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 |
|---|---|---|---|---|
1,234,726 | <p>How many lattice paths are there from $(0, 0)$ to $(10, 10)$ that do not pass to the point $(5, 5)$ but do pass to $(3, 3)$?</p>
<p>What I have so far:</p>
<p>The number of lattice paths from $(0,0)$ to $(n,k)$ is equal to the binomial coefficient $\binom{n+k}n$ (according to Wikipedia). So the number of lattic... | imok1948 | 468,333 | <p>Lets count,<br>
T = Total number of paths from (0,0) to (n,k) = (n!k!)/(n!+k!).<br>
A = Toatl number of paths from (0,0) to (a,b) = (a!b!)/(a!+b!).(In your question a,b =3,3)<br>
B = Total number of paths from (a,b) to (n,k) = ((n-a)!(k-b)!)/((n-a)!+(k-b)!)<br>
Ans = T-(A*B)</p>
<p>If there are more than one path w... |
2,589,938 | <p>Suddenly I am confused by a very elementary question:</p>
<blockquote>
<p>Let $a, b, c$ be the sides of a triangle. How about this inequality:
$$ a+b > c. $$</p>
</blockquote>
<p>Is it the definition of a triangle or is it a theorem?</p>
| lhf | 589 | <p>This follows from the law of cosines:
$$
c^2 = a^2+b^2 - 2ab \cos \hat C
\le a^2+b^2 + 2ab
= (a+b)^2
$$</p>
|
3,244,649 | <ul>
<li>Show the set <span class="math-container">$A=\{(m,n)\in N\times N : m\leq n\}$</span> is countably infinite.</li>
</ul>
<p>If <span class="math-container">$A$</span> is countable then we need to show that there is a bijection between <span class="math-container">$A$</span> and <span class="math-container">$\m... | qualcuno | 362,866 | <p>Note that <span class="math-container">$A$</span> can be written as </p>
<p><span class="math-container">$$
A = \bigsqcup_{n \in \mathbb{N}}\{(m,n) : m \leq n\}.
$$</span></p>
<p>That is, <span class="math-container">$A$</span> is the disjoint <strong>countable</strong> union of sets <span class="math-container">$... |
2,258,139 | <p>A natural number $n>1$ is called <em>good</em> if$$n \mid 2^n+1.$$ For example, $n=3$ is good, as $3 \mid 2^3+1=9$. Prove that if $N_1$ and $N_2$ are good, then:</p>
<ul>
<li>$\mathrm{lcm}(N_1,N_2)$ and $\gcd(N_1,N_2)$ are good,</li>
<li>$N_1\cdot N_2$ is good. </li>
</ul>
<p>This seems pretty difficult for me.... | TBTD | 175,165 | <p>Here is a proof of part 1. Assume,
$$
n_1 \mid 2^{n_1}+1 \ \text{and} \ n_2 \mid 2^{n_2}+1.
$$
Denote, $d=gcd(n_1,n_2)$. We have,
$$
2^{n_1}\equiv -1 \pmod{d} \ \text{and} \ 2^{n_2}\equiv -1 \pmod{d}.
$$
Now, from <a href="https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity" rel="nofollow noreferrer">Bézout's id... |
4,000,062 | <p>A sequence <span class="math-container">$\left\{a_n\right\}$</span> is defined as <span class="math-container">$a_n=a_{n-1}+2a_{n-2}-a_{n-3}$</span> and <span class="math-container">$a_1=a_2=\frac{a_3}{3}=1$</span></p>
<blockquote>
<p>Find the value of <span class="math-container">$$a_1+\frac{a_2}{2}+\frac{a_3}{2^2}... | Ekaveera Gouribhatla | 31,458 | <p>Let <span class="math-container">$$S=\sum_{n=1}^{\infty}\frac{a_n}{2^{n-1}}$$</span>
We have,
<span class="math-container">$$S=\frac{1}{1}+\frac{1}{2}+\frac{3}{4}+P$$</span>
Where <span class="math-container">$$P=\sum_{n=4}^{\infty}\frac{a_n}{2^{n-1}}$$</span>
Using the given recurrence we have,
<span class="math-co... |
2,213,626 | <p>How can you prove that if the gcd(a,b) = 1 then gcd(a,bi) = 1 in the Gaussian integers? I know that $i$ is a unit in the ring, but how can you rigorously prove this?</p>
| QuantumEyedea | 213,634 | <p>Integrate both sides of the second form of your DE with respect to $x$ and then you're left with an equation
$$
\tfrac{1}{2} (y')^2 + a y^{-2}/2 = b y
$$</p>
<p>Then you just are left with a 1-st order DE from there</p>
|
2,358,490 | <p>Let $V$ be a finite dimensional vector space over the field $K$, and let $W_1$ and $W_2$ be subspaces. Express $(W_1+W_2)^{\perp}$ in terms of $W_1^{\perp}$ and $W_2^{\perp}$. Also, express $(W_1\cap W_2)^{\perp}$ in terms of $W_1^{\perp}$ and $W_2^{\perp}$.</p>
<p>I have no idea what this exercise is asking. Remar... | Itay4 | 385,242 | <p>I will prove: $(W_1+W_2)^{\perp}=W_1^\perp\cap W_2^\perp.$</p>
<p>$W_1,W_2\subset W_1+W_2,$ and we know $(W_1+W_2)^{\perp} \subset W_1^{\perp},W_2^{\perp}$ </p>
<p>So we get: $$(W_1+W_2)^{\perp} \subset W_1^{\perp}\cap W_2^{\perp}$$</p>
<p>In the other direction,</p>
<p>Let $v\in W_1^{\perp}\cap W_2^{\perp},$ so... |
550,230 | <p>If 2 vectors form a basis for $\mathbb{R}^2$, must these 2 vectors always be orthogonal to each other?</p>
<p>For instance, the standard bases in $\mathbb{R}^2$ are definitely orthogonal (easily drawn). How about other bases?</p>
| Ted Shifrin | 71,348 | <p>No, they definitely need not be orthogonal, just non-parallel.</p>
|
137,435 | <p>Consider the braid group on n strands given in the usual Artin presentation. Then add extra relations: each Artin generator has order d. For example, if d=2, one recovers the symmetric group. I would like to know what the order of the group is for arbitrary n and d. Even knowing the name of such groups would be help... | Julián Aguirre | 4,791 | <p>Let $A=\{(x,y):y\ge e^x\}$ and $B=\{(x,0)\}$. Then $A+B=\{(x,y):y>0\}$.</p>
|
1,353,432 | <p>I know two proofs about the approximation of Euler-Mascheroni constant $\gamma$ that are very technical. So I would like to know if someone has a strategic proof to show that $0.5<\gamma< 0.6$.</p>
<blockquote>
<p>Let be $\gamma\in \mathbb{R}$ such that</p>
<p>$$\large\gamma= \lim_{n\to +\infty}\left[... | Jaume Oliver Lafont | 134,791 | <p>Setting $n=1$ and $m=8$ into the following inequality involving harmonic numbers</p>
<p>$$
2H_n-H_{n(n-1)}<\gamma<2H_m-H_{m^2}
$$</p>
<p>gives</p>
<p>$$
0.5<\gamma<0.692
$$</p>
|
1,643,013 | <p>I have just started learning about differential equations, as a result I started to think about this question but couldn't get anywhere. So I googled and wasn't able to find any particularly helpful results. I am more interested in the reason or method rather than the actual answer. Also I do not know if there even ... | E.H.E | 187,799 | <p>the question means:
$$y-y'-y''-y'''-......y^n=0$$
the differential equation is homogeneous and the characteristics equation is
