qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
778,605 | <blockquote>
<p>Let $f,g,h : D \to \mathbb{R}$ be functions, $D \subset \mathbb{R}$. Let c be an accumulation point of $D$. Suppose that $$f(x) \le g(x) \le h(x)$$
for all $x \in D$ with $x \neq c$ and suppose
$$\lim_{x \to c} f(x) = \lim_{x \to c} h(x) = L \in \mathbb{R}$$
Prove that $\lim_{x \to c}g(x) = L... | André Nicolas | 6,312 | <p><strong>Outline:</strong> This can be viewed as a geometry problem. Draw the $2\times 2$ square with corners $(0,0)$, $(2,0)$, $(2,2)$, and $(0,2)$. Draw the lines $x-y=\frac{1}{4}$ and $x-y=-\frac{1}{4}$. Draw the line $x=\frac{1}{4}$.</p>
<p>We want the probability that the pair $(X,Y)$ lands in the part of the s... |
55,404 | <p>I have been searching for a version of the isoperimetric inequality which is something like:</p>
<p>$P(\Omega) - 2\sqrt{\pi} Vol(\Omega)^{1/2} \geq \pi (r_{out}^2 - r_{in}^2)$ where $r_{out}$ and $r_{in}$ are the inner and outer radius of a given set. There are of course details which I am missing such as what kind... | Gil Kalai | 1,532 | <p>A very good source of Bonnesen type inequalties is the paper by Rovert Osserman entitled <a href="http://www.jstor.org/stable/2320297" rel="nofollow">Bonnesen style isoprimetric inequalities</a>, Americam Math Monthly 86(1979) 1-29. <a href="http://mathdl.maa.org/images/upload_library/22/Ford/RobertOsserman.pdf" rel... |
3,176,629 | <p>In the evening, pizza was ordered nine people sat around a round table, 50 slices of pizza were served to these nine people. Prove that there were two people sitting next to each other who ate at least 12 pizza slices.</p>
<p>I used the pigeon hole principle to determine 50/9 = 5.5 => 6</p>
<p>Therefore, at least ... | MarianD | 393,259 | <p>Let <span class="math-container">$K_1, K_2, \dots, K_9$</span> be number of pizzas for individual persons. Then</p>
<p><span class="math-container">$$K_1+ K_2+ K_3+K_4+K_5+K_6+K_7+K_8+ K_9 = 50$$</span></p>
<p>There are <span class="math-container">$9$</span> pairs of people sitting one next to other (neighbors). ... |
1,754,931 | <p>If a sequence has a pattern where +2 is the pattern at the start, but 1 is added each time, like the sequence below, is there a formula to find the 125th number in this sequence? It would also need to work with patterns similar to this. For example if the pattern started as +4, and 5 was added each time.</p>
<block... | Ross Millikan | 1,827 | <p>When you have a polynomial expressed as a sequence, you can find it by taking differences
$$ \begin {array} {l l l l l l} 2&4&7&11&16&22\\2&3&4&5&6\\1&1&1&1 \end {array}$$
where the first line is your sequence and subsequent entries are the difference between the one u... |
1,197,654 | <p>Prove that if $G$ is an $r$-regular, $(r-2)$-edge-connected graph $(r>3)$ of even order containing at most $r-1$ distinct edge cuts of cardinality $r-2$ then $G$ has a $1$-factor</p>
<blockquote>
<blockquote>
<p><strong>Tutte's theorem</strong>: A non trivial graph $G$ has a $1$-factor if and only if $k_0 ... | Tyler Seacrest | 172,823 | <p>Here are some hints and answers for you: first off, <em>perfect matching</em> and <em>$1$-factor</em> are synonyms, and can be used interchangeably.</p>
<p>I don't think we can conclude $|S| = r-2$ or $r-1$, unless I'm missing some suppositions in your thinking. $S$ can be any size.</p>
<p>But you're right, I th... |
1,197,654 | <p>Prove that if $G$ is an $r$-regular, $(r-2)$-edge-connected graph $(r>3)$ of even order containing at most $r-1$ distinct edge cuts of cardinality $r-2$ then $G$ has a $1$-factor</p>
<blockquote>
<blockquote>
<p><strong>Tutte's theorem</strong>: A non trivial graph $G$ has a $1$-factor if and only if $k_0 ... | Tyler Seacrest | 172,823 | <p>Assume for sake of contradiction that $G$ does not have a perfect matching. By Tutte's theorem, there exists a set $S$ such that $G - S$ has more odd components than there are vertices in $S$. Let $A_1, \ldots, A_t$ be the odd components of $G - S$. Notice that $t + |S|$ is even, since $G$ has an even number of v... |
20,567 | <p>Excuse me if the language is a bit off I'm not a native English speaker. I've been studying(self study) computers and programming for a little over two years, but do not have much education. I've been taking a math class for the last 11 weeks and start an new one next week. I've been using Unix/Linux and Stackoverfl... | Lord_Farin | 43,351 | <p>Taking a look at your StackOverflow questions, I predict your questions here will be well-received.</p>
<p>Generally, any question that expresses an earnest desire to learn and is reasonably scoped can expect a positive response.</p>
<p>For some further site-specific pointers, please see <a href="https://math.meta... |
3,573,334 | <blockquote>
<p>Given positives <span class="math-container">$a, b, c$</span> such that <span class="math-container">$a + b + c = 3$</span>, prove that <span class="math-container">$$\frac{1}{c^2 + 4a^2 + b^2} + \frac{1}{a^2 + 4b^2 + c^2} + \frac{1}{b^2 + 4c^2 + a^2} \le \frac{1}{2}$$</span></p>
</blockquote>
<p>We ... | Michael Rozenberg | 190,319 | <p>Another way.</p>
<p>We'll prove that our inequality is true for any real <span class="math-container">$a$</span>, <span class="math-container">$b$</span> and <span class="math-container">$c$</span> such that <span class="math-container">$a+b+c=3.$</span></p>
<p>Indeed, let <span class="math-container">$a+b+c=3u$</... |
4,008,152 | <p>Question itself: Throw a coin one million times. What is the expected number of sequences of six tails, if we <strong>do not allow overlap</strong>?</p>
<p>I know when overlap is allowed, the answer is (1,000,000-5)/(2^6). Not sure if we can just do (1,000,000-5)/(2^6) divided by 6 if overlap is not allowed?</p>
<p>... | bof | 111,012 | <p>The expected number <span class="math-container">$a_n$</span> of nonoverlapping runs of <span class="math-container">$6$</span> consecutive tails in a sequence of <span class="math-container">$n$</span> independent fair coin tosses satisfies the nonhomogeneous linear recurrence
<span class="math-container">$$a_n=\fr... |
18 | <p>Some teachers make memorizing formulas, definitions and others things obligatory, and forbid "aids" in any form during tests and exams. Other allow for writing down more complicated expressions, sometimes anything on paper (books, tables, solutions to previously solved problems) and in yet another setting students a... | paul garrett | 63 | <p>It is easy to explain the most immediate disadvantage of allowing "aids" during exams: many students misjudge the situation, thinking that having books and/or papers means they can study less. In particular, they often misjudge information access time.</p>
<p>But many students benefit from some form or degree of op... |
18 | <p>Some teachers make memorizing formulas, definitions and others things obligatory, and forbid "aids" in any form during tests and exams. Other allow for writing down more complicated expressions, sometimes anything on paper (books, tables, solutions to previously solved problems) and in yet another setting students a... | Thomas | 53 | <p>I disagree with one of the other answers when saying that "math is not about memory". Doing math is not only about memory, but remembering your definitions and theorems can be crucial to doing problems. The argument that a mathematician can just look of these things on books disregards the fact that when doing the p... |
18 | <p>Some teachers make memorizing formulas, definitions and others things obligatory, and forbid "aids" in any form during tests and exams. Other allow for writing down more complicated expressions, sometimes anything on paper (books, tables, solutions to previously solved problems) and in yet another setting students a... | DavidButlerUofA | 1,853 | <p>I' m only going to answer one of the questions:<br />
<strong>What are the disadvantages of of allowing aids during tests/exams?</strong></p>
<p>First a point of terminology: Here in Australia, a sheet of notes you are allowed to take into your exam is called a "cheat sheet", and in Asia it's known as a &q... |
2,205,950 | <p>If $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfies $f'(a) \neq 0$ for all $a \in \mathbb{R}$, show that $f$ is one-to-one for all $a\in \mathbb{R}$.</p>
<h2>My attempt</h2>
<p>We know that $f(a)$ is not a constant because $f'(a)\neq 0$.Define $f$ by $f(a)=bx$. $f'(a)=x\neq 0$</p>
<p>If $f(x)=f(v)$ then $$bx=bv$$<... | Bernard | 202,857 | <p>A function which is the derivative of another function has the inttermediate value property. therefore, if $f'(a)\ne 0$ for all $a\in\mathbf R$, $f'$ has a constant sign, and $f$ is increasing or decreasing on $\mathbf R$. In any case, it is one-to-one.</p>
|
121,924 | <p>Why is <span class="math-container">$\mathrm{Hom}_{\mathbb{Z}}\left(\prod_{n \geq 2}\mathbb{Z}_{n},\mathbb{Q}\right)$</span> nonzero?</p>
<p>Context: This is problem <span class="math-container">$2.25 (iii)$</span> of page <span class="math-container">$69$</span> Rotman's Introduction to Homological Algebra:</p>
<... | S.Hamid Hassanzadeh | 636,075 | <p><span class="math-container">$\mathbb{Q}$</span> is an injective <span class="math-container">$\mathbb{Z}$</span>-module.
