qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,328,677 | <p>I wish to find the area between the curves:</p>
<p>$y=\sqrt{x}$</p>
<p>and </p>
<p>$y=x^{2}-3x-2$</p>
<p>and between $x=1, x=9$</p>
<p>Now, I think that the first thing to do is to find the intersection between the two curves, and then to find which curve is the upper one before and after the intersection point... | Michael Rozenberg | 190,319 | <p>We need to solve
$$\sqrt{x}=x^2-3x-2.$$
We have $x\geq0$ and $x^2-3x-2\geq0$, which gives $x\geq\frac{3+\sqrt{17}}{2}$.</p>
<p>Thus, squaring gives
$$(x-4)(x^3-2x^2-3x-1)=0$$
and $x=4$ because for $x\geq\frac{3+\sqrt{17}}{2}$ we see that $$x^3-2x^2-3x-1=x(x^2-3x-2)+x^2-x-1=$$
$$=x(x^2-3x-2)+x^2-3x-2+2x+1>0.$$</... |
3,427,640 | <p>This is a problem that came up as I was learning Hermitian/skew-Hermitian transformations:</p>
<hr>
<p>Let <span class="math-container">$T: V\rightarrow E$</span> be a linear transformation where <span class="math-container">$V$</span> is a subspace of a complex Euclidean space <span class="math-container">$E$</sp... | jazhang | 701,790 | <p>I wasn't fully convinced by comparing coefficients. Here's another way:</p>
<p>Substituting <span class="math-container">$i$</span> and <span class="math-container">$1$</span> for <span class="math-container">$t$</span> in the equation gets:</p>
<p><span class="math-container">$(T(y),x)+(T(x),y) = (y,T(x))+(x,T(y)... |
220,170 | <p>My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that simple invariants as Betti number are not even close to classify ?</p>
<p>For example i would start with :</p>
<ol>
<li... | Louis Deaett | 2,502 | <p>I'm not sure if this is what you have in mind by "without using advanced tools" or "a nice construction which would help understanding the combinatorial complexity of three manifolds," but how about identifying pairs of faces of different polyhedra? I guess this is what convinced Poincaré...</p>
|
2,015,717 | <p>I know that the O notation tells me to find $n_0$ natural and $c>0$ real, so the
$$2^{(n^2)} \le c*2^{2n} $$
Only step that I can think about is this:
$$2^{(n^2)} \le c*2^{n+n} $$
$$2^{(n^2)} \le c*2^n*2^n $$
But I have no clue what to do next. (Some estimate?)</p>
<p>Do you have some hints please?</p>
<p... | mfl | 148,513 | <p>Your claim is false because $$\dfrac{2^{n^2}}{2^{2n}}=2^{n^2-2n}\to \infty$$ as $n\to\infty.$</p>
|
674,620 | <p>What is the slope of the tangent line to the function $$g(x)=x^2 \frac{\cos x}{1+x^3}$$ when $x=\pi/2$?
When $x=a$?</p>
| Heisenberg | 127,460 | <p>The derivative is calculated using the quotient rule,</p>
<p>$$y=\frac{u}{v} \\ y'=\frac{u'v - v'u}{v^2}$$
$$g'(x)=\frac{(1+x^3)(2x\cos x-x^2\sin x)-(x^2\cos x)(3x^2)}{(1+x^3)^2}$$
The slope of the tangent line in calculus is the limit of the rise over run of the tangent at the tangent point - i.e. $dy/dx$ which is... |
2,559,623 | <p>The question is as follows:<p>
Calculate $\lim_{n \to +\infty} \int_{0}^{+\infty} \frac{n \sin(\frac{x}{n})}{1 + x^2} dx$.<p></p>
<p>$\textbf{Some ideas:}$<p>
We can use the fact that $\sin(\frac{x}{n}) \simeq \frac{x}{n} $.But then we find that <p>
$\lim_{n \to +\infty} \int_{0}^{+\infty} \frac{n \sin(\frac{x}{n})... | Parcly Taxel | 357,390 | <p>There is a neat geometric approach that treats the portion of $C_1$ with $0<|z|\le2\sqrt3$ as one side of a regular hexagon whose centre is $2\sqrt3i$, also the centre of the circle $C_2$.
