin x)&\longleftarrow&\\
f''(x)&=&e^x(\cos x -\sin x -\sin x -\cos x) \\ &=& -2e^x\sin x&\longleftarrow&\\
f'''(x)&=&-2e^x(\sin x + \cos x)&\longleftarrow&\\
f''''(x)&=& -2e^x(\si... |
logic | <p>I would like to know more about the <em>foundations of mathematics</em>, but I can't really figure out where it all starts. If I look in a book on <em><a href="http://rads.stackoverflow.com/amzn/click/0387900500">axiomatic set theory</a></em>, then it seems to be assumed that one already have learned about <em>langu... | <p>There are different ways to build a foundation for mathematics, but I think the closest to being the current "standard" is:</p>
<ul>
<li><p>Philosophy (optional)</p></li>
<li><p><a href="http://en.wikipedia.org/wiki/Propositional_logic">Propositional logic</a></p></li>
<li><p><a href="http://en.wikipedia.org/wiki/F... | <p>I strongly urge you to look at Goldrei [9] and Goldrei [10]. I learned about these books by chance in Fall 2011. Among foundational books, I think Goldrei's books must rate as among the best books I've ever come across relative to how little well-known they are. In particular, Goldrei [10] has been invaluable to me ... |
differentiation | <p>I think this is just something I've grown used to but can't remember any proof.</p>
<p>When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then?</p>
| <blockquote>
<p>Radians make it possible to relate a linear measure and an angle
measure. A unit circle is a circle whose radius is one unit. The one
unit radius is the same as one unit along the circumference. Wrap a
number line counter-clockwise around a unit circle starting with zero
at (1, 0). The length ... | <p>To make commenters' points explicit, the "degrees-mode trig functions" functions $\cos^\circ$ and $\sin^\circ$ satisfy the awkward identities
$$
(\cos^\circ)' = -\frac{\pi}{180} \sin^\circ,\qquad
(\sin^\circ)' = \frac{\pi}{180} \cos^\circ,
$$
with all that implies about every formula involving the derivative or anti... |
probability | <p>I was asked this question regarding correlation recently, and although it seems intuitive, I still haven't worked out the answer satisfactorily. I hope you can help me out with this seemingly simple question.</p>
<p>Suppose I have three random variables $A$, $B$, $C$. Is it possible to have these three relationship... | <p>Here's an answer to the general question, which I wrote up a while ago. It's a common interview question.</p>
<p>The question goes like this: "Say you have X,Y,Z three random variables such that the correlation of X and Y is something and the correlation of Y and Z is something else, what are the possible correlati... | <p>Assume without loss of generality that the random variables $A$, $B$, $C$ are standard, that is, with mean zero and unit variance. Then, for any $(A,B,C)$ with the prescribed covariances,
$$\mathrm{var}(A-B+C)=\mathrm{var}(A)+\mathrm{var}(B)+\mathrm{var}(C)-2\mathrm{cov}(A,B)-2\mathrm{cov}(B,C)+2\mathrm{cov}(A,C),
$... |
probability | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/111314/choose-a-random-number-between-0-and-1-and-record-its-value-and-keep-doing-it-u">choose a random number between 0 and 1 and record its value. and keep doing it until the sum of the numbers exceeds 1. ho... | <p>Here is a way to compute $\mathbb E(N)$. We begin by <em>complicating</em> things, namely, for every $x$ in $(0,1)$, we consider $m_x=\mathbb E(N_x)$ where
$$
N_x=\min\left\{n\,;\,\sum_{k=1}^nU_k\gt x\right\}.
$$
Our goal is to compute $m_1$ since $N_1=N$. Assume that $U_1=u$ for some $u$ in $(0,1)$. If $u\gt x$, th... | <p>In fact it turns out that <span class="math-container">$P(N = n) = \frac{n-1}{n!}$</span> for <span class="math-container">$n \ge 2$</span>. Let <span class="math-container">$S_n = \sum_{j=1}^n U_j$</span>, and <span class="math-container">$f_n(s)$</span> the probability density function for <span class="math-conta... |
linear-algebra | <blockquote>
<p>Assume that we are working over a complex space $W$ of dimension $n$. When would an operator on this space have the same characteristic and minimal polynomial? </p>
</blockquote>
<p>I think the easy case is when the operator has $n$ distinct eigenvalues, but what about if it is diagonalizable? Is tha... | <p><strong>Theorem.</strong> <em>Let $T$ be an operator on the finite dimensional complex vector space $\mathbf{W}$. The characteristic polynomial of $T$ equals the minimal polynomial of $T$ if and only if the dimension of each eigenspace of $T$ is $1$.</em></p>
<p><em>Proof.</em> Let the characteristic and minimal po... | <p>The following equivalent criteria, valid for an arbitrary field, are short to state. Whether or not any one of the conditions is easy to test computationally may depend on the situation, though 2. is in principle always doable.</p>
<p><strong>Proposition.</strong> <em>The following are equivalent for a linear operat... |
linear-algebra | <blockquote>
<p>Show that the determinant of a matrix $A$ is equal to the product of its eigenvalues $\lambda_i$.</p>
</blockquote>
<p>So I'm having a tough time figuring this one out. I know that I have to work with the characteristic polynomial of the matrix $\det(A-\lambda I)$. But, when considering an $n \times ... | <p>Suppose that <span class="math-container">$\lambda_1, \ldots, \lambda_n$</span> are the eigenvalues of <span class="math-container">$A$</span>. Then the <span class="math-container">$\lambda$</span>s are also the roots of the characteristic polynomial, i.e.</p>
<p><span class="math-container">$$\begin{array}{rcl} \d... | <p>I am a beginning Linear Algebra learner and this is just my humble opinion. </p>
<p>One idea presented above is that </p>
<p>Suppose that $\lambda_1,\ldots \lambda_2$ are eigenvalues of $A$. </p>
<p>Then the $\lambda$s are also the roots of the characteristic polynomial, i.e.</p>
<p>$$\det(A−\lambda I)=(\lambda_... |
game-theory | <blockquote>
<p>Let <span class="math-container">$ N = {1,2,.....,n } $</span> be a set of elements called voters. Let <span class="math-container">$$C=\lbrace S : S \subseteq N \rbrace$$</span> be the set of all subsets of <span class="math-container">$N$</span>. members of <span class="math-container">$C$</span> are ... | <p>Your example with $2^{n-1}$ winning coalitions is a good one. </p>
<p>To prove there are no more than $2^{n-1}$ winning coalitions, note that a subset and its complement cannot both be winning as they do not share a voter. As no more than half the subsets can be winning, the maximum number of winning coalitions ... | <p>Your solution for $2^{n-1}$ winning coalitions is just perfect. Say that $\{1\}$ is a winning coalition, then there are $2^{n-1}$ coalitions that include $1$ and are winners, and $2^{n-1}$ that do not and are losers. These respect all the conditions:</p>
<p>(a) $N$, including $1$, is a winning coalition</p>
<p>(b)... |
geometry | <p>I'm searching to develop the intuition (rather than memorization) in relating the two forms of a dot product (by an angle theta between the vectors and by the components of the vector ). </p>
<p>For example, suppose I have vector $\mathbf{a} = (a_1,a_2)$ and vector $\mathbf{b}=(b_1,b_2)$. What's the physical or... | <p>Start with the following definition:</p>
<p><a href="https://i.sstatic.net/o2kgw.jpg" rel="noreferrer"><img src="https://i.sstatic.net/o2kgw.jpg" alt="enter image description here"></a></p>
<p>(with a negative dot product when the projection is onto $-\mathbf{b}$)</p>
<p>This implies that the dot product of perpe... | <p>I found a reasonable proof using polar coordinates. Lets suppose the point "$a$" points is $(|a|\cos(r)$ , $|a|\sin(r) )$ and the point vector "$b$" points is ($|b|\cos(s),|b|\sin(s) $).