The exact sequence
<span class="math-container">$$0\rightarrow \mathbb{Z} \rightarrow \prod \mathbb{Z}_n\rightarrow C\rightarrow 0$$</span> yields the exact sequence
<span class="math-container... |
629,275 | <p>A function $f$ is defined on an open set $D$ of $\mathbb R^{2}$ is called a differentiable at a point $x\in D$ if there is a vector $m \in \mathbb R^{2} $ such that
$$\lim_{h\to 0} \frac{f(x+h)-f(x)-m\cdot h}{|h|}=0.$$</p>
<p><strong>My questions are</strong>:
(1) What is a geometric interpretation of $f:\mathbb R... | Andrej Bauer | 30,711 | <p>The geometric interpretation of $f : \mathbb{R}^2 \to \mathbb{R}$ is a surface in $\mathbb{R}^3$. The geometric interpretation of the differentiability of $f$ at a point $x$ is that the surface has a tangent plane at $x$. The components of the vector $m$ are the partial derivatives of $f$.</p>
<p>In the multidimens... |
629,275 | <p>A function $f$ is defined on an open set $D$ of $\mathbb R^{2}$ is called a differentiable at a point $x\in D$ if there is a vector $m \in \mathbb R^{2} $ such that
$$\lim_{h\to 0} \frac{f(x+h)-f(x)-m\cdot h}{|h|}=0.$$</p>
<p><strong>My questions are</strong>:
(1) What is a geometric interpretation of $f:\mathbb R... | Christian Blatter | 1,303 | <p>Given a function ${\bf f}:\>{\mathbb R}^n\to{\mathbb R}^m$ and a "working point" ${\bf p}$ in the domain of ${\bf f}$ one may ask how the value of ${\bf f}$ changes when one moves from ${\bf p}$ to a nearby point ${\bf p}+{\bf X}$, $\>|{\bf X}|\ll1$. This means that we are interested in the auxiliary function ... |
4,590,677 | <p>This is perhaps a silly question related to calculating with surds. I was working out the area of a regular pentagon ABCDE of side length 1 today and I ended up with the following expression :</p>
<p><span class="math-container">$$\frac{\sqrt{5+2\sqrt5}+\sqrt{10+2\sqrt{5}}}{4}$$</span></p>
<p>obtained by summing the... | Jean-Claude Arbaut | 43,608 | <p>One way to (try to) simplify is squaring the sum and see where it leads. Then you get a product of radicals instead of a sum, and it's easier to simplify.</p>
<p><span class="math-container">$$4A=\sqrt{5+2\sqrt5}+\sqrt{10+2\sqrt5}$$</span>
<span class="math-container">$$16A^2=15+4\sqrt5+2\sqrt{(5+2\sqrt5)(10+2\sqrt5... |
2,426,361 | <p>What would be the best mathematical tool/concept to measure how far a matrix is from being singular? Could it be the condition number?</p>
| José Carlos Santos | 446,262 | <p>I am assuming that your matrix is a $n\times n$ matrix. You could take the rank of the matrix. Its possible values are $0,1,\ldots,n$. The matrix is singular if and only if its rank is smaller than $n$. The rank is $0$ if and only if the matrix is the null matrix, which is the most singular of all matrices.</p>
|
728,495 | <p>I'm taking Discrete Math this semester. While I understand the mechanics of proofs, I find that I must refine my understanding of how to work them. To that end, I'm working through some extra problems on spring break. Please read over this proof I did from an exercise from the book. I apologize in advance for po... | user138320 | 138,320 | <p>Why is $ A^c \cap B = B $? I think there is a typo/mistake in that line. What I believe you want to say is that $ (A \cap B) \cup ( A^c \cap B ) = ( A \cup A^c) \cup B $, by the distributive law, and that $ = ( A \cup A^c) \cup B = U \cup B = B $.</p>
|
728,495 | <p>I'm taking Discrete Math this semester. While I understand the mechanics of proofs, I find that I must refine my understanding of how to work them. To that end, I'm working through some extra problems on spring break. Please read over this proof I did from an exercise from the book. I apologize in advance for po... | dsm | 129,367 | <p>Formatting looks good to me. There is a typo in part 1, as pointed out by user138320, but there is also a typo in what he mentioned: The distributive law yields $$(A \cap B) \cup ( A^c \cap B ) = ( A \cup A^c) \cap B = U \cap B = B$$
Part 2 is sufficient, but it lacks necessary formalism. Additionally, you made a t... |
3,862,408 | <p>This is the second example of 1. in <a href="http://www-personal.umich.edu/%7Ebhattb/teaching/mat679w17/lectures.pdf" rel="nofollow noreferrer">Ex. 2.0.3 </a> of Bhatt's notes in perfectoid space.</p>
<p>We define <span class="math-container">$R^{perf}:= \varprojlim ( \cdots R \xrightarrow{\phi} R)$</span> where <sp... | Qiaochu Yuan | 232 | <p>Since two people have already provided concrete answers, an abstract way to see this is to observe that the functor <span class="math-container">$X \mapsto X^H$</span> is represented by <span class="math-container">$G/H$</span>, meaning that <span class="math-container">$\text{Hom}_G(G/H, X) \cong X^H$</span>; in ot... |
19,815 | <p>Problem:</p>
<blockquote>
<p>Prove that if gcd( a, b ) = 1, then gcd( a - b, a + b ) is either 1 or 2.</p>
</blockquote>
<p>From Bezout's Theorem, I see that am + bn = 1, and a, b are relative primes. However, I could not find a way to link this idea to a - b and a + b. I realized that in order to have gcd( a, b ) =... | Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$\rm\,\ a\!-\!b + (a\!+\!b)\ {\it i}\ =\ (1\!+\!{\it i})\ (a\!+\!b\!\ {\it i})\ \ $</span> yields a slick proof using Gaussian integers. This reveals the arithmetical essence of the matter and, hence, suggests <a href="https://math.stackexchange.com/a/33104/242">obv... |
1,757,260 | <p>A little box contains $40$ smarties: $16$ yellow, $14$ red and $10$ orange.</p>
<p>You draw $3$ smarties at random (without replacement) from the box.</p>
<p>What is the probability (in percentage) that you get $2$ smarties of one color and another smarties of a different color?</p>
<p>Round your answer to the ne... | André Nicolas | 6,312 | <p>We imagine taking out the candies one at a time.</p>
<p>Your $\frac{16}{40}\cdot\frac{15}{39}\cdot \frac{24}{38}$ calculates the provability of getting Yellow, Yellow, Other <em>in that order</em>. However, two Yellow and one Other can happen in two additional orders, Yellow, Other, Yellow or Other, Yellow, Yellow.... |
2,699,621 | <p>To show $1 + \frac12 x - \frac18 x^2 < \sqrt{1+x}$ is it enough to tell that the taylor series expansion of $\sqrt{1+x}$ around $0$ has more positive terms?</p>
| John Falvey | 311,586 | <p>8 + 4x -x^2 < 8[1 + x]^.0.5</p>
<p>8 + x[ 4 - x] < 8[1 + x ]^0.5 , square both sides</p>
<p>64 + 16x[4-x] +x^2 [4 -x]^2 < 64[ 1 + x]</p>
<p>64 + 64 x - 16x^2 + x^2 [ x^2 -8x + 16 ] < 64 + 64x , cancel 64 + 64x</p>
<p>then - 16x^2 + x^4 - 8x^3 + 16x^2 < 0 , cancel - 16x^2 + 16x^2</p>
<p>x^4... |
2,174,061 | <p>in $\Delta ABC$ if the $AD\perp BC$,$D\in BC$,and such $$|BC|=2|AD|$$
show that
$$\dfrac{|AB|}{|AC|}\le\sqrt{2}+1$$
<a href="https://i.stack.imgur.com/SXDvI.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SXDvI.png" alt="enter image description here"></a></p>
<p>since
$$\cot{B}+\cot{C}=\dfrac{BD}... | Rafa Budría | 362,604 | <p>I think you have only to check the worst of the cases, so is, prove that $\sqrt{5}\le\sqrt 2 +1$</p>
|
2,174,061 | <p>in $\Delta ABC$ if the $AD\perp BC$,$D\in BC$,and such $$|BC|=2|AD|$$
show that
$$\dfrac{|AB|}{|AC|}\le\sqrt{2}+1$$
<a href="https://i.stack.imgur.com/SXDvI.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SXDvI.png" alt="enter image description here"></a></p>
<p>since
$$\cot{B}+\cot{C}=\dfrac{BD}... | Blue | 409 | <p>Taking <span class="math-container">$|AC| = 1$</span>, we have
<span class="math-container">$$|AD| = \sin C = \frac12|BC| \qquad |CD|=\cos C \quad\text{(as a signed length)}$$</span>
so that
<span class="math-container">$$\begin{align}
|AB|^2 &= |AD|^2+|BD|^2 = |AD|^2+\left(|BC|-|CD|\right)^2 \\[4pt]
&= \sin... |
3,363,944 | <p>A group consisting of <span class="math-container">$3$</span> men and <span class="math-container">$6$</span> women attends a prizegiving ceremony. If <span class="math-container">$ 5$</span> prizes are awarded at random to members of the group, find the probability that exactly <span class="math-container">$3 $</sp... | Oliver Kayende | 704,766 | <p>If the prizes are not identical and each prize is distinct then the total # of outcomes in a) becomes <span class="math-container">$${9\choose 5}*5!$$</span> whereas in b) it becomes <span class="math-container">$9^5$</span> which counts the # of functions from a 5-set to a 9-set. The # of desirable outcomes in a) i... |
4,170,940 | <blockquote>
<p><a href="https://www.isical.ac.in/%7Eadmission/IsiAdmission2017/PreviousQuestion/BStat-BMath-UGA-2016.pdf" rel="nofollow noreferrer">Question 36</a>: Finding graph corresponding to <span class="math-container">$\int_0^{\sqrt{x} } e^{ -\frac{u^2}{x} } du$</span>
<a href="https://i.stack.imgur.com/KIVRA.p... | Tuvasbien | 702,179 | <p>Substitute <span class="math-container">$v=\frac{u^2}{x}$</span>,
<span class="math-container">$$ \int_0^{\sqrt{x}}e^{-\frac{u^2}{x}}du=\frac{\sqrt{x}}{2}\int_0^1\frac{e^{-v}}{\sqrt{v}}dv $$</span>
Now the derivative is easier to calculate, it is <span class="math-container">$>0$</span> and diverges to <span clas... |