<img src="https://i.stack.imgur.com/TCBKE.jpg" alt="">
For the two curves to intersect in precisely two points, the exclusive... |
2,743,827 | <p>This came up on page 40 of Hubbard's <em>Teichmuller Theory</em> (Vol 1), in the context of parametrizing geodesics in the upper half.</p>
<p>Where it is essentially implied that
$$\frac{\sinh(t)+i}{\cosh(t)}$$
is an arc-length parametrization of the unit circle.</p>
<p><strong>This is my first time seeing such a... | user | 505,767 | <p>Note that</p>
<p>$$\frac{\sinh t+i}{\cosh t}=\frac{\sinh t}{\cosh t}+i\frac{1}{\cosh t}$$</p>
<p>and</p>
<p>$$\left(\frac{\sinh t}{\cosh t}\right)^2+\left(\frac{1}{\cosh t}\right)^2=\frac{\sinh^2t+1}{\cosh^2 t}=1$$</p>
|
2,743,827 | <p>This came up on page 40 of Hubbard's <em>Teichmuller Theory</em> (Vol 1), in the context of parametrizing geodesics in the upper half.</p>
<p>Where it is essentially implied that
$$\frac{\sinh(t)+i}{\cosh(t)}$$
is an arc-length parametrization of the unit circle.</p>
<p><strong>This is my first time seeing such a... | Rob Arthan | 23,171 | <p>At least the first part of the following argument is geometric.</p>
<p>Here are two facts about the <a href="https://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model" rel="nofollow noreferrer">Poincaré half-plane model</a> with $d$ denoting the metric:</p>
<ol>
<li>if $x$ and $y$ are positive reals, $d(ix, iy)... |
175,664 | <p>How can I have a region plot with the logarithmic axis? In the following, I have brought the original case which I want to be logarithmic. In this case, I have a density plot and a region plot that shows the realm of validity of the theory. I want to show this figure in logarithmic scales in order to be more clear.<... | kglr | 125 | <p>You can post-process <code>RegionPlot</code> output to rescale the desired coordinate of graphics primitives:</p>
<pre><code>rp = RegionPlot[Sin[x] Sin[y] > 1/10, {x, Pi, 4 Pi}, {y, Pi, 4 Pi},
ColorFunction -> "DarkRainbow", ImageSize -> 300];
Row[{rp,
Show[rp /. GraphicsComplex[c_, prims___] :> ... |
1,014,614 | <p>That Theorem 3.2 says: Every finite orthogonal set of nonzero vectors is linearly independent.</p>
<p>The proof is simple, but it seems to me that the finiteness is redundant, for the argument in the proof applies to an infinite set. Am I right?</p>
<p>The proof runs as follows: If $k > 0$ is an integer, if $a_... | Hagen von Eitzen | 39,174 | <p><em>Hint:</em> Pick any $\epsilon>0$. Then pick $\delta>0$ according to the definition of unifom continuity. Then what can you say by induction about $f(\pm k\frac\delta2)-f(0)$ with $k\in\mathbb N$? What about the values inbetween?</p>
|
1,576,561 | <p>I'd like to show that $$\sum\limits_{n = 1}^\infty {{{{x^{n + 1}}} \over {n(n + 1)}}} $$ absolutely converges for $|x| < 1$</p>
| SchrodingersCat | 278,967 | <p>By <a href="https://en.wikipedia.org/wiki/Ratio_test" rel="nofollow noreferrer">Ratio test</a>, we can write for the given series,
<span class="math-container">$$\lim_{n\to\infty} \frac{u_{n+1}}{u_n}=\lim_{n\to\infty} \frac{\frac{x^{n + 2}} {(n+1)(n+2)}}{\frac{x^{n + 1}}{n(n + 1)}}$$</span>
<span class="math-contain... |
927,815 | <p>Find an equation of the plane that passes through the points $A(0, 1, 0)$, $B(1, 0, 0)$ and $C(0, 0, 1)$.</p>
| georg | 144,937 | <p>I would say</p>
