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 |
|---|---|---|---|---|
184,361 | <p>I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one:</p>
<blockquote>
<p><strong>DEFINITION</strong> Given $x\in \Bbb R$, the integer part of $x$ is the unique $z\in \Bbb Z$ such that $$z\leq x < z+1$$ and we d... | N. S. | 9,176 | <p>This is one of my favourite exercises, because of the following neat solution:</p>
<p>Fix $n$. Let </p>
<p>$$ f(x) := \sum\limits_{k=0}^{n-1} \Biggl[x + \frac{k}{n}\Biggr] - [nx] \,.$$</p>
<p>Then $f(x) =0 \forall x \in [0,\frac{1}{n})$ since all terms are zero, and it is easy to prove that $f(x+\frac{1}{n})=f(x)... |
184,361 | <p>I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one:</p>
<blockquote>
<p><strong>DEFINITION</strong> Given $x\in \Bbb R$, the integer part of $x$ is the unique $z\in \Bbb Z$ such that $$z\leq x < z+1$$ and we d... | Community | -1 | <p>The trick for the inductive proof is that you want to do induction on $x$, not $n$, and you want to take steps of size $\frac{1}{n}$, not steps of size $1$.</p>
<p>For the base case, observe that it's true for $x \in [0, \frac{1}{n})$, since all of the terms are zero.</p>
<p>Then for the inductive step, you can se... |
70,143 | <p>Is there a good way to find the fan and polytope of the blow-up of $\mathbb{P}^3$ along the union of two invariant intersecting lines?</p>
<p>Everything I find in the literature is for blow-ups along smooth invariant centers.</p>
<p>Thanks!</p>
| David E Speyer | 297 | <p>The question is local near the intersection of the two lines, so the more basic question is to work this out for $\mathbb{A}^3$. </p>
<p>So we want to blow up $k[x,y,z]$ at $\langle z, xy \rangle$. There are two maximal charts:</p>
<p>$$\mathrm{Spec} \ k[x,y,z, (xy)/z] \ \mbox{and} \ \mathrm{Spec} \ k[x,y,z,z/(xy)... |
1,073,947 | <p>If I'm given a value $n$. And I know its of the form $p_{1}^x + p_{2}^y$, can I be sure that there is a unique solution for $x$ and $y$ and Can I determine values of $x$ and $y$, If I know the primes $p_{1}$ and $p_{2}$ (and $p_{1} \ne p_{2}$) </p>
<p>If yes, how to prove it?</p>
| ajotatxe | 132,456 | <p>From
$$p^x+q^y=p^s+q^t$$
if $(x,y)\neq(s,t)$, we have
$$p^s(p^{x-s}-1)=q^y(q^{t-y}-1)$$</p>
<p>so $p\equiv 1\pmod q$ and $q\equiv 1\pmod p$. This is not possible.</p>
<p>THIS IS WRONG: PLEASE UNACCEPT (sorry for caps, only for visibility)</p>
<p>To find the exponents, if $n$ is not <em>very</em> big, the fastest ... |
1,073,947 | <p>If I'm given a value $n$. And I know its of the form $p_{1}^x + p_{2}^y$, can I be sure that there is a unique solution for $x$ and $y$ and Can I determine values of $x$ and $y$, If I know the primes $p_{1}$ and $p_{2}$ (and $p_{1} \ne p_{2}$) </p>
<p>If yes, how to prove it?</p>
| Mike Bennett | 59,326 | <p>The short answer to the question is "no". For each of
$$
n \in \{ 11, 35, 133, 259, 2200 \},
$$
we can find distinct primes $p$ and $q$ for which we have
$$
p^a+q^b=p^c+q^d = n
$$
with $(a,b) \neq (c,d)$ and all of $a, b, c$ and $d$ positive integers. Probably this property is not true for other values of $n$, but t... |
756,735 | <blockquote>
<p>Let $n>0$ be a positive integer. For all $x\not=0$, prove that $f(x) = 1/x^n$ is differentiable at $x$ with $f^\prime(x) = -n/x^{n+1}$ by showing that the limit of the difference quotient exists.</p>
</blockquote>
<p>I am having trouble seeing how I can manipulate the difference quotient in order ... | J.R. | 44,389 | <p>Let $x\not=0$. Use the binomial theorem:</p>
<p>\begin{align}
\frac{\frac{1}{(x+h)^n}-\frac{1}{x^n}}{h} &=-\frac{(x+h)^n - x^n}{(x+h)^n x^n h}\\
&=-\frac{1}{h (x+h)^n x^n} \left(\sum_{k=0}^n {n\choose k} x^k h^{n-k} - x^n\right)\\
&=-\frac{1}{h (x+h)^n x^n} \sum_{k=0}^{n-1} {n\choose k} x^k h^{n-k}\\
&a... |
3,035,228 | <p>Show that two cardioids <span class="math-container">$r=a(1+\cos\theta)$</span> and <span class="math-container">$r=a(1-\cos\theta)$</span> are at right angles.</p>
<hr>
<p><span class="math-container">$\frac{dr}{d\theta}=-a\sin\theta$</span> for the first curve and <span class="math-container">$\frac{dr}{d\theta}... | Nosrati | 108,128 | <p>The angle between tangent line and a ray from pole of a polar curve is
<span class="math-container">$$\tan\psi=\dfrac{r}{r'}$$</span>
then for these curves in every <span class="math-container">$\theta$</span> on curves
<span class="math-container">$$\tan\psi_1=\dfrac{r_1}{r'_1}=\dfrac{a(1-\cos\theta)}{a\sin\theta}... |
1,333,994 | <p>We have a function $f: \mathbb{R} \to \mathbb{R}$ defined as</p>
<p>$$\begin{cases} x; \ \ x \notin \mathbb{Q} \\ \frac{m}{2n+1}; \ \ x=\frac{m}{n}, m\in \mathbb{Z}, n \in \mathbb{N} \ \ \ \text{$m$ and $n$ are coprimes} \end{cases}.$$</p>
<p>Find where $f$ is continuous</p>
| Landon Carter | 136,523 | <p>If you know the AM-GM inequality, this is easy.</p>
<p>$a=\log_{\frac{1}{2}}(3)=-\log_23$ and $b=\log_3(\dfrac{1}{2})=-\log_32$.</p>
<p>Now $-a>0,-b>0$ so $-a-b>2\sqrt{|ab|}=2$ implying $a+b<-2$.</p>
|
4,354,133 | <p>In general, if <span class="math-container">$K$</span> is a field, it could be that exists <span class="math-container">$f(x)\in K[x]$</span> such that <span class="math-container">$f(a)=0$</span> for all <span class="math-container">$a\in K$</span>; for example, set <span class="math-container">$K:=\mathbb Z/(2)$</... | Jonas Linssen | 598,157 | <p>For <span class="math-container">$K$</span> a field <span class="math-container">$K[X_1,…,X_n]$</span> is an integral domain, in particular there are no nilpotent elements besides <span class="math-container">$0$</span>. Your argument shows that if <span class="math-container">$f(x_1,…,x_n)=0\in K$</span> for all ch... |
4,354,133 | <p>In general, if <span class="math-container">$K$</span> is a field, it could be that exists <span class="math-container">$f(x)\in K[x]$</span> such that <span class="math-container">$f(a)=0$</span> for all <span class="math-container">$a\in K$</span>; for example, set <span class="math-container">$K:=\mathbb Z/(2)$</... | Hank Scorpio | 843,647 | <p>What you ask happens exactly when <span class="math-container">$K$</span> is infinite. If <span class="math-container">$K=\Bbb F_q$</span> is finite, just take <span class="math-container">$\prod_{a\in K} (x_1-a)$</span> which vanishes on every element of <span class="math-container">$\Bbb F_q^n$</span> but not on <... |
345,766 | <p>I'm trying to calculate this limit expression:</p>
<p>$$ \lim_{s \to \infty} \frac{ab + (ab)^2 + ... (ab)^s}{1 +ab + (ab)^2 + ... (ab)^s} $$</p>
<p>Both the numerator and denominator should converge, since $0 \leq a, b \leq 1$, but I don't know if that helps. My guess would be to use L'Hopital's rule and take the ... | Cameron Buie | 28,900 | <p><strong>Edit</strong>: Since you've added in the assumption that $a,b\in[0,1]$, then $0\le ab\le 1$, and so the numerator and denominator only diverge in the case that $a=b=1$. In that case, L'Hopital's rule does indeed yield a limit of $1$...which is precisely $ab$.</p>
<p>Otherwise, we have $$1+ab+\cdots+(ab)^s=\... |
345,766 | <p>I'm trying to calculate this limit expression:</p>
<p>$$ \lim_{s \to \infty} \frac{ab + (ab)^2 + ... (ab)^s}{1 +ab + (ab)^2 + ... (ab)^s} $$</p>
<p>Both the numerator and denominator should converge, since $0 \leq a, b \leq 1$, but I don't know if that helps. My guess would be to use L'Hopital's rule and take the ... | André Nicolas | 6,312 | <p><strong>Hint:</strong> Use the closed form expression
$$1+r+r^2+\cdots +r^{n}=\frac{1-r^{n+1}}{1-r}.$$
Note that this only applies for $r\ne 1$. </p>
|
2,666,425 | <p>I'm working through Vakil's excellent The Rising Sea notes, and in an exercise, the following question is posed:</p>
<p>If $X$ is a topological space, show that the fibered product always exists in the category of open sets of $X$, by describing what a fibered product is. </p>
<p>Now I know intuitively the fibered... | lush | 499,195 | <p>So lets take two open sets $U,V \subseteq Y \subseteq X$, where $Y$ is open as well.</p>
<p>That means we have two morphisms $U \to Y$, $V \to Y$.