3,467,523 | <p><a href="https://i.stack.imgur.com/G47bX.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/G47bX.png" alt="Attached is the picture of the problem."></a></p>
<p>I was doing some trig problems for leisure. This one particularly seems not trivial. So I thought someone may be interested to take a look.... | Sameer Nilkhan | 710,265 | <p>By using following identities,
<span class="math-container">$$\cos(2x)=2\cos^2(x) - 1$$</span> and
<span class="math-container">$$\sin^2(x)=\dfrac{1-\cos(2x)}{2}$$</span> we can simplify the given expression as,
<span class="math-container">$$2\cos^2 x-1-m \cos(2x)+3=2 m \left({\dfrac{1-\cos(2x)}{2}}\right)^2$$</s... |
320,937 | <p>It is easy to think of $\mathbb{C}^2$ as an ordered pair. I just wonder if it is possible to put $\mathbb{C}^2$ into illustration, since $\mathbb{C}$ has taken the role of two dimensional Euclidean Space.</p>
| Zev Chonoles | 264 | <p>Let
$$P=\{n\in\mathbb{N}\mid n\text{ is a counterexample to the claim}\}.$$
The well-ordering principle implies that, if $P$ is non-empty, then there is a minimum element of $P$. Assuming that $P$ is non-empty, let's call the minimum element $m$. Usually one then proceeds in the same way as one would by induction: s... |
1,859,719 | <blockquote>
<p>Let be $U (x,y) = x^\alpha y^\beta$. Find the maximum of the function $U(x,y)$ subject to the equality constraint $I = px + qy$.</p>
</blockquote>
<p>I have tried to use the Lagrangian function to find the solution for the problem, with the equation</p>
<p>$$\nabla\mathscr{L}=\vec{0}$$</p>
<p>where... | smcc | 354,034 | <h2>The solution</h2>
<p>The answer can be been found on the internet in any number of places. The function $U$ is a Cobb-Douglas utility function. The Cobb-Douglas function is one of the most commonly used utility functions in economics. </p>
<p>The demand functions you should get are:</p>
<p>$$x(p,I)=\frac{\alpha ... |
1,369,482 | <p>I'm have problem proving: Law for Scalar Multiplication :</p>
<p>Vector spaces possess a collection of specific characteristics and properties. Use the definitions in the attached “Definitions” to complete this task.</p>
<p>Define the elements belonging to $\mathbb{R}^2$ as $\{(a, b) | a, b \in \mathbb{ R}\}$. Com... | P Vanchinathan | 28,915 | <p>Take the examples of all directed arrows (every possible length and every possible angle) originating from a single point.</p>
<p>Now in this set vector addition is like addition of forces in physics: parallelogram law. In this set internally there is addition. Also there is an external operation. Any vector can be... |
1,878,884 | <p>I recently figured out my own algorithm to factorize a number given we know it has $2$ distinct prime factors. Let:</p>
<p>$$ ab = c$$</p>
<p>Where, $a<b$</p>
<p>Then it isn't difficult to show that:</p>
<p>$$ \frac{c!}{c^a}= \text{integer}$$</p>
<p>In fact, </p>
<p>$$ \frac{c!}{c^{a+1}} \neq \text{integer}... | TonyK | 1,508 | <p>The best factoring algorithm currently available is the <a href="https://en.wikipedia.org/wiki/General_number_field_sieve" rel="nofollow">General Number Field Sieve</a>. Numbers of more than 200 decimal digits have been factored using this method.</p>
<p>The factorial of such a number would have more than $10^{200}... |
1,878,884 | <p>I recently figured out my own algorithm to factorize a number given we know it has $2$ distinct prime factors. Let:</p>
<p>$$ ab = c$$</p>
<p>Where, $a<b$</p>
<p>Then it isn't difficult to show that:</p>
<p>$$ \frac{c!}{c^a}= \text{integer}$$</p>
<p>In fact, </p>
<p>$$ \frac{c!}{c^{a+1}} \neq \text{integer}... | Charles | 1,778 | <p>The basic algorithm takes about $p$ divisions to find the smallest prime factor $p$ of your number, which in the worst case is around $\sqrt{c}$. Each step requires dividing a huge number* by $c$, which takes about $c\log^2 c$ time, for a total runtime of about $cp\log^2c$. This is much worse than trial division!</p... |
3,854,446 | <p>I am reading a textbook on representation theory which says the following.</p>
<p><span class="math-container">$G$</span> is a finite group with irreducible representation <span class="math-container">$\rho:G\to GL(V)$</span> over field <span class="math-container">$k$</span> (possibly algebraically closed, there's ... | Angina Seng | 436,618 | <p>I presume you are in characteristic zero.</p>
<p>As you say, <span class="math-container">$T$</span> is a scalar matrix (Schur's lemma).</p>
<p>The trace of <span class="math-container">$T$</span> is
<span class="math-container">$$\frac1{|G|}\sum_{g\in G}\phi(g^{-1})\textrm{Tr}(\rho_g)=\frac1{|G|}\sum_{g\in G}\phi(g... |
1,103,624 | <p>Ashamed to admit that I cannot aid my friend's niece with her second grade homework problem. So much for that college education, eh? Here's the problem.</p>
<p>Using only the natural numbers 1 through 9 without repeating any of them (natural because there cannot occur any rationals anywhere in this process, i.e. th... | David Zhang | 80,762 | <p>Assuming that the usual order of operations is to be ignored, there are precisely 32 solutions to your problem in which fractions do not appear in any intermediate step. Unfortunately, I have no idea how one might arrive at such a solution by hand; I produced these with an exhaustive computer search.</p>
<p>$$4\div... |
108,253 | <p>I would like to assign 'x' individuals to 'y' groups, randomly. For example, I would like to divide 50 individuals into 100 groups randomly. Of course, with more groups than individuals many of the groups will have zero individuals, while some groups will have multiple individuals. That is fine. With random assignme... | Coolwater | 9,754 | <p>This does it:</p>
<pre><code>x = 10000;
y = 50;
groups = PositionIndex[RandomChoice[Range[y], x]]
</code></pre>
<p>And you get the poisson:</p>
<pre><code>QuantilePlot[PadLeft[Values[Map[Length, groups]], y],
PoissonDistribution[x/y], Method -> {"ReferenceLineMethod" -> "Diagonal"}]
</code></pre>
<p><a h... |
3,699,105 | <p>If <span class="math-container">$T$</span> is normal operator and <span class="math-container">$T^3=T^2$</span>,then show that <span class="math-container">$T$</span> is idempotent .</p>
<ol>
<li><span class="math-container">$TT*=T*T$</span> </li>
<li><span class="math-container">$T^3=T^2$</span></li>
<li>We are to... | paul blart math cop | 571,438 | <p>This is not true. Here's an explicit example. Let <span class="math-container">$A = \{0, 1\}$</span>, <span class="math-container">$R = \{(0, 0)\}$</span>, <span class="math-container">$S = \{(1, 1)\}$</span>. Then <span class="math-container">$R \cup S = \{(0, 0), (1, 1)\}$</span>, which is an equivalence relation... |
510,814 | <p>I've seen in several places without further comment that if an equalizer is epic, it's an isomorphism. I've only proved one half of this:</p>
<p>Suppose $e:X \rightarrow A$ is an epimorphism and an equalizer for $f$ and $g$. Then $f \circ e = g \circ e \implies f = g$. Then any function $e': X' \rightarrow A$ trivi... | Community | -1 | <p>$id_A$ is an equalizer of $f$ and $g$.</p>
|
1,102,885 | <p>I have exams in Machine Learning coming up and I need help answering this question.</p>
<blockquote>
<p>There are a million identical fish in a lake, one of which has
swallowed the One True Ring. You must get it back! After months of
effort, you catch another random fish and pass your metal detector
over it... | KSmarts | 192,747 | <p>I think you have mixed up a few of your conditional probabilities. If the detector <em>fails</em> to beep over the ring only one in a billion times, then it <em>does</em> beep the rest of the billion times. Similarly, if it beeps over he wrong fish one in ten thousand times, then it doesn't beep on the rest of those... |
2,413,891 | <blockquote>
<p><strong>Question :</strong> Evaluate - $$\int_{0}^{1}2^{x^2+x}\mathrm dx$$</p>
</blockquote>
<p><strong>My Attempt :</strong> First I tried to evaluate the indefinite integral of $2^{x^2+x}$ in order to put the limits $0$ and $1$ later on, but couldn't integrate it. Then I checked on WA and came to k... | Bill Donald | 476,771 | <p>Hint:</p>
<p>Using $u=\frac{2x+1}{2}$ yields an <a href="http://mathworld.wolfram.com/Erfi.html" rel="nofollow noreferrer">imaginary error function</a>.</p>
|
1,347 | <p>Sometimes I check how many users of <code>mathematica.stackexchange.com</code> there are.<br>
I remember that a few weeks ago there were about 15 thousand and recently I've been surprised seeing that the <a href="https://mathematica.stackexchange.com/users?tab=NewUsers&sort=creationdate">new users</a> are signe... | Mr.Wizard | 121 | <p>Now that the technical side of this has been answered I'll attempt to address:</p>
<blockquote>
<p>So what is the reliable number of all users (including unregistered) and of those who are registered? </p>
</blockquote>
<p>The <a href="http://stackexchange.com/leagues/177/alltime/mathematica">Stack Exchange User... |
373,958 | <p>Is $\sum_{n=1}^\infty(2^{\frac1{n}}-1)$ convergent or divergent?
