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 |
|---|---|---|---|---|
2,461,820 | <p>I am a bit confused with the following question, I get that P(T|D) = 0.95 and P(D) = 0.0001 however because i'm unable to work out P(T|~D) i'm struggling to apply the theorem, am i missing something? Also i'm unsure about what to do with the information relating to testing negative when you don't have the disease co... | Alekos Robotis | 252,284 | <p>$\ln x:= n$ such that $e^n=x$. So, $\ln e=n$ such that $e^n=e.$ So, $\ln e=1$. Now, $\ln 1=n$ such that $e^n=1$, which implies that $\ln 1=0$.</p>
<p>$$ \ln(\ln e)=\ln(1)=0.$$</p>
|
54,541 | <p>Apparently, Mathematica has no real sprintf-equivalent (unlike any other high-level language known to man). <a href="https://mathematica.stackexchange.com/questions/970/sprintf-or-close-equivalent-or-re-implementation">This has been asked before</a>, but I'm wondering if the new <code>StringTemplate</code> function ... | Taliesin Beynon | 7,140 | <p>I do want to support something like this with some custom syntax, though I can't guarantee when that will actually happen.</p>
<p>In the meantime, you can do this:</p>
<pre><code>StringTemplate["Pi is <*NumberForm[N[#],3]*>!"][Pi]
</code></pre>
<p>which produces:</p>
<pre><code>"Pi is 3.14!"
</code></pre>... |
3,413,253 | <p>Can anyone give me some examples and non examples of Lindelöf or second countable space and spaces that is Lindelöf but not second countable? And I understand the definition but find it is hard to visualize and imagine.
I have tried google it but it turns out I only found some silly examples like finite set or emp... | Mirko | 188,367 | <p><a href="https://en.wikipedia.org/wiki/Lower_limit_topology" rel="noreferrer">The Sorgenfrey line</a>, also called the lower-limit topology on the real line. It has a basis of intervals <span class="math-container">$[a,b)$</span> (or some authors prefer <span class="math-container">$(a,b]$</span>, upper-limit topolo... |
72,854 | <p>Hi everybody,</p>
<p>Does there exist an explicit formula for the Stirling Numbers of the First Kind which are given by the formula
$$
x(x-1)\cdots (x-n+1) = \sum_{k=0}^n s(n,k)x^k.
$$</p>
<p>Otherwise, what is the computationally fastest formula one knows?</p>
| Gottfried Helms | 7,710 | <p>In Pari/GP; one could simplify for either readability, speed or memory organisation for big matrices:</p>
<pre>
{ makemat_St1(dim=n) = local(f, M);
M=matid(dim);
f=1;
for(r=2,dim, \\ comp diagonal and first column
M[r,1]=f;f*=(r)
);
for(c=2,dim, \\ compute core entries
for(r=c... |
635,301 | <p>I need some help with the following problem: </p>
<blockquote>
<p>Let $f:\Bbb C \to \Bbb C$ be continuous satisfying that $f(\Bbb C)$ is an open set and that $|f(z)| \to \infty$ as $z\to \infty$. Prove that $f(\Bbb C)=\Bbb C$. </p>
</blockquote>
<p>My idea on this one is to prove by contradiction and assume that... | Robert Israel | 8,508 | <p>Hint: Suppose $w$ is in the boundary of $f(\mathbb C)$. Take a sequence $z_n$ with $f(z_n) \to w$. There is a subsequence ...</p>
|
2,793,077 | <blockquote>
<p>Suppose $R$ is a Boolean ring. Prove that $a+a=0$ for all $a\in R$.
Also prove that $R$ is commutative. Give an example (with explanation)
of a Boolean ring.</p>
</blockquote>
<p>From what I know, a Boolean ring is a ring for which $a^2=a$ for all $a\in R$.</p>
<p>Under addition a ring is a comm... | Kempa | 825,616 | <p><span class="math-container">$a+a=(a+a)^2=a^2+a^2+a^2+a^2=a+a+a+a\Rightarrow 0=2a$</span></p>
|
2,793,077 | <blockquote>
<p>Suppose $R$ is a Boolean ring. Prove that $a+a=0$ for all $a\in R$.
Also prove that $R$ is commutative. Give an example (with explanation)
of a Boolean ring.</p>
</blockquote>
<p>From what I know, a Boolean ring is a ring for which $a^2=a$ for all $a\in R$.</p>
<p>Under addition a ring is a comm... | Gustavo Andres Pava Parra | 714,992 | <p>Take <span class="math-container">$x \in R$</span>, with <span class="math-container">$R$</span> boolean ring, in particular is a ring, therefore <span class="math-container">$(x+x) \in R$</span> and <span class="math-container">$(x+x)^{2}=(x+x)$</span>. Thus
<span class="math-container">\begin{align*}
(x+x)... |
387,096 | <blockquote>
<p>Suppose $B$ is an $m \times n$ matrix. Prove that $BB^T$ is positive semidefinite.</p>
</blockquote>
<p>Can someone give a fairly good proof?</p>
<p>Inputs are greatly appreciated. The question is listed above.</p>
| Berci | 41,488 | <p><strong>Hint:</strong> For column vectors $a,b$, their inner (or scalar or dot) product is
$$\langle a,b\rangle = a^Tb$$
using matrix product, and $a^Ta=\langle a,a\rangle=\|a\|^2$.</p>
|
387,096 | <blockquote>
<p>Suppose $B$ is an $m \times n$ matrix. Prove that $BB^T$ is positive semidefinite.</p>
</blockquote>
<p>Can someone give a fairly good proof?</p>
<p>Inputs are greatly appreciated. The question is listed above.</p>
| robjohn | 13,854 | <p>$BB^T$ is positive semidefinite if for any vector $x$
$$
xBB^Tx^T\ge0
$$
Let $u=xB$. Then,
$$
xBB^Tx^T=uu^T=|u|^2\ge0
$$</p>
|
1,878,806 | <p>I am a graduate school freshman.</p>
<p>I did not take a probability lecture.</p>
<p>So I don't have anything about Probability.</p>
<p>Could you suggest Probability book No matter What book level?</p>
| Community | -1 | <p>I think the book <em>Probability Theory</em> by Heinz Bauer is a very good text on probability theory. It contains an extensive discussion of all the basic parts of the theory and is very readable. The book requires, however, a modest background in measure theory. </p>
<p>The original version of the book from 1973,... |
1,878,806 | <p>I am a graduate school freshman.</p>
<p>I did not take a probability lecture.</p>
<p>So I don't have anything about Probability.</p>
<p>Could you suggest Probability book No matter What book level?</p>
| Jean Marie | 305,862 | <p>@EHH I completely agree for Grimmett ! A very valuable book also by Grimmett and a coauthor (Stirzacker): (<a href="http://www.oupcanada.com/catalog/9780198572220.html" rel="nofollow">http://www.oupcanada.com/catalog/9780198572220.html</a>). It has a somewhat larger scope (including stochastic processes). A very goo... |
848,415 | <p>If the limit of one sequence $\{a_n\}$ is zero and the limit of another sequence $\{b_n\}$ is also zero does that mean that $\displaystyle\lim_{n\to\infty}(a_n/b_n) = 1$?</p>
| M. Vinay | 152,030 | <p>No. Let $a_n = \dfrac 1 n$ and $b_n = \dfrac 1 {n^2}$.</p>
|
3,537,226 | <p>A deck contains six cards, one pair labelled '1', another pair labelled '2' and the last labelled '3'. The deck is shuffled and you a pair of cards at a time until there are no cards left. A pair of cards <span class="math-container">$(i,j)$</span> is called acceptable if <span class="math-container">$|i-j|\leq1$</s... | Christopher Wells | 720,910 | <p>I'm afraid your answer to the first half of the question is incorrect.</p>
<p>For one thing, there are 720 ways to shuffle a six-card deck, so whatever the probability is it must be a fraction with 720 on the denominator. Or at least a factor of 720.</p>
<p>You're correct that <span class="math-container">$2 \time... |
406,437 | <p>Calculus the extreme value of the $f(x,y)=x^{2}+y^{2}+xy+\dfrac{1}{x}+\dfrac{1}{y}$</p>
<p>pleasee help me.</p>
| Vishal Gupta | 60,810 | <p><strong>EDIT:</strong> As pointed out in the comments, the following example is not associative.</p>
<p>What about the ring of matrices? It is non-commutative and perhaps as easy to understand as the real numbers. You may also consider the subset of invertible matrices.</p>
<p>You can consider the abelian group of... |
2,377,598 | <p>I am a Ph.D student in computer science, and I work on graph isomorphism. My research work requires some level of mathematics (mostly group theory ). I have done basic level abstract algebra course. I try to write down the theorems on peace of paper and try to understand them; I usually repeat this process four five... | Ethan Bolker | 72,858 | <p>I think the best way to understand abstractions intuitively is to study lots of examples. To understand group actions, write some down. Then think about whether each is transitive, whether it's primitive, look at the stabilizers of elements. Verify that the theorem is true; try to understand why in each particular c... |
2,377,598 | <p>I am a Ph.D student in computer science, and I work on graph isomorphism. My research work requires some level of mathematics (mostly group theory ). I have done basic level abstract algebra course. I try to write down the theorems on peace of paper and try to understand them; I usually repeat this process four five... | Santana Afton | 274,352 | <p>One way to build intuition is to reframe results in terms of objects and relationships that you know of and are familiar with. Since it seems as though you're looking at group action in the context of graph theory, this might be a good route to go.</p>
<p>For example, consider the first theorem you state about the ... |
217,244 | <p>I would like to know some of the most important definitions and theorems of definite and semidefinite matrices and their importance in linear algebra.