Now the fibered product of those morphisms (lets call it $W$) is an object of the same category - hence it is an open subset of $X$.
Furthermore we need some maps $W \to... |
195,150 | <p>Of all the possible combinations of positive numbers that sum to 10, which has the largest multiplication?</p>
<p>I had also got a clue: it's related to <a href="http://en.wikipedia.org/wiki/E_%28mathematical_constant%29" rel="nofollow"><code>e</code></a>.</p>
<p>Please help! (I need explanation aswell)</p>
<p><s... | Seyhmus Güngören | 29,940 | <p>It is a simple sort of constrained optimization problem. Assume we have two positive numbers adding upto $10$, $x+y=10$, find $x$ and $y$ subject to $\max_{\forall x,y}\, xy$. If you rewrite this $\max_{\forall x,y}x(10-x)$ we have $\frac{d(10x-x^2)}{dx}=0$ then we have $x=5$. If we have 3 numbers or four numbers e... |
195,150 | <p>Of all the possible combinations of positive numbers that sum to 10, which has the largest multiplication?</p>
<p>I had also got a clue: it's related to <a href="http://en.wikipedia.org/wiki/E_%28mathematical_constant%29" rel="nofollow"><code>e</code></a>.</p>
<p>Please help! (I need explanation aswell)</p>
<p><s... | Community | -1 | <p>The answers above give the usual methods. Here's a method I've come up with taking up your clue of a link with $e$.</p>
<p>You know that $$x+y+z+t = 10 \:\:\;\:\;\;\;\;\; x,y,z,t > 0$$ and wish to maximise $P=xyzt$. Take the natural logarithm to get $$\log P = \log x + \log y + \log z + \log t.$$ Now $P$ is maxi... |
3,953,674 | <p>Here is a common argument used to prove that the sum of an infinite geometric series is <span class="math-container">$\frac{a}{1-r}$</span> (where <span class="math-container">$a$</span> is the first term and <span class="math-container">$r$</span> is the common ratio):
<span class="math-container">\begin{align}
S &... | A.J. | 654,406 | <p>I think the proofs generally start (or at least they should) with something like "Suppose an infinite geometric series converges to the sum <span class="math-container">$S \,$</span>", after which all the steps would be justified.</p>
<p>The 'cancelling' you mentioned can be avoided by including an extra s... |
13,705 | <p>Let $m$ be a positive integer. Define $N_m:=\{x\in \mathbb{Z}: x>m\}$. I was wondering when does $N_m$ have a "basis" of two elements. I shall clarify what I mean by a basis of two elements: We shall say the positive integers $a,b$ generate $N_m$ and denote $N_m=<a,b>$ if every element $x\in N_m$ can be wri... | Jason DeVito | 331 | <p>This can only happen for the triple you found.</p>
<p>For, if $m=1$, then we must be able to generate 2, so we must take 2 in the basis. Likewise, we must be able to generate 3, so we must allow 3 in the basis.</p>
<p>If $m>1$, then we see by the same reasoning that $N_m$ must be generated by $m+1$ and $m+2$.<... |
1,530,406 | <p>How to multiply Roman numerals? I need an algorithm of multiplication of numbers written in Roman numbers. Help me please. </p>
| Sandith | 689,495 | <p>It is conversion to Hindu numerals, Not Arabic. Just because Europeans learnt from Arabs does not mean the founders change!</p>
<p><a href="https://rbutterworth.nfshost.com/Tables/romanmult" rel="nofollow noreferrer">https://rbutterworth.nfshost.com/Tables/romanmult</a> is one place multiplication is explained</p>
|
514,517 | <p>So this is what my book states:</p>
<p>Random variables $X,Y, and Z$ are said to form a Markov chain in that order denoted $X\rightarrow Y \rightarrow Z$ if and only if:</p>
<p>$p(x,y,z)=p(x)p(y|x)p(z|y) $</p>
<p>That's great and all but that doesn't give any intuition as to what a Markov chain is or what it impl... | Prahlad Vaidyanathan | 89,789 | <p>Hint: Use intermediate value theorem on $g(x) = f(x) - x$</p>
|
1,480,511 | <p>I have included a screenshot of the problem I am working on for context. T is a transformation. What is meant by Tf? Is it equivalent to T(f) or does it mean T times f?</p>
<p><a href="https://i.stack.imgur.com/LtRS1.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LtRS1.png" alt="enter image desc... | YoTengoUnLCD | 193,752 | <p>Using: <a href="https://math.stackexchange.com/q/4468">Square root of integer is integer if rational</a>:</p>
<p>Suppose <span class="math-container">$\sqrt {15}$</span> is rational, then we'd have a positive integer <span class="math-container">$x$</span> such that <span class="math-container">$x^2=15$</span>, by t... |
720,823 | <p>I was solving some linear algebra problems and I have a quick question about one problem. I'm given the matrix $A = \{a_1,a_2\}$ where $a_1=[1,1]$ and $a_2=[-1,1]$. I need to solve for the eigenvalues of the matrix $A$ over the complex numbers $\mathbb{C}$. I solved and got the eigenvalue $1-i$ and $1+i$. Now this i... | Amzoti | 38,839 | <p>The $[A- \lambda I]v$ versus $[\lambda I - A]v$ is just a matter of choice. When we have $Av = \lambda v$, we can choose to subtract from either side, so just a convention. Some people hate negating each term of the matrix as this leaves more room for error. </p>
<p>We are given:</p>
<p>$$A = \begin{bmatrix}1 &... |
1,904,553 | <blockquote>
<p>$$\displaystyle\lim_{x\to0}\left(\frac{1}{x^5}\int_0^xe^{-t^2}\,dt-\frac{1}{x^4}+\frac{1}{3x^2}\right)$$</p>
</blockquote>
<p>I have this limit to be calculated. Since the first term takes the form $\frac 00$, I apply the L'Hospital rule. But after that all the terms are taking the form $\frac 10$. S... | Olivier Oloa | 118,798 | <p>One may recall that, as $t \to 0$, by the Taylor series expansion
$$
e^{-t^2}=1-t^2+\frac{t^4}2+O(t^6)
$$ giving, as $x \to 0$,
$$
\int_0^xe^{-t^2}dt=x-\frac{x^3}3+\frac{x^5}{10}+O(x^7)
$$ and, as $x \to 0$,</p>
<blockquote>
<p>$$
\frac1{x^5}\int_0^xe^{-t^2}dt-\frac1{x^4}+\frac1{3x^2}=\frac1{10}+O(x^2)
$$</p>
</b... |
625,608 | <p>The question is to show that the function $\phi$ given by $\phi(\lambda)=\frac{\lambda}{1+|\lambda|}$ is 1-1 on the complex plane. I would be grateful for a hint on how to start.</p>
| user118829 | 118,829 | <p>Assume that $\phi(\lambda) = \phi(\mu)$. First show that $\lambda$ and $\mu$ must have the same argument. Then show that they have the same modulus.</p>
|
250,496 | <p>Please help me out.. Is there some appropriate method to draw Hasse diagram</p>
<blockquote>
<p>My question is $L=\{1,2,3,4,5,6,10,12,15,30,60\}$</p>
</blockquote>
<p>Please explain me by step by step solution... Thanks for help..</p>
| user2205316 | 221,340 | <p>You have <span class="math-container">$L=\{1,2,3,4,5,6,10,12,15,20,30,60\}$</span>. <span class="math-container">$|L|$</span> = 12. This tells you the first important bit of information: there will be <span class="math-container">$12$</span> points in your Hasse diagram. (Please note that 20 was missing from the s... |
3,875,643 | <p>I am studying the nonlinear ordinary differential equation</p>
<p><span class="math-container">$$\frac{d^2y}{dx^2}=\frac{1}{y}-\frac{x}{y^2}\frac{dy}{dx}$$</span></p>
<p>I have entered this equation into two different math software packages, and they produce different answers.</p>
<p>software 1:</p>
<p><span class="... | Lutz Lehmann | 115,115 | <p>As a symbolic solution exists, it should be possible to transform the equation and integrate it with relatively elementary means. And indeed, by careful examination one finds that the right side is the derivative of <span class="math-container">$\frac xy$</span>, so that a direct integration to
<span class="math-con... |
3,168,130 | <p><a href="https://math.stackexchange.com/questions/2690416/mathematical-proof-of-uniform-circular-motion">Here</a> is a mathematical proof that any force <span class="math-container">$F(t)$</span>, which affects a body, so that <span class="math-container">$\vec{F(t)} \cdot \vec{v(t)} = 0$</span>, where <span class="... | G.Carugno | 572,366 | <p>Try to think like this: the component of the force parallel to the velocity changes the module of the velocity, while component perpendicular to the velocity changes its direction. Since you have only a perpendicular component, the velocity will stay constant in module while changing its direction. Now, if your forc... |
3,168,130 | <p><a href="https://math.stackexchange.com/questions/2690416/mathematical-proof-of-uniform-circular-motion">Here</a> is a mathematical proof that any force <span class="math-container">$F(t)$</span>, which affects a body, so that <span class="math-container">$\vec{F(t)} \cdot \vec{v(t)} = 0$</span>, where <span class="... | Siddharth Bhat | 261,373 | <p>Let the velocity vector be defined as <span class="math-container">$v(t) = (s \cos( \theta(t)), s \sin (\theta(t))$</span>, where <span class="math-container">$\theta(t)$</span> is a time varying quantity. Note that <span class="math-container">$s$</span> is a constant, since we have already proved that the <em>spe... |
1,320,112 | <p>Using the following identity $$\int_{0}^{\infty}\frac{f\left ( t \right )}{t}dt= \int_{0}^{\infty}\mathcal{L}\left \{ f\left ( t \right ) \right \}\left ( u \right )du$$
I rewrote $$\int_{0}^{\infty}\frac{\sin^{2}\left ( t \right )}{t^{2}}dt$$
as $$\int_{0}^{\infty}\frac{\sin^{2}\left ( t \right )}{t\cdot t}dt$$
And... | Mark Viola | 218,419 | <p>We have </p>
<p>$$\begin{align}
\int_0^{\infty}\frac{\sin^2t}{t}e^{-ut}&=\frac{1}{2}\int_0^{\infty}\frac{1-\cos t}{t}e^{-(u/2)t}dt\\\\
&=\frac12\log\left(\frac{(u/2)^2+1}{(u/2)^2}\right)\\\\
&=\frac12 \log\left(\frac{u^2+4}{u^2}\right)
\end{align}$$</p>
|
78,243 | <p>A positive integer $n$ is said to be <em>happy</em> if the sequence
$$n, s(n), s(s(n)), s(s(s(n))), \ldots$$
eventually reaches 1, where $s(n)$ denotes the sum of the squared digits of $n$.</p>
<p>For example, 7 is happy because the orbit of 7 under this mapping reaches 1.