$$\lim_{n\to\infty}(2^{\frac1{n}}-1) = 0$$
I can't think of anything to compare it against. The integral looks too hard:
$$\int_1^\infty(2^{\frac1{n}}-1)dn = ?$$
Root test seems useless as $\left(2^{\frac1{n}}\right)^{\frac1{n}}$ is probably even harde... | chanp | 46,291 | <p>Yet another elementary method</p>
<p>$1=2-1=(2^\frac{1}{n})^n-1=(2^\frac{1}{n}-1)(2^\frac{n-1}{n}+2^\frac{n-2}{n}+\cdots+2^\frac{1}{n}+1) < (2^\frac{1}{n}-1)*(2+2+\cdots+2)$ .</p>
<p>Therefore, $(2^\frac{1}{n}-1) > \frac{1}{2n} $.</p>
|
2,619,185 | <p>Let $$P=(X+2)^m+(X+3)^{2m+3}$$ and $$Q=X^2+5X+7.$$ I need to show that $Q$ divides $P$ for any $m$ natural. </p>
<p>I said like this: let $a$ be a root of $X^2+5X+7=0$. Then $a^2+5a+7=0$. </p>
<p>Now, I know I need to show that $P(a)=0$, but I do not know if it is the right path since I have not found any way to d... | Hagen von Eitzen | 39,174 | <p>Note that
$$(X+3)^3=X^3+9X^2+27X+27=(X^2+5X+7)(X+4)-1 $$
and
$$(X+3)^2 = X^2+6X+9=(X^2+5X+7)+(X+2).$$</p>
|
2,619,185 | <p>Let $$P=(X+2)^m+(X+3)^{2m+3}$$ and $$Q=X^2+5X+7.$$ I need to show that $Q$ divides $P$ for any $m$ natural. </p>
<p>I said like this: let $a$ be a root of $X^2+5X+7=0$. Then $a^2+5a+7=0$. </p>
<p>Now, I know I need to show that $P(a)=0$, but I do not know if it is the right path since I have not found any way to d... | nonuser | 463,553 | <p>Write $t=x+3$. Now we have to prove that $Q=t^2-t+1$ divides $P=(t-1)^m+t^{2m+3}$. Note that if $a$ is root for $Q$ then we have $a^3=-1$. Note that $a-1 = a^2$. Now plug $a$ in to $P$ and we get:</p>
<p>\begin{eqnarray} (a-1)^m +a^{2m}\cdot a^3 &=& (a-1)^m -a^{2m} \\
&=& (a^2)^m-a^{2m} \\
&=&am... |
3,760,594 | <p>Is there a proper notation to <em>compose</em> sets and produce a set of sets? (<em>I am referring to this as compose due to ignorance of a proper manner to call it</em>)</p>
<p>To illustrate what I want, let me <em>suppose</em> that <span class="math-container">$\otimes$</span> does the job, so that</p>
<p><span cl... | Sameer Baheti | 567,070 | <p><strong>HINT:</strong> Integrate both sides of</p>
<p><span class="math-container">$df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\cdots= \vec{\nabla f}\cdot \vec{dl}=\vec{\nabla g}\cdot \vec{dl}=dg$</span></p>
<p><strong>EDIT:</strong> As pointed out by @peek-a-boo, this only works if the funct... |
3,760,594 | <p>Is there a proper notation to <em>compose</em> sets and produce a set of sets? (<em>I am referring to this as compose due to ignorance of a proper manner to call it</em>)</p>
<p>To illustrate what I want, let me <em>suppose</em> that <span class="math-container">$\otimes$</span> does the job, so that</p>
<p><span cl... | alphaomega | 775,794 | <p>Yes that's correct. Solve each (<span class="math-container">$1$</span>-dimensional) equation <span class="math-container">$\partial{f}/\partial{x_i} = \partial{g}/\partial{x_i}$</span></p>
|
3,760,594 | <p>Is there a proper notation to <em>compose</em> sets and produce a set of sets? (<em>I am referring to this as compose due to ignorance of a proper manner to call it</em>)</p>
<p>To illustrate what I want, let me <em>suppose</em> that <span class="math-container">$\otimes$</span> does the job, so that</p>
<p><span cl... | littleO | 40,119 | <p>[Spoiler warning, this is more than a hint. I wanted to show this method because it avoids working with components.]</p>
<hr />
<p>First suppose that <span class="math-container">$h:\mathbb R^n \to \mathbb R$</span> is differentiable and that <span class="math-container">$\nabla h(x) = 0$</span> for all <span class=... |
2,699,170 | <p>How to evaluate
$$ \int \frac{1}{ \ln x} \ \mathrm{d} x, $$
where $\ln x$ denotes the natural logarithm of $x$? </p>
<p>My effort: </p>
<blockquote>
<p>We note that
$$ \int \frac{1}{ \ln x} \ \mathrm{d} x = \int \frac{x}{x \ln x} \ \mathrm{d} x = \int x \frac{ \mathrm{d} }{ \mathrm{d} x } \left( \ln \ln x \r... | Bernard | 202,857 | <p>It cannot be expressed with the <em>elementary functions</em>. Actually, it is a <em>special function</em>, the <em>integral logarithm</em>:
$$\operatorname{li}(x)=\int_0^x\frac{\mathrm dt}{\ln t}\biggl(=\lim_{\varepsilon\to 0}\int_0^{1-\varepsilon}\frac{\mathrm dt}{\ln t}+\lim_{\varepsilon\to 0}\int_{1+\varepsilon}... |
181,367 | <p>It is well known that compactness implies pseudocompactness; this follows from <a href="https://secure.wikimedia.org/wikipedia/en/wiki/Heine%E2%80%93Borel_theorem">the Heine–Borel theorem</a>. I know that the converse does not hold, but what is a counterexample?</p>
<p>(A <a href="https://secure.wikimedia.org/wikip... | Alex Ravsky | 71,850 | <p>Some exotic examples of pseudocompact and non-compact spaces are constructed in my paper “<a href="http://arxiv.org/abs/1003.5343" rel="nofollow">Pseudocompact paratopological groups that are topological</a>”: </p>
<p>Example 1 (p.6). A $T_1$ space having each power countably pracompact (and, hence pseudocompact). ... |
1,692,757 | <p>I was required to find the derivative of $2\sqrt{\cot(x^2)}$.</p>
<p><strong>My solution</strong></p>
<p><a href="https://i.stack.imgur.com/N98SM.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/N98SM.jpg" alt="enter image description here"></a></p>
<p>I can't find any mistake in my solution but... | TSU | 320,683 | <p>I still think the two answers are identical. Maybe there is something wrong with your plot ;) (If I haven't misread your writing, that is...)</p>
<p>$$ \frac{-2x csc^2(x^2)}{\sqrt{cot(x^2)}} = \frac{-2x \left(\frac{1}{sin^2(x^2)} \right)}{\sqrt{\frac{cos(x^2)}{sin(x^2)}}} = \frac{-2x}{sin(x^2) \sqrt{sin(x^2)cos(... |
63,015 | <p>Why does assigning a DownValue using <code>Apply</code>, e.g.,</p>
<pre><code>Remove[a]
index={3,4};
(a @@ index) = 5;
a @@ index
(*Set::write: Tag Apply in a @@ {3, 4} is Protected. >>*)
(*a[3,4]*)
</code></pre>
<p>not work, while an assignment such as</p>
<pre><code>Remove[a]
a[Sequence @@ index] = 5
a @@... | WReach | 142 | <p>The behaviour we see here is due to <code>Set</code> performing limited evaluation of its left-hand-side prior to the assignment.</p>
<p><strong><code>Set</code> Evaluates the Left-Hand-Side</strong></p>