$$1-r-r^2-r^3-.......r^n=0$$
$$r(1+r+r^2+r^3+....)=1$$
by using the geometric series (r<1)
$$\frac{r}{1-r}=1$$
$$r=1-r$$
$$r=\frac{1}{2}$$
so the function is
$$y=Ce^{\frac{x}{2}}$$</p>
|
1,867,401 | <p>I refer to this derivation of the gradient in polar coordinates: <a href="http://www.math.jhu.edu/~js/Math202/polar.grad.chain.pdf" rel="nofollow">http://www.math.jhu.edu/~js/Math202/polar.grad.chain.pdf</a></p>
<p>I can understand all parts except why the unit gradient $$\hat{e_r}=\langle\cos\theta,\sin\theta\rang... | Hans Lundmark | 1,242 | <p>Consider a point with polar coordinates $(r,\theta)$. It lies, of course, at the distance $r$ from the origin. A change of $d\theta$ in the value of $\theta$ will move this point a distance $r \, d\theta$ along the circle of radius $r$. (Notice the factor $r$; it says that the farther out you are, the bigger is the ... |
129,261 | <p>I need to prove several inequalities trivially. (i.e. without using AM-GM, re-arrangement etc). I just keep hitting a blank. Could anyone help?</p>
<p>$$x^{4}+y^{4}+z^{4}\geq x^{2}yz+xy^{2}z+xyz^{2}$$</p>
| Brian M. Scott | 12,042 | <p>The righthand side is clearly $xyz(x+y+z)$. Assume that $x,y,z>0$. For fixed $x+y+z$, $xyz$ is maximized when $x=y=z$, and $x^4+y^4+z^4$ is minimized when $x=y=z$. This is easy to see if you can visualize the surfaces $x+y+z=k$, $xyz=k$, and $x^4+y^4+z^4=k$ for a positive constant $k$. Thus, for a fixed value of ... |
3,491,978 | <blockquote>
<p>Let (X,d) be a compact metric space. For every open cover, show there exists ε > 0 such that for every x ∈ X, B(x,ε) is contained in some member of the cover.</p>
</blockquote>
<p>My attempt:</p>
<p>(X,d) is compact. Therefore there exists a finite subcover of X.</p>
<p>Any element x in X must lie ... | fleablood | 280,126 | <p>The problem with that proof is you have an <span class="math-container">$\epsilon$</span> defined for each <span class="math-container">$x$</span>. You need to prove there is an <span class="math-container">$\epsilon$</span> that works for all <span class="math-container">$x$</span> but is independent of the value ... |
2,379,955 | <p>Assume I want to minimise this:
$$ \min_{x,y} \|A - x y^T\|_F^2$$
then I am finding best rank-1 approximation of A in the squared-error sense and this can be done via the SVD, selecting $x$ and $y$ as left and right singular vectors corresponding to the largest singular value of A.</p>
<p>Now instead, is possible t... | Rodrigo de Azevedo | 339,790 | <p><span class="math-container">$$\| \mathrm A - \mathrm x \mathrm b^{\top} \|_{\text{F}}^2 = \cdots = \| \mathrm b \|_2^2 \, \| \mathrm x \|_2^2 - \langle \mathrm A \mathrm b , \mathrm x \rangle - \langle \mathrm x , \mathrm A \mathrm b \rangle + \| \mathrm A \|_{\text{F}}^2$$</span></p>
<p>Taking the gradient of thi... |
3,767,452 | <blockquote>
<p>Suppose we have
<span class="math-container">$$\begin{align}
\cos x + \cos y + \cos z &= \frac{3}{2}\sqrt{3} \\[4pt]
\sin x + \sin y + \sin z &= \frac{3}{2}
\end{align}$$</span></p>
<p>How can we solve for <span class="math-container">$x$</span>, <span class="math-container">$y$</span> and <sp... | lab bhattacharjee | 33,337 | <p>Hint</p>
<p><span class="math-container">$$(\cos x+\cos y+\cos z)^2+(\sin x+\sin y+\sin z)^2=?$$</span></p>
<p><span class="math-container">$$\implies\cos(x-y)+\cos(y-z)+\cos(z-x)=3$$</span></p>
<p>As for <span class="math-container">$A,\cos A\le1$</span></p>
<p>each of the cosine ratio will be <span class="math-con... |
136,264 | <p>I have a question concerning the stability analysis for a kind of differential equation taking the form
$$\dot x=Ax+Bw,$$
where $A\in \mathbb{R}^{n \times n}$, $B\in \mathbb{R}^{n \times m}$ are constant matrices and
$w \in \mathbb{R}^m$ is a normal random variable, i.e., $w\sim \mathcal{N}(0,W)$ with $W$ ... | just-someone | 100,295 | <p>Here are the rankings of (not just math) journals employed by some countries:</p>
<ul>
<li><p><a href="https://sucupira.capes.gov.br/sucupira/public/" rel="nofollow noreferrer">Brazil</a> (in Portuguese)</p></li>
<li><p><a href="http://www.julkaisufoorumi.fi/en" rel="nofollow noreferrer">Finland</a></p></li>
<li><p... |
3,555,084 | <blockquote>
<p>Let
<span class="math-container">$$f(z) = e^z (1+\cos\sqrt{z} ) $$</span>
<span class="math-container">$\Omega=\{z\in\Bbb C: |z|\gt r\}$</span>, <span class="math-container">$r\gt 0$</span>. What is <span class="math-container">$f(\Omega)$</span>?</p>
<p>where <span class="math-container">$... | Allawonder | 145,126 | <p>If you write the expression as <span class="math-container">$$x^4\left(1-\sqrt{1+6/x^2}+3/x^2\right)=\frac{1-\sqrt{1+6/x^2}+3/x^2}{1/x^4},$$</span> then you may be able to apply the Marquis de L'hopital's method.</p>
<p>After exactly two iterations, you get <span class="math-container">$$\frac92\frac{\frac{1}{\sqrt... |
3,555,084 | <blockquote>
<p>Let
<span class="math-container">$$f(z) = e^z (1+\cos\sqrt{z} ) $$</span>
<span class="math-container">$\Omega=\{z\in\Bbb C: |z|\gt r\}$</span>, <span class="math-container">$r\gt 0$</span>. What is <span class="math-container">$f(\Omega)$</span>?</p>
<p>where <span class="math-container">$... | Barry Cipra | 86,747 | <p>Just to give yet another approach, let <span class="math-container">$u=x^2+3$</span>. Then, for <span class="math-container">$x\ge0$</span>,</p>
<p><span class="math-container">$$x^2(x^2-x\sqrt{x^2+6}+3)=(u-3)\left(u-\sqrt{u^2-9}\right)={9u\over u+\sqrt{u^2-9}}-{27\over u+\sqrt{u^2-9}}\to{9\over1+1}-0={9\over2}$$</... |
3,336,870 | <p>Suppose <span class="math-container">$S= \{x_1+x_5\}$</span> is a vector space in <span class="math-container">$R^5$</span>.</p>
<p>Then what is the orthogonal complement for <span class="math-container">$S$</span>?</p>
<p><em>My interpretation:</em></p>
<p>We can represent as <span class="math-container">$[1, 0,... | aryan bansal | 698,119 | <p>1)let's say the apples are identical
W1 + 2 + w2 +2 + w3 +2 = 10 (w is whole no.) And wi represents no of apple ith person gets
W1 + w2 + w3 = 4 implying 6c2 ways(beggar's method)</p>
<p>2) they are not identical
10= 2+2+6 or 2+3+5 or 2+ 4 +4 or 3+3+4.