Thanks for your help</p>
| EuYu | 9,246 | <p>There are many uses for definite and semi-definite matrices. I can give just a few examples although undoubtedly I will be missing many.</p>
<ol>
<li><p>Positive-definite matrices are the matrix analogues to positive <em>numbers</em>. It is generally not possible to define a consistent notion of "positive" for matr... |
2,536,185 | <p><strong>Lemma 3.21</strong>. </p>
<p>Let $S$ be a subset of $R$. Then $\bar{S}$ is a closed set. $\bar{S}$ denotes the closure of $S$.</p>
<p>The following is a proof of Lemma 3.21:</p>
<p><em>Proof</em>. </p>
<blockquote>
<p>By Corollary 3.16 it is enough to show that $\bar{S^c}$
is open.</p>
</blockquote>
... | rtybase | 22,583 | <p>From the set theory perspective, let's note
$$P \overset{\text{def}}{=}\{p \in \mathbb{N} \mid p \text{ - prime}\}$$
$$P_{\leq n} \overset{\text{def}}{=} \{p \in P \mid p \leq n \}$$
$$M_n \overset{\text{def}}{=} \{p \in P \mid p \mid n\}$$
$\color{red}{M_n \ne \varnothing, \forall n\geq2}$. Then, it's easy to show... |
4,085,635 | <p>What is the volume of an n-dimension cube? Consider the length of each side to be <span class="math-container">$a$</span>. How to solve this problem?</p>
| User12345 | 133,683 | <p>This can be rigorously proved by induction (which may or may not be intuitive to you).</p>
<p>I assume the result is clear for <span class="math-container">$n=3$</span>, i.e. the triangle (let me know if it's not). For <span class="math-container">$n \geq 4$</span>, let <span class="math-container">$l_1 \leq \cdots... |
1,582,275 | <p>Suppose that $B = S^{-1}AS$ for some $n \times n$ matrices $A$, $B$, and $S$.</p>
<ol>
<li>Show that if $x \in \ker(B)$ then $Sx \in \ker(A)$.</li>
</ol>
<p>Proof: $B = S^{-1}AS$ implies that $SB = AS$ which implies that $SBx = ASx = 0$, that is $Sx \in \ker(A)$.</p>
<ol start="2">
<li>Show that the linear transf... | egreg | 62,967 | <p>The proof of 1 is good. Now, the map $T$ is well defined and obviously linear as it's the multiplication by a matrix.</p>
<p>If $x\in\ker T$, then $Sx=0$, so $x=0$ because $S$ is invertible. Therefore $T$ is injective. Since $A$ and $B$ have the same rank, their null spaces (kernels, in your terminology) have the s... |
638,164 | <p>Let $\textbf{F}\left ( x, y \right )=\left ( -\frac y{x^2+y^2},\frac x{x^2+y^2} \right )$ be a vector field in $\mathbb{R}^2-\left \{ \textbf{0} \right \}$.</p>
<p>I know that the potential function of $\textbf{F}$ on $x>0$ is $\arctan \left ( \frac yx \right )$.</p>
<p>But I want to know the potential function... | ShreevatsaR | 205 | <p>Here's a proof along the lines of your agument.</p>
<p>Consider your special number (in your notation)
$$S = [1, 2, 3, \dots, (n-3), (n-2), n]_{n+1}.$$
This number has $n$ in its units $(= (n+1)^0)$ place, $n-2$ in its $n+1$ $(= (n+1)^1)$ place, $n-3$ in its $(n+1)^2$ place, and so on, until $1$ in its $(n+1)^{n-... |
2,468,155 | <p>This problem is from Challenge and Thrill of Pre-College Mathematics:
Prove that $$ (a^3+b^3)^2\le (a^2+b^2)(a^4+b^4)$$</p>
<p>It would be really great if somebody could come up with a solution to this problem.</p>
| DeepSea | 101,504 | <p>To be honest, I do not look at the answers before mine and tried it myself with the most innovative thought I am able to have: $(a^3+b^3)^2 = (a^2\cdot a + b^2\cdot b)^2 \le (a^4+b^4)(a^2+b^2)$ by Cauchy-Schwarz inequality. As you can see how "beautiful" the CS inequality is....</p>
|
16,290 | <p>Hi I am new here and have a calculus question that came up at work.</p>
<p>Suppose you have a $4' \times 8'$ piece of plywood. You need 3 circular pieces all equal diameter. What is the maximum size of circles you can cut from this piece of material?
I would have expected I could write a function for the area of th... | Christian Blatter | 1,303 | <p>Let $Q:=[-2,2]\times[-1,1]$ be the given rectangle and let $r_0$ be the radius computed by Isaac for this rectangle. I shall prove that 3 circles of radius $r>r_0$ cannot be placed into $Q$ without overlap. The midpoints of these circles would have to lie in the smaller rectangle $Q':=[-2+r, 2-r]\times[-1+r,1... |
2,725,831 | <blockquote>
<p>If we define $$f(x)=\left\lfloor \frac {x^{2x^4}}{x^{x^2}+3}\right\rfloor$$ and we have to find unit digit of $f(10)$</p>
</blockquote>
<p>I had tried approximation, factorization and substitutions like $x^2=u$ but it proved of no use. Moreover the sequential powers are feeling the hell out of me. C... | CY Aries | 268,334 | <p>When $x=10$, $\displaystyle \frac {x^{2x^4}}{x^{x^2}+3}=\frac {10^{20000}}{10^{100}+3}$.</p>
<p>Let $y=10^{100}$. $\displaystyle \frac {10^{20000}}{10^{100}+3}=\frac{y^{200}}{y+3}$.</p>
<p>When $y^{200}$ (as a polynomial) is divided by $y+3$, the remainder is $(-3)^{200}=3^{200}$.</p>
<p>So, $\displaystyle \frac... |
4,646,650 | <p>I believe there will be values of <span class="math-container">$x$</span> for which the inequality <span class="math-container">$x^3 - 2x + 2 \ge 3 - x^2$</span> is true and values for which it is not true, because:</p>
<ul>
<li><em>LHS asymptotically increases but RHS decreases for increasingly positive values of <... | David G. Stork | 210,401 | <p>Solve the cubic to find <span class="math-container">$x = -1.8, -.445, 1.25$</span>.</p>
<p><a href="https://i.stack.imgur.com/fkVsf.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fkVsf.png" alt="enter image description here" /></a></p>
|
4,467,763 | <p>I have the equation</p>
<p><span class="math-container">$A\vec{x} = \vec{b} \tag{1}.$</span></p>
<p>where <span class="math-container">$A$</span> is an <span class="math-container">$m\times n$</span> matrix of rank <span class="math-container">$m$</span>, so that <span class="math-container">$m<n$</span> and the ... | user1551 | 1,551 | <p>Suppose <span class="math-container">$x=u$</span> is a solution to the equation <span class="math-container">$Ax=b$</span>. Let <span class="math-container">$u=u_0+u_1$</span> where <span class="math-container">$u_0\in\ker A$</span> and <span class="math-container">$u_1\in(\ker A)^\perp$</span>. If <span class="math... |
2,933,753 | <p>Given two finite groups <span class="math-container">$G, H$</span>, we are going to say that <span class="math-container">$G<_oH$</span> if either</p>
<p>a. <span class="math-container">$|G|<|H|$</span></p>
<p>or </p>
<p>b. <span class="math-container">$|G|=|H|$</span> and <span class="math-container">$\dis... | Hw Chu | 507,264 | <p>Feed the following code to Magma(<a href="http://magma.maths.usyd.edu.au/calc/" rel="nofollow noreferrer">http://magma.maths.usyd.edu.au/calc/</a>):</p>
<pre><code>for grouporder in [1..24] do
printf "Checking sum of order for groups or order %o:", grouporder;
numgroup := NumberOfSmallGroups(grouporder);
sse... |
114,487 | <p>I have a stack of images (usually ca 100) of the same sample. The images have intrinsic variation of the sample, which is my signal, and a lot of statistical noise. I did a principal components analysis (PCA) on the whole stack and found that components 2-5 are just random noise, whereas the rest is fine. How can I ... | Anton Antonov | 34,008 | <h2>Start data</h2>
<p>First let us get some images. I am going to use the MNIST dataset for clarity. (And because <a href="https://mathematicaforprediction.wordpress.com/2013/08/26/classification-of-handwritten-digits/" rel="nofollow noreferrer">I experimented with similar data some time ago</a>.)</p>
<pre><code>MNI... |
575,232 | <p>Consider all strings whose letters belong to the set:</p>
<p>$A = \{ a, b, c, d, e\}$</p>
<p>How many strings of length $6$ are there that contain exactly one $a$?</p>
<p>Attempt:</p>
<p>Since we are only using $\frac{4}{5}$ letters for the rest of the string, </p>
<p>There are $1* 4^5$ strings that contain exa... | Will Orrick | 3,736 | <p>This answer is a small attempt to address your third question. You may find the article,</p>
<p>Jethro van Ekeren, <a href="http://webpages.math.luc.edu/%7Eptingley/oldseminars/QuantumGroupsSpring2011/lecture6b.pdf" rel="nofollow noreferrer"><em>The six-vertex model, <span class="math-container">$R$</span>-matrices... |
3,336,742 | <p>Can we evaluate <span class="math-container">$\displaystyle\sum_{n=1}^\infty\frac{H_n^2}{n^32^n}$</span> ?</p>
<p>where <span class="math-container">$H_n=\sum_{k=1}^n\frac1n$</span> is the harmonic number.</p>
<p>A related integral is <span class="math-container">$\displaystyle\int_0^1\frac{\ln^2(1-x)\operatorna... | Myunghyun Song | 609,441 | <p>Using the generating function of <span class="math-container">$\displaystyle\{H_k^2\}_{k=1}^\infty$</span>:
<span class="math-container">$$
\frac{\text{Li}_2(x) +\ln^2(1-x)}{1-x} = \sum_{k=1}^\infty H_k^2 x^k
$$</span> we can observe that
<span class="math-container">\begin{align*}
S =& \sum_{k=1}^\infty {H_k^... |
422,084 | <p>$\mathbb{R}^2$ with different topologies on it are homeomorphic as a topological space?
