tag stringclasses 9
values | question_body stringlengths 61 12.9k | accepted_answer stringlengths 38 36.4k | second_answer stringlengths 63 33k |
|---|---|---|---|
geometry | <p>I am trying to understand the notion of an orientable manifold.<br>
Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ such that $\textrm{det}(J(\phi_{\alpha} \circ \phi_{\beta}^{-1}))> 0$ (where defined). My question is:<br>
Using ... | <p>If you had an orientation, you'd be able to define at each point $p$ a unit vector $n_p$ normal to the strip at $p$, in a way that the map $p\mapsto n_p$ is continuous. Moreover, this map is completely determined once you fix the value of $n_p$ for some specific $p$. (You have two possibilities, this uses a tangent ... | <p>Let $M:=\{(x,y)|x\in\mathbb R, -1<y<1\}$ be an infinite strip and choose an $L>0$. The equivalence relation $(x+L,-y)\sim(x,y)$ defines a Möbius strip $\hat M$. Let $\pi: M \to \hat M$ be the projection map. The Möbius strip $\hat M$ inherits the differentiable structure from ${\mathbb R}^2$. We have to p... |
linear-algebra | <p>Given two square matrices <span class="math-container">$A$</span> and <span class="math-container">$B$</span>, how do you show that <span class="math-container">$$\det(AB) = \det(A) \det(B)$$</span> where <span class="math-container">$\det(\cdot)$</span> is the determinant of the matrix?</p>
| <p>Let's consider the function <span class="math-container">$B\mapsto \det(AB)$</span> as a function of the columns of <span class="math-container">$B=\left(v_1|\cdots |v_i| \cdots | v_n\right)$</span>. It is straight forward to verify that this map is multilinear, in the sense that
<span class="math-container">$$\det... | <p>The proof using elementary matrices can be found e.g. on <a href="http://www.proofwiki.org/wiki/Determinant_of_Matrix_Product" rel="noreferrer">proofwiki</a>. It's basically the same proof as given in Jyrki Lahtonen 's comment and Chandrasekhar's link.</p>
<p>There is also a proof using block matrices, I googled a b... |
probability | <p>This sounds more like a brain teaser, but I had some kink to think it through :( Suppose you're parking at a non-parking zone, the probability to get a parking ticket is 80% in 1 hour, what is the probability to get a ticket in half an hour? Please show how you deduce the answer. Thanks!</p>
| <p>It really depends on what model is assumed. However, if the idea is that no matter how long you leave your car there, you have a $20$% chance of getting through any given hour unscathed, you can treat it as an exponential decay problem. Let $p(t)$ be the probability that you do <strong>not</strong> get a ticket in t... | <p>If my probability of getting a parking ticket in a half hour is T, then</p>
<p>$$T+(1-T)T$$</p>
<p>are the odds I got one in 1 hour. The first term says that I got it in the 1st half hour (so it doesn't matter what happens after that), and the second term says that I got it in the 2nd half hour (so not in the firs... |
game-theory | <p>Players A and B alternate writing one digit to make a six-figure number. That means A writes digit $a$, B writes digit $b$, ... to make a number $\overline{abcdef}$. </p>
<p>$a,b,c,d,e,f$ are distinct, $a\neq 0$.</p>
<p>A is the winner if this number is composite, otherwise B is. Is there any way to help A or B al... | <p>Player A has a winning strategy. First, Player A picks $a = 3$. </p>
<p>Case I: Player B picks $b = 9$. </p>
<p>Then, Player A picks $c = 1$. If Player B picks $d = 7$, Player A can pick $e$ arbitrarily. If Player B picks $d \neq 7$, Player A picks $e = 7$. Following this strategy, $\{1,3,7,9\} \subseteq \{a,b,c,d... | <p>This is community wiki because it is JimmyK4542's answer with different reasoning.</p>
<p>First note that if the final digit is 0, 2, 4, 5, 6, or 8 then the result is composite and A wins. In other words B can only win if the final digit is 1, 3, 7, or 9 and even then it is not guaranteed.</p>
<p>A gets to choose ... |
game-theory | <p>Yesterday at work we had a staff day, where we were asked to play an interesting game as an icebreaker.</p>
<p>We (50 or so people) were told to stand in a circle and choose 2 people at random out of the group. We were then asked to walk to a point so that we would be equidistant to those two people. After a few ey... | <p>Doesn't this game converge to a point, assuming smallest possible equidistance? Assume that it doesn't. Instead it converges to some equilibrium position L. Then there is a unique convex polygon joining the points on the outside of L. But on the next step , each of these points on the outside must move inside L,... | <p>Following the OP's explanation in the comments, I would like to add some considerations, leaving for clarity's sake my initial clarification requests at the end.</p>
<p>Firsty, I believe the algorithm used for the simulations might introduce some complications compared to the description given, as the distance ste... |
combinatorics | <p>I want to know if the popular Sudoku puzzle is a Cayley table for a group. </p>
<p>Methods I've looked at: Someone I've spoken to told me they're not because counting the number of puzzle solutions against the number of tables with certain permutations of elements, rows and columns, the solutions are bigger than th... | <p>A Cayley table for a group can never be a sudoku. Assume you have a $9 \times 9$ Cayley table for a group of order $9$, and say your identity element is at index $1 \le i \le 9$. Then row $i$ and column $i$ are symmetric to each other because they correspond to multiplication with the identity. In particular, if you... | <p>It is easy to construct a Sudoku that cannot be a Cayley table. <strong>Not even if you label the rows and columns independently from each other.</strong> Consider the following.
$$
\begin{array}{ccc|ccc|ccc}
1&2&3&4&5&6&7&8&9\\
4&5&6&7&8&9&1&3&2\\
7&am... |
matrices | <p>I found out that there exist positive definite matrices that are non-symmetric, and I know that symmetric positive definite matrices have positive eigenvalues.</p>
<p>Does this hold for non-symmetric matrices as well?</p>
| <p>Let <span class="math-container">$A \in M_{n}(\mathbb{R})$</span> be any non-symmetric <span class="math-container">$n\times n$</span> matrix but "positive definite" in the sense that: </p>
<p><span class="math-container">$$\forall x \in \mathbb{R}^n, x \ne 0 \implies x^T A x > 0$$</span>
The eigenvalues of <spa... | <p>I am answering the first part of @nukeguy's comment, who asked:</p>
<blockquote>
<p>Is the converse true? If all of the eigenvalues of a matrix <span class="math-container">$𝐴$</span> have
positive real parts, does this mean that <span class="math-container">$𝑥^𝑇𝐴𝑥>0$</span> for any
<span class="math-contain... |
linear-algebra | <p>I am relatively new to the world of academical mathematics, but I have noticed that most, if not all, mathematical textbooks that I've had the chance to come across, seem completely oblivious to the existence of lambda notation. </p>
<p>More specifically, in a linear algebra course I'm taking, I found it a lot easi... | <p>As Derek already said, there is no essential difference between functions $A\times B \to C$ and functions $A\to (B \to C)$ via Currying (this is also more abstractly expressed by the universal property of an <a href="https://en.wikipedia.org/wiki/Exponential_object" rel="noreferrer">exponential</a> which unifies the... | <p>Lambda calculus is related with computer science through and through. To quote Wikipedia:</p>
<blockquote>
<p>Lambda calculus (also written as λ-calculus) is a formal system in mathematical logic for expressing <em>computation</em> based on function abstraction and <em>application</em> using variable binding and ... |
combinatorics | <p>So, the full problem goes like this:</p>
<blockquote>
<p>There are $25$ people at a party. Assuming that among any three people, at least two of them know each other, prove that there exists one person who must know at least twelve people.</p>
</blockquote>
<p>I've been stuck on this problem for a while and have... | <p>If everyone knows everyone, then you are done.</p>
<p>Otherwise choose two people, A and B say, who don't know each other. These two people are part of $23$ triples. In each of these triples, either A knows the third person, or B knows the third person.</p>
<p>Thus one of A or B knows (at least) $12$ people.</p>... | <p>Pick a vertex <span class="math-container">$v$</span>. If <span class="math-container">$\deg(v) \geq 12$</span> you are done.</p>
<p>Otherwise <span class="math-container">$v$</span> is connected with at most 11 vertices. Let <span class="math-container">$C$</span> be the vertices connected to <span class="math-cont... |
geometry | <p>I'm a fifteen year old who is currently studying circle geometry (if that is the appropriate term) and our teacher stated that concentric circles are similar. I thought about this, and it doesn't make sense to me. The reason is because of proportionality. For example, similar triangles are similar because they have ... | <p>You're right: any two circles are similar (and so there's not much point of talking about "similar circles")! In general, two shapes are "congruent" if you can turn one into the other by translations (moving around in the plane), rotations, and reflections. Two shapes are "similar" if you can rescale ("zoom in or ... | <p>Yes indeed. Every circle is similar. You can always scale one of them to match the other. Actually, this is the definition of similarity. In case of triangles, this definition yields the result that the sides are proportional. "The sides of one triangle are proportional to the other" is not the actual definition of ... |
combinatorics | <p>Yesterday the aliens took 100 scientists from Earth as prisoners. They want to test how smart the humans are.</p>
<p>The aliens made 101 headbands, numbered from 1 to 101. On the contest day, they throw away one of the headbands, and then from the remaining headbands randomly put one on each of the scientists. Then... | <p>You can save 99.5 scientists.</p>
<p>Imagine that the 101st headband was left on the ground behind the first scientist, making a line of 101 headbands.</p>
<p>The strategy is that everyone should assume that the headbands are laid out in an even permutation of the numbers {1, 2, ..., 101}. Each scientist will alwa... | <p>The scientist in the rear states the sum of all 99 visible numbers, $S_{99}$. He or she will die. The second to last scientist is hopefully clever enough to understand why the prior scientist clearly picked a faulty number. This scientist can now can sum the 98 visible numbers, $S_{98}$, to uniquely determine his or... |