$$7 \to 49 \to 97 \to 130 \to 10 \to 1$$
B... | Joe Silverman | 11,926 | <p>Helen Grundman has a written a number of articles about happy numbers. (I first heard of them at a talk of hers at a JMM.) References for her articles are listed below. I don't know if they discuss densities. One can also look at happy numbers to other bases, of course. According to Wikipedia: "The origin of happy n... |
78,243 | <p>A positive integer $n$ is said to be <em>happy</em> if the sequence
$$n, s(n), s(s(n)), s(s(s(n))), \ldots$$
eventually reaches 1, where $s(n)$ denotes the sum of the squared digits of $n$.</p>
<p>For example, 7 is happy because the orbit of 7 under this mapping reaches 1.
$$7 \to 49 \to 97 \to 130 \to 10 \to 1$$
B... | Justin Gilmer | 18,635 | <p>The answer is almost certainly that the limiting density does not exist. Without going into the details of the proof allow me to give a heuristic argument which is based on how the OP likely generated his graph of the relative frequency of happy numbers. </p>
<p>Let $Y_n$ be the r.v. uniformly distributed amongst i... |
320,355 | <p>Show that $$\nabla\cdot (\nabla f\times \nabla h)=0,$$
where $f = f(x,y,z)$ and $h = h(x,y,z)$.</p>
<p>I have tried but I just keep getting a mess that I cannot simplify. I also need to show that </p>
<p>$$\nabla \cdot (\nabla f \times r) = 0$$</p>
<p>using the first result.</p>
<p>Thanks in advance for any help... | Muphrid | 45,296 | <p>While we're here having fun, you can prove this easily enough with geometric calculus. Instead of using the cross product, we use the wedge product instead and Hodge duality.</p>
<p>$$\nabla \cdot (\nabla f \times \nabla h) = \nabla \cdot [-i (\nabla f \wedge \nabla h)] = -i [\nabla \wedge (\nabla f \wedge \nabla ... |
2,072,666 | <p>I have a set as : <b> {∀x ∃y P(x, y), ∀x¬P(x, x)}. </b>. In order to satisfy this set I know that there should exist an interpretation <b> I </b> such that it should satisfy all the elements in the set. For instance my interpretation for x is 3 and for y is 4. Should I apply the same numbers (3,4) to ∀x¬P(x, x) as ... | Olivier Oloa | 118,798 | <p><strong>Hint</strong>. We have
$$
x \to 0 \Longleftrightarrow -\frac{2x}{e} \to 0.
$$</p>
|
332,993 | <p>How do I approach the problem?</p>
<blockquote>
<p>Q: Let $ \displaystyle z_{n+1} = \frac{1}{2} \left( z_n + \frac{1}{z_n} \right)$ where $ n = 0, 1, 2, \ldots $ and $\frac{-\pi}{2} < \arg (z_0) < \frac{\pi}{2} $. Prove that $\lim_{n\to \infty} z_n = 1$.</p>
</blockquote>
| NECing | 60,869 | <p>If the limit exists, then
$$\lim_{n\to\infty}z_{n+1}=\lim_{n\to\infty}z_n$$
Substitute it in, you can get
$$\lim_{n\to\infty}z_n=\pm1$$
Since there is one more constrain on arg$(z)$, you can conclude that the limit must be $1$.</p>
<p>It is not hard to prove the existence of limit.</p>
<p>Assume $|z_n|>1$, th... |
2,699,753 | <p>$a,b$ and $c$ are all natural numbers, and function $f(x)$ always returns a natural number. If$$
\sum_{n=b}^{a} f(n) = c,$$
in terms of $b,c$ and $f$, how would you solve for $a$? Do I require more information to solve for $a$?</p>
<p>EDIT: If $x$ increases $f(x)$ increases</p>
| Alexander Burstein | 499,816 | <p>Your solution to this problem continues as follows. Divide the recurrence relation through by $9^n$ to obtain $g(n)=g(n-1)-\frac{14}{9^n}$ for $n\ge 1$, and $g(0)=3$. Then
$$
g(n)=3-\sum_{k=1}^{n}{\frac{14}{9^k}}=3-14\frac{\frac{1}{9}-\frac{1}{9^{n+1}}}{1-\frac{1}{9}}=3-\frac{14}{8}\left(1-\frac{1}{9^n}\right)=\frac... |
1,690,854 | <p>Solve the equation
$$-x^3 + x + 2 =\sqrt{3x^2 + 4x + 5.}$$
I tried. The equation equavalent to
$$\sqrt{3x^2 + 4x + 5} - 2 + x^3 - x=0.$$
$$\dfrac{3x^2+4x+1}{\sqrt{3x^2 + 4x + 5} + 2}+x^3 - x=0.$$
$$\dfrac{(x+1)(3x+1)}{\sqrt{3x^2 + 4x + 5} + 2}+ (x+1) x (x-1)=0.$$
$$(x+1)\left [\dfrac{3x+1}{\sqrt{3x^2 + 4x + 5} + 2}+... | Claude Leibovici | 82,404 | <p><em>This is not an answer since based on approximations.</em></p>
<p>Consider the function $$f(x)=\frac{-x^3 + x + 2 -\sqrt{3x^2 + 4x + 5}}{x+1}$$ Around $x=0$ it looks as a parabola; so the function can be approximated by a Taylor series to third order. So, around $x=0$, we have $$f(x)\approx \left(2-\sqrt{5}\righ... |
228,437 | <p>The ODE in question: <code>y'' + 3y' + 2y = 8t + 8</code></p>
<p>But I get something like this for my solution:</p>
<p><a href="https://i.stack.imgur.com/6QFoV.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/6QFoV.png" alt="enter image description here" /></a></p>
<p>I also tried getting the solut... | Ulrich Neumann | 53,677 | <p>You should use <code>==</code> instead of <code>=</code> to define the equations:</p>
<pre><code>DSolve[y''[t] + 3 y'[t] + 2 y[t] == 8 t + 8, y, t]
(*{{y -> Function[{t}, 2 (-1 + 2 t) + E^(-2 t) C[1] + E^-t C[2]]}}*)
</code></pre>
|
315,235 | <p>I am learning about vector-valued differential forms, including forms taking values in a Lie algebra. <a href="http://en.wikipedia.org/wiki/Vector-valued_differential_form#Lie_algebra-valued_forms">On Wikipedia</a> there is some explanation about these Lie algebra-valued forms, including the definition of the operat... | Adittya Chaudhuri | 311,277 | <p>I studied it from <em>Differential Geometry Connections,Curvature and Characteristic Classes</em> by <strong>Loring W.Tu</strong> in the Chapter 4, Section 21(specially subsection 21.5). The whole section 21 (Vector-valued forms) is very well treated. As a Special case Lie-Algebra valued differential forms are discu... |
1,930,558 | <p>I want a good textbook covering elemenents of discrete mathematics Average level.Im a mathematics undergraduate so i dont want it to be towards Computer science that much.Interested in combinatorics and graph theory .But also covering enumeration and other stuff.One book i found is <a href="http://rads.stackoverflow... | TheProofIsTrivial | 384,536 | <p>The Book of Proof is a free PDF that is fairly good and easy to read, lots of examples. 'Discrete Mathematics with Graph Theory' By Goodaire and Parmenter may have what you're looking for and is decent, however, I feel it's too much of a 'light read' and the authors' humour is questionable. But there are lots of exa... |
2,089,619 | <p>I have proven that if $X $ is a finite set and $Y$ is a proper set of $X$, then does not exist $f:X \rightarrow Y $ such that $f$ is a bijection.</p>
<p>I'm pretending to show that the naturals is an infinite set. Let $P=\{2,4,6...\} $. So, by contraposity i just have to show that $f: \mathbb{N} \rightarrow P$, gi... | Arthur | 15,500 | <p>The multiplicative inverse of $[a]\in \Bbb Z_n$ only exists if $\gcd(a,n)=1$. That's not the case for your $[2]\in \Bbb Z_4$.</p>
<p>As for how to calculate something like, say, $[2]^{-1}\cdot [3]$ in $\Bbb Z_5$, you first have to <em>find</em> $[2]^{-1}$, which happens to be $[3]$. This makes $[2]^{-1}\cdot[3]=[3]... |
2,089,619 | <p>I have proven that if $X $ is a finite set and $Y$ is a proper set of $X$, then does not exist $f:X \rightarrow Y $ such that $f$ is a bijection.</p>
<p>I'm pretending to show that the naturals is an infinite set. Let $P=\{2,4,6...\} $. So, by contraposity i just have to show that $f: \mathbb{N} \rightarrow P$, gi... | Joshua Mundinger | 106,317 | <p>Not every element has a multiplicative inverse modulo $n$.