for example with discrete topology and usual topology, what I need is a continous bijection with inverse is continous, from usual to discrete any continous map is finally constant map,so I think they are not homeomorphic.
thank... | Dylan Yott | 62,865 | <p>Suppose there exists a homeomorphism $f$ between $\mathbb R^{2}$ with the usual topology, and $\mathbb R^{2}$ with the discrete topology. Then $f(S^{1})$ is compact, as continuous functions preserve compactness, and it is also infinite, as $S^{1}$ has infinitely many points. However in a discrete space, only finite ... |
791,020 | <p>From my textbook. </p>
<p>$$\sum\limits_{k=0}^\infty (-\frac{1}{5})^k$$</p>
<p>My work:</p>
<p>So a constant greater than or equal to $1$ raised to ∞ is ∞.</p>
<p>A number $n$ for $0<n<1$ is $0$. So when taking the limit of this series you get 0 but when formatting the problem a different way $(-1)^k/(5^k)... | user84413 | 84,413 | <p>One way to do this is to rewrite the expression as $(1+(x-3))^{1/2}$, and then substitute $u=x-3$ and expand $(1+u)^{1/2}$ using the Binomial Series
$\displaystyle(1+u)^r=\sum_{n=0}^{\infty}\binom{r}{n}u^n$.</p>
|
1,332,420 | <p>Solution of this task is ${7\pi\over2}$ but I don't know how to get to this solution.I used the formula for double angle for $\sin{2x}$ and $\cos{2x}$ and moved everything on one side of the equation and made $2\cos{x}$ as common factor to get this :$$2\cos{x}(\sin{x} - {\sqrt{2}\over2} - \cos{x}) = 0$$
Then I can g... | Mythomorphic | 152,277 | <p>$$\sin{x} - {\sqrt{2}\over2} - \cos{x} = 0$$
$$\sin x-\cos x=\frac{\sqrt2} 2$$
$$(\sin x-\cos x)^2=\frac12$$
$$\sin^2x+\cos^2x-2\sin x\cos x=\frac12$$
$$1-\sin 2x=\frac12$$
$$\sin 2x=\frac12$$</p>
|
1,985,653 | <p>I want to show that {$\sqrt 2$} $\cup$ {$\sqrt 2$ + $1/n$ : $n$ $∈$ $\mathbb{N}$} is closed. I'm having trouble. I've been trying to show that the complement is open, but the presence of {$\sqrt 2$} is confusing me.</p>
| Evan Aad | 37,058 | <p>A set is closed iff it contains all its limit points. The only limit point of the given set is $\sqrt{2}$.</p>
|
147,378 | <p>I have the following equation:</p>
<p>$$\frac{dx}{dt}+x=4\sin(t)$$</p>
<p>For solving, I find the homogenous part as:
$$f(h)=C*e^{-t}$$</p>
<p>Then finding $f(a)$ and $df(a)$:
$$f(a)=4A\sin(t)+4B\cos(t)$$
$$df(a)=4A\cos(t)-4B\sin(t)$$</p>
<p>Substituting in orginal equation:</p>
<p>$$4A\cos(t)-4B\sin(t)+4A\sin(... | Robert Mastragostino | 28,869 | <p>multiples of $\cos(t)$ can't add to make $\sin(t)$, and vice versa. So this can be split into two equations:
$$4A\cos(t)+4B\cos(t)=0$$
$$4A\sin(t)-4B\sin(t)=4\sin(t)$$
simplifying we get
$$A+B=0$$
$$A-B=1$$
and so
$$A=\frac{1}{2}, B=-\frac{1}{2}$$</p>
<p>So your final solution is $$x=f(h)+f(a)=Ce^{-t}+2\sin(t)-2\c... |
147,378 | <p>I have the following equation:</p>
<p>$$\frac{dx}{dt}+x=4\sin(t)$$</p>
<p>For solving, I find the homogenous part as:
$$f(h)=C*e^{-t}$$</p>
<p>Then finding $f(a)$ and $df(a)$:
$$f(a)=4A\sin(t)+4B\cos(t)$$
$$df(a)=4A\cos(t)-4B\sin(t)$$</p>
<p>Substituting in orginal equation:</p>
<p>$$4A\cos(t)-4B\sin(t)+4A\sin(... | Milo Wielondek | 5,877 | <p>Another approach would be to consider the given equation as one of the form</p>
<p>$$
\frac{dx}{dt}+Px=Q
$$</p>
<p>where $P$ and $Q$ are functions of $t$. Such equation can be solved by multiplying both sides by the integrating factor $e^{\int P\,dt}$.</p>
<p>Applying this to your equation, we get
$$
\begin{align... |
4,274,085 | <p>A linear equation in one variable <span class="math-container">$x$</span>, <span class="math-container">$$ax+b=k$$</span> has only one non-negative integer solution. For example, <span class="math-container">$2x+3=85$</span> has a solution 41.</p>
<p>How to find the number of non-negative integer solutions of a line... | Kuifje | 273,220 | <p>Or, you could solve a linear program with a solver (as the tag <code>linear-programming</code> is in your question), or more precisely an integer program (MIP), and add a no good cut to exclude the current solution, until there are no more left.</p>
<p>The variables are <span class="math-container">$(x,y)\in \mathbb... |
335,483 | <p>Let $N$ be a set of non-negative integers. Of course we know that $a+b=0$ implies that $a=b=0$ for $a, b \in N$.</p>
<p>How do (or can) we prove this fact if we don't know the subtraction or order?</p>
<p>In other words, we can only use the addition and multiplication.</p>
<p>Please give me advise.</p>
<p>EDIT</... | Thomas Andrews | 7,933 | <p>There are three axioms that you need to prove this:</p>
<ol>
<li>For all non-negative integers $n$, $n+1\neq 0$</li>
<li>For all non-negative integers $a,b$, $a+(b+1)= (a+b)+1$</li>
<li>Induction</li>
</ol>
<p>Theorem: For all $b$, either $b=0$ or $a+b\neq 0$.</p>
<p>Proof by induction: If $b=0$ we are done.</p>
... |
335,483 | <p>Let $N$ be a set of non-negative integers. Of course we know that $a+b=0$ implies that $a=b=0$ for $a, b \in N$.</p>
<p>How do (or can) we prove this fact if we don't know the subtraction or order?</p>
<p>In other words, we can only use the addition and multiplication.</p>
<p>Please give me advise.</p>
<p>EDIT</... | Sean Eberhard | 23,805 | <p>Use the Mazur swindle! Namely, if $a+b=0$ then</p>
<p>\begin{align*}
0 &= 0 + 0 + 0 + \cdots\\
&= (a+b)+(a+b)+(a+b)+\cdots\\
&= a+(b+a)+(b+a)+\cdots \\
&= a + 0 + 0 + \cdots\\
&= a.