<p>Despite the fact that <code>Set</code> has the attribute <code>HoldFirst</code>, it performs a special kind ... |
3,414,197 | <p>I have to model/simulate a moving iron meter with Simulink, more specifically, I need to build a Simulink model for the equation of motion, wich is given as:
<span class="math-container">$$
\theta\ddot{\alpha} = T_\phi - T_S
$$</span>
where <span class="math-container">$\theta$</span> denotes the pointers moment ... | Pilotf4 | 272,311 | <p>As far as I know, I can't post images in comments, so here we go.</p>
<p>This is the Simulink model and a screenshot of the scope. I renamed <span class="math-container">$_0$</span> as <span class="math-container">$f$</span> and <span class="math-container">$$</span> as <span class="math-container">$D$</span>. Thei... |
1,384,735 | <p>What is the ODE satisfied by $y=y(x)$ </p>
<p>given that $$\frac{dy}{dx} = \frac{-x-2y}{y-2x}$$</p>
<p>I understand that I need to get it in some form of $\int \cdots \;dy = \int \cdots \; dx$, but am not sure how to go about it.</p>
| haqnatural | 247,767 | <p>\begin{align}
\frac{dy}{dx} = \frac{ -x-2y }{ y-2x } \\
\frac{ dy }{ dx } =\frac{ -1-2\frac{ y }{ x } }{ \frac{ y }{ x } -2 } \\ \frac{ y }{ x } = t \to \quad y=xt \quad \Rightarrow \quad dy=t+x \frac{ dt }{ dx } \\
t+x\frac{ dt }{ dx } =\frac{ -1-2t }{ t-2 } \\
x\frac{ dt }{ dx } =\frac{ -1-2t }{ t-2 } -t=\frac... |
3,956,292 | <p>Consider the Euclidean ball <span class="math-container">$B^n(x,r)$</span> in <span class="math-container">$\mathbb{R}^n$</span> given by:
<span class="math-container">$$B^n(x,r) = \{z\in\mathbb{R}^n : ||z-x||_2 \leq r\}$$</span> with centre <span class="math-container">$x\in\mathbb{R}^n$</span> and radius <span cla... | Hagen von Eitzen | 39,174 | <p>Along every (parametrised by <span class="math-container">$t$</span>) line, the squared Euclidean norm is a quadratic function in <span class="math-container">$t$</span> and does not attain a value <span class="math-container">$\ge1$</span> between distinct points where it is <span class="math-container">$\le1$</spa... |
1,924,568 | <p>This is a question that a friend asked me (has the final answer too).</p>
<p>The pdf of a random variable $X$ is</p>
<p>$$ f(x) = 0.5,\quad -1 < x < 1 $$</p>
<p>The random variable Y is defined as </p>
<p>$$ Y = \begin{cases}
-2X, & -1 < X < 0 \\
X+1, & 0 < X <1
\end{cases}$$</p>
<p>I... | Rizky Reza Fujisaki | 310,950 | <p>You are right intuitively, but I think it is not rigorous</p>
<p>how about this</p>
<p>for $[0,1)$, the inverse is $x=-\frac{y}{2}$ with absolute jacobian is $\frac{1}{2}$, hence</p>
<p>\begin{eqnarray*}
f_Y(y)=f_X\left(-\frac{y}{2}\right)\frac{1}{2}=\frac{1}{4}
\end{eqnarray*}</p>
<p>and for $[1,2)$, we get two... |
4,640,732 | <p>Find roots of:
<span class="math-container">$$x^{6}\ -\ \left(x-1\right)^{6}=0 \tag {1}$$</span></p>
<p>I know this equation has <span class="math-container">$4$</span> complex roots and exactly one real roots of value <span class="math-container">$0.5$</span>.</p>
<p>However, my first instinct was to do this:
<span... | NoChance | 15,180 | <p>Thanks for all the posted comments above. At the moment, no one had posted an answer, but I understood the following, which combined may provide an answer.</p>
<p><span class="math-container">$$|x|=|x-1|$$</span></p>
<p>can't always be written as <span class="math-container">$x=x-1$</span>. I need to learn how to so... |
4,115,417 | <p>I'm stuck on this problem for quite some time:</p>
<blockquote>
<p>Call a triangle a <em>Special Rational triangle</em> if it's area is rational, and the side lengths are consecutive positive integers, Can we find a closed form which generates all <em>Special Rational triangles</em>?</p>
</blockquote>
<p>I have trie... | Brian M. Scott | 12,042 | <p>You’re trying to apply a method for solving recurrences of fixed order to one that is not of fixed order. You can, however, rewrite it: <span class="math-container">$u_{n-1}=\sum_{k=1}^{n-2}u_k$</span>, so</p>
<p><span class="math-container">$$u_n=u_{n-1}+\sum_{k=1}^{n-2}u_k=2u_{n-1}\,,$$</span></p>
<p>and you now h... |
4,115,417 | <p>I'm stuck on this problem for quite some time:</p>
<blockquote>
<p>Call a triangle a <em>Special Rational triangle</em> if it's area is rational, and the side lengths are consecutive positive integers, Can we find a closed form which generates all <em>Special Rational triangles</em>?</p>
</blockquote>
<p>I have trie... | OttR - A. Yu | 321,096 | <p>You dropped the 1. And lets also set the solution more precisely <span class="math-container">$u_n = kx^{n-1}$</span> since your power of two solution is also <span class="math-container">$n-1$</span>. Because then the <span class="math-container">$u_1$</span> term would add another 1 by being power of 0.</p>
<p><sp... |
777,186 | <p>My equation is the following, and I would like to find which $k$ can make it a circle.</p>
<p>$$x^2+y^2+4x-6y+k=0$$</p>
<p>My naive approach is to have $k$ to be $-4x+6y+c$ where c is any number, so that I can have any circle that is in 0. However k is a parameter and I can't really figure that out if I am missing... | lab bhattacharjee | 33,337 | <p>Completing the square $$(x+2)^2+(y-3)^2=2^2+3^2-k$$</p>
<p>For real circle, $9+4-k\ge0$</p>
|
777,186 | <p>My equation is the following, and I would like to find which $k$ can make it a circle.</p>
<p>$$x^2+y^2+4x-6y+k=0$$</p>
<p>My naive approach is to have $k$ to be $-4x+6y+c$ where c is any number, so that I can have any circle that is in 0. However k is a parameter and I can't really figure that out if I am missing... | Anastasiya-Romanova 秀 | 133,248 | <p>\begin{align}
x^2+y^2+4x-6y+k&=0\\
x^2+4x+y^2-6y&=-k\\
x^2+4x+4+y^2-6y+9&=-k+4+9\\
(x+2)^2+(y-3)^2&=13-k
\end{align}
Compare with equation of the circle where its center on $(a,b)$ and radius $r$.
\begin{align}
(x-a)^2+(y-b)^2=r^2
\end{align}
We get $r^2=13-k$. In order to make a circle, then $r>0... |
1,969,169 | <p>We have to do the following integral.
$$\int_1^{\frac{1+\sqrt{5}}{2}}\frac{x^2+1}{x^4-x^2+1}\ln\left(1+x-\frac{1}{x}\right)dx$$
I tried it a lot.