Indeed, since $[2] \cdot [2] = 0$, if $[2]$ had an inverse than $[0] = [2] \cdot [2] \cdot [2]^{-1} = [2]$, a contradiction.</p>
<p>In general, in $\mathbb Z/ n\mathbb Z$ a residue class $[m]$ has a multiplicative inverse if and only if $m$ and $n$ are rela... |
227,618 | <p>I'm creating AI for a card game, and I run into problem calculating the probability of passing/failing the hand when AI needs to start the hand. Cards are A, K, Q, J, 10, 9, 8, 7 (with A being the strongest) and AI needs to play to not take the hand.</p>
<p>Assuming there are 4 cards of the suit left in the game an... | Dan Shved | 47,560 | <p>Denote $A=\{0,1\}$. You can use the fact that $\mathbb{R} \simeq A^{\mathbb{N}}$, where $\simeq$ means "have the same cardinality". So we have
$$
\mathbb{R}^{\mathbb{R}} \simeq \left(A^{\mathbb{N}}\right)^{\mathbb{R}} \simeq A^{\mathbb{N} \times \mathbb{R}}.
$$
Now we use the fact that $\mathbb{N}\times\mathbb{R} \s... |
3,895,275 | <p>I have a matrix <span class="math-container">$A= \begin{pmatrix} 5 & 3 \\ 2 & 1 \end{pmatrix} $</span> and I should find <span class="math-container">$m$</span>, <span class="math-container">$n$</span>, <span class="math-container">$r$</span> in case that <span class="math-container">$A^2+nA+rI=0$</span> (<... | Bernard | 202,857 | <p><strong>Hint</strong>: Compute the characteristic polynomial and apply <em>Hamilton-Cayley</em>.</p>
|
3,651,287 | <p>I would be very grateful if you could help me, I have a question about the Cauchy sequences, they have given me the definition that a Cauchy sequence if:</p>
<p>A sequence <span class="math-container">$(r_{n})_{n\in \mathbb{N}}$</span> is of Cauchy if:</p>
<p><span class="math-container">$\forall \epsilon>0,$</... | Guangyi Zou | 780,777 | <p>To prove the second definition from the first, you just need to fix <span class="math-container">$m=N+1,$</span>and then <span class="math-container">$|a_n-a_m|<\epsilon$</span> for all <span class="math-container">$n>m$</span>.So the second holds if the first holds.</p>
<p>To prove the first from the second,... |
2,843,822 | <p>How do I rewrite $(1\,2)(1\,3)(1\,4)(1\,5)$ as a single cycle? I have tried questions in the form: $(1\,4\,3\,5\,2)(4\,5\,3\,2\,1)$.</p>
| Arnaud Mortier | 480,423 | <p>Keep in mind that as a composition of functions, $(1\,2)(1\,3)(1\,4)(1\,5)$ is to be read from right to left. </p>
<p>Now what happens to $1$? It's mapped to $5$ by the first transposition, and then all other transpositions fix $5$. So overall $$1\longrightarrow 5$$</p>
<p>What happens to $5$ now? The first trans... |
1,564,962 | <p>I need to calculate the following:</p>
<p>$$x=8 \pmod{9}$$
$$x=9 \pmod{10}$$
$$x=0 \pmod{11}$$</p>
<p>I am using the chinese remainder theorem as follows:</p>
<p>Step 1:</p>
<p>$$m=9\cdot10\cdot11 = 990$$</p>
<p>Step 2:</p>
<p>$$M_1 = \frac{m}{9} = 110$$</p>
<p>$$M_2 = \frac{m}{10} = 99$$</p>
<p>$$M_3 = \fra... | Will Jagy | 10,400 | <p>$x \equiv -1 \pmod 9$ and $x \equiv -1 \pmod {10}.$ So $x \equiv -1 \pmod {90}$ and $x = 90 n - 1.$ But $90 = 88 + 2,$ so $90 \equiv 2 \pmod {11}.$</p>
<p>$$ x = 90 n - 1 \equiv 2n - 1 \pmod {11}. $$
$$ 2n \equiv 1 \pmod {11}, $$
$$ n \equiv 6 \pmod {11}. $$
Start with $n=6,$ so $x = 540 - 1 = 539.$</p>
<p>$$ \big... |
1,497,232 | <p>Prove or disprove. If $f(A) \subseteq f(B)$ then $A \subseteq B$</p>
<p>Let y be arbitrary. </p>
<p>$f(A)$ means $\exists a \in A (f(a)=y)$</p>
<p>$f(B)$ means $\exists b \in B (f(b)=y)$ </p>
<p>but $\forall a \in A \exists ! y \in f(a)(f(a)=y)$ </p>
<p>and $\forall b \in B \exists ! y \in f(b)(f(b)=y)$</p>
<p... | Alan | 175,602 | <p>HINT: Consider a constant function, for instance, $f(x)=0$, with domain $\mathbb R$.</p>
|
1,497,232 | <p>Prove or disprove. If $f(A) \subseteq f(B)$ then $A \subseteq B$</p>
<p>Let y be arbitrary. </p>
<p>$f(A)$ means $\exists a \in A (f(a)=y)$</p>
<p>$f(B)$ means $\exists b \in B (f(b)=y)$ </p>
<p>but $\forall a \in A \exists ! y \in f(a)(f(a)=y)$ </p>
<p>and $\forall b \in B \exists ! y \in f(b)(f(b)=y)$</p>
<p... | Balloon | 280,308 | <p>As @Alan said, this is false. Take $E=\{0,1\}$ and $F=\{0\}$, and define $f:E\to F$ by $f(0)=f(1)=0$. Then you have $f(\{0\})=\{0\}\subset f(\{1\})=\{0\}$ whereas $\{0\}\not\subset\{1\}$.</p>
|
2,665,723 | <p>Find value of $k$ for which the equation $\sqrt{3z+3}-\sqrt{3z-9}=\sqrt{2z+k}$ has no solution.</p>
<p>solution i try $\sqrt{3z+3}-\sqrt{3z-9}=\sqrt{2z+k}......(1)$</p>
<p>$\displaystyle \sqrt{3z+3}+\sqrt{3z-9}=\frac{12}{\sqrt{2z+k}}........(2)$</p>
<p>$\displaystyle 2\sqrt{3z+3}=\frac{12}{\sqrt{2z+k}}+\sqrt{2z+k... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
125,503 | <p>Completeness Properties of $\mathbb{R}$: Least Upper Bound Property, Monotone Convergence Theorem, Nested Intervals Theorem, Bolzano Weierstrass Theorem, Cauchy Criterion.</p>
<p>Archimedean Property: $\forall x\in \mathbb{R}\forall \epsilon >0\exists n\in \mathbb{N}:n\epsilon >x$</p>
<p>I can show that LUB ... | George Chailos | 320,942 | <p>Cauchy Completeness does not imply Archimedean Property</p>
<p>Counterexample: the field of formal Laurent series over R
Every Cauchy sequence converges, but the field is Non Archimedean </p>
<p>For a proof see: Counterexamples in Analysis by Bernard R. Gelbaum and John M. H. Olmsted, Chapter 1, Example</p>
|
984,232 | <p>We consider everything in the category of groups. It is known that monomorphisms are stable under pullback; that is, if
$$\begin{array}
AA_1 & \stackrel{f_1}{\longrightarrow} & A_2 \\
\downarrow{h} & & \downarrow{h'} \\
B_1 & \stackrel{g_1}{\longrightarrow} & B_2
\end{array}
$$ is a pullba... | Martin Brandenburg | 1,650 | <p>There is an exact sequence
$0 \to \ker(f_1) \cap \ker(h) \to \ker(f_1) \to \ker(g_1) \to 0$. I doubt that more can be said.</p>
|
104,170 | <p>I am trying to solve a fundamental problem in analytical convective heat transfer: laminar free convection flow and heat transfer from a flat plate parallel to the direction of the generating body force.</p>
<p><strong>Brief History of the problem</strong></p>
<p>Effectively: a flat plate is vertical and parallel ... | george2079 | 2,079 | <p>The problem with your system of equation is that the "zero" far field solution is unstable, hence extremely sensitive to the initial conditions. Posing the problem as an initial value problem, with the "known" conditions from the successful solution:</p>
