blesspearl/math-stackexchange
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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 ... |
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Mathematics Q&A Dataset
This dataset contains questions and answers focused on various topics within the field of mathematics. The dataset includes columns for tags, question bodies, accepted answers, and secondary answers.
Dataset Overview
The Mathematics Q&A Dataset consists of questions and their corresponding answers from various mathematical topics. This dataset is valuable for research and development in natural language processing (NLP), machine learning (ML), and educational applications.
Topics Covered
The dataset covers a variety of mathematical topics:
- Combinatorics
- Differentiation
- Game Theory
- Geometry
- Linear Algebra
- Logic
- Matrices
- Number Theory
- Probability
Data Columns
The dataset is structured into the following columns:
tag: The category or topic of the question.question_body: The full text of the question.accepted_answer: The accepted answer to the question.second_answer: An additional answer to the question, providing another perspective or solution.
Example
| tag | question_body | accepted_answer | second_answer |
|---|---|---|---|
| probability | What is the probability of getting heads in a coin toss? | The probability is 0.5, as there are two possible outcomes. | The probability of heads is 50%. |
| linear-algebra | How do you find the determinant of a matrix? | To find the determinant, you can use the Laplace expansion. | You can also use row reduction. |
Usage
The dataset can be used for various purposes, including but not limited to:
- Training machine learning models for question answering systems.
- Developing educational tools and applications.
- Conducting research in NLP and text mining.
Accessing the Data
To access the dataset, download the file from the repository:
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