We will make groups and arrange
10!/(2!2!6!2!)×3!(extra divis... |
3,360,912 | <p>Why if we have strictly increasing, continuous and onto function its inverse must be continuous? could anyone explain this for me please?</p>
| bsbb4 | 337,971 | <p>In short, the number of lattice points of given bounded magnitude grows linearly, giving a contribution proportional to <span class="math-container">$n \cdot \frac{1}{n^2} = \frac{1}{n}$</span>, making the series diverge.</p>
<p>I give a proof that <span class="math-container">$\sum_{\omega \in \Lambda^*} \omega^{-... |
2,548,177 | <p>I'd like to define <code>sumdiv</code> in Maple such that this:</p>
<pre><code>with(numtheory);
f:=x->x^2;
sumdiv(f(d)*mobius(100/d), d=1..100);
</code></pre>
<p>would do a sum on all divisors <code>d</code> of $100$.</p>
<p><strong>How to do such a sum over divisors in Maple?</strong></p>
<p>Here's what I've... | acer | 12,448 | <p>You can do this by adding up the results from the Maple command <code>numtheory[divisors]</code>, or you could go more directly to the Maple command <code>numtheory[sigma]</code>.</p>
<pre><code>restart;
`+`(op(numtheory[divisors](100)));
217
numtheory[sigma](100);
... |
331,710 | <p>I need to find the area of a parallelogram with vertices $(-1,-1), (4,1), (5,3), (10,5)$.</p>
<p>If I denote $A=(-1,-1)$, $B=(4,1)$, $C=(5,3)$, $D=(10,5)$, then I see that $\overrightarrow{AB}=(5,2)=\overrightarrow{CD}$. Similarly $\overrightarrow{AC}=\overrightarrow{BD}$. So I see that these points indeed form a p... | Berci | 41,488 | <p><strong>Hints:</strong></p>
<ol>
<li>The area of a parallelogram with side vectors $\bf a$ and $\bf b$ is $\det(\bf a\ \bf b)$.</li>
<li>For a parallelogram $A,B,C,D$ its side vectors are e.g. $B-A$ and $C-A$.</li>
</ol>
|
331,710 | <p>I need to find the area of a parallelogram with vertices $(-1,-1), (4,1), (5,3), (10,5)$.</p>
<p>If I denote $A=(-1,-1)$, $B=(4,1)$, $C=(5,3)$, $D=(10,5)$, then I see that $\overrightarrow{AB}=(5,2)=\overrightarrow{CD}$. Similarly $\overrightarrow{AC}=\overrightarrow{BD}$. So I see that these points indeed form a p... | lab bhattacharjee | 33,337 | <p>As the diagonal of a parallelogram $A(-1,-1),B(4,1),C(5,3),D(10,5)$ divides into two congruent triangles, which implies the triangles have same area.</p>
<p>So, the area of the parallelogram will be $2\cdot$ area of any one of $\triangle ABC,\triangle ADC, \triangle ABD, \triangle BCD$</p>
<p>For example, the <a... |
3,413,837 | <p>Jerry the mouse is hungry and according to some confidential information, there is a tempting piece of cheese at the end of one of the three paths after the junction he just found himself!</p>
<p>Fortunately, Tom is standing right there and Jerry hopes he can get some useful information as to which path he must get... | antkam | 546,005 | <p>The problem with these knights (always truthful) / knaves (always lying) puzzles is that many of them can be solved <em>the same way</em>. Simply ask:</p>
<blockquote>
<p>"If I were to ask you is door X correct, what would you say?" </p>
</blockquote>
<p>A knave would have to lie twice, and therefore tell the ... |
2,180,398 | <p>I am not sure if I got the correct answers to these basic probability questions.</p>
<blockquote>
<p>You and I play a die rolling game, with a fair die. The die is equally likely to land on any of its $6$ faces. We take turns rolling the die, as follows. </p>
<p>At each round, the player rolling the die w... | Student | 304,018 | <p>$\color{red}{\text{REMARK: I wrote this answer supposing that one round is one person roling because }}$
$\color{red}{\text{of the formulation of your second question. If this is not the case, look at N.F. Taussig's}}$
$\color{red}{\text{answer.}}$</p>
<p>First of all: you answered the first question correct! </p>... |
249,107 | <p>Im working on my thesis about semidirect products and splitting lemma. I got the following theorems to prove and Im a not sure how to start. I would appreciate any help.</p>
<p>$\\$
1. Let $f:A\to B$ be a map.</p>
<p>Show:</p>
<p>a) if $g:B\to A$ so that $gf=id_{A}$ then $f$ is injective</p>
<p>b) if $g:B\to A$ ... | Hagen von Eitzen | 39,174 | <p>Finding the proof is practically unavoidable as soon as you make yourself clear what you are given and what you want to show:</p>
<p>1a) Assume $f(y)=f(y)$. You want to show that this implies $x=y$. You are given the fact that $g(f(t))=t$ for all $t\in A$. Aha: $f(x)=f(y)$ imlpies $x=g(f(x))=g(f(y))=y$.</p>
<p>1b)... |
157,823 | <p>Check whether function series is convergent (uniformly):</p>
<p>$\displaystyle\sum_{n=1}^{+\infty}\frac{1}{n}\ln \left( \frac{x}{n} \right)$ for $x\in[1;+\infty)$</p>
<p>I don't know how to do that.</p>
| Unoqualunque | 17,703 | <p>The series doesn't converge. Use integral test or Cauchy-condensation-test via monotonicity of the general term</p>
|
3,912,734 | <p>My text book in linear algebra - out of the blue - claims that:</p>
<p><span class="math-container">$|\lambda u|=|\lambda||u|$</span></p>
<p>Where u is a vector and <span class="math-container">$\lambda$</span> is a constant.</p>
<p>I would understand if || were used to denote absolute numbers, but in this book, || ... | lhf | 589 | <p>Perhaps this makes it clear: <span class="math-container">$|\lambda u|_V=|\lambda|_{\mathbb R}\,|u|_V$</span>.</p>
<p>It reads: the length of <span class="math-container">$\lambda u$</span> in the vector space <span class="math-container">$V$</span> is the product to the absolute value of <span class="math-container... |
3,912,734 | <p>My text book in linear algebra - out of the blue - claims that:</p>
<p><span class="math-container">$|\lambda u|=|\lambda||u|$</span></p>
<p>Where u is a vector and <span class="math-container">$\lambda$</span> is a constant.</p>
<p>I would understand if || were used to denote absolute numbers, but in this book, || ... | Hagen von Eitzen | 39,174 | <p>You may know about the dot product that <span class="math-container">$(\lambda u)\cdot (\mu v)=\lambda\mu (u\cdot v)$</span>.
Hence
<span class="math-container">$$ |\lambda u|=\sqrt{\lambda u\cdot \lambda u)}=\sqrt{\lambda ^2 (u\cdot u)}=|\lambda|\sqrt{u\cdot u}=|\lambda|\,|u|.$$</span></p>
|
1,047,263 | <p>I used to do this on my calculators and it never worked! I think it's because you can't multiply any number by itself to get a negative number. Is that even true? I think it is! I've tried it out and it never worked! Look here:$$0.5\cdot0.5=0.25$$$$0\cdot0=0$$$$-1\cdot-1=1$$$$2.1\cdot2.1=4.41$$$$-7.5\cdot-7.5=5... | k170 | 161,538 | <p>For every real $x$ such that $x\gt 0$, we have
$$ \sqrt{-x}=\sqrt{(-1)x}=\sqrt{-1}\sqrt{x} =i\sqrt{x} $$</p>
|
691,112 | <p>Suppose $A$ and $B$ are finite sets and $f:A\rightarrow B$. Prove that if $|A|>|B|$, then $f$ is not one-to-one.</p>
<p>Scratch work:</p>
<p>Since the goal is in negation, I try to prove it by contradiction and assume that $f$ is one-to-one. Since $A$ has more elements than $B$, it can't be the case that $f$ is... | naslundx | 130,817 | <p>Select a subset $A' \subset A$ such that $|A'| = |B|$. Let $A' = \{a_1, a_2, \dots, a_n\}$ and $B = \{b_1, b_2, \dots, b_n\}$. There is a trivial bijection (one-to-one and onto) $f:A' \to B$.</p>
<p>Now assume $g$ is a one-to-one mapping $A \to B$ which when restricted to $A'$ is $f$.</p>
<p>There exists $a_{n+1} ... |
691,112 | <p>Suppose $A$ and $B$ are finite sets and $f:A\rightarrow B$. Prove that if $|A|>|B|$, then $f$ is not one-to-one.</p>
<p>Scratch work:</p>
<p>Since the goal is in negation, I try to prove it by contradiction and assume that $f$ is one-to-one. Since $A$ has more elements than $B$, it can't be the case that $f$ is... | Jose Antonio | 84,164 | <p>We write $A\preceq B$ if there exists an injection $A\to B$.</p>
<p><strong>Theorem:</strong> Let $A, B$ be finite sets. Then, $A \preceq B, \iff \#A \le \#B$.</p>
<p>Proof: ($\Rightarrow$)
Let $\varphi(n)$ be the statement "B is a set of size $\,n\,$ and $A \preceq B \rightarrow \#A \le n$."</p>
<p>$$S = \left\{... |
1,515,775 | <p>So, everyone knows the famous <a href="https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem" rel="nofollow">Lagrange's four-square theorem</a>, which states, that every positive integer can be written down as the sum of $4$ square numbers. Since $4=2^2$, and $2$ represents the square numbers, could this be... | Thomas Delaney | 250,032 | <p>I think yours is an interesting question, and I strongly feel that there is a finite 'n' such that all integers can be expressed as the sum of n cubes. However, isn't it easy to find a counter-example to your conjecture that n=8? What about 23, which can be expressed as 8+8+1+1+1+1+1+1+1. It requires 9 cubes.