Then doing the definition of the scalar product we get : </p>
<p>$a\cdot b = |a||b|\cos(r)\cos(s) + |b||a|\sin(r)\sin(s) = |... |
linear-algebra | <p>I am currently trying to self-study linear algebra. I've noticed that a lot of the definitions for terms (like eigenvectors, characteristic polynomials, determinants, and so on) require a <strong>square</strong> matrix instead of just any real-valued matrix. For example, <a href="http://mathworld.wolfram.com" rel="n... | <p>Remember that an $n$-by-$m$ matrix with real-number entries represents a linear map from $\mathbb{R}^m$ to $\mathbb{R}^n$ (or more generally, an $n$-by-$m$ matrix with entries from some field $k$ represents a linear map from $k^m$ to $k^n$). When $m=n$ - that is, when the matrix is square - we're talking about a map... | <p>Lots of good answers already as to why square matrices are so important. But just so you don't think that other matrices are not interesting, they have analogues of the inverse (e.g., the <a href="https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse" rel="noreferrer">Moore-Penrose inverse</a>) and non-square... |
linear-algebra | <p>Denote by $M_{n \times n}(k)$ the ring of $n$ by $n$ matrices with coefficients in the field $k$. Then why does this ring not contain any two-sided ideal? </p>
<p>Thanks for any clarification, and this is an exercise from the notes of Commutative Algebra by Pete L Clark, of which I thought as simple but I cannot ... | <p>Suppose that you have an ideal $\mathfrak{I}$ which contains a matrix with a nonzero entry $a_{ij}$. Multiplying by the matrix that has $0$'s everywhere except a $1$ in entry $(i,i)$, kill all rows except the $i$th row; multiplying by a suitable matrices on the right, kill all columns except the $j$th column; now yo... | <p>A faster, and more general result, which Arturo hinted at, is obtained via following proposition from Grillet's <em>Abstract Algebra</em>, section "Semisimple Rings and Modules", page 360:</p>
<p><img src="https://i.sstatic.net/bEYxY.jpg" alt="enter image description here"></p>
<p><strong>Consequence:</strong> if ... |
geometry | <p>When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle.</p>
<p>Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is differentiated with respect to $r$, we get $4 \pi r^2$.</p>
<p>Is this just a coincidence, or is there some de... | <p>Consider increasing the radius of a circle by an infinitesimally small amount, $dr$. This increases the area by an <a href="http://en.wikipedia.org/wiki/Annulus_%28mathematics%29" rel="noreferrer">annulus</a> (or ring) with inner radius $2 \pi r$ and outer radius $2\pi(r+dr)$. As this ring is extremely thin, we can ... | <p>$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Bd}{\partial}\DeclareMathOperator{\vol}{vol}$The formulas are no accident, but not especially deep. The explanation comes down to a couple of geometric observations.</p>
<ol>
<li><p>If $X$ is the closure of a bounded open set in the Euclidean space $\Reals^{n}$ (such as ... |
differentiation | <p>Does a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f'(x) > f(x) > 0$ exist?</p>
<p>Intuitively, I think it can't exist.</p>
<p>I've tried finding the answer using the definition of derivative:</p>
<ol>
<li><p>I know that if $\lim_{x \rightarrow k} f(x)$ exists and is finite, then $\lim_{x \ri... | <p><strong>expanded from David's comment</strong> </p>
<p>$f' > f$ means $f'/f > 1$ so $(\log f)' > 1$. Why not take $\log f > x$, say $\log f = 2x$, or $f = e^{2x}$.</p>
<p>Thus $f' > f > 0$ since $2e^{2x} > e^{2x} > 0$.</p>
<p><strong>added:</strong> Is there a sub-exponential solution?<b... | <p>There are multiple mistakes in your proof (i.e. dividing by $f(x)$ is not necessarily okay, since it is not necessarily positive). The most major is that you treat the variable in the limit as if it were not "bound". That is, if you have something like
$$\lim_{h\rightarrow 0^+}1>0$$
which is true, you can't neces... |
geometry | <p>Wait before you dismiss this as a crank question :)</p>
<p>A friend of mine teaches school kids, and the book she uses states something to the following effect: </p>
<blockquote>
<p>If you divide the circumference of <em>any</em> circle by its diameter, you get the <em>same</em> number, and this number is an irr... | <p>If the kids are not too old you could visually try the following which is very straight forward. Build a few models of a circle of paperboard and then take a wire and put it straigt around it. Mark the point of the line where it is meets itself again and then measure the length of it. You will get something like 3.1... | <p>You can try doing what Archimedes did: using polygons inside and outside the circle.</p>
<p><a href="http://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html" rel="nofollow">Here is a webpage which seems to have a good explanation.</a></p>
<p>An other method one can try is to use the fact that the... |
probability | <p>If nine coins are tossed, what is the probability that the number of heads is even?</p>
<p>So there can either be 0 heads, 2 heads, 4 heads, 6 heads, or 8 heads.</p>
<p>We have <span class="math-container">$n = 9$</span> trials, find the probability of each <span class="math-container">$k$</span> for <span class="... | <p>The probability is <span class="math-container">$\frac{1}{2}$</span> because the last flip determines it.</p>
| <p>If there are an even number of heads then there must be an odd number of tails. But heads and tails are symmetrical, so the probability must be <span class="math-container">$1/2$</span>.</p>
|
logic | <p>A bag contains 2 counters, as to which nothing is known except
that each is either black or white. Ascertain their colours without taking
them out of the bag.</p>
<p>Carroll's solution: One is black, and the other is white.</p>
<blockquote>
<h3><a href="https://i.sstatic.net/mlmdS.jpg" rel="noreferrer">Lewis Carroll... | <p>There is a reason this is the last of Lewis Carroll's <em>Pillow Problems</em>. It is a mathematical joke from the author of <em>Alice in Wonderland</em>.</p>
<p>The error (and Lewis Carroll knew it) is the phrase</p>
<blockquote>
<p>We know ... that any <em>other</em> state of things would <em>not</em> give th... | <p>Well, of course the reasoning is flawed, since it's certainly possible to have a bag with two counters of the same color in it!</p>
<p>The facts that are correct are:</p>
<p>The probability of drawing a black counter from a fixed bag with 3 counters is 2/3 iff the bag contains two black counters.</p>
<p>By adding... |
linear-algebra | <p>Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$.</p>
<p>How could we define the following operation? $$\mathsf{A}!$$</p>
<p>Maybe we could make some simple example, admitted it makes any sense, with </p>
<p>$$\mathsf{A} =
\left(\begin{matrix}
1 & 3 \\
2 & 1
\end{matrix}
\right)
... | <p>For any holomorphic function <span class="math-container">$G$</span>, we can define a corresponding matrix function <span class="math-container">$\tilde{G}$</span> via (a formal version of) the Cauchy Integral Formula: We set