<pre><code>max = 50;
Pr = .72;
a = 0.7172594734816521`
b = -... |
131,741 | <p>Take the following example <code>Dataset</code>:</p>
<pre><code>data = Table[Association["a" -> i, "b" -> i^2, "c" -> i^3], {i, 4}] // Dataset
</code></pre>
<p><img src="https://i.stack.imgur.com/PZSgO.png" alt="Mathematica graphics"></p>
<p>Picking out two of the three columns is done this way:</p>
<pr... | WReach | 142 | <p>The following expression might not qualify as <em>elegant</em>, but perhaps it can be scored as <em>less clumsy</em>?</p>
<pre><code>data[All, <| "a" -> "a" /* f, "b" -> "b" /* h |>]
</code></pre>
<p><a href="https://i.stack.imgur.com/q62un.png" rel="noreferrer"><img src="https://i.stack.imgur.com/q62u... |
4,282,006 | <p><strong>Evaluate the limit</strong></p>
<p><span class="math-container">$\lim_{x\rightarrow \infty}(\sqrt[3]{x^3+x^2}-x)$</span></p>
<p>I know that the limit is <span class="math-container">$1/3$</span> by looking at the graph of this function, but I struggle to show it algebraically.</p>
<p>Is there anyone who can ... | David C. Ullrich | 248,223 | <p>Hint: If <span class="math-container">$t=1/x$</span> then <span class="math-container">$$\sqrt[3]{x^3+x^2}-x=\frac{\sqrt[3]{1+t}-1}{t}$$</span></p>
|
3,118 | <p>Can anyone help me out here? Can't seem to find the right rules of divisibility to show this:</p>
<p>If $a \mid m$ and $(a + 1) \mid m$, then $a(a + 1) \mid m$.</p>
| Robin Chapman | 226 | <p>The other answers put this in a general context, but in this example one
can be absolutely explicit. If $a\mid m$ and $(a+1)\mid m$ then there are
integers $r$ and $s$ such that
$$m=ar=(a+1)s.$$
Then
$$a(a+1)(r-s)=(a+1)[ar]-a[(a+1)s]=(a+1)m-am=m.$$
As $r-s$ is an integer, then $a(a+1)\mid m$.</p>
|
4,609,236 | <p>When finding the derivative of <span class="math-container">$f(x) = \sqrt x$</span> via the limit definition, one gets</p>
<p><span class="math-container">$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{\sqrt{x+h} - \sqrt x}{h}$$</span></p>
<p>For this, I could get the answer from applying L'... | gnasher729 | 137,175 | <p>L’Hopital’s rule has been proven, many, many times in the past. It’s true, because you believe hundreds of mathematicians who told you so. You don’t need to know details of the proof of the rule.</p>
<p>So you can use the rule to prove something. And I use what you proved to prove L’Hopital. Yes, there is a circle. ... |
2,043,429 | <p>In my textbook, it states that the general formula for the partial sum </p>
<p>$$\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$$</p>
<p>My question is, if I have the following sum instead:</p>
<p>$$\sum_{i=1}^n \frac{1}{i^2}$$ </p>
<p>Can I just flip the general formula to get this:</p>
<p>$$\sum_{i=1}^n \frac{1}{i... | Olivier Oloa | 118,798 | <p>The answer is no.</p>
<p>Even with <em>just</em> two terms:
$$
\frac1a+\frac1b \neq \frac1{a+b},
$$ for example
$$
\frac11+\frac11=2 \neq \frac1{2}=\frac1{1+1}.
$$</p>
|
4,573,004 | <p>I want to solve <span class="math-container">$y'' +y^3 = 0$</span> with the boundary conditions <span class="math-container">$y(0) = a$</span> and <span class="math-container">$y(k) = b$</span>. My goal is to reduce this problem to <span class="math-container">$y' +y^2 = 0$</span> while solving but I'm not sure it c... | Zuhair | 489,784 | <p>For any set <span class="math-container">$S$</span> and any set <span class="math-container">$C$</span> it is clear that <span class="math-container">$S \cup (S - C)$</span> is <span class="math-container">$S$</span> itself because by the union you are returning back the elements of <span class="math-container">$S$<... |
2,158,369 | <p>prove that :
$$a,b>0\\,0<x<\pi/2$$
$$a\sqrt{\sin x}+b\sqrt{\cos x}≤(a^{4/3}+b^{4/3})^{3/4}$$
my try :</p>
<p>$$a\sqrt{\sin x}+b\sqrt{\cos x}=a(\sqrt{\sin x}+\frac{a}{b}\sqrt{\cos x})$$</p>
<p>$$\frac{a}{b}=\tan y$$</p>
<p>$$a\sqrt{\sin x}+b\sqrt{\cos x}=a(\sqrt{\sin x}+\tan y\sqrt{\cos x})$$</p>
<p>$$\... | Patrick Stevens | 259,262 | <p>Consider a vertex of minimal degree. If the degree was $1$, then we're done by induction if we remove that vertex. Otherwise, the minimal degree in the graph is at least $2$. Take a longest path $v_0 v_1 \dots v_k$; then $v_0$ has degree $2$ or higher, so the only way this can be a <em>longest</em> path is if $v_0$ ... |
2,413,368 | <p>I am to show that if $ w = z + \frac{c}{z} $ and $ |z| = 1 $, then $w$ is an ellipse, and I must find its equation.</p>
<p>Previously, I have solved transformation questions by finding the modulus of the transformation in either the form $ w = f(z) $ or $ z = f(w) $. However, I think the part stumping me here is th... | dxiv | 291,201 | <p>Let $\,z=u^2\,$ with $\,|u| = \sqrt{|z|}=1\,$, and let $\,c=b^2\,$. Then $\displaystyle\,w=u^2+\frac{b^2}{u^2}\,$, and so:</p>
<p>$$
w+2b=u^2+\frac{b^2}{u^2}+ 2b=\left(u+\frac{b}{u}\right)^2 \\ \implies |w+2b|=\left|u+\frac{b}{u}\right|^2 = \left(u+\frac{b}{u}\right)\left(\bar u+\frac{\bar b}{\bar u}\right) = |u|^2... |
16,754 | <p>Let $c$ be an integer, not necessarily positive and not a square. Let $R=\mathbb{Z}[\sqrt{c}]$
denote the set of numbers of the form $$a+b\sqrt{c}, a,b \in \mathbb{Z}.$$
Then $R$ is a subring of $\mathbb{C}$ under the usual addition and multiplication.</p>
<p>My question is: if $R$ is a UFD (unique factorization ... | Bill Dubuque | 242 | <p>Yes, because (quadratic) number rings are easily shown to have dimension at most one (i.e. every nonzero prime ideal is maximal). But <span class="math-container">$\rm PID$</span>s are <em>precisely</em> the <span class="math-container">$\rm UFD$</span>s which have dimension <span class="math-container">$\le 1.\, $<... |
337,252 | <p>I'm guessing that the free group on an empty set is either the trivial group or isn't defined.
Some clarification would be appreciated.</p>
| archipelago | 67,907 | <p>It is defined. Free groups are defined for any sets just by the following universal property:</p>
<p>Let $S$ be a set. A group $F(S)$ with a map $\iota\colon S\rightarrow F(S)$ is called free over $S$, if for any group $H$ and any map $g\colon S\rightarrow H$ there is exactly one group homomorphism $h\colon F(S)\ri... |
19,605 | <p>Let <span class="math-container">$P=(x_1,y_1)$</span> be a non torsion point on an elliptic curve <span class="math-container">$y^2=x^3+Ax+B$</span>.