<p>Sectional plane equation: $\frac{x}{p}+\frac{y}{q}+\frac{z}{r}=1$</p>
<p>In this example p = q = r = 1 --> equation x + y + z =1</p>
|
3,566,083 | <p>I have a game that assigns the probability of finding items in treasure chests by making several independent checks until it fills a quota, with duplicates not allowed. I am trying to figure out how to calculate the resulting probabilities of finding each item - without breaking the bank in terms of calculation brut... | Robert Israel | 8,508 | <p>The equation
<span class="math-container">$\ln(b^y) = y \ln(b)$</span> is not true for negative <span class="math-container">$b$</span>.</p>
|
214,705 | <p>If $a(x) + b(x) = x^6-1$ and $\gcd(a(x),b(x))=x+1$ then find a pair of polynomials of $a(x)$,$b(x)$.</p>
<p>Prove or disprove, if there exists more than 1 more distinct values of the polynomials.</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $\ \ $ If $\rm\:a\!+\!b = c\:(x\!+\!1)\:$ then </p>
<p>$$\rm\:x\!+\!1 = (a,b) = (a,c\,(x\!+\!1)\!-\!a) = (a,c\,(x\!+\!1))\iff a = d(x\!+\!1),\,\ (d,c) = 1$$</p>
<p>Thus for $\rm\:c(x\!+\!1) = x^6\!-\!1\:$ it's true $\rm\iff a = d(x\!+\!1),\:$ for $\rm\,d\,$ coprime to $\rm\:c = (x^6\!-\!1)/(... |
975,207 | <p><strong>Question:</strong></p>
<blockquote>
<p>Solve the following system for $a,b,c\in \mathbb{R}$:
$$\begin{cases}
b^2-6=2\sqrt{2a+6}\\
c^2-6=2\sqrt{2b+6}\\
a^2-6=2\sqrt{2c+6}
\end{cases}$$</p>
</blockquote>
<p>I found the following:$$ (b^2-6)^2=4(2a+6)$$
$$(c^2-6)^2=4(2b+6)$$
$$(a^2-6)^2=4(2c+6)$$
Then mayb... | Tito Piezas III | 4,781 | <p>Surprisingly, this system <strong><em>does</em></strong> have solutions in $\mathbb{R}$ where $a\neq b\neq c$. Unfortunately, it involves a deg-$54$ equation. Given,</p>
<p>$$ (b^2-6)^2=4(2a+6)\tag1$$
$$(c^2-6)^2=4(2b+6)\tag2$$
$$(a^2-6)^2=4(2c+6)\tag3$$</p>
<p>Do the substitutions:</p>
<p>$$a =\tfrac{1}{2}(-6+x^... |
2,842,481 | <p>If the minimum and the maximum values of $y = (x^2-3x+c)/(x^2+3x+c)$ are $7$ and $1/7$ , then the value of $c$ is?</p>
<p>I cross multiplied the equation and tried to find it's discriminant, but I don't think it gets me anywhere. A little hint would be appreciated!</p>
| Maverick | 171,392 | <p>Solving <span class="math-container">$$y=\frac{x^2-3x+c}{x^2+3x+c}$$</span> and <span class="math-container">$$y=7$$</span></p>
<p>we put <span class="math-container">$$\frac{x^2-3x+c}{x^2+3x+c}=7$$</span></p>
<p>to get <span class="math-container">$$x^2-4x+c=0$$</span> which should have exactly one root.</p>
<p>So ... |
40,618 | <p>Let $f_{=}$ be a function from $\mathbb{R}^{2}$ be defined as follows:
(1) if $x = y$ then $f_{=}(x,y) = 1$;
(2) $f_{x,y} = 0$ otherwise.</p>
<p>I would like to have a proof for / a reference to a textbook proof of the following theorem (if it indeed is a theorem):</p>
<p>$f_{=}$ is uncomputable even if one restr... | Gerald Edgar | 454 | <p>Aren't computable functions of reals automatically continuous? And isn't your function discontinuous? Of course you need a <em>definition</em> of computable in this setting to make sense of this...</p>
|
40,618 | <p>Let $f_{=}$ be a function from $\mathbb{R}^{2}$ be defined as follows:
(1) if $x = y$ then $f_{=}(x,y) = 1$;
(2) $f_{x,y} = 0$ otherwise.</p>
<p>I would like to have a proof for / a reference to a textbook proof of the following theorem (if it indeed is a theorem):</p>