<span class="math-container">$$\tilde{G}(B) := \frac{1}{2 \pi i} \oint_C G(z) (z I - B)^{-1... | <p>The <a href="https://en.wikipedia.org/wiki/Gamma_function">gamma function</a> is analytic. Use the <a href="https://en.wikipedia.org/wiki/Matrix_function">power series</a> of it.</p>
<p>EDIT: already done: <a href="http://www.sciencedirect.com/science/article/pii/S0893965997001390">Some properties of Gamma and Beta... |
geometry | <p>I learned that the volume of a sphere is <span class="math-container">$\frac{4}{3}\pi r^3$</span>, but why? The <span class="math-container">$\pi$</span> kind of makes sense because its round like a circle, and the <span class="math-container">$r^3$</span> because it's 3-D, but <span class="math-container">$\frac{4}... | <p>In addition to the methods of calculus, Pappus, and Archimedes already mentioned, <a href="https://en.wikipedia.org/wiki/Cavalieri%27s_principle" rel="nofollow noreferrer">Cavalieri's Principle</a> can be useful for these kinds of problems.</p>
<p>Suppose you have two solid figures lined up next to each other, each ... | <p>The volume of a sphere with radius <span class="math-container">$a$</span> may be found by evaluating the triple integral<span class="math-container">$$V=\iiint \limits _S\,dx\,dy\,dz,$$</span>where <span class="math-container">$S$</span> is the volume enclosed by the sphere <span class="math-container">$x^2+y^2+z^2... |
combinatorics | <p>The Collatz Conjecture is a famous conjecture in mathematics that has lasted for over 70 years. It goes as follows:</p>
<p>Define $f(n)$ to be as a function on the natural numbers by:</p>
<p>$f(n) = n/2$ if $n$ is even and
$f(n) = 3n+1$ if $n$ is odd</p>
<p>The conjecture is that for all $n \in \mathbb{N}$, $n$ e... | <p>You shouldn't expect this to be true. Here is a nonrigorous argument. Let $n_k$ be the sequence of odd numbers you obtain. So (heuristically), with probability $1/2$, we have $n_{k+1} = (5n_k+1)/2$, with probability $1/4$, we have $n_{k+1} = (5 n_k+1)/4$, with probability $1/8$, we have $n_{k+1} = (5 n_k+1)/8$ and s... | <p>In Part I of Lagarias' <a href="http://arxiv.org/abs/math/0309224">extensive, annotated bibliography</a> of the 3x+1 problem, he notes a 1999 paper by Metzger (reference 112) regarding the 5x+1 problem:</p>
<blockquote>
<p>For the 5x + 1 problem he shows that on the positive integers there is no cycle of size 1, ... |
number-theory | <p>I've solved for it making a computer program, but was wondering there was a mathematical equation that you could use to solve for the nth prime?</p>
| <p>No, there is no known formula that gives the nth prime, except <a href="https://www.youtube.com/watch?v=j5s0h42GfvM" rel="nofollow noreferrer">artificial ones</a> you can write that are basically equivalent to "the <span class="math-container">$n$</span>th prime". But if you only want an approximation, the... | <p>Far better than sieving in the large range ShreevatsaR suggested (which, for the 10¹⁵th prime, has 10¹⁵ members and takes about 33 TB to store in compact form), take a good first guess like Riemann's R and use one of the advanced methods of computing pi(x) for that first guess. (If this is far off for some reason—i... |
linear-algebra | <p>If the matrix is positive definite, then all its eigenvalues are strictly positive. </p>
<p>Is the converse also true?<br>
That is, if the eigenvalues are strictly positive, then matrix is positive definite?<br>
Can you give example of $2 \times 2$ matrix with $2$ positive eigenvalues but is not positive definite?... | <p>I think this is false. Let <span class="math-container">$A = \begin{pmatrix} 1 & -3 \\ 0 & 1 \end{pmatrix}$</span> be a 2x2 matrix, in the canonical basis of <span class="math-container">$\mathbb R^2$</span>. Then A has a double eigenvalue b=1. If <span class="math-container">$v=\begin{pmatrix}1\\1\end{pmatr... | <p>This question does a great job of illustrating the problem with thinking about these things in terms of coordinates. The thing that is positive-definite is not a matrix $M$ but the <em>quadratic form</em> $x \mapsto x^T M x$, which is a very different beast from the linear transformation $x \mapsto M x$. For one t... |
geometry | <p><strong>Background:</strong> Many (if not all) of the transformation matrices used in $3D$ computer graphics are $4\times 4$, including the three values for $x$, $y$ and $z$, plus an additional term which usually has a value of $1$.</p>
<p>Given the extra computing effort required to multiply $4\times 4$ matrices i... | <p>I'm going to copy <a href="https://stackoverflow.com/questions/2465116/understanding-opengl-matrices/2465290#2465290">my answer from Stack Overflow</a>, which also shows why 4-component vectors (and hence 4×4 matrices) are used instead of 3-component ones.</p>
<hr>
<p>In most 3D graphics a point is represent... | <blockquote>
<p>Even though 3x3 matrices should (?) be sufficient to describe points and transformations in 3D space.</p>
</blockquote>
<p>No, they aren't enough! Suppose you represent points in space using 3D vectors. You can transform these using 3x3 matrices. But if you examine the definition of matrix multiplica... |
probability | <p>Given $n$ independent geometric random variables $X_n$, each with probability parameter $p$ (and thus expectation $E\left(X_n\right) = \frac{1}{p}$), what is
$$E_n = E\left(\max_{i \in 1 .. n}X_n\right)$$</p>
<hr>
<p>If we instead look at a continuous-time analogue, e.g. exponential random variables $Y_n$ with rat... | <p>There is no nice, closed-form expression for the expected maximum of IID geometric random variables. However, the expected maximum of the corresponding IID exponential random variables turns out to be a very good approximation. More specifically, we have the hard bounds</p>
<p>$$\frac{1}{\lambda} H_n \leq E_n \le... | <p>First principle:</p>
<blockquote>
<p>To deal with maxima $M$ of independent random variables, use as much as possible events of the form $[M\leqslant x]$.</p>
</blockquote>
<p>Second principle:</p>
<blockquote>
<p>To compute the expectation of a nonnegative random variable $Z$, use as much as possible the com... |
probability | <p>I have been using Sebastian Thrun's course on AI and I have encountered a slightly difficult problem with probability theory.</p>
<p>He poses the following statement:</p>
<p>$$
P(R \mid H,S) = \frac{P(H \mid R,S) \; P(R \mid S)}{P(H \mid S)}
$$</p>
<p>I understand he used Bayes' Rule to get the RHS equation, bu... | <p>Taking it one step at a time:
$$\begin{align}
\mathsf P(R\mid H, S) & = \frac{\mathsf P(R,H,S)}{\mathsf P(H, S)}
\\[1ex] & =\frac{\mathsf P(H\mid R,S)\,\mathsf P(R, S)}{\mathsf P(H, S)}
\\[1ex] & =\frac{\mathsf P(H\mid R,S)\,\mathsf P(R\mid S)\,\mathsf P(S)}{\mathsf P(H, S)}
\\[1ex] & =\frac{\mathsf ... | <p>You don't really need Bayes' Theorem. Just apply the definition of conditional probability in two ways. Firstly,</p>
<p>\begin{eqnarray*}
P(R\mid H,S) &=& \dfrac{P(R,H\mid S)}{P(H\mid S)} \\
&& \\
\therefore\quad P(R,H\mid S) &=& P(R\mid H,S)P(H\mid S).