Let <span class="math-container">$(x_n,y_n)=P^{2^n}. x_n,y_n$</span> are rationals with heights growing rapidly. Can <span class="math-container">${x_n} {y_n}$</span... | Ben Webster | 66 | <p>From the discussion above it looks like the answer is yes (<strong>EDIT</strong>: if you allow real numbers; the OP was unclear, perhaps they wanted a rational point, in which case I'm uncertain. Does anybody know anything about the binary expansion of complex numbers with rational Weierstrass p-values?). Let the o... |
1,176,938 | <p>How do you show that $-1+(x-4)(x-3)(x-2)(x-1)$ is irreducible in $\mathbb{Q}$?</p>
<p>I don't think you can use the eisenstein criterion here</p>
| Bill Kleinhans | 73,675 | <p>Actually, the obvious generalization is also true. Let $P$ and $Q$ be polynomial factors, so that the given expression equals $PQ$. Then $PQ=-1$ at each of the integer values, $1,2,3,4$ for this case. So $P,Q=\pm1$ at each of these integers, and whenever $P$ equals one, $Q$ equals minus one, and vice versa. But thes... |
3,483,260 | <p>Given the set <span class="math-container">$\{1,2,3,4,5,6,7\}$</span>.</p>
<p>We would like to create a string of size 8 so that each of the elements of the set appears at least once in the result. How many ways are there to create such a set?</p>
<p>I think that the answer should be: order 7 elements <span class=... | Robert Lewis | 67,071 | <p>With</p>
<p><span class="math-container">$0 \le f''(x) \le f(x), \; \forall x \in \Bbb R, \tag 1$</span></p>
<p>and</p>
<p><span class="math-container">$f'(x) \ge 0, \; \forall x \in \Bbb R, \tag 2$</span></p>
<p>we have</p>
<p><span class="math-container">$0 \le f'(x) f''(x) \le f(x) f'(x), \; \forall x \in \B... |
3,295,318 | <p><span class="math-container">$$\int _{ c } \frac { \cos(iz) }{ { z }^{ 2 }({ z }^{ 2 }+2i) } dz$$</span></p>
<p>I used residue caculus for solving the problem but i am not pretty sure if the approach is right.</p>
<p>The attempt has been annexed in the pictures.</p>
<p><a href="https://i.stack.imgur.com/ZSv27.jpg" r... | fleablood | 280,126 | <p>Notice if you add an even number of odd numbers you get an even sum. If you add an odd number of consecutive numbers you get an odd sum. So to get an odd sum you must have an odd number of terms.</p>
<p>Suppose you have <span class="math-container">$2k + 1$</span> terms and the middle term is <span class="math-co... |
3,249,809 | <p>What’s the explicit rule for for this number sequence?</p>
<p><span class="math-container">$$\displaystyle{1 \over 100},\ -{3 \over 95},\ {5 \over 90},\
-{7 \over 85},\ {9 \over 80}$$</span></p>
<p>The numerator changes to negative every other term, while the denominator subtracts <span class="math-container"... | MarianD | 393,259 | <p>The general formula for an (infinite) sequence of (e. g. real) numbers from the finite number <span class="math-container">$n$</span> of its (first) members is <strong>in principle impossible,</strong> as the next (not listed) <span class="math-container">$(n+1)^\mathrm{th}$</span> member may be an <em>arbitrary</em... |
196,092 | <p>Are there any examples of a semigroup (which is <strong>not a group</strong>) with <strong>exactly one</strong> left(right) identity (which is <strong>not the two-sided identity</strong>)? Are there any “real-world” examples of these (semigroups of some more or less well-known mathematical objects) or they could onl... | Nobody | 17,111 | <p>Take a finite semigroup $S$. Then $S$ has an idempotent element $e$ since $S$ is finite.</p>
<p>Let $T = \{se : s \in S\}$. Then $T$ is a subsemigroup of $S$. We have $e \in T$ because $e = ee$. And $e$ is a right identity of $T$ since $(se)e = s(ee) = se$ for all $s \in S$.</p>
<p>My problem with this example is ... |
196,092 | <p>Are there any examples of a semigroup (which is <strong>not a group</strong>) with <strong>exactly one</strong> left(right) identity (which is <strong>not the two-sided identity</strong>)? Are there any “real-world” examples of these (semigroups of some more or less well-known mathematical objects) or they could onl... | Tara B | 26,052 | <p>Consider for example the semigroup consisting of all constant functions on a set $X$ [acting on the right], together with one non-constant idempotent function $f$ (for example, let $f$ fix some point $x\in X$ and send every other point to some $y\neq x$). Then $f$ is a unique left identity, and $f$ is not a right i... |
196,092 | <p>Are there any examples of a semigroup (which is <strong>not a group</strong>) with <strong>exactly one</strong> left(right) identity (which is <strong>not the two-sided identity</strong>)? Are there any “real-world” examples of these (semigroups of some more or less well-known mathematical objects) or they could onl... | J.-E. Pin | 89,374 | <p>Take the semigroup $S = \{a, b, 0\}$ with $a^2 = a$, $ab = b$ and every other product equal to $0$. Then $a$ is a left identity (since $ax = x$ for all $x \in S$) but it is not an identity (since $ba = 0$). You can define this semigroup in three other equivalent ways:</p>
<p>(1) As a transformation semigroup on $\{... |
2,291,310 | <p>I'm seeking an alternative proof of this result:</p>
<blockquote>
<p>Given $\triangle ABC$ with right angle at $A$. Point $I$ is the intersection of the three angle lines. (That is, $I$ is the incenter of $\triangle ABC$.) Prove that
$$|CI|^2=\frac12\left(\left(\;|BC|-|AB|\;\right)^2+|AC|^2\right)$$</p>
</bloc... | Lazy Lee | 430,040 | <p>Let $AB = c, AC = b, BC = a$, and let $IT\perp AC $ at $I$. Then, $CI$ can be written as $$\begin{split}CI^2 &= CT^2 + IT^2 \\&= \left(\frac{a+b-c}{2}\right)^2+\left(\frac{b+c-a}{2}\right)^2\\&= \frac{2a^2+2b^2+2c^2-4ac}{4}\\&=\frac{(a-c)^2+b^2}{2} \\&=\frac{(BC-AB)^2+AC^2}{2}\end{split}$$</p>
|
19,285 | <p>Is anyone aware of Mathematica use/implementation of <a href="http://en.wikipedia.org/wiki/Random_forest">Random Forest</a> algorithm?</p>
| image_doctor | 776 | <p>As of version 10.0 there is a built in implementation of Random Forests which is accessible through the <em>Classify</em> function.</p>
<pre><code>trainingset = {1 -> "A", 2 -> "A", 3.5 -> "B", 4 -> "B"};
classifier = Classify[trainingset, Method->"RandomForest"];
</code></pre>
|
3,053,386 | <p>This might be a very basic question for some of you. Indeed in <span class="math-container">$\textbf Z$</span>, it's very easy. For example, <span class="math-container">$\textbf Z / \langle 2 \rangle$</span> consists of <span class="math-container">$\langle 2 \rangle$</span> and <span class="math-container">$\langl... | Kenny Lau | 328,173 | <p><span class="math-container">$$\Bbb Z[\sqrt{14}]/(4+\sqrt{14}) = \Bbb Z[X]/(X^2-14,X+4) = \Bbb Z[X]/(2,X+4) = \Bbb Z[X]/(2,X) = \Bbb F_2$$</span></p>
<p><span class="math-container">$$\Bbb Z[\sqrt{10}]/(2,\sqrt{10}) = \Bbb Z[X]/(X^2-10, 2, X) = \Bbb Z[X]/(2, X) = \Bbb F_2$$</span></p>
<p>The trick is to move the s... |
540,227 | <p>I was try to understand the following theorem:-</p>
<p><strong>Let $X,Y$ be two path connected spaces which are of the same homotopy type.Then their fundamental groups are isomorphic.</strong></p>
<p><strong>Proof:</strong> The fundamental groups of both the spaces $X$ and $Y$ are independent on the base points si... | Stefan Hamcke | 41,672 | <p>I suppose $σ$ is a path from $x_0$ the a point $x_1$ and the map $\sigma_\#$ is defined as:
$$σ_\#:\pi_1(X,x_0)\to π_0(X,x_1)\\σ_\#([p])=[σ\cdot p\cdot\barσ]$$ where $σ⋅p$ is the path which first traverses the loop $p$ and then $σ$, and $\bar σ$ is the reversed path.</p>
<p>This is a bijection because it has the in... |
1,494,409 | <blockquote>
<p>Let <span class="math-container">$\Bbb R^+$</span> denote the real numbers. Suppose <span class="math-container">$\phi:\Bbb R^+\to\Bbb R^+$</span> is an automorphism of the group <span class="math-container">$\Bbb R^+$</span> under multiplication with <span class="math-container">$\phi(4)=7$</span>.</p>... | Travis Willse | 155,629 | <p><strong>Hint</strong> If $a \in \Bbb R^+$ satisfies $a^2 = 4$, then because $\phi$ is a homomorphism, we have
$$\phi(a)^2 = \phi(a^2) = \phi(4) .$$</p>
<p>It's interesting to note that we cannot, however, conclude the value of, e.g., $\phi(3)$ from the given information.</p>