\end{align*}</p>
<p>Regrouping the infinite sum is justified because everything is nonnegative. I leave i... |
32,997 | <p>I have made a rather obvious yet peculiar observation while calculating with quadratic inequalities. Take a simple quadratic inequality like the one below</p>
<p>$\frac{x^2+1}{x}>1$</p>
<p>by multiplying both sides by $x$, then subtracting $x$ from both sides we get</p>
<p>$x^2-x+1>0$</p>
<p>Hence, both th... | abel | 9,252 | <p>the method i describe below helps finding a complete solution by sticking to the simple principle that for polynomials and rational functions only candidates for the sign change is across the zeros of the numerator and denominator.</p>
<p>instead of looking at $\frac{x^2 + 1}{x} > 1$ let us look at $\frac{x^2 + ... |
1,811,581 | <p>When discussing the order relation on $\mathbb{C}$, it is said that such a statement as $z_1 < z_2$ where $z_1, z_2 \in \mathbb{C}$ is meaningless, unless $z_1$ and $z_2$ are real.</p>
<p>My question is, when will a complex number $z$ be real? I know that if $\bar{z}$ is the conjugate of $z$, then</p>
<p>$$z + ... | AndreasS | 32,580 | <p>Yes, the (linear) span is a vector space. By definition it is the smallest vector space that contains all the elements in the set. In particular it will contain all linear combinations of those elements (and will in fact contain exactly all linear combinations that can be formed with those elements).</p>
<p>So in y... |
367,885 | <p>Denote the set of all bounded sequences in $\mathbb{R}$ by $l^{\infty}$, endowed with the sup norm $\lVert \rVert _{\infty}$. Define a map $T:l^{\infty} \to l^{\infty}$ as follows:</p>
<p>$$(x_n)_n \mapsto \Big(\frac{x_n}{n}\Big)_n.$$</p>
<p>Then the question is:</p>
<blockquote>
<p>Show that the image of $l^{\... | vadim123 | 73,324 | <p>Consider $t_0=(\frac {1}{\sqrt{n}})_n$. This too is not in the image of $T$, since its preimage is $(\sqrt{n})_n$. Now define $t_m=(\frac{1}{\sqrt{1}},\frac{1}{\sqrt{2}},\ldots,\frac{1}{\sqrt{m}},\frac{\sqrt{m}}{m+1},\frac{\sqrt{m}}{m+2},\ldots)$, the image of $(\sqrt{1},\sqrt{2},\ldots,\sqrt{m},\sqrt{m},\sqrt{m},... |
367,885 | <p>Denote the set of all bounded sequences in $\mathbb{R}$ by $l^{\infty}$, endowed with the sup norm $\lVert \rVert _{\infty}$. Define a map $T:l^{\infty} \to l^{\infty}$ as follows:</p>
<p>$$(x_n)_n \mapsto \Big(\frac{x_n}{n}\Big)_n.$$</p>
<p>Then the question is:</p>
<blockquote>
<p>Show that the image of $l^{\... | Diego | 41,463 | <p>Let $t_0=(1,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{3}},\ldots)$ and $t_m=(1,\frac{1}{\sqrt{2}},\ldots,\frac{1}{\sqrt{m}},0,0,\ldots)$. Clearly $t_m\rightarrow t_0$, $t_m$ is in the image of $T$ and $t_0$ is not in the image of $T$.</p>
|
2,151,491 | <p>For which prime numbers $p$ does the decimal for $\frac{1}{p}$ have cycle length $p-1$? I started with some simple examples to find an idea to solve:</p>
<p>$\frac{1}{2}=0.5,\frac{1}{3}=\overline{3},\frac{1}{5}=0.2,\frac{1}{7}=0.\overline{142857},\frac{1}{11}=0.\overline{09},\frac{1}{13}=0.\overline{0769230}$</p>
... | Peter | 82,961 | <p>The primes upto $1000$ with period length $p-1$ are :</p>
<pre><code>? q=0;forprime(s=1,1000,if(gcd(s,10)==1,if(znorder(Mod(10,s))==s-1,print1(s," ")
;q=q+1;if(Mod(q,18)==0,print))))
7 17 19 23 29 47 59 61 97 109 113 131 149 167 179 181 193 223
229 233 257 263 269 313 337 367 379 383 389 419 433 461 487 491 499 503... |
1,502,309 | <p>The initial notation is:</p>
<p>$$\sum_{n=5}^\infty \frac{8}{n^2 -1}$$</p>
<p>I get to about here then I get confused.</p>
<p>$$\left(1-\frac{3}{2}\right)+\left(\frac{4}{5}-\frac{4}{7}\right)+...+\left(\frac{4}{n-3}-\frac{4}{n-1}\right)+...$$</p>
<p>How do you figure out how to get the $\frac{1}{n-3}-\frac{1}{n-... | Michael Medvinsky | 269,041 | <p>\begin{align}
\sum_{n=5}^\infty \frac{8}{n^2 -1}
&=\sum_{n=5}^\infty \frac{8}{(n -1)(n+1)}\\
&=\sum_{n=5}^\infty \left(\frac{4}{n-1}-\frac{4}{n+1} \right)\\
&=\left(\frac 44 - \color{red}{\frac 46} \right)
+ \left(\frac 45 - \color{blue}{\frac 47} \right)
+ \left(\color{red}{\frac 46} -... |
1,502,309 | <p>The initial notation is:</p>
<p>$$\sum_{n=5}^\infty \frac{8}{n^2 -1}$$</p>
<p>I get to about here then I get confused.</p>
<p>$$\left(1-\frac{3}{2}\right)+\left(\frac{4}{5}-\frac{4}{7}\right)+...+\left(\frac{4}{n-3}-\frac{4}{n-1}\right)+...$$</p>
<p>How do you figure out how to get the $\frac{1}{n-3}-\frac{1}{n-... | ncmathsadist | 4,154 | <p>$$S = \sum_{n=5}^\infty \frac{8}{n^2 -1}=
\lim_{N\to\infty} \sum_{n=5}^N \frac{8}{n^2 -1}
= \lim_{N\to\infty} \sum_{n=5}^N \left\{ {4\over n-1} - {4\over n+1}\right\} = \lim_{N\to\infty} \sum_{n=5}^N {4\over n-1} - \sum_{n=5}^N {4\over n+1}$$</p>
<p>Now reindex to get </p>
<p>$$S = \lim_{N\to\infty} \sum_{n=4}^{N... |
3,256,881 | <p>How to prove that?
I try to use the comparasion test, but i don't know with that function compare.</p>
| XT Chen | 618,783 | <p>As Azif said this function is bounded. It's continuous and consequently integrable on <span class="math-container">$[δ,1]$</span> and |<span class="math-container">$\int_o^{\delta} \sin(x+\frac{1}{x}) \mathrm{d}x| \le \delta$</span> which converges to <span class="math-container">$0$</span> as <span class="math-cont... |
3,501,052 | <p>I want to find the number of real roots of the polynomial <span class="math-container">$x^3+7x^2+6x+5$</span>.
Using Descartes rule, this polynomial has either only one real root or 3 real roots (all are negetive). How will we conclude one answer without doing some long process?</p>
| Claude Leibovici | 82,404 | <p>If you do not want to use the cubic discriminant, the easy solution has been proposed by Ross Millikan.</p>
<p>Consider that you lookf for the zero's of function
<span class="math-container">$$f(x)=x^3+7x^2+6x+5$$</span> for which
<span class="math-container">$$f'(x)=3x^2+14x+6 \qquad \text{and} \qquad f''(x)=6x+14... |
3,501,052 | <p>I want to find the number of real roots of the polynomial <span class="math-container">$x^3+7x^2+6x+5$</span>.