\end{eqnarray*}</p>
<p>Secondly,</p>
... |
probability | <p>I was watching the movie <span class="math-container">$21$</span> yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the wrong answer) which revolves around theoretical probability.</p>
<p>... | <p>This problem, known as the Monty Hall problem, is famous for being so bizarre and counter-intuitive. It is in fact best to switch doors, and this is not hard to prove either. In my opinion, the reason it seems so bizarre the first time one (including me) encounters it is that humans are simply bad at thinking about ... | <p>To understand why your odds increase by changing door, let us take an extreme example first. Say there are $10000$ doors. Behind one of them is a car and behind the rest are donkeys. Now, the odds of choosing a car is $1\over10000$ and the odds of choosing a donkey are $9999\over10000$. Say you pick a random door, w... |
logic | <p>Let's say that I prove statement $A$ by showing that the negation of $A$ leads to a contradiction. </p>
<p>My question is this: How does one go from "so there's a contradiction if we don't have $A$" to concluding that "we have $A$"?</p>
<p>That, to me, seems the exact opposite of logical. It sounds like we say "so... | <p>Proof by contradiction, as you stated, is the rule$\def\imp{\Rightarrow}$ "$\neg A \imp \bot \vdash A$" for any statement $A$, which in English is "If you can derive the statement that $\neg A$ implies a contradiction, then you can derive $A$". As pointed out by others, this is not a valid rule in intuitionistic log... | <p>A contradiction isn't a “problem”. A contradiction is an impossibility. This isn't a matter of saying “Gee, if I have fewer than 20 dollars in the back I won't be able to go out to dinner and I want to so badly, I'll just assume I have more than 20 dollars.” This is a matter of walking into the bank and saying "I... |
number-theory | <p>In <a href="https://math.stackexchange.com/a/373935/752">this recent answer</a> to <a href="https://math.stackexchange.com/q/373918/752">this question</a> by Eesu, Vladimir
Reshetnikov proved that
$$
\begin{equation}
\left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1}
\end{equation}
$$</p... | <p>The solutions are of the form $\displaystyle(p, q)= \left(\frac{3t^2nr+n^3}{8},\,\frac{3n^2t+t^3r}{8}\right)$, for any rational parameter $t$. To prove it, we start with $$\left(p+q\sqrt{r}\right)^{1/3}+\left(p-q\sqrt{r}\right)^{1/3}=n\tag{$\left(p,q,n,r\right)\in\mathbb{N}^{4}$}$$
and cube both sides using the iden... | <p>Here's a way of finding, at the very least, a large class of rational solutions. It seems plausible to me that these are all the rational solutions, but I don't actually have a proof yet...</p>
<p>Say we want to solve $(p+q\sqrt{3})^{1/3}+(p-q\sqrt{3})^{1/3}=n$ for some fixed $n$. The left-hand side looks an awful ... |
logic | <p>Recently I have started reviewing mathematical notions, that I have always just accepted. Today it is one of the fundamental ones used in equations: </p>
<blockquote>
<p>If we have an equation, then the equation holds if we do the same to both sides. </p>
</blockquote>
<p>This seems perfectly obvious, but it mus... | <p>This axiom is known as the <em>substitution property of equality</em>. It states that if $f$ is a function, and $x = y$, then $f(x) = f(y)$. See, for example, <a href="https://en.wikipedia.org/wiki/First-order_logic#Equality_and_its_axioms" rel="nofollow noreferrer">Wikipedia</a>.</p>
<p>For example, if your equati... | <p>"Do the same to both sides" is rather vague. What we can say is that if $f:A \rightarrow B$ is a <em>bijection</em> between sets $A$ and $B$ then, by definition</p>
<p>$\forall \space x,y \in A \space x=y \iff f(x)=f(y)$</p>
<p>The operation of adding $c$ (and its inverse subtracting $c$) is a bijection in groups... |
number-theory | <p>Our algebra teacher usually gives us a paper of $20-30$ questions for our homework. But each week, he tells us to do all the questions which their number is on a specific form.</p>
<p>For example, last week it was all the questions on the form of $3k+2$ and the week before that it was $3k+1$. I asked my teacher wh... | <blockquote>
<p>If we prove that for every <em>x</em> there exists a prime number between $x^2$ and $x^2+2x+1$, we are done.</p>
</blockquote>
<p>This is <a href="http://en.wikipedia.org/wiki/Legendre%27s_conjecture">Legendre's conjecture</a>, which remains unsolved. Hence <a href="http://static1.wikia.nocookie.net/... | <p>Any of the accepted conjectures on sieves and random-like behavior of primes would predict that the chance of finding counterexamples to the conjecture in $(x^2, (x+1)^2)$ decrease rapidly with $x$, since they correspond to random events that are (up to logarithmic factors) $x$ standard deviations from the mean, and... |
linear-algebra | <p>The rotation matrix
$$\pmatrix{ \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta}$$
has complex eigenvalues $\{e^{\pm i\theta}\}$ corresponding to eigenvectors $\pmatrix{1 \\i}$ and $\pmatrix{1 \\ -i}$. The real eigenvector of a 3d rotation matrix has a natural interpretation as the axis of rotation.... | <p><a href="https://math.stackexchange.com/a/241399/3820">Tom Oldfield's answer</a> is great, but you asked for a geometric interpretation so I made some pictures.</p>