<p>$f_{=}$ is uncomputable even if one restr... | Paul Taylor | 2,733 | <p>It is entirely reasonable to ask for a continuous and computable equality-test function on~$\mathbb R$. The unreasonable thing is to expect it to take values in the <em>discrete</em> two-element space $\mathbf 2$.</p>
<p>The truth-value space that we need to use is the <strong>Sierpinski space</strong>, for which I... |
265,549 | <p>$$ \left\{ x \in\mathbb{R}\; \middle\vert\; \tfrac{x}{|x| + 1} < \tfrac{1}{3} \right\}$$</p>
<p>What is the supremum and infimum of this set? I thought the supremum is $\frac{1}{3}$. But can we say that for any set $ x < n$ that $n$ is the supremum of the set? And for the infimum I have no idea at all. Also,... | André Nicolas | 6,312 | <p>Let $x$ be negative. then $\dfrac{x}{|x|+1}$ is negative, and in particular $\lt \dfrac{1}{3}$.</p>
<p>So there is no infimum. (But some people allow the symbol $-\infty$ as an infimum.)</p>
<p>For the supremum, note that there are positive $x$ such that $\dfrac{x}{x+1}\lt \dfrac{1}{3}$.</p>
<p>Since $\dfrac{x}{x... |
944,992 | <p>I'm trying to understand each part of this completed proof that my professor did, here is my interpretation in parentheses, please advise as necessary.</p>
<p>Proof: Assume that $2^{1/2}$ is rational, then $2^{1/2}$ = $p/q$ for some integer $p$ and $q$ with $q \neq 0$ and $\gcd(p,q) = 1$</p>
<p>$$(p/q)^2 = 2 = p^2... | NicNic8 | 24,205 | <p>Unfortunately not. Physical explanations like that are a great way to develop intuition that might lead to a proof, but it's not a mathematical proof. A mathematical proof is a set of logical statements that lead from axioms or theorems to the desired conclusion.</p>
<p>In fact, physical intuition can lead to som... |
1,918,885 | <p>I would like to show that $\mathbb{Q}$ is the smallest field between $\mathbb{R}$ and $\mathbb{Z}$. In other words, if there is a field $\mathbb{F}$ where $\mathbb{Z} \subset \mathbb{F} \subset \mathbb{R}$, then it must also be that $\mathbb{Q} \subset \mathbb{F}$.</p>
<p>My proof is by contradiction, assume that i... | fleablood | 280,126 | <p>You can be pretty straightforward.</p>
<p>Let $\mathbb Z \subset \mathbb F$ be a field. Then $\{a*(1/b)|a\in \mathbb Z; b \in \mathbb Z; b \ne 0\}\subset \mathbb F$. But $\{a*(1/b)|a\in \mathbb Z; b \in \mathbb Z; b \ne 0\}= \mathbb Q$. So $\mathbb Q \subset \mathbb F$.</p>
<p>That's it. Nothing more needs to... |
657,801 | <p>So this is a simple problem but I'm just getting stumped. The question is:<br><br>
A particle not connected to a spring, moving in a straight line, is subject to a retardation force of magnitude $\beta(\frac{dx}{dt})^n$, with $\beta > 0$.<br>
a) Show that if 0 < n < 1, the particle will come to rest in a fi... | Robert Lewis | 67,071 | <p>I too think</p>
<p>$m\dfrac{d^2x}{dt^2} + \beta(\dfrac{dx}{dt})^n = 0, \tag{1}$</p>
<p>is the right equation with which to start. Then observe that (1) implies</p>
<p>$m(\dfrac{dx}{dt})^{-n}\dfrac{d^2x}{dt^2} + \beta = 0, \tag{2}$</p>
<p>and that</p>
<p>$\dfrac{d}{dt}(\dfrac{dx}{dt})^{1 - n} = (1 - n)(\dfrac{d... |
657,801 | <p>So this is a simple problem but I'm just getting stumped. The question is:<br><br>
A particle not connected to a spring, moving in a straight line, is subject to a retardation force of magnitude $\beta(\frac{dx}{dt})^n$, with $\beta > 0$.<br>