matrices | <p>In one of my exams I'm asked to prove the following </p>
<blockquote>
<p>Suppose <span class="math-container">$A,B\in \mathbb R^{n\times n}$</span>, and <span class="math-container">$AB=BA$</span>, then <span class="math-container">$A,B$</span> share the same eigenvectors. </p>
</blockquote>
<p>My attempt is let... | <p>The answer is in the book <em>Linear Algebra and its Application</em> by Gilbert Strang. I'll just write down what he said in the book.</p>
<blockquote>
<p>Starting from <span class="math-container">$Ax=\lambda x$</span>, we have</p>
<p><span class="math-container">$$ABx = BAx = B \lambda x = \lambda Bx$$</s... | <p>Commuting matrices do not necessarily share <em>all</em> eigenvector, but generally do share <em>a</em> common eigenvector.</p>
<p>Let <span class="math-container">$A,B\in\mathbb{C}^{n\times n}$</span> such that <span class="math-container">$AB=BA$</span>. There is always a nonzero subspace of <span class="math-con... |
combinatorics | <p>Aside from <span class="math-container">$1!\cdot n!=n!$</span> and <span class="math-container">$(n!-1)!\cdot n! = (n!)!$</span>, the only product of factorials known is <span class="math-container">$6!\cdot 7!=10!$</span>.</p>
<p>One might naturally associate these numbers with the permutations on <span class="math... | <p>This family of bijections (of sets) <span class="math-container">$S_6\times S_7 \to S_{10}$</span> has already been suggested in comments and linked threads, but it is so pretty I wanted to spell it out:</p>
<p>There are <span class="math-container">$10$</span> ways of partitioning the numbers <span class="math-cont... | <p>It may be connected with, of all things, the <span class="math-container">$3-4-5$</span> right triangle! This triangle and its multiples stand out as having the sides in arithmetic progression. Such an arithmetic progression leads to factorial expressions when the sides are multiplied together.</p>
<p>As a prelimi... |
matrices | <p>I can follow the definition of the transpose algebraically, i.e. as a reflection of a matrix across its diagonal, or in terms of dual spaces, but I lack any sort of geometric understanding of the transpose, or even symmetric matrices.</p>
<p>For example, if I have a linear transformation, say on the plane, my intui... | <p>To answer your second question first: an orthogonal matrix $O$ satisfies $O^TO=I$, so $\det(O^TO)=(\det O)^2=1$, and hence $\det O = \pm 1$. The determinant of a matrix tells you by what factor the (signed) volume of a parallelipiped is multipled when you apply the matrix to its edges; therefore hitting a volume in ... | <p>yoyo has succinctly described my intuition for orthogonal transformations in the comments: from <a href="http://en.wikipedia.org/wiki/Polarization_identity">polarization</a> you know that you can recover the inner product from the norm and vice versa, so knowing that a linear transformation preserves the inner produ... |
logic | <p>Consider propositional logic with primitive connectives $\{{\to},{\land},{\lor},{\bot}\}$. We view $\neg \varphi$ as an abbreviation of $\varphi\to\bot$ and $\varphi\leftrightarrow\psi$ as an abbreviation of $(\varphi\to\psi)\land(\psi\to\varphi)$, etc.</p>
<p>The <em>classical sequent calculus</em> LK has rules su... | <p>This rings a bell. And ah yes, on p. 48 of Sara Negri and Jan van Plato's admirable book <em>Structural Proof Theory</em> (CUP, 2001), they write</p>
<blockquote>
<p>It is not the [general] feature of having a multiset as a succedent that leads
to classical logic, but the unrestricted $R\!\supset$ rule,</p>
</b... | <p>Peter Smith found a nice reference which more or less settles the question. Here are some concluding remarks that won't fit as comments:</p>
<p>Knowing that it's true gave me the courage to try proving it, which turned out to be easier than expected. In fact, each of the classical rules <em>except</em> ${\to}R$ (bu... |
logic | <p>In philosophical contexts, the <em>Principia Mathematica</em> is sometimes held in high regard as a demonstration of a logical system.</p>
<p>But what did Whitehead and Russell's <em>Principia Mathematica</em> achieve for mathematics?</p>
| <p>I'll try to answer referring to the <em>Introduction</em> to the 1st edition of W&R's <a href="http://plato.stanford.edu/entries/principia-mathematica/" rel="nofollow noreferrer">Principia</a> (3 vols, 1910-13); see :</p>
<ul>
<li>Alfred North Whitehead & Bertrand Russell, <a href="https://books.google.it/bo... | <p>Whilst Whitehead and Russell failed in their main aim of deriving mathematics from logic, the logic that they developed is the mathematical logic that is used universally today.</p>
|
linear-algebra | <p>I’m learning multivariate analysis and I have learnt linear algebra for two semesters when I was a freshman.</p>
<p>Eigenvalues and eigenvectors are easy to calculate and the concept is not difficult to understand. I found that there are many applications of eigenvalues and eigenvectors in multivariate analysis. For... | <p>Personally, I feel that intuition isn't something which is easily explained. Intuition in mathematics is synonymous with experience and you gain intuition by working numerous examples. With my disclaimer out of the way, let me try to present a very informal way of looking at eigenvalues and eigenvectors.</p>
<p>Fir... | <p>First let us think what a square matrix does to a vector. Consider a matrix <span class="math-container">$A \in \mathbb{R}^{n \times n}$</span>. Let us see what the matrix <span class="math-container">$A$</span> acting on a vector <span class="math-container">$x$</span> does to this vector. By action, we mean multip... |
linear-algebra | <p>It seems, at times, that physicists and mathematicians mean different things when they say the word "tensor." From my perspective, when I say tensor, I mean "an element of a tensor product of vector spaces." </p>
<p>For instance, here is a segment about tensors from Zee's book <em>Einstein Gravity in a Nutshell</em... | <p>What a physicist probably means when they say "tensor" is "a global section of a tensor bundle." I'll try and break it down to show the connection to what mathematicians mean by tensor.</p>
<p>Physicists always have a manifold <span class="math-container">$M$</span> lying around. In classical mec... | <p>Being a physicist by training maybe I can help.</p>
<p>The "physicist" definition of a vector you allude to, in more mathematicians-friendly terms would become something like</p>
<blockquote>
<p>Let $V$ be a vector space and fix a reference frame $\mathcal{F}$ (mathematicians lingo: a basis.) A collection $\{v^1... |
matrices | <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!} ... |
number-theory | <p><strong>Conjecture:</strong> <span class="math-container">$100$</span> is the only square number of the form <span class="math-container">$a^b+b^a$</span> for integers <span class="math-container">$b>a>1$</span>.</p>
<p>In other words, <span class="math-container">$(a,b)=(2,6)$</span> is the only solution. Ca... | <p><span class="math-container">$\newcommand{\eps}{\varepsilon}$</span>
<span class="math-container">$\newcommand{\rad}{\mathrm{rad}}$</span></p>
<p>At least, under the <a href="https://en.wikipedia.org/wiki/Abc_conjecture" rel="noreferrer">abc conjecture</a>, there can be only finitely many pairs <span class="math-co... | <p><strong>(21-03-2020) Update:</strong> As no unconditional answer has yet been given, I will include a few more conditions that any solution must satisfy. The results used are rather advanced, so I will only include references, not proofs.</p>
<p><strong>Lemma 5:</strong> Let <span class="math-container">$a$</span> ... |
logic | <p>I am trying to understand what mathematics is really built up of. I thought mathematical logic was the foundation of everything. But from reading a book in mathematical logic, they use "="(equals-sign), functions and relations.</p>
<p>Now is the "=" taken as undefined? I have seen it been defined in terms of the id... | <p>Most set theories, such as ZFC, require an underlying knowledge of first-order logic formulas (as strings of symbols). This means that they require acceptance of facts of string manipulations (which is essentially equivalent to accepting arithmetic on natural numbers!) First-order logic does not require set theory, ... | <p>What you are butting your head against here IMO is the fact that you need a meta-language at the beginning. Essentially at some point you have to agree with other people what your axioms and methods of derivation are and these concepts cannot be intrinsic to your model. </p>
<p>Usually I think we take axioms in pro... |
geometry | <p>A hypothetical (and maybe practical) question has been nagging at me.</p>
<p>If you had a piece of paper with dimensions 4 and 3 (4:3), folding it in half along the long side (<strong><em>once</em></strong>) would result in 2 inches and 3 inches (2:3), which wouldn't retain its ratio. For example, here is a piece o... | <p>The $1:\sqrt{2}$ ratio ensures exactly that. That is the idea behind the <a href="http://en.wikipedia.org/wiki/ISO_216">ISO 216</a> standard for paper sizes, which was adopted from the German <a href="https://en.wikipedia.org/wiki/Deutsches_Institut_f%C3%BCr_Normung">DIN</a> 476 standard.</p>
<p>Its most common usa... | <p>A ratio of $1:\sqrt{2}$ will do the trick!</p>
<p>The original rectangle will have a ratio of $x:y$, where $y$ is the longer side and $x$ the shorter side. Then the folded rectangle will have a ratio of $\frac{1}{2}y:x$ and we want</p>
<p>$$\begin{align}
\frac{x}{y} & =\frac{\frac{1}{2}y}{x} \\
x^2 & = \tf... |
logic | <p>I've been reading a book on Gödel's incompleteness theorems and it makes the following claim regarding provability of statements in Peano arithmetic (paraphrased):</p>
<blockquote>
<p>There exists a formula $A(x)$ such that the statements $A(0), A(1), A(2), \dots$ are all provable, but $\forall x\, A(x)$ is not p... | <p>Great question! Yes, there are specific examples. One of the most famous is <a href="https://en.wikipedia.org/wiki/Goodstein%27s_theorem">Goodstein's theorem</a>. If $A(n)$ is the statement that Goodstein's sequence starting at $n$ terminates, then it is known (via stronger theories than PA, namely use of a clever o... | <p>Possibly the easiest example is to let $A(x)$ say that $x$ is not the Gödel number of a proof of $0=1$ from the axioms of PA. </p>