<blockquote class="spoiler">
<p><stro... |
37,804 | <p>I'm trying to gain some intuition for the usefullness of the spectral theory for bounded self adjoint operators. I work in PDE and any interesting applications/examples I've ever encountered are concerning <em>compact operators</em> and <em>unbounded operators</em>. Here I have the examples of $-\Delta$, the laplaci... | Newbie | 9,032 | <p>Maybe the corresponding functional calculus (FC) addresses your question about why it is interesting to know the spectrum of an operator? Many applications use it, see for example Pedersen's book on Functional Analysis for the various types (Holomorphic, Continuous, Measurable FC). </p>
<p>For an intuitive approach... |
2,201,085 | <p>Let
$$x_{1},x_{2},x_{3},x_{5},x_{6}\ge 0$$ such that
$$x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}=1$$
Find the maximum of the value of
$$\sum_{i=1}^{6}x_{i}\;x_{i+1}\;x_{i+2}\;x_{i+3}$$
where
$$x_{7}=x_{1},\quad x_{8}=x_{2},\quad x_{9}=x_{3}\,.$$</p>
| Yuri Negometyanov | 297,350 | <p>Let $x_1=a,\ x_2=b,\ x_3=c,\ x_4=d,\ x_5=e,\ x_6=f.$ </p>
<p>Objective function is
$$Z(a,b,c,d,e,f) = abcd + bcde + cdef + defa + efab + fabc.$$</p>
<p>Let maximize $Z(a,b,c,d,e,f)$ using Lagrange mulptiplyers method for the function
$$F(a,b,c,d,e,f,\lambda) = abcd + bcde + cdef + defa + efab + fabc + \lambda(1-a... |
203,114 | <p>If we have a pair of coordinates <span class="math-container">$(x,y)$</span>, let's say</p>
<pre><code>pt = {1,2}
</code></pre>
<p>then we can easily rotate the coordinates, by an angle <span class="math-container">$\theta$</span>, by using the rotation matrix</p>
<pre><code>R = {{Cos[\[Theta]], -Sin[\[Theta]]}, ... | Alx | 35,574 | <p>This should give what you want:</p>
<pre><code>data2=data/.{x_?NumericQ,y_?NumericQ}:>RotationMatrix[\[Theta]].{x,y}
</code></pre>
|
198,555 | <p>I am having difficulties understanding how do I perform set operation like union or intersection on Relations. </p>
<p>In a question, I am asked to prove/disprove: </p>
<ul>
<li>If R & S are symmetric, is $R \cap S$ symmetric? </li>
<li>If R & S are transitive, is $R \cup S$ transitive?</li>
</ul>
<p>How ... | Austin Mohr | 11,245 | <p>Other answers have already discussed symmetry.</p>
<p>An example of transitive $R$ and $S$ with non-transitive $R \cup S$ can be obtained by taking
$$
R = \{(1,2),(2,1)\}
$$
and
$$
S = \{(2,3),(3,2)\}.
$$
In $R \cup S$, we have $(1,2)$ and $(2,3)$, but not $(1,3)$ (that is, $1$ is related to $2$ and $2$ is related ... |
332,380 | <p>The following is an excerpt from Sharpe's <em>Differential Geometry - Cartan's Generalization of Klein's Erlangen Program</em>.</p>
<blockquote>
<p>Now we come to the question of higher derivatives. As usual in modern
differential geometry, we shall be concerned only with the skew-symmetric
part of the high... | Jesper Göransson | 112,784 | <p>Here's some calculations that may help. (I'll leave it to you to decide if the arguments are valid or not.) We'll work in a two-dimensional manifold with local coordinates <span class="math-container">$x$</span> and <span class="math-container">$y$</span>. Let <span class="math-container">$d$</span> be the covariant... |
332,380 | <p>The following is an excerpt from Sharpe's <em>Differential Geometry - Cartan's Generalization of Klein's Erlangen Program</em>.</p>
<blockquote>
<p>Now we come to the question of higher derivatives. As usual in modern
differential geometry, we shall be concerned only with the skew-symmetric
part of the high... | Mozibur Ullah | 35,706 | <p>Although higher derivatives are best thought through with jet bundles where we actually differentiate a bundle and not just a manifold, there is a useful description of second derivatives using secondary tangent bundles, this is just iterating the tangent bundle, ie <span class="math-container">$TTM$</span>. This is... |
132,591 | <p>Let $f(x)$ be a positive function on $[0,\infty)$ such that $f(x) \leq 100 x^2$. I want to bound $f(x) - f(x-1)$ from above. Of course, we have $$f(x) - f(x-1) \leq f(x) \leq 100 x^2.$$ This is not good for me though. I need a bound which is linear (or at worst linear-times-root) in $x$.</p>
<p>Is there an inequali... | Michael Hardy | 11,667 | <p>Suppose $f(x) = 100x^2\sin^2(\pi x/2)$. Then $f(x) = 0$ when $x$ is an even integer and $f(x) = 100x^2$ when $x$ is an odd integer. So $f(x)-f(x-1)\ge 100(x-1)^2$, with equality when $x$ is even.</p>
|
548,470 | <p>Prove $$(x+1)e^x = \sum_{k=0}^{\infty}\frac{(k+1)x^k}{k!}$$ using Taylor Series.</p>
<p>I can see how the $$\sum_{k=0}^{\infty}\frac{x^k}{k!}$$ plops out, but I don't understand how $(x+1)$ can become $(k+1)$.</p>
| André Nicolas | 6,312 | <p>Note that
$$xe^x=\sum_{k=0}^\infty \frac{x^{k+1}}{k!}.$$
Differentiate both sides with respect to $x$. </p>
|
96,377 | <p>I have a polynomial with the coefficients of {a1, b1, b2}</p>
<pre><code>x = 1/8 (a1^4 E^(4 I τ ω) - 2 a1^2 E^( 2 I τ ω) (b2 Sqrt[1 - t] + b1 Sqrt[t])^2 + (b2 Sqrt[ 1 - t] + b1 Sqrt[t])^4);
</code></pre>
<p><a href="https://i.stack.imgur.com/6W2Ob.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com... | Alexei Boulbitch | 788 | <p>Try this:</p>
<pre><code>Collect[Expand[x], {a1^2 b2^2, b2^4, b1 b2^3, a1^2 b1 b2} ]
</code></pre>
<p>I do not give here the result, since it is much too long, but it looks like what you want up to reordering. Concerning the reordering, it is more complex in Mma. There are few approaches, but I do not recommend to... |
4,250,292 | <p>I am stuck trying to solve the following integral:</p>
<p><span class="math-container">$$\int_{0}^{3}\frac{1}{3}e^{3t-2}dt$$</span></p>
<p>I understand that I can take out <span class="math-container">$\frac{1}{3}$</span> of the integral, and that the integral of <span class="math-container">$e^{3t-2}$</span> is <sp... | Parthib Ghosh | 921,782 | <p>We have,
<span class="math-container">$\displaystyle\mathsf{\int^{3}_{0}\,\dfrac{1}{3}\,e^{3t-2}\,dt}$</span>
<span class="math-container">$\mathtt{Put\,\,\,3t-2=u}$</span>
<span class="math-container">$\mathtt{\implies\,3\,dt=du}$</span>
<span class="math-container">$\mathtt{\implies\,dt=\dfrac{du}{3}}$</span>
So, ... |
1,717,149 | <p>Is it true or false that if $V$ is a vector space and $T:V \to W$ is a linear transformation such that $T^2 = 0$, then $Im(T) \subseteq Ker(T)$ ?<br>
I don't understand it that much. It doesn't seem related... I can have a vector $v$ from $V$ that its power by 2 equals zero but $T(v) \neq 0_{v}$ </p>
| Deusovi | 256,930 | <p>I recommend Carter's <em>Visual Group Theory</em>. It makes heavy use of pictures and diagrams (hence the name) and I found it very clear.</p>
|
1,650,881 | <p>I found this problem in a book on undergraduate maths in the Soviet Union (<a href="http://www.ftpi.umn.edu/shifman/ComradeEinstein.pdf" rel="nofollow">http://www.ftpi.umn.edu/shifman/ComradeEinstein.pdf</a>):</p>
<blockquote>
<p>A circle is inscribed in a face of a cube of side a. Another circle is circumscribed... | John Alexiou | 3,301 | <p>I placed the inscribed circle on the top face (+y direction) and the circumscribed circle on the front face (+z direction). Their locus of points is</p>
<p>$$ \begin{align}
\vec{r}_1 & = \begin{bmatrix} r_1 \cos \theta_1 & \frac{a}{2} & r_1 \sin \theta_1 \end{bmatrix} \\
\vec{r}_2 & = \begin{bmatri... |
201,807 | <p>I heard this problem, so I might be missing pieces. Imagine there are two cities separated by a very long road. The road has only one lane, so cars cannot overtake each other. $N$ cars are released from one of the cities, the cars travel at constant speeds $V$ chosen at random and independently from a probability di... | Berci | 41,488 | <p>One lane? No overtaking? Simultaneous arrival? Then they must arrive one by one, so $N$ is the number of group of cars. What have I misunderstood?</p>
|
201,807 | <p>I heard this problem, so I might be missing pieces. Imagine there are two cities separated by a very long road. The road has only one lane, so cars cannot overtake each other. $N$ cars are released from one of the cities, the cars travel at constant speeds $V$ chosen at random and independently from a probability di... | Aditya Bansal | 304,867 | <p>This is not drastically different from the answers above but a slightly different way to arrive at the recurrence relation.</p>