a) Show that if 0 < n < 1, the particle will come to rest in a fi... | Qmechanic | 11,127 | <p>Hint: Split the problem in two:</p>
<ol>
<li><p>First solve the 1st order ODE $m\frac{dv}{dt}=-\beta v^n$ for the velocity $v(t)$ as a function of time $t$. (The exercise formulation doesn't say so, but there are physical reasons to believe that the velocity $v\geq 0$ should be assumed to be non-negative).</p></li>... |
64,653 | <p>Consider the two (inequivalent) $\mathbb{Z}$-representations $\phi,\psi$ of the symmetric group $S=S_3$ given by</p>
<p>$(1,2)^\phi=\left(\begin{array}{rr}0 &-1\\\ -1 & 0\end{array}\right), \qquad
(1,2,3)^\phi=\left(\begin{array}{rr}0 &1\\\ -1 & -1\end{array}\right);$</p>
<p>$(1,2)^\psi=\left(\begi... | DavidLHarden | 12,610 | <p>I have wondered about such lifts myself, and I want to give what I hope is a tantalizing hint of what such lifts may be able to tell us:</p>
<p>The lift you give of $S_{3}$ from $GL_{2}( \mathbb{Z} )$ to $Aut(F_{2})$ also gives an embedding of $C_{3} = A_{3}$ into $Aut(F_{2})$. Why is this useful? It gives a charac... |
463,539 | <p>I've transformed the number 11 to:</p>
<p>$11^e = 677.32$</p>
<p>Given the exponent and the transformed value, how can I solve for the original number?</p>
<p>I know that $x = y^z$ and that $z = \log_y(x)$, but I don't know how to solve for $x$? Can anyone explain how I can use the exponent $e$ and $677.32$ to fi... | cuneiformium | 500,021 | <p>This is equation with constant e (2.71) as exponent and no obvious unknown quantities. We can set equation with one unknown from Rasmus's answer (the general solution). However, the terms can be manipulated with the rules of logs and exponents, assuming work uses natural logarithms for base e.</p>
<pre><code> 1.... |
1,628,386 | <p>I had following limit of two variables as a problem on my calculus test. How does one show whether the limit below exists or does not exist? I think it does not exist but I was not able to show that rigorously. There was a hint reminding that $\lim_{t\to 0}\sin t / t=1$.</p>
<p>$$\lim_{(x,y)\to (0,0)} \frac{3x^2\si... | Brad A.M. | 296,774 | <p>Consider various paths that approach $(0,0)$. For instance,</p>
<p>$$\lim_{(x,y)\to (0,0)}\frac{3x^2\sin^2y}{2x^4+2\sin(y^4)}$$ along the line $y=0$ becomes</p>
<p>$$\lim_{x\to 0}\frac{3x^2\cdot 0}{2x^4}=0$$</p>
<p>However, along the line $y=x$ the limit is </p>
<p>$$\lim_{x\to 0}\frac{3x^2\sin^2x}{2x^4+2\sin(x^... |
1,407,944 | <p>Can this be proved that $\log(n)$ is irrational for every $n=1,2,3,\dots$ ?</p>
<p>I find that question in my mind in searching for if $\log(x)$ is irrational for every rational number $x\gt0$. </p>
| Ben Sheller | 250,221 | <p>Let $n\in\mathbb{N}$, $n\neq 1$. Suppose that there exist $a,b\in\mathbb{Z}$ such that $\log(n)=\frac{a}{b}$. Then we have that $n=e^{a/b}$ is a trancendental number since $e$ is a trancendental number and $\frac{a}{b}$ is rational. This is a contradiction. Therefore, $\log(n)$ is irrational for every $n=2,3,4,....$... |
2,266,342 | <p>So on my review for my final exam there is this question: </p>
<p>Is there a linear transformation from $P_2$ to $P_2$ with the following properties? In each case, either give an example of such a transformation or prove that no such transformation exist.</p>