I'm gu... |
2,331,191 | <p>Use either direct proof, proof by contrapositive, or proof by contradiction.</p>
<p>Using proof by contradiction method</p>
<blockquote>
<p>Assume n is a perfect square and n+3 is a perfect square (proof by
contradiction)</p>
<p>There exists integers x and y such that <span class="math-container">$n = x^2$</span> an... | Steven Alexis Gregory | 75,410 | <p>First, as mentioned by others, $1$ is a perfect square and $1+3$ is a perfect square. So you need to prove "If $N > 1$ is a perfect square, then $n+3$ is not a perfect square.</p>
<p>Let $n = m^2$ where $m$ and $n$ are positive integers and $n > 1$. Then we must have $m > 1$. Since $m$ is an integer, we ca... |
461 | <p>There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called <em>Steenrod squaring</em>: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from here on out.) Its notable axiom (besides things like naturality), and the reason for its name, is that if $a\i... | Andy Putman | 317 | <p>For the Steenrod squares, I highly recommend the first couple of chapters of the book "Cohomology operations and applications in homotopy theory" by Mosher and Tangora. It's beautifully written (and now available in a cheap Dover edition).</p>
|
461 | <p>There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called <em>Steenrod squaring</em>: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from here on out.) Its notable axiom (besides things like naturality), and the reason for its name, is that if $a\i... | Dev Sinha | 4,991 | <p>Here's how I explain Steenrod squares to geometers. First, if $X$ is a manifold of dimension $d$ then one can produce classes in $H^n(X)$ by proper maps $f: V \to X$ where $V$ is a manifold of dimension $d-n$ through many possible formalisms - eg. intersection theory (the value on a transverse $i$-cycle is the coun... |
1,643,579 | <p>I've an homework problem that i'm unable to find the right answer.</p>
<p>The problem is:</p>
<p>The line $tx + sy = 2$ goes through point $(2,1)$ and is parallel to line $y = 8 -3x$, find the value of $t^2 + s^2$. </p>
<p>$ A. {32\over49}$ $B.{18\over49}$ $C.{36\over49}$ $D.{25\over49} $ $E.{40\over49} $</p>
<p... | Mark Viola | 218,419 | <p><strong>HINT:</strong></p>
<p>Using $\cos(x)=1-\frac12 x^2+O(x^4)$ we have</p>
<p>$$\frac{1-\prod_{k=1}^n\cos(kx)}{x^2}=\frac12\sum_{k=1}^{n}k^2+O(x^2)$$</p>
|
1,643,579 | <p>I've an homework problem that i'm unable to find the right answer.</p>
<p>The problem is:</p>
<p>The line $tx + sy = 2$ goes through point $(2,1)$ and is parallel to line $y = 8 -3x$, find the value of $t^2 + s^2$. </p>
<p>$ A. {32\over49}$ $B.{18\over49}$ $C.{36\over49}$ $D.{25\over49} $ $E.{40\over49} $</p>
<p... | Mark Viola | 218,419 | <p>Since it has yet to be posted, I thought it would be instructive to present the approach suggested by @DanielFischer. We note that we can write </p>
<p>$$a_{n+1}-a_n=\frac12(n+1)^2 \tag 1$$</p>
<p>Summing $(1)$ we find that </p>
<p>$$\begin{align}
\sum_{k=1}^{n-1}(a_{k+1}-a_k)&=a_{n}-a_1\\\\
a_n&=a_1+\fr... |
51,757 | <p>I'm trying to find closed form for</p>
<p>$$\sum_{k=1}^{n}\sin\frac{1}{k}$$</p>
<p>I typed it in Mathematica 6.0 and WolframAlpha, but no result what i expected.</p>
<p>Any hints will be appreciated, thank you.</p>
| Andrew | 11,265 | <p>The sum can be expanded in the asymptotic series, several first members being
$$
\sum_{k=1}^n\sin\frac{1}{k}=
\log n+a+\frac1{2n}-\frac1{12n^3}+O\left(\frac1{n^3}\right),
$$
where
$$
a=\gamma+\sum_{k=1}^\infty (-1)^k\frac{\zeta(2k+1)}{(2k+1)!}
$$
and $\gamma$ is the Euler constant. The value of $a$ is $0.38...$... |
4,069,053 | <blockquote>
<p>If a finite field has characteristic 2 why is every element a square?</p>
</blockquote>
<p>I've attempted this problem by calculating the square root as follows: <span class="math-container">$\mathbb{F}$</span> has <span class="math-container">$q = 2^m$</span> elements, so if <span class="math-container... | Mark Bennet | 2,906 | <p>You can do it like this.</p>
<p>If <span class="math-container">$a^2=b^2$</span> then <span class="math-container">$a=\pm b$</span>, but in characteristic <span class="math-container">$2$</span> we have <span class="math-container">$b=-b$</span> so that if <span class="math-container">$a^2=b^2$</span> then <span cla... |
569,927 | <p>$p^{\; \left\lfloor \sqrt{p} \right\rfloor}\; -\; q^{\; \left\lfloor \sqrt{q} \right\rfloor}\; =\; 999$</p>
<p>How do you find positive integer solutions to this equation?</p>
| Dietrich Burde | 83,966 | <p>The exponents must be less than $4$ (we have $17^4-16^4=17985$ and
$16^4-15^3=62161$). There is an easy solution with $q=1$. The other solution arises from a difference of two cubes equal to $999$. In fact, $37=4^3-3^3$ is the difference of two consecutive cubes, and $999=3^3\cdot 37$. This gives
$$
12^3-9^3=999.
$$... |
367,643 | <p>A Norman window has the shape of a rectangle with a semi circle on top; diameter of the semicircle exactly matches the width of the rectangle. Find the dimensions of the Norman window whose perimeter is 300 in that has maximal area.</p>
<p><a href="https://i.stack.imgur.com/WBPm7.png" rel="nofollow noreferrer"><im... | colormegone | 71,645 | <p>OK, I spotted the error: when you wrote the expression (which Anil Baseski repeated) for the perimeter, you used the circumference of a circle written as $C = \pi d$. However, the "lunette" of the Norman window is only a semi-circle, so the perimeter equation should be $p = 2h + w + \frac{\pi}{2} w = 300$ . The... |
802,014 | <p>For all sets $A$, $B$, $C$, if $A$ is subset of $B$, $B$ is subset of $C$, and $C$ is subset of $A$, then $A = B = C$.</p>
<p>This is a true statement and I need to provide a proof? Thus, when a statement is false I need to provide it with counterexample whereas if it is true then it has to be provided by a proof?<... | Rene Schipperus | 149,912 | <p>Yes what you say is correct. The statement is an application of the Extensionality Axiom of set theory. Extensionality says that if two sets have the same elements then they are equal. It is easy to see from $A \subseteq B$ and $B \subseteq A$ that an element belongs to $A$ if and only if it belongs to $B$, so by ex... |
1,407,700 | <p>I am stuck on solving the following systems of equations with 3 variables. The textbook asks to use the addition method so can we please stick to that.</p>
<p>${5x -y = 3}$</p>
<p>${3x + z = 11}$</p>
<p>${y - 2z = -3}$</p>
<p>I am used to systems of equations where each equation has at least one instance of the... | Alex1357 | 137,382 | <p>$f$ is not uniformly continuous.
Short explanation:
For $x \to \infty$ we have asymptotically $\sin \frac 1x \approx \frac 1x$, so $f(x) \approx x^2$. </p>
<p>A bit longer explanation: There is an $y_0 > 0$ such that $0<\sin(y) < 2y$ for $0<y<y_0$. Consequently, $\sin(\frac 1x) > \frac 1 {2x}$ and... |
1,390,382 | <p>I have a problem that comes from absorbing random walks on a connected undirected graph $G$ with two types of nodes, absorbing nodes and free nodes. We randomly pick a node to start, once the random walk reaches an absorbing node, it will never leave the node again. But if we are at a free node, we will pick an outg... | Clement C. | 75,808 | <p>I may be missing something on the conditions: what about $$A=\begin{pmatrix} 0 & 1 & 0\\ 1& 0 & 0\\ 0&0&0\end{pmatrix}?$$ You have $A^3 = A$, so its powers cannot converge to $0$.</p>
|
3,360,879 | <p>Given the localised ring <span class="math-container">$\mathbb{Z}_{(2)}=\{\frac{a}{b}:a,b \in \mathbb{Z}, 2 \nmid b \}$</span>, I want to show that this is an integral domain.</p>
<p>We choose some fraction <span class="math-container">$ \frac{a}{b}\in \mathbb{Z}_{(2)}$</span>,where <span class="math-container">$... | Olórin | 187,521 | <p>Let <span class="math-container">$A$</span> be a commutative ring with unit and <span class="math-container">$S$</span> multiplicative subset of <span class="math-container">$A$</span> (contains <span class="math-container">$1$</span> by definition), and <span class="math-container">$S^{-1}A$</span> the localization... |
19,996 | <p>In 1556, Tartaglia claimed that the sums<br>
1 + 2 + 4<br>
1 + 2 + 4 + 8<br>
1 + 2 + 4 + 8 + 16<br>