<p>The pictures will use what I called a "phased bar chart", which shows complex values as bars that have been rotated. Each bar corresponds to a vecto... | <p>Lovely question!</p>
<p>There is a kind of intuitive way to view the eigenvalues and eigenvectors, and it ties in with geometric ideas as well (without resorting to four dimensions!). </p>
<p>The matrix, is unitary (more specifically, it is real so it is called orthogonal) and so there is an orthogonal basis of ei... |
linear-algebra | <p>I'm doing a raytracing exercise. I have a vector representing the normal of a surface at an intersection point, and a vector of the ray to the surface. How can I determine what the reflection will be?</p>
<p>In the below image, I have <code>d</code> and <code>n</code>. How can I get <code>r</code>?</p>
<p><img src... | <p>$$r = d - 2 (d \cdot n) n$$
</p>
<p>where $d \cdot n$ is the dot product, and
$n$ must be normalized.</p>
| <p>Let $\hat{n} = {n \over \|n\|}$. Then $\hat{n}$ is the vector of magnitude one in the same direction as $n$. The projection of $d$ in the $n$ direction is given by $\mathrm{proj}_{n}d = (d \cdot \hat{n})\hat{n}$, and the projection of $d$ in the orthogonal direction is therefore given by $d - (d \cdot \hat{n})\hat{n... |
logic | <p>I have recently been reading about first-order logic and set theory. I have seen standard set theory axioms (say ZFC) formally constructed in first-order logic, where first-order logic is used as an object language and English is used as a metalanguage. </p>
<p>I'd like to construct first-order logic and then const... | <p>I think there are two (very interesting) questions here. Let me try to address them.</p>
<blockquote>
<p>First, the title question: do we presuppose natural numbers in first-order logic?</p>
</blockquote>
<p>I would say the answer is definitely <em>yes</em>. We have to assume something to get off the ground; on ... | <p>There are three inter-related concepts:</p>
<ul>
<li>The natural numbers</li>
<li>Finite strings of symbols</li>
<li>Formulas - particular strings of symbols used in formal logic.</li>
</ul>
<p>If we understand any one of these three, we can use that to understand all three. </p>
<p>For example, if we know what s... |
number-theory | <p>We have,</p>
<p>$$\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$$</p>
<p>$$2\sum_{k=1}^n k^5 = -\Big(\sum_{k=1}^n k\Big)^2+3\Big(\sum_{k=1}^n k^2\Big)^2$$</p>
<p>$$2\sum_{k=1}^n k^7 = \Big(\sum_{k=1}^n k\Big)^2-3\Big(\sum_{k=1}^n k^2\Big)^2+4\Big(\sum_{k=1}^n k^3\Big)^2$$</p>
<p>and so on (apparently). </p>
<p>... | <p>This is a partial answer, it just establishes the existence.</p>
<p>We have
$$s_m(n) = \sum_{k=1}^n k^m = \frac{1}{m+1}
\left(\operatorname{B}_{m+1}(n+1)-\operatorname{B}_{m+1}(1)\right)$$
where $\operatorname{B}_m(x)$ denotes the monic
<a href="http://mathworld.wolfram.com/BernoulliPolynomial.html">Bernoulli polyn... | <p><strong>Not an answer, but only some results I found.</strong></p>
<p>Denote $s_m=\sum_{k=1}^nk^m.$
Suppose that
$$s_{2m+1}=\sum_{i=1}^m a_is_i^2,a_i\in\mathbb Q.$$</p>
<p>Use the method of undetermined coefficients, we can get a list of $\{m,\{a_i\}\}$:
$$\begin{array}{l}
\{1,\{1\}\} \\
\left\{2,\left\{-\frac{... |
probability | <p>If you are given a die and asked to roll it twice. What is the probability that the value of the second roll will be less than the value of the first roll?</p>
| <p>There are various ways to answer this. Here is one:</p>
<p>There is clearly a $1$ out of $6$ chance that the two rolls will be the same, hence a $5$ out of $6$ chance that they will be different. Further, the chance that the first roll is greater than the second must be equal to the chance that the second roll is... | <p>Here another way to solve the problem
$$
\text{Pr }[\textrm{second} > \textrm{first}] + \text{Pr }[\textrm{second} < \textrm{first}] + \text{Pr }[\textrm{second} = \textrm{first}] = 1
$$
Because of symmetry $\text{Pr }[\text{second} > \text{first}] = \text{Pr }[\text{second} < \text{first}]$, so
$$
\text... |
logic | <p>A bag contains 2 counters, as to which nothing is known except
that each is either black or white. Ascertain their colours without taking
them out of the bag.</p>
<p>Carroll's solution: One is black, and the other is white.</p>
<blockquote>
<h3><a href="https://i.sstatic.net/mlmdS.jpg" rel="noreferrer">Lewis Carroll... | <p>There is a reason this is the last of Lewis Carroll's <em>Pillow Problems</em>. It is a mathematical joke from the author of <em>Alice in Wonderland</em>.</p>
<p>The error (and Lewis Carroll knew it) is the phrase</p>
<blockquote>
<p>We know ... that any <em>other</em> state of things would <em>not</em> give th... | <p>Well, of course the reasoning is flawed, since it's certainly possible to have a bag with two counters of the same color in it!</p>
<p>The facts that are correct are:</p>
<p>The probability of drawing a black counter from a fixed bag with 3 counters is 2/3 iff the bag contains two black counters.</p>
<p>By adding... |
linear-algebra | <p>What is the difference between sum of two vectors and direct sum of two vector subspaces? </p>
<p>My textbook is confusing about it. Any help would be appreciated.</p>
| <p><em>Direct sum</em> is a term for <em>subspaces</em>, while <em>sum</em> is defined for <em>vectors</em>.