<p>Because there is no such proof, $A(0)$ is true, $A(1)$ is true, etc., and because of their syntactic form each of these is provable in PA.</p>
<p>However, $(\forall x)A(x)$ is the s... |
differentiation | <p>In calculus of one variable I read:</p>
<p>A function <span class="math-container">$f$</span> is differentiable on an <strong>open interval</strong> if it is differentiable at every number of the interval.</p>
<p>I wonder why in the definition we suppose that the interval is open. This is the case in Rolle theorem... | <p>The problem is one of <em>consistent definitions</em>. Intuitively we can make sense of differentiable on a closed interval: but it requires a slightly more careful phrasing of the definition of "differentiable at a point". I don't know which book you are using, but I am betting that it contains (some version of the... | <p>A function is differentiable on a set $S$, if it is differentiable at every point of $S$. This is the definition that I seen in the beginning/classic calculus texts, and this mirrors the definition of continuity on a set. </p>
<p>So $S$ could be an open interval, closed interval, a finite set, in fact, it could be ... |
probability | <h1>Problem</h1>
<p>The premise is <em>almost</em> the same as in <a href="https://math.stackexchange.com/questions/29242/probability-that-a-quadratic-polynomial-with-random-coefficients-has-real-roots?rq=1">this question</a>. I'll restate for convenience.</p>
<blockquote>
<p>Let <span class="math-container">$A$</span>... | <p>I would probably start by breaking into cases based on <span class="math-container">$A$</span> and <span class="math-container">$C$</span>.</p>
<p>Conditioned on <span class="math-container">$A$</span> and <span class="math-container">$C$</span> having different signs, there are always real roots (because <span clas... | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
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... |
geometry | <p>I was just wondering why we have 90° degrees for a perpendicular angle. Why not 100° or any other number?</p>
<p>What is the significance of 90° for the perpendicular or 360° for a circle?</p>
<p>I didn't ever think about this during my school time.</p>
<p>Can someone please explain it mathematically? Is it due t... | <p>360 is an incredibly <a href="http://en.wikipedia.org/wiki/Abundant_number">abundant</a> number, which means that there are many factors. So it makes it easy to divide the circle into $2, 3, 4, 5, 6, 8, 9, 10, 12,\ldots$ parts. By contrast, 400 gradians cannot even be divided into 3 equal whole-number parts. Whi... | <p>I have heard that the ancient Babylonians used a base-$60$ numeral system with sub-base $10$.</p>
<p>Certainly such a system was used by Ptolemy in the second century AD. See Gerald Toomer's translation of Ptolemy's <em>Almagest</em>. In particular Ptolemy divided the circle into $360$ degrees. See <a href="http... |
probability | <p>This morning, I wanted to flip a coin to make a decision but only had an SD card:</p>
<p><img src="https://i.sstatic.net/T7fUT.png" alt="enter image description here"></p>
<p>Given that <em>I don't know</em> the bias of this SD card, would flipping it be considered a "fair toss"?</p>
<p>I thought if I'm just as l... | <p>Here's a pragmatic answer from an engineer. You can always get a fair 50/50 outcome with <strong>any</strong> "coin" (or SD card, or what have you), <em>without having to know whether it is biased, or how biased it is</em>:</p>
<ul>
<li>Flip the coin twice. </li>
<li>If you get $HH$ or $TT$, discard the trial and
r... | <p>That is a very good question!</p>
<p>There are (at least) two different ways to define probability: as a measure of frequencies, and as a measure of (subjective) knowlegde of the result.</p>
<p>The frequentist definition would be: the probability of the sd card landing "heads" is the proportion of times it lands "... |
linear-algebra | <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... |
differentiation | <p>I am a senior in high school so I know I am simply misunderstanding something but I don't know what, please have patience.</p>
<p>I was tasked to find the derivative for the following function:</p>
<p><span class="math-container">$$ y = \frac{ (4x)^{1/5} }{5} + { \left( \frac{1}{x^3} \right) } ^ {1/4} $$</span></p... | <p>The <span class="math-container">$(\cdot)^{\frac{1}{4}}$</span> operation has to be understood as a function. A function can only have one image for any argument. Depending upon how you interpret the fourth root, the image could be positive or negative. But once you set how you interpret your function (positive or n... | <p>You are confused about what <span class="math-container">$y^{1/4}$</span> actually means.</p>
<p>Suppose that <span class="math-container">$x^4=1$</span>. We could raise both sides to the <span class="math-container">$1/4$</span> power: <span class="math-container">$$\left(x^4\right)^{1/4}=1^{1/4}$$</span></p>
<p>... |
probability | <p>I found this math "problem" on the internet, and I'm wondering if it has an answer:</p>
<blockquote>
<p><strong>Question:</strong> If you choose an answer to this question at random, what is the probability that you will be correct?</p>
<p><strong>a.</strong> <span class="math-container">$25\%$</span></p>
... | <p>No, it is not meaningful. 25% is correct iff 50% is correct, and 50% is correct iff 25% is correct, so it can be neither of those two (because if <em>both</em> are correct, the only correct answer could be 75% which is not even an option). But it cannot be 0% either, because then the correct answer would be 25%. So ... | <p>The question is underspecified since it doesn't say which distribution is used in choosing an answer at random. Any of the answers could be correct:</p>
<p>If I choose a. with probability 25% and b. with probability 75%, a and d are correct.</p>
<p>If I choose a. with probability 50% and b. with probability 50%, b... |
probability | <blockquote>
<p>If a $1$ meter rope is cut at two uniformly randomly chosen points (to give three pieces), what is the average length of the smallest piece?</p>
</blockquote>
<p>I got this question as a mathematical puzzle from a friend. It looks similar to MathOverflow question <a href="https://mathoverflow.net/q/2... | <p>Update: The current version of this answer is more intuitive (IMHO) than the previous one. See also <a href="https://math.stackexchange.com/questions/14190/average-length-of-the-longest-segment/14194#14194">this similar answer</a> to a similar question.</p>
<p>The generalization of this problem to <span class="mat... | <p>My approach is maybe more naive than the others posted.</p>
<p>Break the unit interval at $x$ and $y$ where $x < y$. Our lengths are then $x$, $y - x$, and $1 - y$. It's not hard to show that they all have probability $1/3$ of being the shortest. In any case, our joint PDF is given by $f(x,y) = 6$ (since $x$ ... |
probability | <p>Choose a random number between $0$ and $1$ and record its value. Do this again and add the second number to the first number. Keep doing this until the sum of the numbers exceeds $1$. What's the expected value of the number of random numbers needed to accomplish this?</p>
| <p>Here is a (perhaps) more elementary method. Let $X$ be the amount of numbers you need to add until the sum exceeds $1$. Then (by linearity of expectation):</p>
<p>$$ \mathbb{E}[X] = 1 + \sum_{k \geq 1} \Pr[X > k] $$</p>
<p>Now $X > k$ if the sum of the first $k$ numbers $x_1,\ldots,x_k$ is smaller than $1$. ... | <p>Assuming the numbers come from a uniform distribution over $[0,1]$ and that the trials are independent, here is an outline (this is example 7.4 4h. in Sheldon Ross' <i>A First Course in Probability</i>, sixth edition): </p>
<ol>
<li><p>Let $X_i$ be the number obtained on the $i$'th trial. </p></li>
<li><p>For $0\l... |
combinatorics | <p>A friend of mine was doodling with numbers arranged somewhat reminiscent of Pascal's Triangle, where the first row was $ 1^{n-1} \ \ 2^{n-1} \ \cdots \ n^{n-1} $ and subsequent rows were computed by taking the difference of adjacent terms. He conjectured that the number we get at the end is $ n! $ but I've not been... | <p>The top rows are indeed made of the powers $i^n=P_n(i)$, which are polynomials of degree $n$, with the leading coefficient $1$.</p>
<p>On the next row you take the first order difference. By the binomial formula, we have</p>
<p>$$P_{n-1}(i)=P_n(i+1)-P_n(i)=i^n+ni^{n-1}+\cdots-i^n=ni^{n-1}+\cdots$$</p>
<p>which is... | <p>Here is another way to show that
$$ \sum_{k=0}^n (-1)^{n-k} \binom{n}{k} k^n = n!$$</p>
<p>We consider the set $S$ consisting of all strings of length $n$ consisting of the symbols $a_1, a_2, \cdots, a_n$, with repetition allowed.</p>
<p>Then we clearly have that $|S|=n^n$ because there are $n$ choices for each sy... |
linear-algebra | <p>I have a couple of questions about tensor products:</p>
<p>Why is $\text{Hom}(V,W)$ the same thing as $V^* \otimes W$? </p>
<p>Why is an element of $V^{*\otimes m}\otimes V^{\otimes n}$ the same thing as a multilinear map $V^m \to V^{\otimes n}$? </p>
<p>What is the general formulation of this principle?</p>
| <p>The result is generally wrong for infinite-dimensional spaces: see <a href="https://math.stackexchange.com/questions/573378/u-otimes-v-versus-lu-v-for-infinite-dimensional-spaces/573416#573416">this question</a>.</p>
<p>For finite dimensional space $V$, let's build an isomorphism $f : V^* \otimes W \to \hom(V,W)$ b... | <p>One general form of that assertion (noting that it cannot be quite as simple as one might imagine) is the Cartan-Eilenberg adjunction
<span class="math-container">$$
\mathrm{Hom}(X\otimes Y,Z)\;\approx\;\mathrm{Hom}(X,\mathrm{Hom}(Y,Z))
$$</span>
in some reasonable additive (or whatever) category, where, significant... |
differentiation | <p>I thought I had a good idea on why/how implicit differentiation works until I read the following passage in my Calculus book:</p>
<blockquote>
<p>Furthermore, implicit differentiation works just as easily for equations such as
$$x^5+5x^4y^2+3xy^3+y^5=1$$
which are actually <i>impossible</i> to solve for $y$ i... | <p>At a basic level, I think this question is really about the difference between saying that something <em>exists</em>, on the one hand, and being able to <em>write a formula for it</em> on the other. It's important to distinguish between three different (but closely related) ideas:</p>
<ol>
<li>It may be that the re... | <p>That's an excellent question. </p>