are alternative prime and composite. Show that his conjecture is false. </p>
<p>With a simple counter example, $1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255$, apparently it's false. However, I want to prove it in gen... | Per Alexandersson | 934 | <p>If $2^{n}-1$ is prime, then $n$ must be odd, otherwise, we could factor
$2^{n}-1$ as $(2^{n/2}+1)(2^{n/2}-1).$</p>
<p>So <i>at least</i> every other number in the series $2^n-1,n=1,2,3...$
must be composite, by the conjugacy rule.</p>
<p>EDIT: Reread the question, but you can do the same for $a^3-b^3 = (a-b)(a^2+a... |
1,641,076 | <p>Let's say I have the following decomposition: </p>
<p>$$\{100,10011,00110\}^*$$</p>
<p>How would I determine if the decomposition is ambiguous or unambiguous?</p>
| Maria | 308,011 | <p>We know that the two planes hit at an intersection, and thus their
intersection should be orthogonal to the "facing" of said planes. Hence,
you can find the (orthogonal) vector of intersection by taking the
cross product the two normal vectors of these planes, since the cross
product of two vectors produces a vector... |
2,521,710 | <p>I am trying to do a proof for convergence. But I am stuck in my proof not getting any further... What is missing to finish that proof?</p>
<p>$$a_n = \frac{1}{(n+1)^2}$$
Show that: $$\lim_{n \to \infty}a_n=0$$</p>
<p>Let $e > 0$ and $\forall n \ge n_0 = \lceil \frac{1}{\sqrt{\epsilon}}\rceil+1 \in \mathbb Z^+... | boaz | 83,796 | <p>Let $\varepsilon>0$. We need to find $n_0$, such that
$$
\left|\frac{1}{(n+1)^2}-0\right|<\varepsilon
$$
for all $n_0\leq n$. </p>
<p>Set $n_0=\lceil 1/\varepsilon \rceil$. Note that if $n_0\leq n$, then
$$
\left|\frac{1}{(n+1)^2}-0\right|=\frac{1}{(n+1)^2}\leq\frac{1}{n+1}<\frac{1}{n}\leq\frac{1}{n_0}=\... |
1,921,562 | <p>Couldn't solve this indefinite integral, can someone help me? $$\int \frac {x^3+4x^2+6x+1}{x^3+x^2+x-3} dx$$</p>
| B. Goddard | 362,009 | <p>First, do long division to change your improper rational expression into a polynomial plus a proper rational expression. $$\int 1 +\frac{3x^2+5x+4}{x^3+x^2+x-3} \; dx.$$ Then factor the denominator by noting that if you plug in $x=1$ you get $0$, so $x-1$ is a factor. Then do partial fractions to get $$\int 1 + \... |
1,179,843 | <p>Proving $\sum_{n=1}^\infty \frac{\xi ^n}{n}$ is not uniformly convergent for $\xi \in (0,1)$.</p>
<p>I am trying to do the above. I have attempted to show it is not a cauchy sequence by considering $||\frac{\xi ^n}{n} ||_{\sup}$ but no avail. Any help please</p>
| user2566092 | 87,313 | <p>Hint: Show for $z$ close to $1$ that the convergence of the series becomes arbitrarily slow. More formally, show the negation of uniform convergence: There exists $\epsilon > 0$ such that for all positive integers $N$, there exists $z \in (0,1)$ and $n \geq N$ such that $|f_n(z) - f(z)| \geq \epsilon$. Here $f_n$... |
275,310 | <p>I am a bit confused. What is the difference between a linear and affine function? Any suggestions will be appreciated</p>
| kozenko | 145,312 | <p>An affine function is the composition of a linear function followed by a translation.
$ax$ is linear ; $(x+b)\circ(ax)$ is affine.
see Modern basic Pure mathematics : C.Sidney</p>
|
3,580,293 | <blockquote>
<p>The value of <span class="math-container">$$\lim\limits_{x \rightarrow \infty} \left(5^x + 5^{3x}\right)^{\frac{1}{x}}$$</span> is...</p>
</blockquote>
<p>My approach :</p>
<blockquote>
<p><span class="math-container">$$\lim\limits_{x \rightarrow \infty} \left(5^x + 5^{3x}\right)^{\frac{1}{x}}$$</... | Z Ahmed | 671,540 | <p><span class="math-container">$$(5^{3x})^{1/x}<(5^x+5^{3x})^{1/x} < 5^3(1+5^{-2x})^{1/x}$$</span>
So by sandwich theorem the required limit is 125.</p>
|
77,089 | <p>Fix a field $k$. For a singular variety $X$, I understand that the Grothendieck group $K^0(X)$ of vector bundles on $X$ is not necessarily isomorphic to the Grothendieck group $K_0(X)$ of coherent sheaves on $X$. </p>
<p>I am curious to learn what is known about these two groups in one family of examples: $\mathb... | Georges Elencwajg | 450 | <p>If $X$ is a noetherian separated scheme and $X_{red}$ its reduction , we have $K_0(X)=K_o(X_{red})$: in other words $K_o$ doesn't see nilpotents .<br>
Much more generally and profoundly, Quillen has proved that for all his $K$-theory groups, $K_i(X)=K_i(X_{red})$.<br>
In your particular case you thus have (in the ... |
1,234,661 | <p>I lost my baby Rudin book on real analysis book but I recall a pair of results in homework exercises that he seemed to indicate that there is no "boundary" between convergent and divergent series of positive decreasing terms. One result was that if $a_n$ is positive decreasing, and $\sum_n a_n$ is divergent, then $\... | zhw. | 228,045 | <p>If $\sum a_n = \infty,$ then we can find $n_1 < n_2 < \dots $ such that $\sum_{n_k\le n < n_{k+1}} a_n > 1$ for all $k.$ Define $b_n = a_n/k$ for $n_k\le n < n_{k+1}.$ Then $\sum b_n =\infty,$ and $b_n/a_n \to 0.$</p>
<p>If $\sum a_n < \infty,$ then we can find $n_1 < n_2< \dots $ such that ... |
1,475,235 | <p>Why doesn't $e^x$ have an inverse in the complex plane? Can someone please clarify it?</p>
| Jack D'Aurizio | 44,121 | <p>Here comes the overkill: by <a href="https://en.wikipedia.org/wiki/Picard_theorem" rel="nofollow">Great Picard's Theorem</a>, any analytic function with an essential singularity at infinity takes every complex value, with at most one exception, an infinite number of times. $e^z$ clearly has an essential singularity ... |
1,406,535 | <p>Let $ f$ be a function such that $|f(u)-f(v)|\leq|u-v|$ for all real $u$ and $v$ in an interval $[a,b]$.Then:<br>
$(i)$Prove that $f$ is continuous at each point of $[a,b]$.<br></p>
<p>$(ii)$Assume that $f$ is integrable on $[a,b]$.Prove that,$|\int_{a}^{b}f(x)dx-(b-a)f(c)|\leq\frac{(b-a)^2}{2}$,where $a\leq c \leq... | Jyrki Lahtonen | 11,619 | <p>I begin by trying also to answer your question:</p>
<blockquote>
<p>Why would we exclude $a$ such that $\gcd(a, n) = 1$ when the test works well <em>without</em> such condition? If we know that $\gcd(a, n) \not= 1$, we already that $n$ is not prime, thus the test is pointless.</p>
</blockquote>
<p>The answer is ... |
102,357 | <p>Let $G=PSL(2,q)$ where $q$ is prime power. What is Aut$(G\times G)$ and Aut$(G\times G\times G)$? Also if $G=A_{n}$ where $A_{n}$ is the alternating group of degree $n$, then what is Aut$(G\times G)$? </p>
<p>Thanks in advance</p>
| Igor Rivin | 11,142 | <p>See <a href="http://en.wikipedia.org/wiki/Direct_product_of_groups#Automorphisms_and_endomorphisms" rel="nofollow">this wikipedia article</a>. The result mentioned there implies that the automorphism group is the wreath product power of the automorphism group of $G.$</p>
|
181,855 | <p>In the latest <a href="http://what-if.xkcd.com/113/" rel="noreferrer">what-if</a> Randall Munroe ask for the smallest number of geodesics that intersect all regions of a map. The following shows that five paths of satellites suffice to cover the 50 states of the USA:
<img src="https://i.stack.imgur.com/gyfYt.png" al... | Gerhard Paseman | 3,206 | <p>Here is a suggestion following the idea of the original poster to show for the given instance that four is too low a bound. Assume that four geodesics suffice and aim for a contradiction as follows:</p>
<p>Consider five states sharing or almost sharing a common longitude, e.g. the five states north of Texas or tho... |
360,063 | <p>Let <span class="math-container">$\mathbb{N}$</span> denote the set of positive integers. For any prime <span class="math-container">$p\in\mathbb{N}$</span> let <span class="math-container">$p\mathbb{N} = \{np: n\in \mathbb{N}\}$</span>. Is there a partition <span class="math-container">${\cal P}$</span> of <span cl... | LSpice | 2,383 | <p>Recursively define a sequence of <span class="math-container">$B$</span>'s as follows. Initially, each is empty. At each step <span class="math-container">$n > 1$</span>, place <span class="math-container">$n$</span> in the first <span class="math-container">$B$</span> that contains only elements coprime to <sp... |
3,276,984 | <p>Why we divide the small difference 'd√x' by the difference in area 'dx' Where we normally divide the difference in area by the small difference ?<a href="https://i.stack.imgur.com/LDKB7.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LDKB7.jpg" alt="enter image description here"></a></p>