<blockquote>
<p>$T(t^2+t+1) = t^2+t+1, T(t^2+2t+3) = ... | freshman | 217,694 | <p>Hint: $\{t^2+t+1, t^2+2t+3,t^2+2t+2\}$ is independent in $P_2$. Thus there exists such liner map $T$ and is unique. Find $T(t^2), T(t)$ and $T(1)$ solving the system
$$T(t^2)+T(t)+T(1)=t^2+t+1\\
T(t^2)+2T(t)+3T(1)=3t^2+2t+1\\
T(t^2)+2T(t)+2T(1)=2t^2+2t+1$$</p>
|
925,941 | <p>Solve:</p>
<p>$xy=-30$<br>
$x+y=13$</p>
<p>{15, -2} is a particular solution, but, how would I know if is the only solution, or what would be the way to solve this without "guessing" ?</p>
| Jose Arnaldo Bebita Dris | 28,816 | <p>The sum</p>
<p>$$x + y = 13 = -a$$</p>
<p>and the product</p>
<p>$$xy = -30 = b$$</p>
<p>gives rise to the quadratic equation</p>
<p>$$z^2 + az + b = 0.$$</p>
<p>Consequently, we have to solve the quadratic equation</p>
<p>$$z^2 - 13z - 30 = 0$$</p>
<p>for $z$, in order to get $x$ and $y$.</p>
<p>Since the ... |
6,990 | <p>The Fourier transform of periodic function $f$ yields a $l^2$-series of the functions coefficients when represented as countable linear combination of $\sin$ and $\cos$ functions.</p>
<ul>
<li><p>In how far can this be generalized to other countable sets of functions? For example, if we keep our inner product, can ... | David Hansen | 1,464 | <p>If you are interested in more general Fourier transforms, then the two things which spring immediately to my mind are:</p>
<ol>
<li><p>Titchmarsh's book <em>Fourier Integrals</em> contains a detailed treatment of what he calls "generalized kernels", which vaguely are pairs of functions $h(x),k(x)\in L^2(\mathbb{R})... |
2,083,475 | <p>This is the conic $$x^2+6xy+y^2+2x+y+\frac{1}{2}=0$$
the matrices associated with the conic are:
$$
A'=\left(\begin{array}{cccc}
\frac{1}{2} & 1 & \frac{1}{2} \\
1 & 1 & 3 \\
\frac{1}{2} & 3 & 1
\end{array}\right),
$$</p>
<p>$$
A=\left(\begin{array}{cccc}
1 & 3 \\
3 & 1
\end{array}\r... | MvG | 35,416 | <p>It seems you're homogenizing by introducing a first coordinate equal to one, instead of a last coordinate <a href="http://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections" rel="nofollow noreferrer">conventionally used</a>. So your conic is essentially described as</p>
<p>$$
(1,x,y)\cdot\begin{pmatrix}
... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| J. M. ain't a mathematician | 498 | <p>Two related integrals:</p>
<p>$$\int_0^\infty\sin\;x\quad\mathrm{d}x=1$$</p>
<p>$$\int_0^\infty\ln\;x\;\sin\;x\quad \mathrm{d}x=-\gamma$$</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Douglas S. Stones | 139 | <p>\[\sqrt{n^{\log n}}=n^{\log \sqrt{n}}\]</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| J126 | 2,838 | <p>$$\sec^2(x)+\csc^2(x)=\sec^2(x)\csc^2(x)$$</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| 2'5 9'2 | 11,123 | <p>\begin{align}
\frac{\mathrm d}{\mathrm dx}(x^x) &= x\cdot x^{x-1} &\text{Power Rule?}&\ \text{False}\\
\frac{\mathrm d}{\mathrm dx}(x^x) &= x^{x}\ln(x) &\text{Exponential Rule?}&\ \text{False}\\
\frac{\mathrm d}{\mathrm dx}(x^x) &= x\cdot x^{x-1}+x^{x}\ln(x) &\text{Sum of these?}&... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Noam D. Elkies | 93,983 | <p>$$ 71 = \sqrt{7! + 1}. $$</p>
<p>Besides the amusement of reusing the decimal digits $7$ and $1$,
this is conjectured to be the last solution of $n!+1 = x^2$ in integers.