Using Descartes rule, this polynomial has either only one real root or 3 real roots (all are negetive). How will we conclude one answer without doing some long process?</p>
| 2'5 9'2 | 11,123 | <p>If you take <span class="math-container">$$f(x)=x^3+7x^2+6x+5$$</span> and subtract <span class="math-container">$5$</span>, you have <span class="math-container">$$g(x)=(x+6)(x+1)x$$</span> a cubic with roots at <span class="math-container">$-6$</span>, <span class="math-container">$-1$</span>, and <span class="mat... |
646,010 | <p>So i kinda think i have figured this out, i'm not very good at math, and need a formula to figure out some stats for a game i'm playing.</p>
<p>I have a Weapon with a reload speed of X sec.. however, i also have a modifier attached, that will make the weapon reload faster by +Y%</p>
<p>i made this formula, mostly ... | TheOscillator | 95,893 | <p>Consider the Gamma function
$$\Gamma(s)=\int_{0}^{\infty}{t^{s-1}\cdot e^{-t}dt} ,s\in \mathbb{R^{+}}$$ now consider the substitution $t=iu^{2}$ which yields the following $$\Gamma(s)=2e^{i\frac{\pi}{2}s}\int_{i0}^{i\infty}{u^{2s-1}\cdot e^{-i u^{2}}du}=_{(1)} 2e^{i\frac{\pi}{2}s}\int_{0}^{\infty}{u^{2s-1}\cdot e^{-... |
4,518,908 | <p>For sufficiently large integer <span class="math-container">$m$</span>, in order to prove</p>
<p><span class="math-container">$\frac{(m+1)}{m}<\log(m)$</span></p>
<p>is it sufficient to point out that</p>
<p><span class="math-container">$ \displaystyle\lim_{m \to \infty} \frac{(m+1)}{m}=1 $</span></p>
<p>while</p... | Taladris | 70,123 | <p>Yes, <span class="math-container">$\forall x {\in} \mathbb{R} \left(P(x) \implies \left(\forall y {\in}\mathbb{R} \;P(x + y) \right)\right)$</span> implies that <span class="math-container">$\left(\forall x {\in}\mathbb{R}\; P(x)\right) \text{ or } \left(\forall x {\in}\mathbb{R}\;\lnot P(x)\right)$</span>.</p>
<p>T... |
467,301 | <p>I'm reading Intro to Topology by Mendelson.</p>
<p>The problem statement is in the title.</p>
<p>My attempt at the proof is:</p>
<p>Since $X$ is a compact metric space, for each $n\in\mathbb{N}$, there exists $\{x_1^n,\dots,x_p^n\}$ such that $X\subset\bigcup\limits_{i=1}^p B(x_i^n;\frac{1}{n})$. Let $K=\frac{2p}... | Dan Rust | 29,059 | <p>Suppose no such $K$ exists. Let $x$ be in $X$. Then for all $n\geq 1$, there exists an $N$ such that $B_N(x)\setminus B_n(x)\neq\emptyset$. let $\Lambda=\{B_n(x)\mid n\in\mathbb{N}\}$. $\Lambda$ is an open cover of $X$ but no finite subset of $\Lambda$ will cover $X$ because a finite subset has as union its largest ... |
982,938 | <p>I was doing the integration
$$
\int\frac{1}{(u^2+a^2)^2}du
$$
and I had a look at the lecturer's steps where I got stuck in the following step:
$$
\int\frac{1}{(u^2+a^2)^2}du=\frac{1}{2a^2}\left(\frac{u}{u^2+a^2}+\int\frac{1}{u^2+a^2}du\right).
$$
I guess it is integrating this by parts, but I could't see the trick.... | Community | -1 | <p>The trick is using the substitution $u = a\tan x $, Then we have</p>
<p>$$ \int \frac{du}{(u^2 +a^2)^2 } = \int \frac{du}{(a^2 \tan^2 x + a^2)^2}= \int \frac{du}{(a \sec x)^2}$$</p>
<p>and $ du = a \sec^2 x dx $</p>
|
846,980 | <p>Assume we need to choose $k$ numbers out of $[1...n]$ so no two numbers are consecutive. I know the number of such combinations is given by$\binom {n-k+1}{k}$. Assume the numbers are given by $r_1<r_2...<r_k$ when the difference between $r_{i+1}$ and $r_i$ is always higher than $2$. Now, I want to restrict the... | Asinomás | 33,907 | <p>We will show the particular case when $p=k-1$. </p>
<p>Any selection of $k$ numbers from $\{1,2,\dots ,n-(t-1)(k-1)\}$ gives you a suitable combination when you add $t-1$ to the second smallest element, $2(t-1)$ to the third smallest element$\dots$ $(t-1)(k-1)$ to the largest alement. Using the inverse process (sub... |
846,980 | <p>Assume we need to choose $k$ numbers out of $[1...n]$ so no two numbers are consecutive. I know the number of such combinations is given by$\binom {n-k+1}{k}$. Assume the numbers are given by $r_1<r_2...<r_k$ when the difference between $r_{i+1}$ and $r_i$ is always higher than $2$. Now, I want to restrict the... | Geoffrey Critzer | 139,257 | <p>Table[Table[Binomial[n - 2 (k - 1), k], {k, 0, Floor[(n + 2)/3]}], {n,
0, 10}] // Grid</p>
<p>Table[Table[
Length[Select[Tuples[{0, 1}, n],
FreeQ[#, {<strong>_, 1, 0, 1, <em></strong>}] && FreeQ[#, {<strong></em>, 1, 1, _</strong>}] &&
Count[#, 1] == k &]], {k, 0, Floor[n + ... |
396,713 | <p>I'm attempting to derive a formula for the sum of all elements of an arithmetic series, given the first term, the limiting term (the number that no number in the sequence is higher than), and the difference between each term; however, I am unable to find one that works. Here is what I have so far:</p>
<p>Let $a_0$ ... | Sandkar | 232,349 | <p>This may be a bit late but I stumbled across this question myself and I thought about this not too long ago, so I might as well post what I managed to derive on here.</p>
<p>We know that a term, can be expressed thusly:</p>
<p>$$a_n = a_{n-1} + d$$
$$a_{n-1} = a_{n-2} + d$$
$$...$$
$$a_{n-k} = a_{n-k-1} + d$$</p>
... |
42,617 | <p>Let $f(x,y)$ define a surface $S$
in $\mathbb{R}^3$ with a unique local minimum at $b \in S$.
Suppose <a href="http://en.wikipedia.org/wiki/Gradient_descent" rel="noreferrer">gradient descent</a> from any start point $a \in S$
follows a geodesic on $S$ from $a$ to $b$.
(<b>Q1</b>.)
What is the class of functions/sur... | Joseph O'Rourke | 6,094 | <p>Here is a function $f(x,y)$ which is 0 inside the square $C=[\pm1,\pm1]$,
and outside that square
has value equal to the Euclidean distance $d( p, C )$ from $p=(x,y)$ to the boundary of $C$.
[I am trying to follow Pietro Majer's suggestion, as far as I understand it.]
It is not a surface of revolution
(but it is cen... |
280,156 | <p>I have a code as below:</p>
<pre><code>countpar = 10;
randomA = RandomReal[{1, 10}, {countpar, countpar}];
randomconst = RandomInteger[{0, 1}, {countpar, 1}];
For[i = 1, i < countpar + 1, i++,
If[randomconst[[i, 1]] != 0,
randomA[[All, i]] = 0.; randomA[[i, All]] = 0.;
randomA[[i, i]] = 1;
];
];
</code></pre... | E. Chan-López | 53,427 | <p>Following Syed's idea, another way to do this is to use <code>FoldList</code>:</p>
<pre><code>With[{x = 2, y = 1, n = 2}, FoldList[(#^(x - 1) + n)/(#^(x - 1) + #^(x - 2)) &, y, Range[6]]]
(*{1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169}*)
</code></pre>
<p>Also, you can use <code>FixedPointList</code>:</p>
<pre><co... |
2,563,966 | <p>I'm reading a proof in a linear algebra book. It mentions
$$p(x) -p(c)= (x - c) h(x),$$
where $c$ is a constant, and $p(x)$ and $h(x)$ are polynomials.</p>
<p>Can we always factor $p(x) - p(c)$ in this way? </p>
<p>Please give a proof.</p>
| Chirantan Chowdhury | 337,567 | <p>Let $ p(x) = \sum_{i=0}^n p_ix^i $ Now $$p(x) - p(a) = \sum_{i=o}^n p_i(x^i-a^i)$$ Use the formula $$x^i-a^i= (x-a)( \sum_{j=0}^{i-i}x^ja^{i-1-j} ) $$(This comes from a geometric series) Thus now you can see from each term a factor of $x-a$ comes out . Thus it is proved.</p>
|
2,555,463 | <p>Given a line $l$ and two points $p_1$ and $p_2$, identify the point $v$ which is equidistant from $l$, $p_1$, and $p_2$, assuming it exists.</p>
<p>My idea is to: (1) identify the parabolas containing all points equidistant from each point and the line, then (2) intersect these parabolas. As $v$ is equidistant from... | Nominal Animal | 318,422 | <p>I personally prefer to attack such problems first using basic vector algebra, mostly because they tend to be easier to implement for numerical computations.</p>
<p>In other words, this answer assumes the underlying problem to be solved, is to find and implement a stable numerical algorithm for solving specific prob... |
2,178,395 | <p>In how many ways can the letters of the word CHROMATIC be arranged,</p>
<p>find the probability that the string of letters begins with the letter M</p>
<p>I don't understand how to single out M so the possibilities would only begin with M?</p>
| Graham Kemp | 135,106 | <p>There are two methods.</p>
<p>As joeb states, you can compare the size of the favoured event to the sample space by <a href="https://math.stackexchange.com/questions/2176973/in-how-many-ways-can-the-letters-of-the-word-chromatic-be-arranged">counting arrangements of all the letters</a>.</p>
<p>Alternatively, only ... |
18,493 | <p>Is there a way to split an external file using Mathematica? (Other than importing it, dividing it internally and exporting the parts)</p>
<p>For instance like some File Split utility do.</p>
| Sjoerd C. de Vries | 57 | <p>From the <a href="http://reference.wolfram.com/mathematica/howto/CreateAComputableDocumentFormatFile.html" rel="nofollow">docs</a>: "Because CDF Player cannot load custom data at runtime, you must ensure that all necessary information is embedded within the interactive elements of your .cdf file. This can be done wi... |
18,493 | <p>Is there a way to split an external file using Mathematica? (Other than importing it, dividing it internally and exporting the parts)</p>
<p>For instance like some File Split utility do.</p>
| Murta | 2,266 | <p>If you use Enterprise Mathematica edition, or some kind of signed version, you can do what you want.