<p>My h... | Henry | 6,460 | <p>You are trying to find the rate of change in the side of a square relative to the change in area. So you have a division, similar to finding other instantaneous rates of change. </p>
<p>If <span class="math-container">$x$</span> is the area and <span class="math-container">$s=\sqrt{x}$</span> the side then you... |
28,955 | <p>I need to crack a stream cipher with a repeating key.</p>
<p>The length of the key is definitely 16. Each key can be any of the characters numbered 32-126 in ASCII.</p>
<p>The algorithm goes like this:</p>
<p>Let's say you have a plain text:</p>
<p>"Welcome to Q&A for people studying math at any level and pr... | Dave Hull | 254,289 | <p>There's a good description of an approach that can be used to crack this in the <a href="http://cryptopals.com" rel="nofollow">cryptopals.com</a> challenges, see set 1, problem 6 for details.</p>
<p>I've written a PowerShell script that can crack repeating xor key crypto based on the info in cryptopals.com's descri... |
32,849 | <p>I am trying to simulate a signal that randomly increases its phase, so far I have tried two thing but neither worked. I usually use matlab but I want to learn some <em>Mathematica</em> so I thought I would try this in <em>Mathematica</em>.</p>
<p>My first try was</p>
<pre><code>times = Table[i, {i, 0, 2, 0.05}];
f... | ssch | 1,517 | <p>Your problem can be reduced to creating an increasing function, <code>phase</code>, and then use <code>Sin[t + phase[t]]</code>.</p>
<p>Here is one way to do this by interpolating a sorted list of random numbers:</p>
<pre><code>tmax = 40;
phase = Interpolation[Sort[RandomReal[10, tmax]]];
Plot[phase[t], {t, 1, tma... |
130,914 | <p>I dont know how to proceed with solving $$\sum_{i=1}^{n}i^{k}(n+1-i).$$ Please give advise.</p>
| Ross Millikan | 1,827 | <p>You can factor out the $(n+1)$ to give $(n+1)\sum_{i=1}^n i^k-\sum_{i=1}^n i^{k+1}$ For positive integral $k$ you can use <a href="http://en.wikipedia.org/wiki/Faulhaber%27s_formula" rel="nofollow">Faulhaber's formulas</a>. What kind of $k$ are you considering?</p>
|
4,567,410 | <p>I know that when calculating the pre-image of a function <span class="math-container">$f:X \rightarrow Y$</span> in a given subset <span class="math-container">$B$</span>, that is, <span class="math-container">$f^{-1}(B)=\{x \in X\mid f(x) \in B\}$</span>,the <span class="math-container">$f^{-1}$</span> simbol is ju... | FShrike | 815,585 | <p>It is indeed false. It's true when <span class="math-container">$f(x)\in B$</span> implies <span class="math-container">$x\in A$</span> - equivalently, <span class="math-container">$f(X\setminus A)\subseteq Y\setminus B$</span>. This is true, for example, if <span class="math-container">$f$</span> is injective. But ... |
4,567,410 | <p>I know that when calculating the pre-image of a function <span class="math-container">$f:X \rightarrow Y$</span> in a given subset <span class="math-container">$B$</span>, that is, <span class="math-container">$f^{-1}(B)=\{x \in X\mid f(x) \in B\}$</span>,the <span class="math-container">$f^{-1}$</span> simbol is ju... | Arturo Magidin | 742 | <p><strong>Theorem.</strong> Let <span class="math-container">$f\colon X\to Y$</span> be a function. Then <span class="math-container">$f$</span> is one-to-one if and only if for every <span class="math-container">$A\subseteq X$</span>, if <span class="math-container">$f(A)=B$</span> then <span class="math-container">$... |
2,952,028 | <p>The question asks: Find the values of k for which the line</p>
<p><span class="math-container">$y=2x-k$</span> is tangent to the circle with equation <span class="math-container">$x^2+y^2=5$</span></p>
<p>So I started by substituting,</p>
<p><span class="math-container">$x^2+(2x-k)^2=5$</span></p>
<p><span class... | Szeto | 512,032 | <p>Rewrite it as
<span class="math-container">$$5x^2-(4k)x+(k^2-5)=0$$</span></p>
<p>Thus,
<span class="math-container">$$\Delta=16k^2-20(k^2-5)=100-4k^2$$</span></p>
<p>Setting <span class="math-container">$\Delta=0$</span>, we obtain <span class="math-container">$k=\pm 5$</span>.</p>
<p>Visualization by Desmos:
<a... |
3,787,111 | <p>Sets can have minimal and least elements, and they are two different things, for example:
given the set <span class="math-container">$A=\{\{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{1,3\},\{1,2,3\}\}$</span> and the subset relation, this set has <span class="math-container">$\{1\}$</span> as a minimal element but not as a l... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>Determine first the image of the interval <span class="math-container">$(-2,3)$</span>. Observe that <span class="math-container">$f$</span> is an even function, increasing on <span class="math-container">$\mathbf R^+$</span>, so that
<span class="math-container">$$f((-2,3))\subset f((-... |
3,787,111 | <p>Sets can have minimal and least elements, and they are two different things, for example:
given the set <span class="math-container">$A=\{\{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{1,3\},\{1,2,3\}\}$</span> and the subset relation, this set has <span class="math-container">$\{1\}$</span> as a minimal element but not as a l... | farruhota | 425,072 | <p>Alternatively, you can draw the graph:</p>
<p><span class="math-container">$\hspace{2cm}$</span><a href="https://i.stack.imgur.com/C1tDI.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/C1tDI.png" alt="enter image description here" /></a></p>
<p>Note that the extreme points of a function occur at e... |
67,630 | <p>I know of a theorem from Axler's <em>Linear Algebra Done Right</em> which says that if $T$ is a linear operator on a complex finite dimensional vector space $V$, then there exists a basis $B$ for $V$ such that the matrix of $T$ with respect to the basis $B$ is upper triangular.</p>
<p>The proof of this theorem is b... | Per Alexandersson | 934 | <p>You can always use gaussian elimination to put a matrix in UT form.
Every step in that process is reversible (invertible).
But, gaussian elimination can be thought of as a basis change,
since every step can also be represented as a matrix multiplication,
by an invertible matrix, (matrices looking like identity matr... |
3,716,619 | <p>Evaluating
<span class="math-container">$$\lim_{x\to 0}\left(\frac{\pi ^2}{\sin ^2\pi x}-\frac{1}{x^2}\right)$$</span>
with L'Hospital is so tedious. Does anyone know a way to evaluate the limit without using L'Hospital? I have no idea where to start.</p>
| robjohn | 13,854 | <p><strong>Pre-Calculus Answer to the Question</strong></p>
<p>Note that since <span class="math-container">$\lim\limits_{x\to0}\frac{\sin(x)}x=1$</span>, as shown in <a href="https://math.stackexchange.com/a/75151">this answer</a>, and <span class="math-container">$\frac1x$</span> is continuous at <span class="math-co... |
4,621,390 | <p>I'm studying Linear Algebra and have come to think of a column vector as an ordered bunch of objects, where each object is the product of a scalar and a basis vector, vis:</p>
<p><span class="math-container">$\begin{bmatrix}
a \\
b \\
\vdots \\
\end{bmatrix} = a \hat{i} + b \hat{j} + \dots$</span></p>
<p>Where <span... | StudentsTea | 170,148 | <p>@mr_e_man got it:</p>
<blockquote>
<p>If <span class="math-container">$\hat{i} = \hat{j}$</span>, then <span class="math-container">$(1) \hat{i} + (−1) \hat{j} = 0$</span>, which contradicts linear independence of <span class="math-container">$\hat{i}$</span> and <span class="math-container">$\hat{j}$</span>. A basi... |
4,651,364 | <p>I am trying to evaluate the following contour integral by evaluating the residue at <span class="math-container">$z=0$</span> of the integrand.</p>
<p><span class="math-container">$$I=\oint_{|z|=1} \frac{z^n+z^{-n}}{2iz(1-rz)(1-rz^{-1})}dz.$$</span></p>
<p>We can manipulate the integrand (which we denote by <span cl... | Pavan C. | 914,078 | <p>Suppose <span class="math-container">$d_X$</span> and <span class="math-container">$d_Y$</span> are the metrics of <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>, respectively.</p>