<p>Part of the <em>conclusion</em> of the Implicit Differentiation Theorem (perhaps better known as the Implicit Function Theorem) is that the equation <strong>locally</strong> defines $y$ as a function of $x$.</p>
<p>The theorem says, roughly, this. Suppose that $(x,y)=(a,b)$ li... |
combinatorics | <p>I need to calculate the possible combinations for 8 characters password. The password must contain at least one of the following: (lower case letters, upper case letters, digits, punctuations, special characters).</p>
<p>Assume I have 95 ascii characters (lower case letters, upper case letters, digits, punctuations... | <p>Start with all $8$-character strings: $95^8$</p>
<p>Then remove all passwords with no lowercase ($69^8$), all passwords with no uppercase ($69^8$), all passwords with no digit ($85^8$) and all passwords with no special character ($62^8$).</p>
<p>But then you removed some passwords twice. You must add back all pass... | <p>The answer I get is 0.3051925477389360000E+16 = 3,051,925,477,389,360. This answer is taken from the answer at <a href="https://math.stackexchange.com/questions/2103361/derive-an-algorithm-for-computing-the-number-of-restricted-passwords-for-the-gen#2103361">Derive an algorithm for computing the number of restricte... |
probability | <p>What is the average number of times it would it take to roll a fair <span class="math-container">$6$</span>-sided die and get all numbers on the die? The order in which the numbers appear does not matter.</p>
<p>I had this questions explained to me by a professor (not math professor), but it was not clear in the ex... | <p>The time until the first result appears is $1$. After that, the random time until a second (different) result appears is geometrically distributed with parameter of success $5/6$, hence with mean $6/5$ (recall that the mean of a geometrically distributed random variable is the inverse of its parameter). After that, ... | <p>Here's the logic:</p>
<p>The chance of rolling a number you haven't yet rolled when you start off is $1$, as any number would work. Once you've rolled this number, your chance of rolling a number you haven't yet rolled is $5/6$. Continuing in this manner, after you've rolled $n$ different numbers the chance of roll... |
linear-algebra | <p>What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is some operation on vectors.</p>
<p>The definition is:</p>
<p>A $n$ × $n$ Hermitian matrix M is called <em>positive-sem... | <p>One intuitive definition is as follows. Multiply any vector with a positive semi-definite matrix. The angle between the original vector and the resultant vector will always be less than or equal $\frac{\pi}{2}$. The positive definite matrix tries to keep the vector within a certain half space containing the vector. ... | <p>First I'll tell you how I think about Hermitian positive-definite matrices. A Hermitian positive-definite matrix $M$ defines a sesquilinear inner product $\langle Mv, w \rangle = \langle v, Mw \rangle$, and in fact every inner product on a finite-dimensional inner product space $V$ has this form. In other words it... |
matrices | <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... |
matrices | <p>How can I understand that $A^TA$ is invertible if $A$ has independent columns? I found a similar <a href="https://math.stackexchange.com/questions/1181271/if-ata-is-invertible-then-a-has-linearly-independent-column-vectors">question</a>, phrased the other way around, so I tried to use the theorem</p>
<p>$$
rank(A^T... | <p>Consider the following:
$$A^TAx=\mathbf 0$$
Here, $Ax$, an element in the range of $A$, is in the null space of $A^T$. However, the null space of $A^T$ and the range of $A$ are orthogonal complements, so $Ax=\mathbf 0$.</p>
<p>If $A$ has linearly independent columns, then $Ax=\mathbf 0 \implies x=\mathbf 0$, so the... | <p>If $A $ is a real $m \times n $ matrix then $A $ and $A^T A $ have the same null space. Proof: $A^TA x =0\implies x^T A^T Ax =0 \implies (Ax)^TAx=0 \implies \|Ax\|^2 = 0 \implies Ax = 0 $. </p>
|
logic | <p>Assuming ZFC has a model, is there a model of ZFC such that in that model, ZFC has no model?</p>
<p>Also, assuming ZFC has a model, is there a model of ZFC such that in that model, ZFC is inconsistent?</p>
| <p>Yes.</p>
<p>Recall that by the Completeness Theorem, having a model and being consistent are the same thing. Also, by Incompleteness, ZFC doesn't prove its own consistency. Finally, ZFC can prove the Soundness Theorem - that an inconsistent theory has no models!</p>
<p>So - assuming ZFC has a model - ZFC is consis... | <p>Noah gave the correct answer. It is true, that if $\sf ZFC$ has any models, then there are models of $\sf ZFC$ in which there are no models of $\sf ZFC$.</p>
<p>But let me point out an important piece of information here.</p>
<p>We formulate statements about consistency in terms of natural numbers. This is because... |
probability | <p>Of course, we've all heard the colloquialism "If a bunch of monkeys pound on a typewriter, eventually one of them will write Hamlet."</p>
<p>I have a (not very mathematically intelligent) friend who presented it as if it were a mathematical fact, which got me thinking... Is this really true? Of course, I've learne... | <p>I found online the claim (which we may as well accept for this purpose) that there are $32241$ words in Hamlet. Figuring $5$ characters and one space per word, this is $193446$ characters. If the character set is $60$ including capitals and punctuation, a random string of $193446$ characters has a chance of $1$ in... | <p>Some references (I am mildly surprised that no one has done this yet). This is called the <a href="http://en.wikipedia.org/wiki/Infinite_monkey_theorem">infinite monkey theorem</a> in the literature. It follows from the second <a href="http://en.wikipedia.org/wiki/Borel%E2%80%93Cantelli_lemma">Borel-Cantelli lemma... |
logic | <blockquote>
<p><em><strong>Theorem 1</strong> [ZFC, classical logic]:</em> If $A,B$ are sets such that $\textbf{2}\times A\cong \textbf{2}\times B$, then $A\cong B$.</p>
</blockquote>
<p>That's because the axiom of choice allows for the definition of cardinality $|A|$ of any set $A$, and for $|A|\geq\aleph_0$ we ha... | <p>There was a paper recently posted to arXiv about this question: Swan, <a href="https://arxiv.org/abs/1804.04490" rel="noreferrer"><em>On Dividing by Two in Constructive Mathematics</em></a>.</p>
<p>It turns out that there are examples of toposes where you can't divide by two.</p>
| <p>(The following is not really an answer, or just a very partial one, but it's definitely relevant and too long for a comment.)</p>
<p>There is a theorem of Richard Friedberg ("<a href="https://eudml.org/doc/169898" rel="nofollow noreferrer">The uniqueness of finite division for recursive equivalence types</a>", <em>... |
logic | <p>Today I had an argument with my math teacher at school. We were answering some simple True/False questions and one of the questions was the following:</p>
<p><span class="math-container">$$x^2\ne x\implies x\ne 1$$</span></p>
<p>I immediately answered true, but for some reason, everyone (including my classmates and ... | <p>The short answer is: Yes, it is true, because the contrapositive just expresses the fact that $1^2=1$.</p>
<p>But in controversial discussions of these issues, it is often (but not always) a good idea to try out non-mathematical examples:</p>
<hr>
<p>"If a nuclear bomb drops on the school building, you die."</p>
... | <p>First, some general remarks about logical implications/conditional statements. </p>
<ol>
<li><p>As you know, $P \rightarrow Q$ is true when $P$ is false, or when $Q$ is true. </p></li>
<li><p>As mentioned in the comments, the <em>contrapositive</em> of the implication $P \rightarrow Q$, written $\lnot Q \righta... |
linear-algebra | <p>Let $A$ be a symmetric $n\times n$ matrix.</p>
<p>I found <a href="https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/posdef/" rel="noreferrer">a method on the web to check if $A$ is <strong>positive definite</strong></a>:</p>
<blockquote>
<p>$A$ is positive-definite if all the diagonal entrie... | <p>These matrices are called (strictly) <a href="http://en.wikipedia.org/wiki/Diagonally_dominant_matrix">diagonally dominant</a>. The standard way to show they are positive definite is with the <a href="http://en.wikipedia.org/wiki/Gershgorin_circle_theorem">Gershgorin Circle Theorem</a>. Your weaker condition does no... | <p>Before continuing, let me add the caution that a symmetric matrix can violate your rules and still be positive definite, give me a minute to check the eigenvalues</p>
<p><span class="math-container">$$ H \; = \;
\left( \begin{array}{rrr}
3 & 2 & 0 \\
2 & 3 & 2 \\
0 & 2 & 3
\end... |
differentiation | <p>How to calculate the gradient with respect to $X$ of:
$$
\log \mathrm{det}\, X^{-1}
$$
here $X$ is a positive definite matrix, and det is the determinant of a matrix.</p>
<p>How to calculate this? Or what's the result? Thanks!</p>
| <p>I assume that you are asking for the derivative with respect to the elements of the matrix. In this cases first notice that</p>
<p>$$\log \det X^{-1} = \log (\det X)^{-1} = -\log \det X$$</p>
<p>and thus</p>
<p>$$\frac{\partial}{\partial X_{ij}} \log \det X^{-1} = -\frac{\partial}{\partial X_{ij}} \log \det X = -... | <p>The simplest is probably to observe that
$$-\log\det (X+tH) = -\log\det X -\log\det(I+tX^{-1}H)
\\= -\log\det X - t \textrm{Tr}(X^{-1}H) + o(t),$$</p>
<p>where is used the "obvious" fact that $\det(I+A) = 1+\textrm{Tr}(A)+o(|A|)$ (all the other terms are quadratic expressions of the coefficients of $A$).</p>
<p>... |
matrices | <p>I gave the following problem to students:</p>
<blockquote>
<p>Two $n\times n$ matrices $A$ and $B$ are <em>similar</em> if there exists a nonsingular matrix $P$ such that $A=P^{-1}BP$.</p>
<ol>
<li><p>Prove that if $A$ and $B$ are two similar $n\times n$ matrices, then they have the same determinant and th... | <p>If <span class="math-container">$A$</span> is a <span class="math-container">$2\times 2$</span> matrix with determinant <span class="math-container">$d$</span> and trace <span class="math-container">$t$</span>, then the characteristic polynomial of <span class="math-container">$A$</span> is <span class="math-contain... | <p>As Eric points out, such $2\times2$ matrices are special.
In fact, there are only two such pairs of matrices.
The number depends on how you count, but the point is that such matrices have a <em>very</em> special form.</p>
<p>Eric proved that the two matrices must have a double eigenvalue.