In this example I use the CDF with package to hide a database connection, so the package can be distributed without someone discover your connection user name and password.</p>
<p>There is the package, that should b... |
1,840,485 | <p>I am an undergraduate really passionate about the mathematics and microbiology. I have few big problems in learning which I would like to seek your advice. </p>
<p>Whenever I study mathematical books (Rudin, Hoffman/Kunze, etc.), I always try to prove every theorem, lemma, corollary, and their relationships in t... | Will R | 254,628 | <p><strong>Overall answer: no.</strong></p>
<p>I struggled with the same problem as you for quite a long time and, in hindsight, I think I could have spent my time more wisely. Here are my current general guidelines at the time of this post. They may or may not work for you. The overall philosophy I employ is that exe... |
1,107,250 | <p><img src="https://i.stack.imgur.com/ILg7L.png" alt="enter image description here"></p>
<p>In above lemma, why $|a'| \leq 1$ still holds? I didn't see how it relates to "algebraic conjugate of a root of unity is also a root of unity", since $a$ is the sum of unity.</p>
<p>(definition of algebraic conjugate: <a href... | Empy2 | 81,790 | <p>The functions which are $f(-1)=-1,f(1)=1$ and all other derivatives are zero are</p>
<p>$$C_n\int_0^x (1-t^2)^n dt$$ </p>
<p>It follows that, pointwise, they reach constants as $n\to\infty$ because the bulk of $(1-t^2)^n$ narrows towards $x=0$.</p>
<p>The OP's functions are </p>
<p>$$D_n\int_0^x (t-t^2)^n \, dt... |
2,356,553 | <p>Why can't we write a simple equation where if we give the value of $x$ as input, we get the value of $\sin(x)$ as output?</p>
<p>By simple, I mean an equation involving just addition, division, subtraction and multiplication and exponentiation and keeping it in the realm of real numbers.</p>
<p>And I'm not necessa... | Daniel | 221,735 | <p>It can $sin(x)=\sum^{\infty}_{k=0}\frac{-(1)^kx^{2k+1}}{2k+1}$, for further information look at <a href="https://en.wikipedia.org/wiki/Taylor_series" rel="nofollow noreferrer">Taylor Series</a></p>
|
2,356,553 | <p>Why can't we write a simple equation where if we give the value of $x$ as input, we get the value of $\sin(x)$ as output?</p>
<p>By simple, I mean an equation involving just addition, division, subtraction and multiplication and exponentiation and keeping it in the realm of real numbers.</p>
<p>And I'm not necessa... | Pedro | 70,305 | <p>Because $\sin(x)$ is not an <a href="https://en.wikipedia.org/wiki/Algebraic_function" rel="nofollow noreferrer"><em>algebraic function</em></a>. It is instead a <a href="https://en.wikipedia.org/wiki/Transcendental_function" rel="nofollow noreferrer"><em>transcendental function</em></a>.</p>
<p><a href="https://ma... |
192,821 | <p>I am using <a href="https://reference.wolfram.com/language/ref/TransformedField.html" rel="noreferrer"><code>TransformedField</code></a> to convert a system of ODEs from Cartesian to polar coordinates:</p>
<pre><code>TransformedField[
"Cartesian" -> "Polar",
{μ x1 - x2 - σ x1 (x1^2 + x2^2), x1 + μ x2 - σ x2... | Moo | 36,141 | <p>We can define our own functions.</p>
<blockquote>
<p>From <span class="math-container">$x',y'$</span> to <span class="math-container">$r',\theta'$</span>, we derive:
<span class="math-container">$$
r' = \left(\sqrt{x^2 +y^2} \right)'
= \frac{(x^2 +y^2)'}{2
\sqrt{x^2 +y^2}}=\frac{xx' +yy'}{r}
$$</span>
and
<... |
139,417 | <p>I have a polygon defined by a list of nodes (x,y). I want to cut the polygon by a horizontal line at position y = a and get the new polygon above the position y = a. I am using the RegionIntersect function, but it seems very slow if I want to combine the function with Manipulate function as well. Is there any way to... | Jason B. | 9,490 | <p>It's a lot faster to find the intersection with a <code>Polygon</code> than with an <code>ImplicitRegion</code>. In your case you can write your region succinctly as a simple <code>Polygon</code></p>
<pre><code>DynamicModule[{R1, R2, R3},
R2 = Polygon[{{0, 0}, {300, 0}, {300, 500}, {0, 750}}];
Manipulate[
R1 =... |
4,069,185 | <p>I got stuck here:</p>
<p>The probability that it will rain today is that it did not rain in the previous two days is <span class="math-container">$0.3$</span>, but if it rained in one of the last two days then the probability of rain today is <span class="math-container">$0.6$</span>.</p>
<p><span class="math-contai... | cos_dm_math21 | 679,837 | <p>First, notice that <span class="math-container">$\text{Ker}(T^i) \subseteq \text{Ker}(T^{i+1})$</span>, for any <span class="math-container">$i\ge 0$</span>.</p>
<p>Remark that if <span class="math-container">$\text{Ker}(T^k) = \text{Ker}(T^{k+1})$</span> for some <span class="math-container">$k\ge 0$</span> then on... |
2,978,605 | <p>Is there a sophisticate way to proof that:</p>
<p><span class="math-container">$$\frac{n}{N}\cdot\frac{n-1}{N-1}\cdot\frac{n-2}{N-2}\cdot\ldots\cdot \frac{1}{N-n+1} = \frac{1}{{N\choose n}}$$</span></p>
<p>where <span class="math-container">${N\choose n}$</span> denotes combinations.</p>
<p>When replacing <span c... | phaedo | 549,654 | <p>This is a definition: <span class="math-container">$ {N \choose n} = \frac{N!}{n!(N-n)!} $</span></p>
<p>Expand all factorials and simplify the tail (N-n)!</p>
|
843,634 | <p>I am wondering whether for any two lines $\mathfrak{L}, \mathfrak{L'}$ and any point $\mathfrak{P}$ in $\mathbf{P}^3$ there is a line having nonempty intersection with all of $\mathfrak{L}, \mathfrak{L'}$, $\mathfrak{P}$. I don't really know how to approach this, because I was never taught thinking about such a prob... | lhf | 589 | <p>Over the reals (or any infinite field), a polynomial induces the zero function iff it is the zero polynomial because a polynomial cannot have an infinite number of roots.</p>
<p>Things are different in finite fields. If the field has $q$ elements, then $X^{q}-X$ induces the zero function but is not the zero polynom... |
2,131,679 | <p>Let $f: \mathbb{R} \to \mathbb{R}$ be continuous and $D \subset \mathbb{R}$ be a dense subset of $\mathbb{R}$. Furthermore, $\forall y_1,y_2 \in D \ f(y_1)=f(y_2)$. Should $f$ be a constant function?</p>
<p>My attempt:
Since $f$ is continuous
$$\forall x_0 \ \forall \varepsilon >0 \ \exists \delta>0 \ \forall... | Stefan4024 | 67,746 | <p>Extend $AO$ to cut $BC$ in $F$ and also let $G$ be the altitude from $C$ to $BC$. Now from Ceva's Theorem we have that $BF:FC = 1:4$. From Menelaus' Theorem on $\triangle BCE$ and $A-O-F$ we have:</p>
<p>$$1 = \frac{BF}{FC} \times \frac{CO}{OE} \times \frac{EA}{BA} \implies \frac{CO}{OE} = 6$$</p>
<p>Now we'll pro... |
726,574 | <p>Ant stands at the end of a rubber string which has 1km of length. Ant starts going to the other end at speed 1cm/s. Every second the string becomes 1km longer. </p>
<p>For readers from countries where people use imperial system: 1km = 1000m = 100 000cm</p>
<p><strong>Will the ant ever reach the end of the string? ... | Fly by Night | 38,495 | <p>As you rightly say, the gradient of the line $2y-x=7$ is $\frac{1}{2}$ because we may rearrange $2y-x=7$ to give $y=\frac{1}{2}x+\frac{7}{2}$. The line you want to find is parallel to this and so also has gradient $\frac{1}{2}$.</p>
<p>Again, you rightly say that the bisector of $(3,1)$ and $(1,-5)$ passes through ... |
4,241 | <p>I was preparing for an area exam in analysis and came across a problem in the book <em>Real Analysis</em> by Haaser & Sullivan. From p.34 Q 2.4.3, If the field <em>F</em> is isomorphic to the subset <em>S'</em> of <em>F'</em>, show that <em>S'</em> is a subfield of <em>F'</em>. I would appreciate any hints on ho... | Jonas Kibelbek | 1,461 | <p>Yuval just posted the function field construction, which is an important example for your question.</p>
<p>You mentioned the algebraic numbers, but just to make it explicit, there are many algebraic number fields (often called just "number fields"). We have $ \bar Q $, the field of <em>all</em> algebraic numbers, ... |
36,756 | <p>In a class, 18 students like to play chess, 23 like to play soccer, 21 like biking, and 17 like jogging. The number of those who like to play both chess and soccer is 9. We also know
that 7 students like chess and biking, 6 students like chess and jogging, 12 like soccer and
biking, 9 like soccer and jogging, and fi... | yoyo | 6,925 | <p>given sets $X_1, X_2, X_3, X_4$ we have
$$|\cup X_i|=\sum_i|X_i|-\sum_{i\neq j}|X_i\cap X_j|+\sum_{i\neq j\neq k}|X_i\cap X_j\cap X_k|-|\cap X_i|$$
so in your example this is
$$
(18+23+21+17)-(9+7+6+12+9+12)+(4+3+5+7)-3=40
$$</p>
|
157,374 | <p>I'm new to Wolfram Mathematica.