<p><span class="math-container">$f$</span> is isometric on <span class="math-container">$X_0$</span... |
945,334 | <p>Here is a lemma whose proof is as under:</p>
<blockquote>
<p>If $S \in L(X,Y)$ and lim$_{r \to 0}\frac{\|Sr\|}{\|r\|}=0$,then $S=0$.</p>
</blockquote>
<p>Proof:</p>
<p>The condition lim$_{r \to 0}\Big(\frac{\|Sr\|}{\|r\|}\Big)=0$means that for each $\epsilon \gt 0$ there is a $\delta \gt 0$ such that
... | Petr Naryshkin | 178,423 | <p>Let $(y_k)$ be a convergent subsequence with a limit $A$. Since $(x_n)$ is monotone, $(y_k)$ is monotone as well. Then every $y$ is smaller than $A$. Then We know that for every $eps > 0$ there exists $K: k > K \Rightarrow y_k > A - eps$. Then for every x after $y_k$ in original sequence it's also true sinc... |
2,929,025 | <p>Bill gave exams for the entrance at some specific gymnasium. <span class="math-container">$602$</span> students took part, which were classified, after the exams, in an ascending order, and the first <span class="math-container">$108$</span> students will be taken, which will accept to enter. Every student that has... | Acccumulation | 476,070 | <p>This is a binomial distribution with <span class="math-container">$p=.02$</span>, <span class="math-container">$n=112$</span>, and five successes required. So the simple way to find the answer is simply to find a binomial calculator. For instance, <a href="https://stattrek.com/online-calculator/binomial.aspx" rel="n... |
2,623,924 | <p>My textbook explains the proof, which I don't understand:</p>
<p>"Consider two bases $v_1...v_p$ and $w_1...w_q$ of V. Since the vectors $v$ are linearly independent and the vectors $w$ span V..."</p>
<p>How exactly does $w$ span V? </p>
<p>The book then says the same for vectors $v$, that $v$ spans V and hence $... | Arnaud Mortier | 480,423 | <p>Yes, you can find such examples, loads of them. However your question seems to be related to the big O notation so I'll answer in the spirit. </p>
<p><em>Just because $\log(f)>c\log(g)$ <strong>for this particular c</strong> doesn't mean that $\log(f)$ is not a $O(\log(g))$.</em></p>
<p>In @dxiv 's example, for... |
300,944 | <p>Show that there are no intergers $x$ and $y$ such that</p>
<p>$P(x,y)=x^2-5y^2=2$</p>
<p>Hint from professor:</p>
<p>Consider the equation in a convenient $\mod (n)$ so that you end up with a polynomial in a single variable. Then proceed as solving number of congruence.</p>
<hr>
<p>Im not sure how to approach t... | Herng Yi | 34,473 | <p>Suppose there exists integers $x$ and $y$ such that $x^2 - 5y^2 = 2$. Use the fact that</p>
<blockquote>
<p>Every square number is congruent to either $0$ or $1$ modulo $4$. $(\ast)$</p>
</blockquote>
<p>Hence, $x^2 - 5y^2 \equiv x^2 - y^2 \equiv 2 \pmod{4}$. However, the difference of two squares $x^2 - y^2 \eq... |
300,944 | <p>Show that there are no intergers $x$ and $y$ such that</p>
<p>$P(x,y)=x^2-5y^2=2$</p>
<p>Hint from professor:</p>
<p>Consider the equation in a convenient $\mod (n)$ so that you end up with a polynomial in a single variable. Then proceed as solving number of congruence.</p>
<hr>
<p>Im not sure how to approach t... | awllower | 6,792 | <p>I think your professor means to divide the equation by $5$, so that it becomes $x²\equiv 2 \pmod 5$. But, by the supplementary law of quadratic reciprocity, this is impossible.</p>
|
3,159,884 | <p>Prove that if <span class="math-container">$|z+w|=|z-w|$</span> then <span class="math-container">$z\overline{w}$</span> is purely imaginary.</p>
<p>To start off, I said let <span class="math-container">$z=a+bi$</span> and let <span class="math-container">$w=p+qi$</span>. Not sure where to go from here after subbin... | Bernard | 202,857 | <p>You don't even need to explicit <span class="math-container">$z$</span> and <span class="math-container">$w$</span>. You just have to show that
<span class="math-container">$$\overline{z\,\overline w}=-z\,\overline w.$$</span></p>
<p>You can use that <span class="math-container">$|a|=|b|\iff a\,\overline a=b\,\ove... |
83,565 | <p>I am learning Mathematica on the fly, one of my tasks is to find the variance of white noise. I followed the tutorial for finding white noise by using the code:</p>
<pre><code>WN = WhiteNoiseProcess[NormalDistribution[0, 10]];
data = RandomFunction[WN, {0, 10000}];
</code></pre>
<p>I know I can use the following c... | bbgodfrey | 1,063 | <p>The specific approach is as follows. Convert data to an ordinary list, eliminate an extra set of <code>{}</code>, and insert the list into your formula:</p>
<pre><code>dta = First@Normal@data;
Last@Total[(dta - Last@Mean[dta])^2]/(Length[dta] - 1)
</code></pre>
<p>which gives the same result as </p>
<pre><code>V... |
83,565 | <p>I am learning Mathematica on the fly, one of my tasks is to find the variance of white noise. I followed the tutorial for finding white noise by using the code:</p>
<pre><code>WN = WhiteNoiseProcess[NormalDistribution[0, 10]];
data = RandomFunction[WN, {0, 10000}];
</code></pre>
<p>I know I can use the following c... | kglr | 125 | <pre><code>SeedRandom[1]
WN = WhiteNoiseProcess[NormalDistribution[0, 10]];
data = RandomFunction[WN, {0, 10000}];
</code></pre>
<h2>CentralMoment</h2>
<pre><code>variance = # /(# - 1)& @ #["PathLength"] CentralMoment[#["Values"], 2]&;
variance[data]
</code></pre>
<blockquote>
<p>97.25341025240807</p>
</bl... |
275,974 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/264889/how-is-this-called-rationals-and-irrationals">How is this called? Rationals and irrationals</a> </p>
</blockquote>
<p>Please help me prove, that
$$\underset{n\rightarrow\infty}{\lim}\left(\underset{... | Calvin Lin | 54,563 | <p><strong>Hint:</strong> Show that if $x \in \mathbb{Q}$, then there exists some $N$ such that for $n > N$, $n! \pi x$ is an integer multiple of $2\pi$. Conclude that it tends to 1.</p>
<p>Show that if $x \not \in \mathbb{Q}$, then $\lim_{k\rightarrow \infty} [\cos (n! \pi x)]^{2k} = 0$.</p>
|
3,059,833 | <blockquote>
<p>If the equation <span class="math-container">$2^{2x} + a*2^{x+1} + a + 1=0$</span> has roots of opposite sign then the exhaustive values of a are?</p>
</blockquote>
<p>I tried taking <span class="math-container">$2^x = t$</span>. But then didn't know what to do.</p>
<p>The equation became, <span cla... | Shubham Johri | 551,962 | <p>In order for distinct real roots, <span class="math-container">$a^2-a-1>0$</span>.</p>
<p>Since <span class="math-container">$x_1>0,2^{x_1}=t_1=-a+\sqrt{a^2-a-1}>1$</span>. </p>
<p>Similarly, <span class="math-container">$x_2<0\therefore 0<2^{x_2}=t_2=-a-\sqrt{a^2-a-1}<1$</span>.</p>
|
3,401,630 | <p>I am trying to prove the inequality
<span class="math-container">$$\frac{1}{n}-\frac{1}{(n+1)^2}>\frac{1}{n+1}\quad \forall \ n>1.$$</span> How would I go about doing this? I've tried solving it on my own but my final answer is <span class="math-container">$1>0$</span>. </p>
| Dr. Sonnhard Graubner | 175,066 | <p>Multiplying by <span class="math-container">$n+1>0$</span> we get
<span class="math-container">$\frac{n+1}{n}-\frac{1}{n+1}>1$</span> and this is
<span class="math-container">$$(n+1)^2-n>n(n+1)$$</span> this is <span class="math-container">$$n^2+n+1>n^2+n$$</span> or <span class="math-container">$1>0$... |
1,765 | <p>Can anyone explain to me this behaviour? I've been having more than a couple of similar doubts these last weeks. </p>
<p>For example</p>
<pre><code>f[_?NumericQ] := 8;
</code></pre>
<p>Now, if I do</p>
<pre><code>With[{a = f[a]}, HoldForm@Block[{NumericQ = True &}, a]]
</code></pre>
<p>I get</p>
<pre><code... | Verbeia | 8 | <p>See the answers to <a href="https://mathematica.stackexchange.com/a/567/8">this question</a>. </p>
<p>The short answer is that, yes, <code>With</code> is designed for creating local <em>constants</em>, so once the local expressions are initialised, they won't be re-evaluated. </p>
<p>Using the <code>{a := f[a]}</c... |
28,892 | <p>I was searching on MathSciNet recently for a certain paper by two mathematicians. As I often do, I just typed in the names of the two authors, figuring that would give me a short enough list. My strategy was rather dramatically unsuccessful in this case: the two mathematicians I listed have written 80 papers toget... | lhf | 532 | <p>Perhaps Hardy and Littlewood?</p>