Let the eigenvalue be $\la... |
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... |
logic | <p>The Continuum Hypothesis says that there is no set with cardinality between that of the reals and the natural numbers. Apparently, the Continuum Hypothesis can't be proved or disproved using the standard axioms of set theory.</p>
<p>In order to disprove it, one would only have to construct one counterexample of a se... | <p>Here's an example axiomatic system:</p>
<ol>
<li>There exist exactly three objects $A, B, C$.</li>
<li>Each of these objects is either a banana, a strawberry or an orange.</li>
<li>There exists at least one strawberry.</li>
</ol>
<p>Let's name the system $X$.</p>
<p><strong>Vincent's Continuum Hypothesis (VCH)</s... | <p>I think the basic problem is in your statement that "In order to disprove it, one would only have to construct one counterexample of a set with cardinality between the naturals and the reals." Actually, to disprove CH by this strategy, one would have to produce a counterexample <strong>and prove</strong> that it act... |
matrices | <p>I have one triangle in <span class="math-container">$3D$</span> space that I am tracking in a simulation. Between time steps I have the previous normal of the triangle and the current normal of the triangle along with both the current and previous <span class="math-container">$3D$</span> vertex positions of the tria... | <p>Suppose you want to find a rotation matrix $R$ that rotates unit vector $a$ onto unit vector $b$.</p>
<p>Proceed as follows:</p>
<p>Let $v = a \times b$</p>
<p>Let $s = \|v\|$ (sine of angle)</p>
<p>Let $c = a \cdot b$ (cosine of angle)</p>
<p>Then the rotation matrix R is given by:
$$R = I + [v]_{\times} + [v]... | <p>Using <a href="https://math.stackexchange.com/a/180436/76513">Kjetil's answer</a> answer, with <a href="https://math.stackexchange.com/users/17349/process91">process91</a>'s comment, we arrive at the following procedure.</p>
<h2>Derivation</h2>
<p>We are given two unit column vectors, $A$ and $B$ ($\|A\|=1$ and $\... |
linear-algebra | <p>I have looked extensively for a proof on the internet but all of them were too obscure. I would appreciate if someone could lay out a simple proof for this important result. Thank you.</p>
| <p>These answers require way too much machinery. By definition, the characteristic polynomial of an $n\times n$ matrix $A$ is given by
$$p(t) = \det(A-tI) = (-1)^n \big(t^n - (\text{tr} A) \,t^{n-1} + \dots + (-1)^n \det A\big)\,.$$
On the other hand, $p(t) = (-1)^n(t-\lambda_1)\dots (t-\lambda_n)$, where the $\lambda... | <p>Let $A$ be a matrix. It has a <a href="http://en.wikipedia.org/wiki/Jordan_normal_form">Jordan Canonical Form</a>, i.e. there is matrix $P$ such that $PAP^{-1}$ is in Jordan form. Among other things, Jordan form is upper triangular, hence it has its eigenvalues on its diagonal. It is therefore clear for a matrix ... |
combinatorics | <p>It is quite easy to calculate the total number of functions from a set $X$ with $m$ elements to a set $Y$ with $n$ elements ($n^{m}$), and also the total number of injective functions ($n^{\underline{m}}$, denoting the falling factorial). But I am thinking about how to calculate the total number of surjective functi... | <p>Consider $f^{-1}(y)$, $y \in Y$. This set must be non-empty, regardless of $y$. What you're asking for is the number of ways to distribute the elements of $X$ into these sets.</p>
<p>The number of ways to distribute m elements into n non-empty sets is given by the <a href="http://en.wikipedia.org/wiki/Stirling_numb... | <p>Here is a solution that does not involve the Stirling numbers of the second kind, $S(n,m)$. The number of surjective functions from a set $X$ with $m$ elements to a set $Y$ with $n$ elements is</p>
<p>$$
\sum_{i=0}^{n-1} (-1)^i{n \choose i}(n-i)^m
$$</p>
<p>or more explicitly
$$
{n \choose 0}n^m - {n \choose 1}(n... |
game-theory | <p>I am from programming background but with very limited knowledge of maths.</p>
<p>I am very much eager to learn and apply <a href="http://www.britannica.com/EBchecked/topic/224893/game-theory" rel="noreferrer">game theory</a> to understand dynamics of International Politics and economics. But I am facing difficulti... | <p>As @JordanMahar mentions, Fudenberg and Tirole is the standard graduate-level text. But I would start with <strong><em>Game Theory for Applied Economists</em></strong> by <strong>Gibbons</strong>. It is very readable. </p>
<p>Prerequisites for Gibbons are minimal. A little algebra and probability will do just fine.... | <p>In economics, the classic source for game theory is:</p>
<p>Fudenberg, D. and Tirole, J. (1991). <a href="http://www.amazon.ca/Game-Theory-Drew-Fudenberg/dp/0262061414/ref=sr_1_1?ie=UTF8&qid=1369889236&sr=8-1&keywords=game+theory+tirole" rel="nofollow">Game Theory</a></p>
<p>At least with Game Theory a... |
logic | <p><em>Background: I'm a logic student with very little background in cohomology etc., so this question is fairly naive.</em></p>
<hr>
<p>Although mathematical logic is generally perceived as sitting off on its own, there are some striking applications of algebraic/geometric/combinatorial ideas to logic. In general, ... | <p>I guess I will take a crack at this.</p>
<p>First of all, it is probably worthwhile for you to learn some cohomology in its original home so that you have some intuition for it, and some knowledge of what theorems there are, how to compute it, etc. You do not give much indication in your answer as to how much know... | <p>As Steven Gubkin and Ryan Budney have already pointed out, cohomology is often used to measure how far a "locally consistent" object is from being "globally consistent". I thought I might describe another example of this in set theory, which it turns out contains a lot of open problems. As far as I know, this exam... |
logic | <p>I am teaching a first-semester course in abstract algebra, and we are discussing group isomorphisms. In order to prove that two group are not isomorphic, I encourage the students to look for a group-theoretic property satisfied by one group but not by the other. I did not give a precise meaning to the phrase "g... | <p>First, let's start with the silly answer. Your language only has countably many different expressions, so can only divvy groups up into continuum-many classes - so there are definitely non-isomorphic groups it can't distinguish! In general this will happen as long as your language has <em>only set-many</em> expressi... | <p>Here are some simple examples where you at least need to make some decisions about what you believe about set theory to determine whether two groups are isomorphic. Assuming the axiom of choice every vector space has a basis, so <span class="math-container">$\mathbb{R}$</span> is isomorphic (as a group) to some dire... |
logic | <p>I want to get this as a tattoo as I love the role maths plays in the universe and the idea that the farthest reaches of what we can ever know, fall short of the limits of what is true, even in mathematics. I've done a lot of research but have yet to find a mathematical symbol version of this theorem.</p>
<p>Is ther... | <p>If you have enough skin and endurance, the following is not exactly Gödel's theorem, but an instance of a statement as predicted by Gödel's theorem, that is - with a suitable interpretation - it states that it is not provable (which means that it must be true, but unprovable - or PA is inconsistent). But don't compl... | <p>Gödel's second incompleteness theorem has a succinct description:</p>
<p>$$ PA \not{\vdash} Con(PA). $$</p>
|
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 | <blockquote>
<p>Let $a,b,c \in \mathbb N$ find integer in the form: $$I=\frac{a}{b+c}
+ \frac{b}{c+a} + \frac{c} {a+b}$$</p>
</blockquote>
<p>Using Nesbitt's inequality: $I \ge \frac 32$</p>
<p>I am trying to prove $I \le 2$ to implies there $\nexists \ a,b,c$ such that $I\in \mathbb Z$: $$I\le 2 \\ \iff c{a}^{2}+... | <p>I find a solution to $n=4,$when</p>
<blockquote>
<p>a=2332797891204725453580403814216955612718693675609518139813675622446336
8530351921955206357565424226029748329737767516130520072674084336131550
2597616224970927979227396663481447506019173462295157784788781420305046
52018159936616800590064485753155232061032... | <p>I found a few solutions $(a,b,c)$ of equatoin
$$
\dfrac{a}{b+c}+\dfrac{b}{a+c}+\dfrac{c}{a+b}=n \qquad\qquad (a,b,c,n\in\mathbb{N})\tag{1}
$$
for even $n$.
<br/>
And I have a doubt whether there are smaller solutions.</p>
<hr>
<p>$\color{#FF4400}{n=4:}$ </p>
<p>$\small
a=4373612677928697257861252602371390152816... |
geometry | <blockquote>
<p><strong>Problem:</strong></p>
<p><em>A vertex of one square is pegged to the centre of an identical square, and the overlapping area is blue. One of the squares is then rotated about the vertex and the resulting overlap is red.</em></p>
<p><em>Which area is greater?</em></p>
</blockquote>
<p><img src="h... | <p><a href="https://i.sstatic.net/mXMhQ.jpg" rel="noreferrer"><img src="https://i.sstatic.net/mXMhQ.jpg" alt="some text"></a> </p>
<p>The four numbered areas are congruent.</p>
<hr>
<p>[Added later] The figure below is from a suggested edit by @TomZych, and it shows the congruent parts more clearly. Given all the up... | <p>I think sketching the two identical triangles marked with green below makes this rather intuitive. This could also be turned into a formal proof quite easily.</p>
<p><a href="https://i.sstatic.net/zilQU.png" rel="noreferrer"><img src="https://i.sstatic.net/zilQU.png" alt="Identical triangles"></a></p>
|
linear-algebra | <p>Singular value decomposition (<a href="http://en.wikipedia.org/wiki/Singular_value_decomposition" rel="noreferrer">SVD</a>) and principal component analysis (<a href="https://en.wikipedia.org/wiki/Principal_component_analysis" rel="noreferrer">PCA</a>) are two eigenvalue methods used to reduce a high-dimensional dat... | <p>(I assume for the purposes of this answer that the data has been preprocessed to have zero mean.)</p>
<p>Simply put, the PCA viewpoint requires that one compute the eigenvalues and eigenvectors of the covariance matrix, which is the product <span class="math-container">$\frac{1}{n-1}\mathbf X\mathbf X^\top$</span>, ... | <p><em><a href="https://arxiv.org/pdf/1404.1100.pdf" rel="nofollow noreferrer">A tutorial on Principal Component Analysis</a></em> by Jonathon Shlens is a good tutorial on PCA and its relation to SVD. Specifically, section VI: A More General Solution Using SVD.</p>
|
logic | <p>In <a href="https://math.stackexchange.com/a/440219/1543">this answer</a>, user18921 wrote that the $\iff$ operation is associative, in the sense that</p>
<ul>
<li>$(A\iff B)\iff C$</li>
<li>$A\iff (B\iff C)$ </li>
</ul>
<p>are equivalent statements. </p>
<p>One can brute-force a proof fairly easily (just write d... | <p>I would say that I personally don't feel that this equivalence is that intuitive. (Maybe it is because I have been thinking too constructively these days.) In classical logic, we have the isomorphism with the Boolean algebra, and that makes it possible to think of $A \leftrightarrow B$ as $A \oplus B \oplus 1$ where... | <p>This might be one of the situations where a more general (and therefore stronger) result is more intuitive (depending on how your intuition works). The result I have in mind is that, if you combine any list $\mathcal L$ of formulas (not just three like the $A,B,C$ in the question) by $\iff$, then no matter how you p... |