<p>Let $E(n)$ be the expected number of clusters in the case where we begin with $n$ cars in the beginning. Now, if we add one more car from the behind (i.e behind the last car of the slow... |
1,851,209 | <p>Let $L:X\to Y$ an linear operator. I saw that $L$ is bounded if $$\|Lu\|_Y\leq C\|u\|_X$$
for a suitable $C>0$. This definition looks really weird to me since such application is in fact not necessary bounded as $f:\mathbb R\to \mathbb R$ defined by $f(x)=x$. So, is there an error in <a href="https://en.wikipedia... | Mike | 268,604 | <p>This definition is not really trying to tell you that the values this operator spits out for each argument are always bounded by a given constant. </p>
<p>This definition says that the size (=norm) of the argument you plugged in in the domain, $\mid\mid u \mid\mid_X$ can only be "enlarged" or "diminished" by this m... |
2,236,862 | <p>$$\frac{1^2}{1!}+ \frac{2^2}{2!}+ \frac{3^2}{3!} + \frac{4^2}{4!} + \dotsb$$</p>
<p>I wrote it as:
$$\lim_{n\to \infty}\sum_{r=1}^n \frac{(r^2)}{r!}.$$</p>
<p>Then I thought of sandwich theorem, it didn't work. Now I am trying to convert it into difference of two consecutive terms but can't. Need hints. </p>
| N. S. | 9,176 | <p><strong>Hint 1:</strong>
$$\sum_{r=1}^n \frac{(r^2)}{r!}=\sum_{r=1}^n \frac{r}{(r-1)!}=\sum_{r=0}^{n-1} \frac{r+1}{r!}$$</p>
<p><strong>Hint 2:</strong> Derivate
$$xe^x=\sum_{r=0}^\infty \frac{x^{r+1}}{r!}$$</p>
|
4,617,031 | <p>How would I order <span class="math-container">$x = \sqrt{3}-1, y = \sqrt{5}-\sqrt{2}, z = 1+\sqrt{2} \ $</span> without approximating the irrational numbers? In fact, I would be interested in knowing a general way to solve such questions if there is one.</p>
<p>What I tried to so far, because they are all positive ... | S.H.W | 325,808 | <p>One way is as follows. It can be proved that <span class="math-container">$f(x) = \sqrt{x}$</span> is an increasing function. So we have:
<span class="math-container">$$x_1\lt x_2 \implies \sqrt{x_1}\lt \sqrt{x_2} \implies \sqrt{x_2} - \sqrt{x_1} \gt 0$$</span>This implies that <span class="math-container">$x,y\gt 0... |
3,406,056 | <p>By the definition of matrix exponentiation,</p>
<p><span class="math-container">$$A^k = \begin{cases}
I_n, & \text{if } k=0 \\[1ex]
A^{k-1}A, & \text{if } k\in \mathbb {N}_0 \\
\end{cases}$$</span></p>
<p>In my book, there's an exercise to do <span class="math-container">$D^k$</span>, where <span class="... | Z Ahmed | 671,540 | <p>Let <span class="math-container">$$D=\begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix}.$$</span> then by the property of diagonal matrix
<span class="math-container">$$f(D)=\begin{bmatrix} f(a) & 0 & 0 \\ 0 & f(b) & 0 \\ 0 & 0 & f(c) \end{bmatrix}$$</sp... |
4,551,497 | <p>How do I find the Taylor series of <span class="math-container">$\cos^{20}{(x)}$</span> for <span class="math-container">$x_0 = 0$</span>, knowing the Taylor series of <span class="math-container">$\cos{(x)}$</span>?</p>
| boojum | 882,145 | <p>We can use the even symmetry of the cosine function to determine that <span class="math-container">$ \ (\cos x)^{20} \ $</span> is likewise an even function, thus the Maclaurin series for this function must contain only terms with even powers of <span class="math-container">$ \ x \ \ . \ $</span> This means that we... |
2,572,884 | <p>Let $X$ be a random variable with values $0$ and $1$.</p>
<p>Let $Y$ be a random variable with values in $\mathbb{N_0}$.</p>
<p>Let $ p \in (0,1)$ and $ P(X=0, Y=n) = p \cdot \frac{e^{-1}}{n!} $ and $ P(X=1, Y=n) = (1-p) \cdot \frac{2^{n}e^{-2}}{n!} $.</p>
<p>Calculate $E(Y)$ and $Var(Y)$. (expected value and ... | Christian Blatter | 1,303 | <p>Given a finite probability distribution $p:=(p_i)_{1\leq i\leq n}$ its <em>entropy</em> is defined by
$$H(p):=-\sum_{i=1}^n p_i \log_2(p_i)\ .$$
If $p$ models the frequencies of the letters of an alphabet then $H(p)$ turns out to be the average number of bits per letter. This is the essential content of Shannon theo... |
2,572,884 | <p>Let $X$ be a random variable with values $0$ and $1$.</p>
<p>Let $Y$ be a random variable with values in $\mathbb{N_0}$.</p>
<p>Let $ p \in (0,1)$ and $ P(X=0, Y=n) = p \cdot \frac{e^{-1}}{n!} $ and $ P(X=1, Y=n) = (1-p) \cdot \frac{2^{n}e^{-2}}{n!} $.</p>
<p>Calculate $E(Y)$ and $Var(Y)$. (expected value and ... | celtschk | 34,930 | <p>The Huffman code is the best you can achieve for encoding single symbols from a given set. To achieve a better encoding, you must encode combinations of several symbols at once.</p>
<p>For example, for two-symbol combinations, you get the probabilities:
$$\begin{aligned}
p(AA) &= \frac14 &
p(AB) &= \fra... |
19,373 | <p>I posted this question earlier today on the Mathematics site (<a href="https://math.stackexchange.com/q/3988907/96384">https://math.stackexchange.com/q/3988907/96384</a>), but was advised it would be better here.</p>
<p>I had a heated argument with someone online who claimed to be a school mathematics teacher of man... | Syntax Junkie | 5,465 | <p>You're both right, depending on the domain of discourse and the rules of engagement.</p>
<p>In pure math, the traditional expectation is that the numbers given are exact unless stated otherwise, and answers are also to be exact unless stated otherwise. So when the mathematician read "22 miles," he's using ... |
19,373 | <p>I posted this question earlier today on the Mathematics site (<a href="https://math.stackexchange.com/q/3988907/96384">https://math.stackexchange.com/q/3988907/96384</a>), but was advised it would be better here.</p>
<p>I had a heated argument with someone online who claimed to be a school mathematics teacher of man... | Eric Duminil | 9,380 | <p>Here's a joke I like to tell when people could use a reminder about precision vs accuracy:</p>
<blockquote>
<p>A tour guide at Giza was explaining how the Pyramids were 4507 years
old. Someone in the crowd asked: "That's oddly specific. How do we know this?"</p>
<p>"Well. I was told they were 4500 yea... |
19,373 | <p>I posted this question earlier today on the Mathematics site (<a href="https://math.stackexchange.com/q/3988907/96384">https://math.stackexchange.com/q/3988907/96384</a>), but was advised it would be better here.</p>
<p>I had a heated argument with someone online who claimed to be a school mathematics teacher of man... | user3067860 | 1,908 | <p>It depends on the level of the class.</p>
<p>I would expect someone who has a recent undergraduate degree in mathematics to have experienced significant figures at at least some point in their life, either in high school or in college. I would also expect common sense to kick in and say that the level of accuracy pr... |
794,389 | <p>Q1. I was fiddling around with squaring-the-square type algebraic maths, and came up with a family of arrangements of $n^2$ squares, with sides $1, 2, 3\ldots n^2$ ($n$ odd). Which seems like it would work with ANY odd $n$. It's so simple, surely it's well-known, but I haven't seen it in my (brief) web travels. I ha... | Adam Ponting | 149,936 | <p>(OP here)</p>
<p>Update: Well, I answered Q2 "Yes" after a couple of years, and found the rules governing when an arrangement "has no gaps" or not. It seems every arrangement actually tiles the plane in 2 layers, but only for some the layers of squares coincide. I wrote <a href="http://www.adamponting.com/wp-conten... |
102,721 | <p>This is probably a very simple question, but I couldn't find a duplicate.</p>
<p>As everybody knows, <code>{x, y} + v</code> gives <code>{x + v, y + v}</code>. But if I intend <code>v</code> to represent a vector, for example if I am going to substitute <code>v -> {vx, vy}</code> in the future, then the result <... | JungHwan Min | 35,945 | <p>One workaround would be <code>ClearAttributes[Plus,Listable]</code> (The <code>Plus</code> is threading because of the <code>Listable</code> attribute). If you need the <code>Listable</code> attribute in another part of your code, you would need to run <code>SetAttributes[Plus,Listable]</code> to put the attribute b... |
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