We can take the sum of subspaces, but then their intersection need not be $\{0\}$.</p>
<p><strong>Example:</strong> Let $u=(0,1),v=(1,0),w=(1,0)$. Then</p>
<ul>
<li>$u+v=(1,1)$ (sum of vectors),</li>
<li>$\ope... | <p>In Axler's Linear Algebra Done Right, he defines the <em>sum of subspaces</em> $U + V$ as </p>
<p>$\{u + v : u \in U, v \in V \}$.</p>
<p>He then says that $W = U \oplus V$ if </p>
<p>(1) $W = U + V$, and</p>
<p>(2) The representation of each $w$ as $u + v$ is <em>unique</em>.</p>
<p>This is a different way... |
probability | <p>This is a neat little problem that I was discussing today with my lab group out at lunch. Not particularly difficult but interesting implications nonetheless</p>
<p>Imagine there are a 100 people in line to board a plane that seats 100. The first person in line, Alice, realizes she lost her boarding pass, so when sh... | <p>Here is a rephrasing which simplifies the intuition of this nice puzzle.</p>
<p>Suppose whenever someone finds their seat taken, they politely evict the squatter and take their seat. In this case, the first passenger (Alice, who lost her boarding pass) keeps getting evicted (and choosing a new random seat) until, b... | <p>This is a classic puzzle!</p>
<p>The answer is that the probability that the last person ends in up in their proper seat is exactly <span class="math-container">$\frac{1}{2}$</span>.</p>
<p>The reasoning goes as follows:</p>
<p>First observe that the fate of the last person is determined the moment either the first ... |
combinatorics | <p>How can we prove that a graph is bipartite if and only if all of its cycles have even order? Also, does this theorem have a common name? I found it in a maths Olympiad toolbox.</p>
| <p>One direction is very easy: if <span class="math-container">$G$</span> is bipartite with vertex sets <span class="math-container">$V_1$</span> and <span class="math-container">$V_2$</span>, every step along a walk takes you either from <span class="math-container">$V_1$</span> to <span class="math-container">$V_2$</... | <blockquote>
<p>does this theorem have a common name?</p>
</blockquote>
<p>It is sometimes called <strong>König's Theorem</strong> (1936), for example in lecture notes <a href="http://www-math.ucdenver.edu/%7Ewcherowi/courses/m4408/gtln5.html" rel="nofollow noreferrer">here</a> (<a href="https://web.archive.org/web/202... |
linear-algebra | <p>A Hermitian matrix always has real eigenvalues and real or complex orthogonal eigenvectors. A real symmetric matrix is a special case of Hermitian matrices, so it too has orthogonal eigenvectors and real eigenvalues, but could it ever have complex eigenvectors?</p>
<p>My intuition is that the eigenvectors are alway... | <p>Always try out examples, starting out with the simplest possible examples (it may take some thought as to which examples are the simplest). Does for instance the identity matrix have complex eigenvectors? This is pretty easy to answer, right? </p>
<p>Now for the general case: if $A$ is any real matrix with real ei... | <p>If $A$ is a symmetric $n\times n$ matrix with real entries, then viewed as an element of $M_n(\mathbb{C})$, its eigenvectors always include vectors with non-real entries: if $v$ is any eigenvector then at least one of $v$ and $iv$ has a non-real entry.</p>
<p>On the other hand, if $v$ is any eigenvector then at lea... |
geometry | <p>My wife came up with the following problem while we were making some decorations for our baby: given a triangle, what is the largest equilateral triangle that can be inscribed in it? (In other words: given a triangular piece of cardboard, what is the largest equilateral triangle that you can cut out of it?)</p>
<p>... | <p>Consider an arbitrary triangle PQR. Excluding the trivial case of an equilateral triangle, either one or two of the angles are at least 60 degrees, and <em>wlog</em> let P be the minimal example of these. I.e. P is the smallest angle of at least 60 degrees, and Q is either the largest angle or the second largest aft... | <p>Let $T$ be the given triangle, having vertices $A_i$, angles $\alpha_i$, and edges $a_i=[A_{i-1},A_{i+1}]$. Use $\triangle$ as variable for equilateral triangles contained in $T$, and denote the vertices of such triangles by $v_j$. Given an instance $I:=(\triangle, T)$ call the number of incidences $v_j\in a_i$ th... |
game-theory | <p>[There's still the strategy to go. A suitably robust argument that establishes what is <em>statistically</em> the best strategy will be accepted.]</p>
<p><strong>Here's my description of the game:</strong></p>
<p>There's a <span class="math-container">$4\times 4$</span> grid with some random, numbered cards on. The ... | <p>The strategy I employ is simply to make the move that leaves the most available moves, and to disregard score entirely. As a natural consequence of playing more moves, the score of the board will increase simply because the only way to continue playing is to make combinations, and combinations generate higher scores... | <p><em>A partial answer:</em> </p>
<p>The highest possible score is $16\times 3^{12}$.</p>
<p>If the game could start with no available moves, then just suppose it starts with $2^{11}3$ everywhere.</p>
<p>Alternatively, suppose you start with $2^{11}3$ in the top left and suppose every new card happens to be $2^{11}... |
probability | <p>I've a confession to make. I've been using PDF's and PMF's without actually knowing what they are. My understanding is that density equals area under the curve, but if I look at it that way, then it doesn't make sense to refer to the "mass" of a random variable in discrete distributions. How can I interpre... | <p>(This answer takes as its starting point the OP's question in the comments, "Let me understand mass before going to density. Why do we call a point in the discrete distribution as mass? Why can't we just call it a point?")</p>
<p>We could certainly call it a point. The utility of the term "probability mass functio... | <p>Probability mass functions are used for discrete distributions. It assigns a probability to each point in the sample space. Whereas the integral of a probability density function gives the probability that a random variable falls within some interval. </p>
|
matrices | <p>I am taking a proof-based introductory course to Linear Algebra as an undergrad student of Mathematics and Computer Science. The author of my textbook (Friedberg's <em>Linear Algebra</em>, 4th Edition) says in the introduction to Chapter 4:</p>
<blockquote>
<p>The determinant, which has played a prominent role in... | <p>Friedberg is not wrong, at least on a historical standpoint, as I am going to try to show it.</p>
<p>Determinants were discovered "as such" in the second half of the 18th century by Cramer who used them in his celebrated rule for the solution of a linear system (in terms of quotients of determinants). Thei... | <p><strong>It depends who you speak to.</strong></p>
<ul>
<li>In <strong>numerical mathematics</strong>, where people actually have to compute things on a computer, it is largely recognized that <strong>determinants are useless</strong>. Indeed, in order to compute determinants, either you use the Laplace recursive ru... |
probability | <p>Wikipedia says:</p>
<blockquote>
<p>The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one.</p>
</blockquote>
<p>and it also says.</p>
<blockquote>
<p>Unlike a probability, a probability density function can take on values greater than one; for examp... | <p>Consider the uniform distribution on the interval from $0$ to $1/2$. The value of the density is $2$ on that interval, and $0$ elsewhere. The area under the graph is the area of a rectangle. The length of the base is $1/2$, and the height is $2$
$$
\int\text{density} = \text{area of rectangle} = \text{base} \cdo... | <p>Remember that the 'PD' in PDF stands for "probability density", not probability. Density means probability per unit value of the random variable. That can easily exceed $1$. What has to be true is that the integral of this density function taken with respect to this value must be exactly $1$.</p>
<p>If we know a PD... |
matrices | <p>In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t have an inverse. I can find the determinant of a <span class="math-container">$2\times 2$</span> matrix by the formul... | <p>Your trouble with determinants is pretty common. They’re a hard thing to teach well, too, for two main reasons that I can see: the formulas you learn for computing them are messy and complicated, and there’s no “natural” way to interpret the value of the determinant, the way it’s easy to interpret the derivatives yo... | <p>You could think of a determinant as a volume. Think of the columns of the matrix as vectors at the origin forming the edges of a skewed box. The determinant gives the volume of that box. For example, in 2 dimensions, the columns of the matrix are the edges of a rhombus.</p>
<p>You can derive the algebraic proper... |
matrices | <p>there is a similar thread here <a href="https://math.stackexchange.com/questions/311580/coordinate-free-proof-of-operatornametrab-operatornametrba">Coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$?</a>, but I'm only looking for a simple linear algebra proof. </p>
| <p>Observe that if $A$ and $B$ are $n\times n$ matrices, $A=(a_{ij})$, and $B=(b_{ij})$, then
$$(AB)_{ii} = \sum_{k=1}^n a_{ik}b_{ki},$$
so
$$
\operatorname{Tr}(AB) = \sum_{j=1}^n\sum_{k=1}^n a_{jk}b_{kj}.