game-theory | <p>A stack of silver coins is on the table. </p>
<p>For each step we can either remove a silver coin and write the number of gold coins on a piece of paper, or we can add a gold coin and write the number of silver coins on another piece of paper. </p>
<p>We stop when only gold coins are left.</p>
<p>Prove that the s... | <p>The state of the game can be desribed by
$$
(g,s,G,S),
$$
where $g$ is the number of golden coins on the table, $s$ is the number of silver coins on the table, $G$ is the sum of the numbers in the first paper, and $S$ is the sum of the numbers in the second paper. The initial state is $(0,n,0,0)$, and we want to sho... | <p>When you add a gold coin, you write $n$ for the number of silver coins left.</p>
<p>Every time you remove one of those $n$ silver coins, that gold coin gets counted once as part of the number of gold coins - a total of $n$ times, since all the silver coins are eventually removed.</p>
|
linear-algebra | <p>When I came across the Cauchy-Schwarz inequality the other day, I found it really weird that this was its own thing, and it had lines upon lines of <a href="https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/proof-of-the-cauchy-schwarz-inequality">proof</a>.</p>
<p>I've always t... | <p><em>Side note: it's actually the Cauchy-Schwarz-Bunyakovsky inequality, and don't let anyone tell you otherwise.</em></p>
<p>The problem with using the geometric definition is that you have to define what an angle is. Sure, in three dimensional space, you have pretty clear ideas about what an angle is, but what do ... | <ol>
<li><p>The inequality is ubiquitous, so some name is needed.</p></li>
<li><p>As there is no cosine in the statement of the inequality, it cannot be called "cosine inequality" or anything like that.</p></li>
<li><p>The geometric interpretation with cosines only works for finite-dimensional real Euclidean space, but... |
combinatorics | <p>At the end of Probability class, our professor gave us the following puzzle:</p>
<blockquote>
<p>There are 100 bags each with 100 coins, but only one of these bags has gold coins in it. The gold coin has weight of 1.01 grams and the other coins has weight of 1 gram. We are given a digital scale, but we can only u... | <p>To understand the formula, it would be easiest to explain how it works conceptually before we derive it.</p>
<p>Let's simplify the problem and say there are only 3 bags each with 2 coins in them. 2 of those bags have the 1 gram coins and one has the 1.01 gram gold coins. Let's denote the bags arbitrarily as $Bag_... | <p>For #1: imagine for a moment that all the coins are fake. If we took 0 coins from bag 0, 1 coin from bag 1, 2 coins from bag 2... we'd have $99\times100/2=4,950$ coins, and those 4,950 coins would weigh a total of 4,950 grams. But now, say that bag 25 were the one with real coins that are slightly heavier: 0.01 gr... |
geometry | <p>I'm an 8th grader. After browsing aops.com, a math contest website, I've seen a lot of problems solved by Cauchy Schwarz. I'm only in geometry (have not started learning trigonometry yet). So can anyone explain Cauchy Schwarz in layman's terms, as if you are explaining it to someone who has just started geo in 8th g... | <p>In geometry terms that you can understand, the Cauchy-Schwarz inequality says that:</p>
<blockquote>
<p>Among all the parallelograms with sides <em>a</em> and <em>b</em>, the rectangle is the one with the largest area.</p>
</blockquote>
<p>Usually you can use this inequality when you are looking for an upper (or... | <p>Since you said you browse AoPS, maybe you're interested in math contests. In math contests, the following forms of Cauchy-Schwarz (C-S) inequality are used:</p>
<p>For all $a_i,b_i\in\Bbb R$:</p>
<p>$$\left(a_1^2+a_2^2+\cdots+a_n^2\right)\left(b_1^2+b_2^2+\cdots+b_n^2\right)\ge (a_1b_1+a_2b_2+\cdots+a_nb_n)^2$$</p... |
probability | <p>I might just be slow (or too drunk), but I'm seeing a conflict in the equations for adding two normals and scaling a normal. According to <a href="http://www.cs.toronto.edu/~yuvalf/CLT.pdf" rel="noreferrer">page 2 of this</a>, if $X_1 \sim N(\mu_1,\sigma_1^2)$ and $X_2 \sim N(\mu_2,\sigma_2^2)$, then $X_1 + X_2 \si... | <p>On the first page of the cited document, $X_1$ and $X_2$ were previously defined to be two (distinct) <em>independent</em>, identically distributed random variables. For your purposes, the "identically distributed" part is not important, but the "independent" part <em>is</em>.</p>
<p>On the second page, where $X_1$... | <p>Expectation is always linear. So for any two variables <span class="math-container">$X,Y$</span>, we have <span class="math-container">$E[X+Y]=E[X]+E[Y]$</span>. And <span class="math-container">$E[\underbrace{X+X+\ldots+X}_{k \text{ times}}]=E[kX]=kE[X]$</span></p>
<p>Variance is linear when the variables are <em>i... |
logic | <p>A friend of mine asked me if I could explain this statement: "It's not logically possible to prove that something can't be done". The actual reason is the understanding of this strip: </p>
<p><img src="https://i.sstatic.net/YtHMp.gif" alt="enter image description here"></p>
<p>Since I'm not an expert on logic, but... | <p>What the strip most likely was (awkwardly) referring to is not that it's impossible to prove something can't be done, but rather that it's impossible to prove within a (sufficiently powerful) formal system that something can't be <em>proven</em> in that system — in other words, letting $P(\phi)$ be the stateme... | <p>Scott Adams may have intended something different, but in the given context and with the given wording, Dan is right and Asok is wrong.</p>
<p>Imagine Asok writing a piece of software that crucially depends on deciding the <a href="http://en.wikipedia.org/wiki/Halting_problem">halting problem</a> (or any other prop... |
logic | <p>An extreme form of constructivism is called <em>finitisim</em>. In this form, unlike the standard axiom system, infinite sets are not allowed. There are important mathematicians, such as Kronecker, who supported such a system. I can see that the natural numbers and rational numbers can easily defined in a finitis... | <p>Set theory with all sets finite has been studied, is a familiar theory in disguise, and is enough for most/all concrete real analysis.</p>
<p>Specifically, Zermelo-Fraenkel set theory with the Axiom of Infinity replaced by its negation (informally, "there is no infinite set") is equivalent to first-order Peano Arit... | <p>Disclaimer: I am not a finitist --- but as a theoretical computer scientist, I have a certain sympathy for finitism. The following is the result of me openly speculating what an "official" finitist response would be, based on grounds of computability.</p>
<p>The short version is this: <strong>(a)</strong> It d... |
game-theory | <p>Imagine Rock Paper Scissors, but where winning with a different hand gives a different reward.</p>
<ul>
<li><p>If you win with Rock, you get \$9. Your opponent loses the \$9.</p>
</li>
<li><p>If you win with Paper, you get \$3. Your opponent loses the \$3.</p>
</li>
<li><p>If you win with Scissors, you get \$5. Your... | <p>Let $(x_1,x_2,x_3)$ be the first player's strategy (i.e., his probabilities of playing rock, paper, and scissors respectively), and let $(y_1,y_2,y_3)$ be the second player's strategy. The expected payoff to the first player is
$$
P(x,y)=9(x_1y_3-x_3y_1)+3(x_2y_1-x_1y_2)+5(x_3y_2-x_2y_3).
$$
To constrain the probab... | <p>This table summarizes the possible outcomes in playing this game once:
$$
\begin{array}{c|c|c|c|c}
\text{Hero Plays} & \text{Villain Plays} & \text{Hero's Earnings} \\ \hline
R & R & +\$0 \\
R & P & -\$3 \\
R & S & +\$9 ... |
matrices | <blockquote>
<p>How can one prove that <span class="math-container">$ \|A\|_2 \le \|A\|_F $</span> without using <span class="math-container">$ \|A\|_2^2 := \lambda_{\max}(A^TA) $</span>?</p>
</blockquote>
<hr />
<p>It makes sense that the <span class="math-container">$2$</span>-norm would be less than or equal to the ... | <p>Write $x=\sum_{j=1}^nc_je_j$, for coefficients $c_1,\ldots,c_n$. Suppose that $\|x\|_2=1$, i.e. $\sum_j |c_j|^2=1$.
Then
\begin{align}
\|Ax\|_2^2&=\left\|\sum_j c_j\,Ae_j\right\|_2^2\leq\left(\sum_j|c_j|\,\|Ae_j\|_2\right)^{2}\\ \ \\ &\leq\left(\sum_j|c_j|^2\right)\sum_j\|Ae_j\|_2^2=\sum_j\|Ae_j\|_2^2=\|A\|... | <p>In fact, the proof from $\left\| \mathbf{A}\right\|_2 =\max_{\left\| \mathbf{x}\right\|_2=1} \left\| \mathbf{Ax} \right\|_2$ to $\left\| \mathbf{A}\right\|_2 = \sqrt{\lambda_{\max}(\mathbf{A}^H \mathbf{A})}$ is straight forward. We can first simply prove when $\mathbf{P}$ is Hermitian
$$
\lambda_{\max} = \max_{\| \m... |
probability | <p>The PDF describes the probability of a random variable to take on a given value:</p>
<p>$f(x)=P(X=x)$</p>
<p>My question is whether this value can become greater than $1$?</p>
<p>Quote from wikipedia:</p>
<p>"Unlike a probability, a probability density function can take on values greater than one; for example, t... | <p>Discrete and continuous random variables are not defined the same way. Human mind is used to have discrete random variables (example: for a fair coin, -1 if it the coin shows tail, +1 if it's head, we have that $f(-1)=f(1)=\frac12$ and $f(x)=0$ elsewhere). As long as the probabilities of the results of a discrete ra... | <p>Your conception of <a href="https://en.wikipedia.org/wiki/Probability_density_function" rel="noreferrer">probability density function</a> is wrong.</p>
<p>You are mixing it up with <a href="https://en.wikipedia.org/wiki/Probability_mass_function" rel="noreferrer">probability mass function</a>.</p>
<p>If <span class=... |
probability | <p>what are some good books on probabilities and measure theory?
I already know basic probabalities, but I'm interested in sigma-algrebas, filtrations, stopping times etc, with possibly examples of "real life" situations where they would be used</p>
<p>thanks</p>
| <p>I'd recommend Klenke's <a href="http://books.google.ch/books?id=tcm3y5UJxDsC&printsec=frontcover&hl=de#v=onepage&q&f=false"><em>Probability Theory</em></a>.</p>
<p>It gives a good overview of the basic ideas in probability theory. In the beginning it builds up the basics of measure theory and set fu... | <p>I think Chung's <a href="http://rads.stackoverflow.com/amzn/click/0121741516" rel="noreferrer">A Course in Probability Theory</a> is a good one that is rigorous. Also Sid Resnick's <a href="http://rads.stackoverflow.com/amzn/click/081764055X" rel="noreferrer">A Probability Path</a> is advanced but easy to read.</p>... |
linear-algebra | <p>How can I understand that $A^TA$ is invertible if $A$ has independent columns? I found a similar <a href="https://math.stackexchange.com/questions/1181271/if-ata-is-invertible-then-a-has-linearly-independent-column-vectors">question</a>, phrased the other way around, so I tried to use the theorem</p>
<p>$$
rank(A^T... | <p>Consider the following:
$$A^TAx=\mathbf 0$$
Here, $Ax$, an element in the range of $A$, is in the null space of $A^T$. However, the null space of $A^T$ and the range of $A$ are orthogonal complements, so $Ax=\mathbf 0$.</p>
<p>If $A$ has linearly independent columns, then $Ax=\mathbf 0 \implies x=\mathbf 0$, so the... | <p>If $A $ is a real $m \times n $ matrix then $A $ and $A^T A $ have the same null space. Proof: $A^TA x =0\implies x^T A^T Ax =0 \implies (Ax)^TAx=0 \implies \|Ax\|^2 = 0 \implies Ax = 0 $. </p>
|