I substitute $t=1+x-(1/x)$, $dt=1+(1/x^2)$</p>
<p>But then I stuck at
$$\int\limits_{1}^{2} \frac{\ln(t)}{(t-1)^{2} + 1} \mathrm{d}t$$</p>
<p>But now how to proceed.</p>... | Robert Z | 299,698 | <p>Hint. We have that
$$
\int_1^2\frac{\log(t)}{(t-1)^2+1}dt
=\int_0^1\frac{\log(1+v)}{v^2+1}dv
=\int_0^{\pi/4}\log(1+\tan(u))\,du\\
=\int_0^{\pi/4}\log(\cos(u)+\sin(u))\,du-\int_0^{\pi/4}\log(\cos(u))\,du\\
=\int_0^{\pi/4}\log(\sqrt{2}\cos(\pi/4-u)))\,du-\int_0^{\pi/4}\log(\cos(u))\,du.$$
Now work on the first integra... |
1,981,360 | <blockquote>
<p>Given function $f:\mathbb{R}_0^+ \to \mathbb{R},~f(x) = x^2 + 4x + 4$ prove that it is injective.</p>
</blockquote>
<p>Using definition of injectivity $(\forall x_1, x_2 \in \mathbb{R}_0^+)(x_1 \neq x_2 \implies f(x_1) \neq f(x_2))$ I'm doing the following:</p>
<p>$$x_1^2 + 4x_1 + 4 = x_2^2 + 4x_2 +... | Piquito | 219,998 | <p>You have $$f(x)=(x+2)^2$$ so if the function has domain $\Bbb R$ then the points
$x=t-2$ and $x=-t-2$ have same image for all $t$ so the function $f$ could not be injective. However with domain $\mathbb{R}_0^+ $ it is in fact injective and strictly increasing on its domain because is of the quadratic form $x^2$.</p>... |
191,548 | <p>Say I have a list:</p>
<pre><code>{{Line[{{-Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}, {0, 1}}],
Line[{{Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}, {0,1}}]},
{Line[{{-Sqrt[5/8 + Sqrt[5]/8],1/4 (-1 + Sqrt[5])}, {Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}}],
Line[{{Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}, {0, ... | Ulrich Neumann | 53,677 | <p>What about <code>/.Line->List</code></p>
<pre><code>lines /. Line -> (Flatten[ List[#], 1] &)
</code></pre>
|
393,580 | <p>Show that $-Z$ is also a standard normal random variable; that is, show that $P[-Z < x] = P[Z < x] \,\forall x.$</p>
| Argha | 35,821 | <p>Since the standard normal is a symmetric distribution about $0$ we have $P[Z<0+x]=P[Z>0-x]\forall x\\ \implies P[Z<x]=P[Z>-x]\forall x$</p>
<p>Again note that $$P[-Z<x] =P[Z>-x]\forall x$$Hence it is proved.</p>
|
533,855 | <p>I need to show that $\{x_{n}\}$ is Cauchy given that there exists $0<C<1$ s.t. $|x_{n+1}-x_{n}|\leq C|x_{n}-x_{n-1}|$. Intuitively, that statement clearly implies $\{x_{n}\}$ is Cauchy, since it implies the sequence terms become arbitrarily close. But how to make it precise? </p>
<p>Couldn't it also be said ... | robjohn | 13,854 | <p>If I understand the question, as soon as one T appears, the next flip is equally likely to be a T or an H, so neither is more likely to appear before the other.</p>
|
2,316,159 | <p>I'm interested in the differences in the groups but also in the Lie algebra associated. I know that two groups can have the same lie algebra if they differ from discrete elements, for instance: $SO(n)$ and $O(n)$ should have the same algebra. But then if I have a group $O(2,2)$, what is the associated Lie algebra? D... | Tsemo Aristide | 280,301 | <p>Consider $J=\pmatrix{1&0&0&0\cr 0&1&0&0\cr 0&0&-1&0\cr 0&0&0&-1}$ $M$ in $SO(2,2)$ i.e $M^t J M=I$. The Lie algebra of $SO(2,2)$ is the set of $4\times 4$ matrices such that $A^t J+JA=0$.</p>
<p>The Lie algebra $so(2)$ of $SO(2)$ is the set of $2\times 2$ matrices suc... |
940,352 | <p>If there is a mapping of $B$ onto $A$, then $2^{|A|} \leq 2^{|B|}$. [Hint: Given $g$ mapping $B$ onto $A$, let $f(X)=g^{-1}(X)$ for all $X \subseteq A$]</p>
<p>I follow the hint and obtain the function $f$. If $f$ is injective, then the statement is proven.</p>
<p>Question: Why does $g^{-1}$ exist in the first pla... | Daniel Fischer | 83,702 | <blockquote>
<p>Question: Why does $g^{−1}$ exist in the first place?</p>
</blockquote>
<p>It exists for all maps. Here, it does not denote the inverse, but the pre-image map,</p>
<p>$$g^{-1}(X) = \{b\in B : g(b)\in X\}.$$</p>
<p>Now use the surjectivity of $g$ to deduce the injectivity of $g^{-1}$.</p>
|
940,352 | <p>If there is a mapping of $B$ onto $A$, then $2^{|A|} \leq 2^{|B|}$. [Hint: Given $g$ mapping $B$ onto $A$, let $f(X)=g^{-1}(X)$ for all $X \subseteq A$]</p>
<p>I follow the hint and obtain the function $f$. If $f$ is injective, then the statement is proven.</p>
<p>Question: Why does $g^{-1}$ exist in the first pla... | Diego Robayo | 177,025 | <p>If there exists a function $f:B \longrightarrow A$ such that $f$ is onto then $\lvert B \rvert \geq \lvert A \rvert$. And this means that there exists an injective function $g: A \longrightarrow B$. Now as we want to see that $2^{\lvert A \rvert} \leq 2^{\lvert B \rvert}$ it's enough to define an injective function ... |
1,774,670 | <p>Among many fascinating sides of mathematics, there is one that I praise, especially for didactic purposes : the parallels that can be drawn between some "Continuous" and "Discrete" concepts.</p>
<p>I am looking for examples bringing a help to a global understanding...</p>
<p>Disclaimer : Being d... | Benjamin Dickman | 37,122 | <p>I like the following problem:</p>
<blockquote>
<p>Decompose a positive integer <span class="math-container">$N$</span> into positive integer addends summing to <span class="math-container">$N$</span> with maximal product.</p>
</blockquote>
<p>In the "discrete" version of this problem, you find that the ans... |
368,114 | <blockquote>
<p>Prove that this set is closed:</p>
<p><span class="math-container">$$ \left\{ \left( (x, y) \right) : \Re^2 : \sin(x^2 + 4xy) = x + \cos y \right\} \in (\Re^2, d_{\Re^2}) $$</span></p>
</blockquote>
<p>I've missed a few days in class, and have apparently missed some very important definitions if they ex... | Community | -1 | <ol>
<li><p>Prove that the function $f(x,y) = \sin(x^2 + 4xy) - x\cos y $ is continuous on all of $\Bbb{R}^2$.</p></li>
<li><p>The point $0 \in \Bbb{R}$ is closed. (Why?)</p></li>
<li><p>Continuous functions take open sets to open sets and so take closed sets to closed sets, because taking the complement of a set commu... |
110,162 | <p>One can use <code>$Epilog</code> to do something when the Kernel is quit or put an <code>end.m</code> file next to the <code>init.m</code>.</p>
<blockquote>
<p>For Wolfram System sessions, <code>$Epilog</code> is conventionally defined to read in a file named end.m.</p>
</blockquote>
<p>But if <code>$Epilog</code> i... | Mr.Wizard | 121 | <p>If you are willing to rely on undocumented behavior you can move <code>$Epilog</code> out of the System context, and give it a definition that evaluates both an internal (default) expression as well as the public expression assigned to <code>System`$Epilog</code>.</p>
<p>Your specialized setup:</p>
<pre><code>Cont... |
2,781,017 | <p>I known that $\sum a_i b_i \leq \sum a_i \sum b_i$ for $a_i$, $b_i > 0$. It seems this inequality will also hold true when $a_i$, $b_i \in (0,1)$. However, I am unable to find out if</p>
<p>$\sum \frac{a_i}{b_i} \leq \frac{\sum a_i}{\sum b_i}$ </p>
<p>holds true for $a_i$, $b_i \in (0,1)$.</p>
| zhw. | 228,045 | <p>Let $(a_n)$ be the sequence</p>
<p>$$1/2,1/4, 1/8,1/16,\dots,$$</p>
<p>and let $(b_n)$ be the same as $(a_n)$ except for $n=1,2,$ where let let $b_1=1/4, b_2= 1/2.$ Then $\sum a_n,\sum b_n$ both equal $1,$ hence so does their quotient, but</p>
<p>$$\sum \frac{a_n}{b_n} = \frac{1/2}{1/4} + \cdots > 2.$$</p>
|