I have an initial image:</p>
<p><a href="https://i.stack.imgur.com/XekQw.png" rel="noreferrer"><img src="https://i.stack.imgur.com/XekQw.png" alt="enter image description here"></a></p>
<p>My goal is to find a radius of the water drop, like so:</p>
<p><a href="https://i.stack.imgur... | b3m2a1 | 38,205 | <p>Here's a way that might work for you:</p>
<pre><code>img = Import["https://i.stack.imgur.com/XekQw.png"]
imClip = ImageClip[img, {.9, 1}, {0, 1}];
coords = Position[MorphologicalComponents[imClip], 0];
possibilities = MinMax[#[[All, 2]]] & /@ GatherBy[coords, #[[1]] &];
Subtract @@
Reverse@MaximalBy[p... |
157,374 | <p>I'm new to Wolfram Mathematica.
I have an initial image:</p>
<p><a href="https://i.stack.imgur.com/XekQw.png" rel="noreferrer"><img src="https://i.stack.imgur.com/XekQw.png" alt="enter image description here"></a></p>
<p>My goal is to find a radius of the water drop, like so:</p>
<p><a href="https://i.stack.imgur... | bill s | 1,783 | <p>One way to proceed (that doesn't require searching through the rows) is to binarize the image, remove the top portion, and then crop. The width of the cropped image is the maximum width of the water drop.</p>
<pre><code>img = Import["https://i.stack.imgur.com/XekQw.png"];
ImageCrop[ImageTake[Binarize[img], -550]] /... |
705,744 | <p>Hello everyone. I have a couple questions this time, but I think if I understand how to do this one, I'll understand the others.</p>
<p>A particular online banking system uses the following rules for its passwords:<br/>
a. Passwords must be 6-8 characters in length<br/>
b. Passwords must use only alphabetical and... | André Nicolas | 6,312 | <p>Count the number of legal passwords of length $6$, $7$, $8$ separately, and then add up.</p>
<p>We do the length $7$ case.</p>
<p>If we are using the standard alphabet, there are $26$ lower case characters, $26$ upper case characters, and $10$ digits, for a total of $62$.</p>
<p>There are $62^7$ words of length $... |
3,162,338 | <p>Consider <span class="math-container">$ x_1, x_2, ..., x_n \in \mathbb{R}$</span>.</p>
<p>We have to prove that each <span class="math-container">$\sqrt x $</span> is rational if the sum of <span class="math-container">$\sqrt x_1 + \ldots + \sqrt x_n $</span> is rational. </p>
<p>I think that I could prove it us... | jmerry | 619,637 | <p>As stated, with real numbers <span class="math-container">$x_i$</span>, it's false. After all, we could take something like <span class="math-container">$x_1=(\sqrt{2})^2$</span>, <span class="math-container">$x_2=(2-\sqrt{2})^2$</span>. For this to make sense, those <span class="math-container">$x_i$</span> must al... |
72,201 | <p>Two people play a game. They play a series of points, each producing a winner and a loser, until one player has won at least 4 points and has won at least 2 more points than the other. Anne wins each point with probability p.
What is the probability that she wins the game on the kth point played for k=4,5,6,...</p>
... | Dilip Sarwate | 15,941 | <p>I am wondering if the interpretation used by caligirl11 and @joriki is the one that was intended by whomsoever devised the problem. </p>
<blockquote>
<p>What is the probability that she wins the game on the kth point played for k=4,5,6,...</p>
</blockquote>
<p>I suggest that the $k$-th point <em>played</em> is ... |
1,171,911 | <p><img src="https://i.stack.imgur.com/3q5iO.png" alt="Taken from khan academy ">
Hi, so this question is taken straight from khan academy help exercises, i know how to do it dynamically meaning using the determinant and the adjugate how i was trying to do it using guass bla bla way with help of RREF but i somehow nev... | Ben Grossmann | 81,360 | <p>Using Gauss-Jordan elimination:
$$
\left[\begin{array}{ccc|ccc}
0&1&2 &1&0&0\\
1&0&1 &0&1&0\\
0&1&0 &0&0&1
\end{array}\right] \to \\
\left[\begin{array}{ccc|ccc}
1&0&1 &0&1&0\\
0&1&0 &0&0&1\\
0&1&2 ... |
97,261 | <p>This semester, I will be taking a senior undergrad course in advanced calculus "real analysis of several variables", and we will be covering topics like: </p>
<p>-Differentiability.
-Open mapping theorem.
-Implicit function theorem.
-Lagrange multipliers. Submanifolds.
-Integrals.
-Integration on surfaces.
-Stokes ... | Community | -1 | <p>Here are some free options:</p>
<p><a href="http://www.whitman.edu/mathematics/multivariable/" rel="nofollow">http://www.whitman.edu/mathematics/multivariable/</a></p>
<p><a href="http://synechism.org/drupal/cfsv/" rel="nofollow">http://synechism.org/drupal/cfsv/</a></p>
<p><a href="http://www.mecmath.net/" rel="... |
1,350,837 | <p>Find all integer numbers $n$, such that, $$\sqrt{\frac{11n-5}{n+4}}\in \mathbb{N}$$</p>
<p>I really tried but I couldn't guys, help please.</p>
| DeepSea | 101,504 | <p><strong>hint</strong>: $\dfrac{11n-5}{n+4} = k^2\to 11n-5=nk^2+4k^2\to n(11-k^2)=5+4k^2= 49+4(k^2-11)\to n = \dfrac{-49}{k^2-11}-4\to k^2-11 = \pm1,\pm7,\pm49.$ From this you can find all possible values of $k$, and then find $n$.</p>
|
526,627 | <p>What is the answer for factoring:</p>
<p>$$10r^2 - 31r + 15$$</p>
<p>I have tried to solve it. This was my prior attempt:</p>
<p>$$10r^2 - 31r + 15\\
= (10r^2 - 25r) (-6r + 15)\\
= -5r(-2r+5) -3 (2r-5) $$ </p>
| Ross Millikan | 1,827 | <p>You lost a $+$ sign in the first line starting with $=$. Then note that in the next line, the expressions inside the parentheses are the same except for multiplying by $-1$. So change the sign inside one set of parentheses and the leading coefficient.