$$
Conclude calculating the term $(BA)_{ii}$ and comparing both traces.</p>
| <blockquote>
<p>For any couple $(A,B)$ of $n\times n$ matrices with complex entries, the following identity holds:
$$ \operatorname{Tr}(AB) = \operatorname{Tr}(BA).$$</p>
</blockquote>
<p><strong>Proof</strong>. Assuming $A$ is an invertible matrix, $AB$ and $BA$ share the same characteristic polynomial, since the... |
logic | <p>Why is this true?</p>
<p><span class="math-container">$\exists x\,\big(P(x) \rightarrow \forall y\:P(y)\big)$</span></p>
| <p>Since this may be homework, I do not want to provide the full formal proof, but I will share the informal justification. Classical first-order logic typically makes the assumption of existential import (i.e., that the domain of discourse is non-empty). In classical logic, the principle of excluded middle holds, i.... | <p>Hint: The only way for $A\implies B$ to be false is for $A$ to be true and $B$ to be false.</p>
<p>I don't think this is actually true unless you know your domain isn't empty. If your domain is empty, then $\forall y: P(y)$ is true "vacuously," but $\exists x: Q$ is not true for any $Q$.</p>
|
logic | <p>maybe this question doesn't make sense at all. I don't know exactly the meaning of all these concepts, except the internal language of a topos (and searching on the literature is not helping at all). </p>
<p>However, vaguely speaking by a logic I mean a pair $(\Sigma, \vdash)$ where $\Sigma$ is a signature (it has ... | <p>The internal logic of a topos is an instance of the internal logic of a category (since toposes are special kinds of categories). The internal logic of toposes (instead of an arbitrary category) can also be interpreted with the Kripke-Joyal semantics. (<strong>Update almost ten years later:</strong> A form of the Kr... | <p>Your definition of logic is pretty much correct. A logic contains both the <em>language</em> which the signature <span class="math-container">$\Sigma$</span> generates and the deductive system defined by <span class="math-container">$\vdash$</span>.</p>
<p>A <em>type theory</em> is a logic with different sorts of i... |
linear-algebra | <p>If I have a covariance matrix for a data set and I multiply it times one of it's eigenvectors. Let's say the eigenvector with the highest eigenvalue. The result is the eigenvector or a scaled version of the eigenvector. </p>
<p>What does this really tell me? Why is this the principal component? What property m... | <p><strong>Short answer:</strong> The eigenvector with the largest eigenvalue is the direction along which the data set has the maximum variance. Meditate upon this.</p>
<p><strong>Long answer:</strong> Let's say you want to reduce the dimensionality of your data set, say down to just one dimension. In general, this me... | <p>Some informal explanation:</p>
<p>Covariance matrix $C_y$ (it is symmetric) encodes the correlations between variables of a vector. In general a covariance matrix is non-diagonal (i.e. have non zero correlations with respect to different variables).</p>
<p><strong>But it's interesting to ask, is it possible to dia... |
geometry | <p>Consider the function <span class="math-container">$f(x)=a_0x^2$</span> for some <span class="math-container">$a_0\in \mathbb{R}^+$</span>. Take <span class="math-container">$x_0\in\mathbb{R}^+$</span> so that the arc length <span class="math-container">$L$</span> between <span class="math-container">$(0,0)$</span> ... | <p>Phrased differently, what we want are the level curves of the function</p>
<p><span class="math-container">$$\frac{1}{2}f(x,y) = \int_0^x\sqrt{1+\frac{4y^2t^2}{x^4}}\:dt = \frac{1}{2}\int_0^2 \sqrt{x^2+y^2t^2}\:dt$$</span></p>
<p>which will always be perpendicular to the gradient at that point</p>
<p><span class="ma... | <p><span class="math-container">$L$</span> being the known arc length, let <span class="math-container">$x_1=\frac t{2a}$</span> and <span class="math-container">$k=4a_1L$</span>; then you need to solve for <span class="math-container">$t$</span> the equation
<span class="math-container">$$k=t\sqrt{t^2+1} +\sinh ^{-1}(... |
logic | <p>In <a href="https://www.youtube.com/watch?v=O4ndIDcDSGc" rel="noreferrer">the most recent numberphile video</a>, Marcus du Sautoy claims that a proof for the Riemann hypothesis must exist (starts at the 12 minute mark). His reasoning goes as follows:</p>
<ul>
<li><p>If the hypothesis is undecidable, there is no pr... | <p>The issue here is how complicated is each statement, when formulated as a claim about the natural numbers (the Riemann hypothesis can be made into such statement).</p>
<hr>
<p>For the purpose of this discussion we work in the natural numbers, with $+,\cdot$ and the successor function, and Peano Axioms will be our ... | <p>The last bullet point, saying that this constitutes a proof it is decidable, does not follow.</p>
<p>$X$ is decidable means either $X$ is provable or $\neg X$ is provable. It's possible that both are provable, in which case the theory is inconsistent and every statement and its negation are provable and every stat... |
probability | <p><strong>Question :</strong> What is the difference between Average and Expected value?</p>
<hr />
<p>I have been going through the definition of expected value on <a href="http://en.wikipedia.org/wiki/Expected_value" rel="noreferrer">Wikipedia</a> beneath all that jargon it seems that the expected value of a distrib... | <p>The concept of expectation value or expected value may be understood from the following example. Let $X$ represent the outcome of a roll of an unbiased six-sided die. The possible values for $X$ are 1, 2, 3, 4, 5, and 6, each having the probability of occurrence of 1/6. The expectation value (or expected value) of $... | <p>The expected value, or mean $\mu_X =E_X[X]$, is a parameter associated with the distribution of a random variable $X$.</p>
<p>The average $\overline X_n$ is a computation performed on a sample of size $n$ from that distribution. It can also be regarded as an unbiased estimator of the mean, meaning that if each $X_i... |
number-theory | <p>Mathematica knows that the logarithm of $n$ is:</p>
<p>$$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$</p>
<p>The von Mangoldt function should then be:</p>
<p>$$\Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \frac{\mu(d)}{d^{(s-1)}}.$$</p>
<p>Setting... | <p>The Laplace transform of a function</p>