158,896 | <p>Being interested in the very foundations of mathematics, I'm trying to build a rigorous proof on my own that $a + b = b + a$ for all $\left[a, b\in\mathbb{R}\right] $. Inspired by interesting properties of the complex plane and some researches, I realized that defining multiplication as repeated addition will lead m... | Potato | 18,240 | <p>Unfortunately, your method is not rigorous. There are too many undefined notions running around. For example, you never define what $\mathbb{R}$ is in your construction, or what properties it has. To answer your question, in the usual construction of the real numbers, the fact that $a+b=b+a$ is taken as axiomatic (a... |
158,896 | <p>Being interested in the very foundations of mathematics, I'm trying to build a rigorous proof on my own that $a + b = b + a$ for all $\left[a, b\in\mathbb{R}\right] $. Inspired by interesting properties of the complex plane and some researches, I realized that defining multiplication as repeated addition will lead m... | Pedro | 23,350 | <p>You should first think how you define $\alpha$, a real number. One of the classical construction is defining it as a set. This is a construction based on Dedkind cuts, defined as follows:</p>
<blockquote>
<p><strong>DEFINITION</strong> (Spivak)</p>
<p>A <strong>real number</strong> is a set of rational numbe... |
41,718 | <p>Currently I am working on creating the package on Mathematica version 9 on Windows 7. Here is my code as follows: </p>
<pre><code>BeginPackage["mypackage`"]
Begin["`Private`"]
getColumn[data_,branch_List]:=
Module[{pos},
pos = Position[data,#][[1,2]]&/@branch;
data[[All,pos]]
];
RemoveMissing[data_]:=Delete... | Nasser | 70 | <p>You had few issues. Try this. It worked now on my system. You needed <code>Usage</code> and better to also add the <code>Unprotect</code> and <code>ClearAll</code></p>
<pre><code>BeginPackage["mypackage`"]
Unprotect @@ Names["mypackage`*"];
ClearAll @@ Names["mypackage`*"];
getColumn::usage = "getColumn[data,bran... |
1,821,437 | <p>I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following exercise, which poses some difficulty. <em>If it's any indication of difficulty, the exercise is only Part 3-A of the sheet, graded for 10%</em></p>
<blockquote>
<p>Solve the equation $z^5=-32$ and draw its s... | Emilio Novati | 187,568 | <p>Let $z=\rho e^{i\theta}$. Representing $-32$ in the Argand plane you see that, in polar form, it is $-32=32e^{i\pi}$, so your equation becomes:
$$
\left(\rho e^{i\theta}\right)^5=32e^{i\pi}
$$
that gives:
$$
\rho e^{i\theta}=\left( 32e^{i\pi}\right)^{1/5}=2\left(e^{i\pi}\right)^{1/5}
$$
Now, since $e^{i\pi}=e^{i(\pi... |
1,987,230 | <p>On Socratica, I saw a video demonstrating writing groups by writing the Cayley's table satisfying three conditions of the desired order. (1) Neutral element row and column are copies of the row and column headers. (2) Every row and column has neutral element once (3) All the elements of the set are present in each ... | Siong Thye Goh | 306,553 | <p>if $a \equiv 0 \mod 5$, then $a^2 \equiv 0 \mod 5$.</p>
<p>if $a \equiv 1 \mod 5$, then $a^2 \equiv 1 \mod 5$.</p>
<p>if $a \equiv 2 \mod 5$, then $a^2 \equiv 4 \mod 5$.</p>
<p>if $a \equiv -2 \mod 5$, then $a^2 \equiv 4 \mod 5$.</p>
<p>if $a \equiv -1 \mod 5$, then $a^2 \equiv 1 \mod 5$.</p>
|
3,584,113 | <p>When we prove things like continuity in real analysis, why do we always aim for the result <span class="math-container">$<\epsilon$</span> when any positive multiple of <span class="math-container">$\epsilon$</span> proves the same result?</p>
| just_floating | 399,291 | <p>We could use an alternative definition, consider the following alternative definition of convergence when it sufficies to prove for any positive multiple K of epsilon:</p>
<p>We say <span class="math-container">$x_n \rightarrow l$</span> when <span class="math-container">$(\exists K > 0)(\forall \epsilon > 0)(... |
1,371,649 | <p>The question is:</p>
<blockquote>
<p>What does the following interation formula do?:
<span class="math-container">$$x_{k+1}=2x_k-cx_{k}^2.$$</span></p>
</blockquote>
<p>I tried to identify this with Newtons method. I.e. I tried to bring that into the form <span class="math-container">$x_{k+1}=x_k-\frac{f(x_0)}{f'(x_... | user137794 | 137,794 | <p>You were on the right track, but stopped a bit early. Instead of</p>
<p>$$f(x) = f'(x) \cdot (cx^2-x)$$</p>
<p>Write that as</p>
<p>$$f'(x) = f(x) \cdot \frac{1}{cx^2-x}$$</p>
<p>So a function that works would be</p>
<p>$$
\begin{align}
f(x) &= e^{\int 1/(cx^2-x)\ dx} \\
&= e^{\ln((1-cx)/x)} \\
&= \... |
2,369,717 | <p>From Jaynes' probability theory: the logic of science, I found this:</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/bnogp.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/bnogp.png" alt="enter image description here"></a></p>
</blockquote>
<p>$p$ here is the joint probability distributi... | Adayah | 149,178 | <p>I think it means that the probability density function $\rho(x, y)$ is given by $$\rho(x, y) = f(x) \cdot g(y).$$</p>
<p>This implies independence, because if $A \subseteq X$ and $B \subseteq Y$, then $$\begin{align*} \Pr(x \in A \wedge y \in B) & = \int \limits_{A \times B} \rho(x, y) \, \mbox{d} x \mbox{d} y ... |
321,916 | <p>In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also adds a layer of complexity). Why do we do it this way? </p>
<p>The first question is to what extent are the notions d... | Nik Weaver | 23,141 | <p>If <span class="math-container">$f: \mathbb{R} \to [0,\infty)$</span> is Borel (or Lebesgue) measurable, then for each rational <span class="math-container">$a > 0$</span> define <span class="math-container">$X_a = f^{-1}([a,\infty)) \times [0,a)$</span>. Then each <span class="math-container">$X_a$</span> is mea... |
829,390 | <p>In a tennis tournament, there are $10$ players. In the first round, $5$ groups(of 2 players) will be formed among them and elimination matches will be held among the two players in each group. In how many ways can pairings be done?</p>
<p>Answer is given as : $\frac{10!}{2^5\times5!}$</p>
<p>My solution :</p>
<p>... | Asimov | 137,446 | <p>The $5!$ comes from the order. You have picked $5$ groups, but you picked them in a certain order, eliminating $2$ at a time the $5!$ accounts for all possible orders of picking</p>
|
3,248,863 | <p>I want to calculate the operator norm of the operator <span class="math-container">$A: L^2[0,1] \to L^2[0,1]$</span> which is defined by <span class="math-container">$$(Af)(x):=i\int\limits_0^x f(t)\,dt-\frac{i}{2} \int\limits_0^1 f(t)\, dt$$</span></p>
<p>I've already shown that this operator is compact and selfad... | Oliver Díaz | 121,671 | <p>Let <span class="math-container">$k(t,x)=i\big(\mathbb{1}(0<t\leq x)-\frac12\big)$</span> and define the operator <span class="math-container">$A_k:f\mapsto\int^1_0k(t,x) f(t)\,dt$</span> in <span class="math-container">$L_2([0,1])$</span>. As pointed out in the statement of the problem, <span class="math-contain... |
230,416 | <p>I'm looking for numerical graph invariants that are bounded by a constant either for a graph $G$ or its complement $\bar{G}$. (The complement graph $\bar{G}$ has the same set of vertices as $G$ but the edges are complemented.) More specifically I’m looking for what numerical “Invariant $X$” is out there for which ... | Tony Huynh | 2,233 | <p>Let $c(G)$ be the number of connected components of $G$. Then for all graphs $G$,
$$
c(G)=1 \text{ or } c(\overline{G})=1.