geometry | <p>The derivative of the volume of a sphere, with respect to its radius, gives its surface area. I understand that this is because given an infinitesimal increase in radius, the change in volume will only occur at the surface. Similarly, the derivative of a circle's area with respect to its radius gives its circumferen... | <p>If you continuously enlarge a solid cone by adding material to the base, then every inch of added height corresponds to an inch-thick layer added to the circular base, so the rate of change of volume is equal to the area of the base times the rate of change of the height. That is, <span class="math-container">$dV/dh... | <p>My intuition tells me that you should be able to find such a formula by choosing the right parameterization. A couple of motivating examples:</p>
<h4>First example: a disc</h4>
<p>Consider a disk of radius <em>r</em>. The area in terms of the radius is <span class="math-container">$\pi r^2$</span> and in terms of th... |
differentiation | <p>I have <span class="math-container">$x= \exp(At)$</span> where <span class="math-container">$A$</span> is a matrix. I would like to find derivative of <span class="math-container">$x$</span> with respect to each element of <span class="math-container">$A$</span>. Could anyone help with this problem?</p>
| <p>Considering the expression <span class="math-container">$x = \exp(tA)$</span> I can think of two derivatives.</p>
<p>First, the derivative with respect to the real variable <span class="math-container">$t$</span> of the matrix-valued function <span class="math-container">$t \mapsto \exp(tA)$</span>. Here the result ... | <p>If $A\in{\mathbb R}^{n\times n}$, then you can use Higham's "Complex Step Approximation" to calculate each component
$$
\frac {\partial f} {\partial A_{jk}} = {\rm Im}\bigg(\frac{f(A+ihE_{jk})}{h}\bigg)
$$
where $f(A)={\rm exp}(tA)$ and $h=10^{-20}$. </p>
|
differentiation | <p>Consider a function <span class="math-container">$g: \mathbb{R^3} \to \mathbb{R}$</span> defined by <span class="math-container">$g(x,y,z) = z^2 -x^3 + 2z + 3y^3$</span> </p>
<p>Find the gradient of <span class="math-container">$g$</span> at the point <span class="math-container">$(2,1,-1)$</span>.</p>
<p>Is the g... | <p>The gradient is a vector; it points in the direction of steepest ascent.</p>
<p>The directional derivative is a number; it is the rate of change when your point in $\Bbb R^3$ moves in that direction. (You can imagine "reducing" your function to a function of a single variable, say $t$, by "slicing" the curve in th... | <p>Be careful that directional derivative of a function is a scalar while gradient is a vector.</p>
<p>The only difference between <em>derivative</em> and <em>directional derivative</em> is the definition of those terms. Remember:</p>
<ul>
<li>Directional derivative is the instantaneous rate of change (which is a sc... |
differentiation | <p>I have a clarifying question about this question:</p>
<p><a href="https://math.stackexchange.com/questions/183325/what-is-the-difference-between-d-and-partial">What is the difference between $d$ and $\partial$?</a></p>
<p>I understand the idea that $\frac{d}{dx}$ is the derivative where all variables are assumed t... | <p>Some key things to remember about partial derivatives are:</p>
<ul>
<li>You need to have a function of one or more variables.</li>
<li>You need to be very clear about what that function is.</li>
<li>You can only take partial derivatives of that function with respect to each of the variables it is a function of.</li... | <p>I hope this answers your question.</p>
<p>The partial derivative notation is used to specify the derivative of a function of more than one variable with respect to one of its variables.</p>
<p>e.g. Let y be a function of 3 variables such that $y(s, t, r) = r^2 - srt$</p>
<p>$$\frac{\partial y}{\partial r} = 2r-s... |
number-theory | <blockquote>
<p>Let <span class="math-container">$k$</span> be a natural number. Then <span class="math-container">$3k+1$</span> , <span class="math-container">$4k+1$</span> and <span class="math-container">$6k+1$</span> cannot all be square numbers.</p>
</blockquote>
<p>I tried to prove this by supposing one of them i... | <p><strong>Complete answer from the literature:</strong></p>
<p>I found a list of solved Pell systems in <a href="http://www.kurims.kyoto-u.ac.jp/EMIS/journals/AMI/2007/ami2007-szalay.pdf">Szalay</a>, Appendix 4, page 84. </p>
<p><a href="http://www.ams.org/journals/mcom/1998-67-222/S0025-5718-98-00918-1/S0025-5718-9... | <p>$\def\Q{\mathbf{Q}}$
$\def\Qbar{\overline{\Q}}$
$\def\Z{\mathbf{Z}}$
$\def\F{\mathbf{F}}$</p>
<p>I want to try to explain why there won't be any slick elementary argument in this case. The usual competition techniques involve a tricky use of congruences. A more advanced trick is to use infinite descent. However, in... |
linear-algebra | <p>What is the "standard basis" for fields of complex numbers?</p>
<p>For example, what is the standard basis for $\Bbb C^2$ (two-tuples of the form: $(a + bi, c + di)$)? I know the standard for $\Bbb R^2$ is $((1, 0), (0, 1))$. Is the standard basis exactly the same for complex numbers?</p>
<p><strong>P.S.</strong... | <p>Just to be clear, by definition, a vector space always comes along with a field of scalars $F$. It's common just to talk about a "vector space" and a "basis"; but if there is possible doubt about the field of scalars, it's better to talk about a "vector space over $F$" and a "basis over $F$" (or an "$F$-vector spac... | <p>The "most standard" basis is also $\left\lbrace(1,0),\, (0,1)\right\rbrace$. You just take complex combinations of these vectors. Simple :)</p>
|
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) = |... |
geometry | <p>For <span class="math-container">$n = 2$</span>, I can visualize that the determinant <span class="math-container">$n \times n$</span> matrix is the area of the parallelograms by actually calculating the area by coordinates. But how can one easily realize that it is true for any dimensions?</p>
| <p>If the column vectors are linearly dependent, both the determinant and the volume are zero.
So assume linear independence.
The determinant remains unchanged when adding multiples of one column to another. This corresponds to a skew translation of the parallelepiped, which does not affect its volume.
By a finite sequ... | <p>Here is the same argument as Muphrid's, perhaps written in an elementary way.<br /><br /></p>
<p>Apply Gram-Schmidt orthogonalization to $\{v_{1},\ldots,v_{n}\}$, so that
\begin{eqnarray*}
v_{1} & = & v_{1}\\
v_{2} & = & c_{12}v_{1}+v_{2}^{\perp}\\
v_{3} & = & c_{13}v_{1}+c_{23}v_{2}+v_{3}^... |
combinatorics | <p>Popular mathematics folklore provides some simple tools
enabling us compactly to describe some truly enormous
numbers. For example, the number $10^{100}$ is commonly
known as a <a href="http://en.wikipedia.org/wiki/Googol">googol</a>,
and a <a href="http://en.wikipedia.org/wiki/Googolplex">googol
plex</a> is
$10^{10... | <p>OK, let's attempt a sorting of the names having at most
three operands. I'll make several observations, and then
use them to assemble the order section by section,
beginning with the part below googol stack.</p>
<ul>
<li><p>googol bang bang bang $\lt$ googol
stack. It seems clear that we shall be able to iterated b... | <p>The following describes a comparison algorithm that will work for expressions where the number of terms is less than googol - 2.</p>
<p>First, consider the situation with only bangs and plexes. To compare two numbers, first count the total number of bangs and plexes in each. If one has more than the other, that n... |
linear-algebra | <p>In my opinion both are almost same. However there should be some differenes like any two elements can be multiplied in a field but it is not allowed in vector space as only scalar multiplication is allowed where scalars are from the field.</p>
<p>Could anyone give me atleast one counter- example where field and vec... | <p>It is true that vector spaces and fields both have operations we often call multiplication, but these operations are fundamentally different, and, like you say, we sometimes call the operation on vector spaces <em>scalar multiplication</em> for emphasis.</p>
<p>The operations on a field <span class="math-container">... | <p>A <em>field</em> is an algebraic structure allowing the four basic operations $+$, $-$, $\cdot$, and $:\,$, such that the usual rules of algebra hold, e.g., $(x+y)\cdot z=(x\cdot z) +(y\cdot z)$, etcetera, and division by $0$ is forbidden. The elements of a given field should be considered as "numbers". The systems ... |
differentiation | <p>Considering many elementary functions have an antiderivative which is not elementary, why does this type of thing not also happen in differential calculus?</p>
| <p>Just think of how we find those elementary functions:</p>
<ul>
<li><p>We start with the constant functions, which have derivative $0$, and the identity function $f(x)=x$ which has derivative $1$.</p></li>
<li><p>We combine functions by means of addition, subtraction, multiplication, division, composition. For all o... | <p>The short answer is that we have differentiation rules for all the elementary functions, and we have differentiation rules for every way we can combine elementary functions (addition, multiplication, composition), where the derivative of a combination of two functions may be expressed using the functions, their deri... |
game-theory | <p>You select 10 numbers from the set $\{2,3,\dots,12\}$.
You then continually roll 2 fair dice and sum them up, until your selection of 10 numbers come up. </p>
<p>For example if your selection was 7,7,7,7,8,8,8,6,6,6 (4 7's, 3 8's and 3 6's), and you roll the dice repeatedly and get 7,7,6,5,8,7,7,9,8,3,5,10,12,6,6... | <p>Using a brute-force exhaustive search, and a recursion for each case, I get the following answer:
<p>
There are two equivalent optimal strategies, namely
$$4,5,6,6,7,7,7,8,8,9$$
$$5,6,6,7,7,7,8,8,9,10$$
yielding, for the expected number of rounds,
$$
e=\frac{a}{b} \approx 28.26676327
$$
where
$$
\begin{align*}
a&... | <p>(too long for a comment, but not a full answer). I'll work it out for $n=2$, $n$ being the number of values you select.</p>
<p>Let $E[a,b]$ be the expected number of throws it take assuming you chose $a,b$. Similarly, let $E[a]$ be the expected number of throws it takes just to see $a$. Of course $E[a]=\frac 1{P... |
differentiation | <p>Here's a statement in Zygmund's Measure and Integral on page 17:</p>
<blockquote>
<p>If $f$ has a continuous derivative on $[a,b]$, then (by the mean-value theorem) $f$ satisfies a Lipschitz condition on $[a,b]$.</p>
</blockquote>
<p>This does not seem obvious to me. How can I show it?</p>
<p>Also, what does a ... | <p>By the mean value theorem, </p>
<p><span class="math-container">$$f(x) - f(y) = f'(\xi)(x-y)$$</span>
for some <span class="math-container">$\xi \in (y,x)$</span>. But since <span class="math-container">$f'$</span> is continuous and <span class="math-container">$[a,b]$</span> is compact, then <span class="math-con... | <p>Since $f'$ is continuous it is bounded on $[a,b]$. Choose $M\in \Bbb R$ such that $|f'(x)|\le M$ for all $x\in [a,b]$. If $x,y\in [a,b]$, then by the MVT there is $\xi\in [a,b]$ with $f(x)-f(y)=f'(\xi)(x-y)$, thus $|f(x)-f(y)|\le M|x-y|$. In fact you can show that a differentiable function on an open interval (not n... |
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 (... |
combinatorics | <blockquote>