<p>$\sum _{i=1}^{\infty } a_i \delta (t-\log (i))$
where $\delta (t-\log (i))$ is the Delta function (i.e Unit impulse) at time $\log(i)$</p>
<p>is</p>
<p>$\int_0^{\infty } e^{-s t} \sum _{i=1}^{\infty } a_i \delta (t-\log (i)) \,
dt$</p>
<p>or</p>
<p>$\sum _{i=1}^{\i... | <p>This is not a complete answer but I want show that there is nothing mysterious going on here.</p>
<p>We want to prove that:</p>
<p>$$\text{Fourier Transform of } \Lambda(1)...\Lambda(k) \sim \sum\limits_{n=1}^{n=\infty} \frac{1}{n} \zeta(s)\sum\limits_{d|n}\frac{\mu(d)}{d^{(s-1)}}$$</p>
<p>The Dirichlet inverse o... |
linear-algebra | <blockquote>
<p>Let <span class="math-container">$ \sigma(A)$</span> be the set of all eigenvalues of <span class="math-container">$A$</span>. Show that <span class="math-container">$ \sigma(A) = \sigma\left(A^T\right)$</span> where <span class="math-container">$A^T$</span> is the transpose matrix of <span class="mat... | <p>The matrix <span class="math-container">$(A - \lambda I)^{T}$</span> is the same as the matrix <span class="math-container">$\left(A^{T} - \lambda I\right)$</span>, since the identity matrix is symmetric.</p>
<p>Thus:</p>
<p><span class="math-container">$$\det\left(A^{T} - \lambda I\right) = \det\left((A - \lambda... | <p>I'm going to work a little bit more generally.</p>
<p>Let $V$ be a finite dimensional vector space over some field $K$, and let $\langle\cdot,\cdot\rangle$ be a <em>nondegenerate</em> bilinear form on $V$.</p>
<p>We then have for every linear endomorphism $A$ of $V$, that there is a unique endomorphism $A^*$ of $V... |
geometry | <p>Given a surface $f(x,y,z)=0$, how would you determine whether or not it's a surface of revolution, and find the axis of rotation?</p>
<p>The special case where $f$ is a polynomial is also of interest.</p>
<p>A few ideas that might lead somewhere, maybe:</p>
<p><strong>(1) For algebra folks:</strong> Surfaces of t... | <p>You can reduce it to an algebraic problem as follows:</p>
<p>The definition of a surface of revolution is that there is some axis such that rotations about this axis leave the surface invariant. Let's denote the action of a rotation by angle $\theta$ about this axis by the map $\vec x\mapsto \vec R_\theta(\vec x)$.... | <p>If you are working numerically you are probably not interested in degenerate cases, so let's assume that the surface is "generic" in a suitable sense (this rules out the sphere, for example). As you point out, it is helpful to use Gaussian curvature $K$ because $K$ is independent of presentation, such as the particu... |
linear-algebra | <p>Today, at my linear algebra exam, there was this question that I couldn't solve.</p>
<hr />
<blockquote>
<p>Prove that
<span class="math-container">$$\det \begin{bmatrix}
n^{2} & (n+1)^{2} &(n+2)^{2} \\
(n+1)^{2} &(n+2)^{2} & (n+3)^{2}\\
(n+2)^{2} & (n+3)^{2} & (n+4)^{2}
\end{bmatrix} = -... | <p>Here is a proof that is decidedly not from the book. The determinant is obviously a polynomial in n of degree at most 6. Therefore, to prove it is constant, you need only plug in 7 values. In fact, -4, -3, ..., 0 are easy to calculate, so you only have to drudge through 1 and 2 to do it this way !</p>
| <p>Recall that $a^2-b^2=(a+b)(a-b)$. Subtracting $\operatorname{Row}_1$ from $\operatorname{Row}_2$ and from $\operatorname{Row}_3$ gives
$$
\begin{bmatrix}
n^2 & (n+1)^2 & (n+2)^2 \\
2n+1 & 2n+3 & 2n+5 \\
4n+4 & 4n+8 & 4n+12
\end{bmatrix}
$$
Then subtracting $2\cdot\operatorname{Row}_2$ from $... |
combinatorics | <p>Source: <a href="http://www.math.uci.edu/%7Ekrubin/oldcourses/12.194/ps1.pdf" rel="noreferrer">German Mathematical Olympiad</a></p>
<h3>Problem:</h3>
<blockquote>
<p>On an arbitrarily large chessboard, a generalized knight moves by jumping p squares in one direction and q squares in a perpendicular direction, p, q &... | <p>Case I: If $p+q$ is odd, then the knight's square changes colour after each move, so we are done.</p>
<p>Case II: If $p$ and $q$ are both odd, then the $x$-coordinate changes by an odd number after every move, so it is odd after an odd number of moves. So the $x$-coordinate can be zero only after an even number of ... | <p>This uses complex numbers.</p>
<p>Define $z=p+qi$. Say that the knight starts at $0$ on the complex plane. Note that, in one move, the knight may add or subtract $z$, $iz$, $\bar z$, $i\bar z$ to his position.</p>
<p>Thus, at any point, the knight is at a point of the form:
$$(a+bi)z+(c+di)\bar z$$
where $a$ and $... |
combinatorics | <p>I know that the sum of squares of binomial coefficients is just <span class="math-container">${2n}\choose{n}$</span> but what is the closed expression for the sum <span class="math-container">${n\choose 0}^2 - {n\choose 1}^2 + {n\choose 2}^2 + \cdots + (-1)^n {n\choose n}^2$</span>?</p>
| <p><span class="math-container">$$(1+x)^n(1-x)^n=\left( \sum_{i=0}^n {n \choose i}x^i \right)\left( \sum_{i=0}^n {n \choose i}(-x)^i \right)$$</span></p>
<p>The coefficient of <span class="math-container">$x^n$</span> is <span class="math-container">$\sum_{k=0}^n {n \choose n-k}(-1)^k {n \choose k}$</span> which is e... | <p>Here's a combinatorial proof. </p>
<p>Since $\binom{n}{k} = \binom{n}{n-k}$, we can rewrite the sum as $\sum_{k=0}^n \binom{n}{k} \binom{n}{n-k} (-1)^k$. Then $\binom{n}{k} \binom{n}{n-k}$ can be thought of as counting ordered pairs $(A,B)$, each of which is a subset of $\{1, 2, \ldots, n\}$, such that $|A| = k$ a... |
differentiation | <p>If $(V, \langle \cdot, \cdot \rangle)$ is a finite-dimensional inner product space and $f,g : \mathbb{R} \longrightarrow V$ are differentiable functions, a straightforward calculation with components shows that </p>
<p>$$
\frac{d}{dt} \langle f, g \rangle = \langle f(t), g^{\prime}(t) \rangle + \langle f^{\prime}(... | <p>Observe that
$$
\begin{align*}
\frac{1}{h}
&
\left[
\langle f(t+h),\, g(t+h)\rangle - \langle f(t),\, g(t) \rangle
\right] \\
& =
\frac{1}{h}
\left[
\langle f(t+h),\, g(t+h)\rangle - \langle f(t),\, g(t+h)\rangle
\right]
+ \frac{1}{h}
\left[
\langle ... | <p>This answer may be needlessly complicated if you don't want such generality, taking the approach of first finding the Fréchet derivative of a bilinear operator.</p>
<p>If $V$, $W$, and $Z$ are normed spaces, and if $T:V\times W\to Z$ is a continuous (real) <a href="http://en.wikipedia.org/wiki/Bilinear_map">bilinea... |
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