$$-5r(-2r+5)-3(2r-5)=5r(2r-5)-3(24-5)=(5r-3)(2r-5)$$</p>
|
1,053,065 | <p>I have a function called $P(t)$ that is the number of the population at time $t$. $t$ being in days.</p>
<p>We know the growth rate is $P'(t) = 2t + 6$</p>
<p>We also know that $P(0) = 100$. How many days till the population doubles?</p>
<p>edit: $P(t) = t^2 + 6t$
edit: $P(t) = t^2 + 6t = 200$
edit: $t^2 + 6t - 2... | Dietrich Burde | 83,966 | <p>A bound for the number of divisors of $n$ is given here: <a href="https://math.stackexchange.com/questions/63687/bound-for-divisor-function">Bound for divisor function</a>. We also have an effective upper bound as follows:
$$
d(n)\le n^{\frac{1.5379 \log(2)}{\log(\log(n))}},
$$
for all $n\ge 3$. This is much better ... |
2,529,533 | <p>Let $f:\mathbf{R}^n \to \mathbf{R}$ be differentiable, $\sum_{i=1}^n y_i \frac{\partial f}{\partial x_i}(y)\geq 0$ for all $y=(y_1,...,y_n)\in \mathbf{R}^n$. How do I show that $f$ is bounded from below by $f(0)$?</p>
| Netivolu | 440,984 | <p>Isn't
$$
f:\mathbb{R}^2 \rightarrow \mathbb{R}: (x_1,x_2) \mapsto x_1 + x_2
$$
a counterexample?</p>
|
3,985,302 | <p>Let <span class="math-container">$L/K$</span> be an extension of local fields. We can find <span class="math-container">$\alpha$</span> such that <span class="math-container">$\mathcal{O}_L=\mathcal{O}_K[\alpha]$</span>. What do we know about this generating element? I think that this <span class="math-container">$\... | reuns | 276,986 | <p><span class="math-container">$O_L=O_K[\pi_L]$</span> iff <span class="math-container">$f(L/K)=1$</span> and <span class="math-container">$O_L=O_K[\zeta_{q_L-1}]$</span> iff <span class="math-container">$e(L/K)=1$</span>. In general it is <span class="math-container">$$O_L=O_K[\zeta_{q_L-1}+\pi_L]$$</span></p>
<p>Hen... |
3,985,302 | <p>Let <span class="math-container">$L/K$</span> be an extension of local fields. We can find <span class="math-container">$\alpha$</span> such that <span class="math-container">$\mathcal{O}_L=\mathcal{O}_K[\alpha]$</span>. What do we know about this generating element? I think that this <span class="math-container">$\... | Torsten Schoeneberg | 96,384 | <p>Hint: If <span class="math-container">$\mathcal{O}_L = \mathcal{O}_K[\alpha]$</span> with <span class="math-container">$\alpha$</span> a uniformiser of <span class="math-container">$\mathcal{O}_L$</span>, then the canonical inclusion of their residue fields <span class="math-container">$\mathcal{O}_K/(\pi_K) \hookri... |
607,264 | <p>Let the directional derivative of a function $f(x,y)$ at a point $P$ in the direction of $(1/\sqrt{5})\mathbf{i}+(2/\sqrt{5})\mathbf{j}$ be $16/\sqrt{5}$ and the partial derivative $\partial f / \partial x$ evaluated at $P$ be $6$. Then what is the directional derivative in the direction of $\mathbf{i}-\mathbf{j}$?<... | Berci | 41,488 | <p>The differential of $f$ at the given point $P$ is a linear map, respresented by a $1\times 2$ matrix, namely $f'(P)=\left(\displaystyle\frac{df}{dx}(P),\ \frac{df}{dy}(P)\right)$. Now, by assumption,
$df/dx(P)=6$, and let $b:=df/dy(P)$ the missing coordinate.</p>
<p>The other piece of information tells us that
$$\p... |
355,552 | <p>How would you compute the first $k$ digits of the first $n$th Fibonacci numbers (say, calculate the first 10 digits of the first 10000 Fibonacci numbers) without computing (storing) the whole numbers ?</p>
<p>A trivial approach would be to store the exact value of all the numbers (with approximately $0.2n$ digits f... | Ross Millikan | 1,827 | <p>How about <a href="http://en.wikipedia.org/wiki/Jacques_Philippe_Marie_Binet" rel="nofollow">Binet's formula</a>? If you want the first $k$ digits, you need to calculate $\sqrt 5$ to not many more than $k$ digits (use the continued fraction, for example) and you are there. The subtraction doesn't cancel, and if $n... |
60,322 | <p>I'm interested in Lie theory and its connections to dynamical systems theory. I am starting my studies and would like references to articles on the subject.</p>
| mathphysicist | 2,149 | <p>One of the most important connections of the two fields can be found in the theory of Hamiltonian dynamical systems with Lie groups being the symmetry groups. The interplay of these leads to many interesting concepts (including, inter alia, the classical R-matrix) and results. For starters you can try the books <a h... |
3,459,106 | <p>I have a function
<span class="math-container">$$
\frac{\ln x}{x}
$$</span> and I wonder, is <span class="math-container">$y=0$</span> an asymptote? I mean it is kinda strange that graph is in some place is going through that asymptote. I know it meets the criterium of asymptote, but its kinda strange if you unders... | Kavi Rama Murthy | 142,385 | <p>Without continuity of <span class="math-container">$f_n$</span>'s this is false. Take any bounded function <span class="math-container">$f$</span> which is not continuous and let <span class="math-container">$f_n =f $</span>. Then <span class="math-container">$f_n \to f$</span> uniformly. </p>
|
3,873,071 | <p>This is my first post and I apologize in advance if I'm not using the right formatting/approach.</p>
<p><strong>Problem</strong></p>
<p>A coin, having probability <span class="math-container">$p$</span> of landing heads, is continually flipped until at least one head and one tail have been flipped.</p>
<p>Find the e... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p><span class="math-container">$$\dfrac{(x^4-1)\sqrt{x^4+1}}{x^8+1}=x\cdot\dfrac{\left(1-\dfrac1{x^4}\right)\sqrt{x^2+\dfrac1{x^2}}}{x^4+\dfrac1{x^4}}$$</span></p>
<p>Set <span class="math-container">$x^2=y$</span> to find</p>
<p><span class="math-container">$$\int\dfrac{(x^4-1)\sqrt{x^4+1}}{x^8+1}dx=\dfr... |
470,617 | <ol>
<li><p>Two competitors won $n$ votes each.
How many ways are there to count the $2n$ votes, in a way that one competitor is always ahead of the other?</p></li>
<li><p>One competitor won $a$ votes, and the other won $b$ votes. $a>b$.
How many ways are there to count the votes, in a way that the first competitor ... | lab bhattacharjee | 33,337 | <p>If $7x=\pi,4x=\pi-3x$</p>
<p>$\implies \sin4x=\sin(\pi-3x)=\sin3x$</p>
<p>$\implies 2\sin2x\cos2x=3\sin x-4\sin^3x$</p>
<p>$\implies 4\sin x\cos x\cos2x=3\sin x-4\sin^3x$</p>
<p>If $\sin x\ne0,$ we have $4\cos x\cos2x=3-4\sin^2x\implies 4\cos x(1-2\sin^2x)=3-4\sin^2x$</p>
<p>On squaring & rearrangement, ... |
112,096 | <p>Does the inequality $2 \langle x , y \rangle \leqslant \langle x , x \rangle + \langle y , y \rangle $, where $$ \langle \cdot, \cdot \rangle $$ denotes scalar product, have a name? </p>
<p>I've tried looking at several inequalities on wikipedia but I didn't find this one. And of course googling doesn't work for ... | Beni Bogosel | 7,327 | <p>It is the development of
$$ \langle x-y,x-y \rangle \geq 0$$
and it follows from the positive definitness of the scalar product.</p>
<hr>
<p>Apart of the above proof of the inequality, and as a response to the comments to the question, here are a few reasons as to why this inequality should be true, at a first gla... |
685,567 | <p>For the function, $f(x,y,z)=\sqrt{x^2+y^2+z^2}$, do directional derivatives exist at the origin? If I use the definition $$lim_{h\to 0}\frac{f(x+hv)-f(x)}{h},$$ then I get $$\frac{|h|}{h}$$ which is without limit. But in some places, I keep reading that the directional derivative is 1. </p>
<p>Also, if I were to wr... | Vadim | 26,767 | <p><strong>Hint</strong> Consider the PGF for this distribution. It is</p>
<p>$$pgf(z)=\mathbb{E}z^X=\sum_{k=0}^\infty\left(e^{-\lambda}\frac{\lambda^k}{k!}\right)z^k=e^{\lambda(z-1)}$$</p>
<p>Then, assuming they are independent, the sum has PGF</p>
<p>$$pgf_n(z)=(pgf(z))^n=e^{n\lambda(z-1)}=\sum_{k=0}^\infty\left(e... |
3,559,167 | <p>I know that it is a bounded below set and the infimum is 4, but I'm unsure of going about how to prove that it is indeed bounded. Any help would be greatly appreciated! </p>
| RishiNandha Vanchi | 633,100 | <p>Consider Positive Values for x alone for now,</p>
<p>AM-GM</p>
<p><span class="math-container">$x + \frac 4x \geq 2 \sqrt {x \frac 4x}$</span></p>
<p><span class="math-container">$f(x) \geq 4$</span></p>
<p>It is Easy to see that it is a monotonically increasing function beyond x = 2 by differentiating,</p>
<p>... |
3,564,476 | <p>I'm stuck on the following problem: </p>
<p><span class="math-container">$$\int_{\frac\pi{12}}^{\frac\pi2}(1-\cos4x)\cos2x\>dx$$</span></p>
<p>I think I can use the double angle formulas here but I'm not sure how to apply it, or even if it's the right approach. I'm also not sure if 1-cos4x can be translated in... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>For this type of integrals, use the linearisation formulæ:
<span class="math-container">$$2\sin a\sin b=\cos(a-b)-\cos(a+b),\qquad2\cos a\cos b=\cos(a-b)+\cos(a+b),$$</span>
<span class="math-container">$$2\sin a\cos b=\sin(a-b)+\sin(a+b.$$</span></p>
<p><em>Some details</em>:</p>
<p... |
24,927 | <p>Does this mean that the first homotopy group in some sense contains more information than the higher homotopy groups? Is there another generalization of the fundamental group that can give rise to non-commutative groups in such a way that these groups contain more information than the higher homotopy groups? </p>
| Qiaochu Yuan | 232 | <p>Thinking about the higher homotopy groups as just groups is in some sense missing the point. The higher homotopy groups are not just abelian groups: they are $\pi_1$-modules, for one thing.</p>
<p>More loftily, from the <a href="http://ncatlab.org/nlab/show/nPOV">n-categorical point of view</a>, the homotopy groups... |
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