$$ </p>
<p>Here is a slightly more interesting family of examples. For a fixed integer $k$, define $d_k(G)$ to be the number of degree-$k$ vertices of $G$. </p>
<p><strong>Claim.</strong>
... |
230,416 | <p>I'm looking for numerical graph invariants that are bounded by a constant either for a graph $G$ or its complement $\bar{G}$. (The complement graph $\bar{G}$ has the same set of vertices as $G$ but the edges are complemented.) More specifically I’m looking for what numerical “Invariant $X$” is out there for which ... | Shahrooz | 19,885 | <p>There are a lot of properties which $G$ and $\overline{G}$ both have them: number of vertices (edges), automorphism group, and etc. So, naturally this question is interesting for me. I want to mention a spectral property. Let $G$ be a graph with $n$ vertices, and $\rho(G)$ denotes the largest eigenvalue of the adjac... |
2,910,301 | <p>Most of the textbooks state that provided a nonzero field $F$, a nonzero polynomial $f\in F[x]$ of degree $n$ has at most $n$ <em>distinct</em> roots. I am wondering whether the word "distinct" can be removed? I guess the answer is yes, but I cannot come up with a nice proof. </p>
<p>Sorry for the confusion. The qu... | Community | -1 | <p>Suppose that $\alpha$ were not a root of g.</p>
<p>Then $\alpha$ is a simple root of f, which is a contradiction.</p>
|
229,703 | <p>Given the function
<span class="math-container">\begin{align*}
f \colon \mathbb{R}^n &\to \mathbb{R}^n\\
v&\mapsto \dfrac{v}{\|v\|},
\end{align*}</span>
I would like to compute the derivative of <span class="math-container">$f$</span>, that is <span class="math-container">$df(v)$</span>. It is possible t... | Michael Seifert | 27,813 | <p>You can abuse the variational derivative functionality in <code>xTensor</code> to do this:</p>
<pre><code><< xAct`xTensor`
DefManifold[M, dim, IndexRange[a, m]];
DefMetric[1, metric[-a, -b], PD, PrintAs -> "\[Delta]",
FlatMetric -> True, SymbolOfCovD -> {",", "\[PartialD]&q... |
1,251,537 | <p>$f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$ with $g(a)=g(b)=0$. Must $f$ vanish identically?</p>
<hr>
<p>Using integration by parts I got the form:
$\int_a^bg(x)f(x)-g'(x)F(x)=0$. Where $F'(x)=f(x)$.</p>
| egreg | 62,967 | <p>Suppose $f(x_0)>0$ for some $x_0\in(a,b)$. Then there exists $\delta>0$ such that, for $|x-x_0|<\delta$, $f(x)>k>0$ (take $k=f(x_0)/2$, for instance), with $(x_0-\delta,x_0+\delta)\subseteq(a,b)$.</p>
<p>Now build a function $g$ by decreeing that
$$
g(x)=\begin{cases}
0 & \text{if $a\le x< x_0... |
4,316,876 | <p>I want to prove that given <span class="math-container">$a,b,c\in\mathbb{R}$</span> we have <span class="math-container">$|a+b|\leq|a|+|b|$</span> using an absurd and reaching a contradiction.</p>
<p>So, I state, by absurd, that <span class="math-container">$|a+b|>|a|+|b|$</span>, but I can't reach the contradict... | Paul Sinclair | 258,282 | <p>It depends on what you already know about absolute values and inequalities.</p>
<p>For instance, have you already shown that if <span class="math-container">$a > b > 0$</span> and <span class="math-container">$c > 0$</span>, then <span class="math-container">$ac > bc$</span>? From that we get <span class... |
976,910 | <p>i'm having a small issue with a certain question. </p>
<p>Given a parametric equation of a plane $x=5-2a-3b$, $y=3-4a+2b$, $z=7-6a-2b$, find a point $P$ on the plane so that the position vector of $P$ is perpendicular to the plane.</p>
<p>How would you go about this for a parametric equation? I think I could conve... | WikiSandhu | 184,910 | <p>You can show that every point of $X-K$ is an interior point. Since for $x \in X-K$, there exist an open set $O \ni x$, such that $O \cap K= \phi$, otherwise $x$ would be the limit point for $K$. and thus for an arbitrary $x \in X-K$, we hav $x \in O \subset X-K$. Hence $X-K$ is open. </p>
|
148,420 | <p>A simple concept but I've not been able to solve it. I'm trying to create a stack of 2D plots in 3D space using Mathematica 9. <strong>This is not a parametric plot</strong>, but I'm creating it from an array of vectors (imported
.csv file). The ListPlot3D function creates a filled mesh but what I want is this typ... | Quantum_Oli | 6,588 | <p>Here's one way, using <code>Graphics3D</code> and starting from a list of $x,y,z$ values for datapoints.</p>
<p>Mock data:</p>
<pre><code>data = Flatten[Table[
{x, y, PDF[MultinormalDistribution[{0, 0}, {{0.2, 0.1}, {0.1, 0.4}}],{x, y}]},
{x, -2, 2, 0.01}, {y, -2, 2, 0.25}],1];
</code></pre>
<p>Group by y-va... |
4,004,978 | <blockquote>
<p>For all <span class="math-container">$a, b, c, d > 0$</span>, prove that
<span class="math-container">$$2\sqrt{a+b+c+d} ≥ \sqrt{a} + \sqrt{b} + \sqrt{c} + \sqrt{d}$$</span></p>
</blockquote>
<p>The idea would be to use AM-GM, but <span class="math-container">$\sqrt{a} + \sqrt{b} + \sqrt{c} + \sqrt{d}... | fleablood | 280,126 | <p>If you square both sides:</p>
<p><span class="math-container">$(\sqrt a + \sqrt b + \sqrt c +\sqrt d)^2 =a + b+c+d + 2(\sqrt{ab} + \sqrt{ac}+\sqrt{ad} + \sqrt{bc}+\sqrt{bd} + \sqrt{cd})$</span></p>
<p>while <span class="math-container">$(2\sqrt {a+b+c+d})^2= 4(a+b+c+d)$</span> so it suffices to prove</p>
<p><span cl... |
1,929,445 | <blockquote>
<p>Is there a solution to the problem $$\left\{\begin{matrix}
y'=y+y^4\\
y(x_0)=y_0
\end{matrix}\right.$$
which is defined on $\mathbb{R}$? ($x_0,y_0$ might be any real numbers)</p>
</blockquote>
<p>It's easy to prove that for all $(x,y)\in\mathbb{R}^2$ there exists an open interval $I$ (with $x_0\in... | Will Jagy | 10,400 | <p>actual solutions, three regions: $y > 0,$ then $0 > y > -1,$ then $y < -1.$ I should emphasize that your ODE is autonomous, no explicit dependence on $x,$ which means that every solution curve is a sideways translate of one of those in the picture. It is sort of accidental that a solution for $y > 0$ ... |
858,952 | <p>related to <a href="https://math.stackexchange.com/questions/830599/one-sided-limit-lim-x-rightarrow-0-fx-where-wolfram-alpha-does-not-hel">this question</a>:</p>
<p>Is there an easy closed-form term for</p>
<p>$$\sum_{j=k}^{\infty} \frac{x^j}{j!}e^{-x},$$</p>
<p>thus when the sum starts at a constant $k$ instead... | JJacquelin | 108,514 | <p>Incomplete Gamma function and Eq.2 in : <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/IncompleteGammaFunction.html</a></p>
<p><img src="https://i.stack.imgur.com/d6SsD.jpg" alt="enter image description here"></p>
|
3,063,053 | <p>I'm a Calculus I student and my teacher has given me a set of problems to solve with L'Hoptial's rule. Most of them have been pretty easy, but this one has me stumped. <br /></p>
<p><span class="math-container">$$\lim\limits_{x\to \infty} \frac{x}{\sqrt{x^2 + 1}}$$</span> </p>
<p>You'll notice that using L'Hopital... | cqfd | 588,038 | <p><strong>Hint</strong>: Divide the numerator and denominator by <span class="math-container">$x $</span> and apply the limit.</p>
<p><span class="math-container">$$\frac{x}{\sqrt{x^2 + 1}}=\frac{1}{\sqrt{1 + \frac{1}{x^2}}}$$</span></p>
|
3,063,053 | <p>I'm a Calculus I student and my teacher has given me a set of problems to solve with L'Hoptial's rule. Most of them have been pretty easy, but this one has me stumped. <br /></p>
<p><span class="math-container">$$\lim\limits_{x\to \infty} \frac{x}{\sqrt{x^2 + 1}}$$</span> </p>
<p>You'll notice that using L'Hopital... | Mostafa Ayaz | 518,023 | <p><strong>Hint</strong></p>
<p>Simply use <span class="math-container">$${x\over x+1}={x\over \sqrt{x^2+2x+1}}<{x\over \sqrt{x^2+1}}<1$$</span>for large enough <span class="math-container">$x>0$</span>.</p>
|
621,409 | <p>I need some help with the following question:</p>
<p>We have $H$ acting by automorphisms on $N$, and let $\rho:H\to Aut(N)$ the associated representation by automorphisms.</p>
<p>Suppose that $G=H[N]_{\rho}$ is a semidirect product, and $K=\ker(\rho)$.</p>
<p>Prove that $K\unlhd G$ and that $G/K$ is also a semid... | Ulrik | 53,012 | <p>Do you know how to find multivariable limits using polar coordinates? Substitute $x^2 + y^2 = r^2$, where $r$ is the distance from the origin to the point $(x,y)$. Then $(x,y)$ approaching the origin is equivalent to $r$ approaching 0. In this way you can reduce the problem to a single variable limit problem, and us... |
1,043,094 | <p>I have to find the limit of following</p>
<p><span class="math-container">$$\lim_{x \to 0}\left(\frac{1}{x} - \frac{1}{x^2}\right)$$</span></p>
<p>I have no idea how to start this one off.
How would I do it?</p>
<p>Do I just substitute the <span class="math-container">$0$</span>? It doesn't look that easy and sim... | Paul | 17,980 | <p>$$\lim_{x\to 0}\frac{x-1}{x^2} = \lim_{x\to 0}\frac{-1}{x^2} = -\infty$$</p>
|
1,676,505 | <p>Let $f:[0,1]\times[0,1]\to \mathbb R$,
$$f(x,y)=
\begin{cases}
\frac1q+\frac1n, & \text{if $(x,y)=(\frac mn,\frac pq) \in \Bbb Q\times\Bbb Q,$ $ (m,n)=1=(p,q)$ } \\
0, & \text{if $x$ or $y$ irrational$ $ or $0,1$}
\end{cases}
$$</p>
<p>Prove that f is integrable over $R=[0,1]\times[0,1]$ and find the va... | Christian Blatter | 1,303 | <p>Let $u\mapsto T(u)$ $\>(0\leq u\leq1)$ be <a href="https://en.wikipedia.org/wiki/Thomae%27s_function" rel="nofollow">Thomae's function</a>. Then
$$0\leq f(x,y)\leq T(x)+T(y)\qquad\bigl((x,y)\in Q:=[0,1]^2\bigr)\ .$$
By "Fubini's theorem" for Riemann integrals one obtains
$$\int_Q T(y)\>{\rm d}(x,y)=\int_0^1 \i... |
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