|
28,892 | <p>I was searching on MathSciNet recently for a certain paper by two mathematicians. As I often do, I just typed in the names of the two authors, figuring that would give me a short enough list. My strategy was rather dramatically unsuccessful in this case: the two mathematicians I listed have written 80 papers toget... | Eric Rowell | 6,355 | <p>E. Cline, B. Parshall, and L. Scott have 28 papers together, spanning 35 years or so. Just one example of a triple... </p>
|
4,272,755 | <p>Calculate the triple integral using spherical coordinates: <span class="math-container">$\int_C z^2dxdydz$</span> where C is the region in <span class="math-container">$R^3$</span> described by <span class="math-container">$1 \le x^2+y^2+z^2 \le 4$</span></p>
<p>Here's what I have tried:</p>
<p>My computation for <s... | Math Lover | 801,574 | <p>The region <span class="math-container">$C$</span> is <span class="math-container">$1 \le x^2+y^2+z^2 \le 4$</span>, which
is the entire region between spherical surfaces <span class="math-container">$x^2 + y^2 + z^2 = 1$</span> and <span class="math-container">$x^2 + y^2 + z^2 = 4$</span>.</p>
<p>So clearly, <span ... |
85,343 | <p>I am looking for a reference to study classical (i.e., not quantized) Yang-Mills theory. </p>
<p>Most of the sources I find focus on mathematical aspects of the theory, like Bleecker's book <em>Gauge theory and variational principles</em>, or Baez & Muniain's <em>Gauge fields, knots and gravity</em>.</p>
<p>Bu... | Steve Huntsman | 1,847 | <p>This is underrepresented in the literature. I have Nakahara and have looked at Frenkel (both listed in other answers) as well as many other "standard" references. The best book reference for classical YM theory that I found was <a href="http://books.google.com/books?id=BxjL6EkIpfUC">Rubakov's <em>Classical Theory of... |
85,343 | <p>I am looking for a reference to study classical (i.e., not quantized) Yang-Mills theory. </p>
<p>Most of the sources I find focus on mathematical aspects of the theory, like Bleecker's book <em>Gauge theory and variational principles</em>, or Baez & Muniain's <em>Gauge fields, knots and gravity</em>.</p>
<p>Bu... | Hollis Williams | 119,114 | <p>My personal suggestion is 'Differential Geometry, Gauge Theories, and Gravity' by M. Gockeler and T. Schucker. However, it assumes a fairly high degree of mathematical sophistication (it's one of the texts in the 'Cambridge Monographs on Mathematical Physics). If you do get it, it is really worth the effort to mas... |
2,666,409 | <blockquote>
<p>If $SL(2)=\{A\in \mathcal{M}_2: \det(A)=1\}$, find a parametrization around the identity matrix $I_2$ and find the first fundamental form. </p>
</blockquote>
<p>I've proved that $SL(2)$ is an hypersurface in $\mathbb{R}^4$, so it has dimension $3$. However, I don't understand very well what does "aro... | Mariano Suárez-Álvarez | 274 | <p>The group is the set of matrices $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ such that $ad-bc=1$. If you fix $a$, $b$ and $c$, then you can solve for $d$ using a somewhat obvious formula, and that formula works in a neighborhood of the point $(a,b,c)=(1,0,0)$ which gives the identity. This gives you a parametriza... |
1,064,091 | <p>I am asked to generate 200 and 1000 points from a bi-variate normal mixture densities.
I am trying to understand the algorithm, not just the matlab code (I have to write it, not use an existing function). I found a code on mathworks: <a href="http://www.mathworks.com/help/matlab/ref/randn.html#bufqioz-2" rel="nofoll... | Robert Israel | 8,508 | <p>The equality is quite definitely wrong. Try any example.</p>
|
226,449 | <p>Many counting formulas involving factorials can make sense for the case $n= 0$ if we define $0!=1 $; e.g., Catalan number and the number of trees with a given number of vetrices. Now here is my question:</p>
<blockquote>
<p>If $A$ is an associative and commutative ring, then we can define an
unary operation on ... | eggcrook | 71,261 | <p>The most elementary way to understand is just to work backwards. If you start with a sum of several elements and subtract them one at a time, you get $0$ after you finish. If you start with a product of several elements and divide them out one at a time, you get $1$ after you finish. Instead of thinking of $0!$ a... |
542,951 | <p>I am trying show that the function $f:[0,1]\to \mathbb{R}$ defined by $f(x)=\sin \dfrac{1}{x}$ if $x\neq 0$ and $f(0)=0$ possesses IVP. Though it looks easy, but I am not getting any clue how to start with. Any help would be appreciated.</p>
| JessicaK | 102,435 | <p>Consider the interval</p>
<p>$$I_{k} = \left[ \frac{1}{\left(2k+\frac{3}{2}\right)}, \frac{1}{\left(2k+\frac{1}{2}\right) \pi}\right]$$</p>
<p>notice that $-1\leq f(I_{k}) \leq 1$ for all $k\in \mathbb{N}$ and that $I_{1}\subseteq I_{2}\subseteq \dots$</p>
|
1,851,084 | <p>I have to solve the following problem:
find the matrix $A \in M_{n \times n}(\mathbb{R})$ such that:
$$A^2+A=I$$ and $\det(A)=1$.
How many of these matrices can be found when $n$ is given?
Thanks in advance.</p>
| quid | 85,306 | <p>The fact $A^2 + A = I$, means that $X^2 + X -1$ is an annihilating polynomial of $A$. </p>
<p>The minimal polynomial of $A$ thus divides that polynomial. Thus all the eigenvalues of $A$ are among the roots of $X^2 + X - 1$, which are<br>
$$\frac{-1 \pm \sqrt{5}}{2}.$$</p>
<p>Since the determinant is the product ... |
1,873,194 | <p>Entropy of random variable is defined as:</p>
<p>$$H(X)= \sum_{i=1}^n p_i \log_2(p_i)$$</p>
<p>Which as far as I understand can be interpreted as how many yes/no questions one would have to ask on average, to find out the value of the random variable $X$.</p>
<p>But what if the log base is changed to for example ... | Chill2Macht | 327,486 | <p>Obviously this interpretation breaks down (at least somewhat) for non-integer bases, but for any logarithm base $b$, not just base $b=2$, we can interpret the information with respect to that base, $$H_b(X) = \sum_{i=1}^n p_i \log_b(p_i)$$ to be the average number of $b-$ary questions one would have to ask on averag... |
4,552,955 | <p>I'm solving a probability problem, and I've ended up with this sum:</p>
<p><span class="math-container">$$\sum\limits_{k=0}^{n-a-b}\binom{n-a-b}{k}(a+k-1)!(n-a-k)!$$</span></p>
<p>WolframAlpha says I should get the answer <span class="math-container">$\frac{n!}{a\binom{a+b}{a}}$</span>, but I don't see how to get th... | Bruno B | 1,104,384 | <p>It seems to be the result of taking <span class="math-container">$x := a$</span>, <span class="math-container">$y := b+1$</span>, <span class="math-container">$z := 1$</span> and "<span class="math-container">$n$</span>" <span class="math-container">$:= n - a - b$</span> in the <a href="https://en.wikipedi... |
2,942,879 | <p>Suppose that <span class="math-container">$lim_{n\rightarrow \infty} a_n = L$</span> and <span class="math-container">$L \neq 0$</span>. Prove there is some <span class="math-container">$N$</span> such that <span class="math-container">$a_n \neq 0$</span> for all <span class="math-container">$n \geq N$</span>.</p>
... | Siong Thye Goh | 306,553 | <p>You might like to choose your <span class="math-container">$\epsilon$</span> explicitly might, for example, take <span class="math-container">$\epsilon = \frac{|L|}2$</span>.</p>
<p>Then we can find <span class="math-container">$N$</span> such that for all <span class="math-container">$n \ge N$</span>, we have <spa... |
3,910,623 | <p>There is a problem that appears in an interview<span class="math-container">$^\color{red}{\star}$</span> with <a href="https://en.wikipedia.org/wiki/Vladimir_Arnold" rel="nofollow noreferrer">Vladimir Arnol'd</a>.</p>
<blockquote>
<p>You take a spoon of wine from a barrel of wine, and you put it into your cup of tea... | Atbey | 327,944 | <p>The volume of spoon, <span class="math-container">$s$</span>, is the conserved quantity. It is also the amount of wine in the cup.<br> When you then take some mixture <span class="math-container">$\mathit{tea}+\mathit{wine} = s$</span> into the spoon, <br><span class="math-container">$s-\mathit{wine}$</span> is the ... |
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