<p>For which <span class="math-container">$n\in \mathbb{N}$</span> can we divide the set <span class="math-container">$\{1,2,3,\ldots,3n\}$</span> into <span class="math-container">$n$</span> subsets each with <span class="math-container">$3$</span> elements such that in each subset <span class="math-conta... | <p>If there is a solution for <span class="math-container">$N$</span>, then there is a solution for <span class="math-container">$7N+5$</span>.<br />
The solution for <span class="math-container">$N$</span> uses up numbers from <span class="math-container">$1$</span> to <span class="math-container">$3N$</span>. Then
<... | <p>Here is the integer linear programming approach I used to find partitions for all such <span class="math-container">$n\le 496$</span> with <span class="math-container">$n \equiv 0,5 \pmod 8$</span>. First enumerate all triples <span class="math-container">$\{x,y,z\}$</span> with <span class="math-container">$x+y=3z... |
differentiation | <p>I have just started learning about differential equations, as a result I started to think about this question but couldn't get anywhere. So I googled and wasn't able to find any particularly helpful results. I am more interested in the reason or method rather than the actual answer. Also I do not know if there even ... | <p>$f(x)=\exp(\frac{1}{2}x)$ is such a function, since $f^{(n)}=2^{-n} f(x)$, you have</p>
<p>$$\sum_{n=1}^\infty f^{(n)}(x)=\sum_{n=1}^\infty 2^{-n}f(x)=(2-1)f(x)=f(x)$$</p>
<p>This is the only function (up to a constant prefactor) for which $\sum_{n}f^{(n)}$ and its derivatives converge uniformly (on compacta), as ... | <p>Differentiate both sides to get $f'(x)=f''(x)+f^{(3)}(x)+...$ </p>
<p>So the starting equation becomes $f(x)=f'(x)+f'(x)\Rightarrow f(x)=2f'(x)$ </p>
<p>Multiply now both sides by $e^{-\frac{x}{2}}$ and this becomes<br>
$$[e^{-\frac{x}{2}}f(x)]'=0$$<br>
So $f(x)=ce^{\frac{x}{2}}$<br>
Done.</p>
|
combinatorics | <p>I checked several thousand natural numbers and observed that $\lfloor n!/e\rfloor$ seems to always be an even number. Is it indeed true for all $n\in\mathbb N$? How can we prove it?</p>
<p>Are there any positive irrational numbers $a\ne e$ such that $\lfloor n!/a\rfloor$ is even for all $n\in\mathbb N$?</p>
| <p>Note that:</p>
<p>$$e^{-1}=\sum_{k=0}^\infty \frac{(-1)^k}{k!}$$</p>
<p>Then:</p>
<p>$$\frac{n!}e=n!e^{-1} = \left(\sum_{k=0}^{n} (-1)^k\frac{n!}{k!}\right) + \sum_{k=n+1}^{\infty} (-1)^{k}\frac{n!}{k!}$$</p>
<p>Show that if $a_n=\sum_{k=n+1}^{\infty} (-1)^{k}\frac{n!}{k!}$ then $0<|a_{n}|<1$ and $a_n>0... | <p>The number of <a href="https://en.wikipedia.org/wiki/Derangement">derangements</a> of $[n]=\{1,\ldots,n\}$ is </p>
<p>$$d_n=n!\sum_{k=0}^n\frac{(-1)^k}{k!}\;,$$</p>
<p>so</p>
<p>$$\frac{n!}e-d_n=n!\sum_{k>n}\frac{(-1)^k}{k!}\;,$$</p>
<p>which is less than $\frac1{n+1}$ in absolute value. Thus for $n\ge 1$, $d... |
probability | <p>Suppose I have a line segment of length $L$. I now select two points at random along the segment. What is the expected value of the distance between the two points, and why?</p>
| <p>Byron has already answered your question, but I will attempt to provide a detailed solution...</p>
<p>Let $X$ be a random variable uniformly distributed over $[0,L]$, i.e., the probability density function of $X$ is the following</p>
<p>$$f_X (x) = \begin{cases} \frac{1}{L} & \textrm{if} \quad{} x \in [0,L]\\ ... | <p>Sorry. I posted a cryptic comment just before running off to class. What I meant was that if $X,Y$ are independent uniform $(0,1)$ random variables, then the triple
$$(A,B,C):=(\min(X,Y),\ \max(X,Y)-\min(X,Y),\ 1-\max(X,Y))$$
is an exchangeable sequence. In particular, $\mathbb{E}(A)=\mathbb{E}(B)=\mathbb{E}(C),$ a... |
number-theory | <p>Some time ago when decomponsing the natural numbers, $\mathbb{N}$, in prime powes I noticed a pattern in their powers. Taking, for example, the numbers $\lbrace 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 \rbrace$ and factorize them, we will get
$$\begin{align} 1&=2^0\times 3^0\times 5^0\times 7^0\times 11^0\times ... | <p>You should take a look at this <a href="https://math.stackexchange.com/questions/44290/can-insight-be-derived-from-direct-formulae-for-prime-number-functions">math stack exchange post</a>, and the answer there, as it applies directly here.</p>
<p>For a formula for $\pi (x)$ you can take a look at <a href="http://en... | <p>These formulas are convoluted and what looks to me as probably useless representations of the p-adic order of an integer. All you seem to have done is taken advantage of the fact that: $$1_{p^j\mid n}=\frac{1}{p^j}\sum_{t=0}^{p^j-1}e^{2\pi i t\frac{n}{p^j}}$$ And then re-wrote it it without the imaginary part of eac... |
linear-algebra | <p>This may be a trivial question yet I was unable to find an answer:</p>
<p>$$\left \| A \right \| _2=\sqrt{\lambda_{\text{max}}(A^{^*}A)}=\sigma_{\text{max}}(A)$$</p>
<p>where the spectral norm $\left \| A \right \| _2$ of a complex matrix $A$ is defined as $$\text{max} \left\{ \|Ax\|_2 : \|x\| = 1 \right\}$$</p>
... | <p>Put <span class="math-container">$B=A^*A$</span> which is a Hermitian matrix. A linear transformation of the Euclidean vector space <span class="math-container">$E$</span> is Hermite iff there exists an orthonormal basis of E consisting of all the eigenvectors of <span class="math-container">$B$</span>. Let <span cl... | <p>First of all,
<span class="math-container">$$\begin{align*}\sup_{\|x\|_2 =1}\|Ax\|_2 & = \sup_{\|x\|_2 =1}\|U\Sigma V^Tx\|_2 = \sup_{\|x\|_2 =1}\|\Sigma V^Tx\|_2\end{align*}$$</span>
since <span class="math-container">$U$</span> is unitary, that is, <span class="math-container">$\|Ux_0\|_2^2 = x_0^TU^TUx_0 = x_0... |
geometry | <p>When I came across the Cauchy-Schwarz inequality the other day, I found it really weird that this was its own thing, and it had lines upon lines of <a href="https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/proof-of-the-cauchy-schwarz-inequality">proof</a>.</p>
<p>I've always t... | <p><em>Side note: it's actually the Cauchy-Schwarz-Bunyakovsky inequality, and don't let anyone tell you otherwise.</em></p>
<p>The problem with using the geometric definition is that you have to define what an angle is. Sure, in three dimensional space, you have pretty clear ideas about what an angle is, but what do ... | <ol>
<li><p>The inequality is ubiquitous, so some name is needed.</p></li>
<li><p>As there is no cosine in the statement of the inequality, it cannot be called "cosine inequality" or anything like that.</p></li>
<li><p>The geometric interpretation with cosines only works for finite-dimensional real Euclidean space, but... |
differentiation | <p>Let $f$ be a real-valued function continuous on $[a,b]$ and differentiable on $(a,b)$.<br>
Suppose that $\lim_{x\rightarrow a}f'(x)$ exists.<br>
Then, prove that $f$ is differentiable at $a$ and $f'(a)=\lim_{x\rightarrow a}f'(x)$. </p>
<p>It seems like an easy example, but a little bit tricky.<br>
I'm not sure whi... | <p>Some hints:</p>
<p>Using the definition of derivative, you need to show that
$$
\lim_{h\rightarrow 0^+} {f(a+h)-f(a)\over h }
$$
exists and is equal to $\lim\limits_{x\rightarrow a^+} f'(x)$.</p>
<p>Note that for $h>0$ the Mean Value Theorem provides a point $c_h$ with $a<c_h<a+h$ such that
$$
{f(a+h)-f(a... | <p>The result is essentially Theorem 5.29 from <a href="http://alpha.math.uga.edu/%7Epete/2400full.pdf" rel="nofollow noreferrer">my honors calculus notes</a>. As I mention, I learned this result from Spivak's <em>Calculus</em>. I say "essentially" because the version discussed in my notes is for an interio... |
logic | <p>Here's the picture I have in my head of Model Theory:</p>
<ul>
<li>a <em>theory</em> is an axiomatic system, so it allows proving some statements that apply to <em>all</em> models consistent with the theory</li>
<li>a <em>model</em> is a particular -- consistent! -- function that assigns every statement to its truth... | <p>This answer only addresses the second part of your question, but you asked many questions so hopefully it's okay.</p>
<p>First, there is in the comments a statement: "If Goldbach is unprovable in PA then it is necessarily true in all models." This is incorrect. If Goldbach were true in all models of PA the... | <p>Let me try to get at the heart of your misunderstanding as concise as possible:</p>
<p><strong>1. We are not deliberately choosing to use a language that permits self-reference, we are forced to do so.</strong></p>
<p>The only choice we made is that of a logic that is sufficiently strong to include integer arithmeti... |
geometry | <p>I have seen some examples of duality. Sometimes applied to theorems, as for example Desargues theorem and Pappus theorem. Sometimes applied to spaces, for example the dual space of a vector space. Sometimes even applied to a method like simplex and dual simplex methods in linear programming.</p>
<p>My question is w... | <p>Duality is a very general and broad concept, without a strict definition that captures all those uses. When applied to specific concepts, there usually is a precise definition for just that context. The common idea is that there are two things which basically are just <strong>two sides of the same coin</strong>.</p>... | <p>It is true that "duality" is a quite broad concept in many fields and the field of mathematics is not the exception. For example the <a href="https://en.wikipedia.org/wiki/Duality_(mathematics)" rel="noreferrer">Wikipedia website</a> shows many forms of duality in Mathematics. As mathematicians we want to abstract ... |
matrices | <p>Are there any open source math programs out there that have an interactive terminal and that work on linux?</p>
<p>So for example you could enter two matrices and specify an operation such as multiply and it would then return the answer or a error message specifying why an answer can't be computed? I am just lookin... | <p><a href="https://en.wikipedia.org/wiki/Sage_%28mathematics_software%29">Sage</a> is basically a Python program/interpreter that aims to be an open-source mathematical suite (ala Mathematica and Magma etc.). There are many algorithms implemented as a direct part of Sage, as well as wrapping many other open-source mat... | <p>I am aware of two packets that might help You:</p>
<ol>
<li><a href="http://www.gnu.org/software/octave/"><strong>Octave</strong></a> - free clone of Matlab, </li>
<li><a href="http://maxima.sourceforge.net/"><strong>maxima</strong></a> - a packet for symbolic computations.</li>
</ol>
<p>Octave is good when you w... |
matrices | <p>I am trying to write a program that will perform <a href="http://en.wikipedia.org/wiki/Optical_character_recognition" rel="noreferrer">OCR</a> on a mobile phone, and I recently encountered this article : </p>
<hr>
<p><img src="https://i.sstatic.net/2jenb.png" alt="article excerpt"></p>
<hr>
<p>Can someone explai... | <p><a href="https://math.stackexchange.com/a/92176/4423">J.M. has given a very good answer</a> explaining singular values and how they're used in low rank approximations of images. However, a few pictures always go a long way in appreciating and understanding these concepts. Here is an example from one of my presentati... | <p>It is rather unfortunate that they used "eigenvalues" for the description here (although it's correct); it is better to look at principal component analysis from the viewpoint of singular value decomposition (SVD).</p>
<p>Recall that any $m\times n$ matrix $\mathbf A$ possesses the singular value decomposition $\ma... |
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