($n=4$ and $n=5$ also work.) Even finiteness of the set of solutions
is not known except using the ABC conjecture.</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Feynman | 196,748 | <p>I have another one, but I'm quite unwilling to post this here because it's MINE, I haven't found it anywhere, so don't steal this.</p>
<p>Let us take the four most important mathematical constants: The Euler number $e$, the Aurea Golden Ratio $\phi$, the Euler-Mascheroni constant $\gamma$ and finally $\pi$. Well we... |
2,505,971 | <blockquote>
<p>Let $\lim_{x\to \infty }x^n f(x)=0$ for any $n \in \mathbb{N}$. Then find the symptote
$$g(x)=\dfrac{x^2+f(x)}{x+1+2f(x)}.$$</p>
</blockquote>
<hr>
<p>My try :
$$\lim_{x \to \infty }\dfrac{x^2+f(x)}{x+1+2f(x)}=\lim_{x \to \infty }\dfrac{x^{n+2}+x^nf(x)}{x^{n+1}+2x^nf(x)}=\infty?$$
Please help me.<... | Marcus Andrews | 97,648 | <p>As a general rule, when you have a recurrence $S(n)$ that is a summation of itself, it usually helps to try $S(n)-S(n-1)$ so you can cancel out most of the summation.</p>
<p>$S(n)=\sum\limits_{i=1}^{n-1}iS(i)$</p>
<p>$S(n)-S(n-1)=\sum\limits_{i=1}^{n-1}iS(i) - \sum\limits_{i=1}^{n-2}iS(i) = (n-1)S(n-1)$</p>
<p>No... |
39,828 | <p>Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.</p>
<p>So after this longish introduction, here goes: Man... | Charles Matthews | 6,153 | <p>Not sure I agree with the whole post in detail. Distinguish "pure algebra" from "applied algebra"; and within "pure algebra" distinguish "structural" issues from "combinatorial" ones such as the Burnside problem. Remembering that "abstract algebra" is the m... |
39,828 | <p>Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.</p>
<p>So after this longish introduction, here goes: Man... | David Corfield | 447 | <p>There are surely no hard and fast rules as to assessing the importance of a generalization of a concept. I once took a <a href="http://books.google.co.uk/books?id=s37rhfWs73QC">look</a> (chap. 9) at debates surrounding the move from groups to groupoids. One important step up for a concept is being deemed essential r... |
39,828 | <p>Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.</p>
<p>So after this longish introduction, here goes: Man... | Gerhard Paseman | 3,402 | <p>I think something is worth studying if it helps one of:</p>
<ul>
<li><p>solving a problem I know about,</p></li>
<li><p>giving a new perspective on something I know, or</p></li>
<li><p>raising interesting questions, some of which are easy to solve and
some of which aren't.</p></li>
</ul>
<p>Especially, I study it ... |
3,843,559 | <p>Can someone please give me a hint why for metric spaces we have</p>
<p><span class="math-container">$d_1(x,y)<d(x,y)\Rightarrow \{x|d(x,y)<\varepsilon\}\subset \{x|d_1(x,y)<\varepsilon\}$</span></p>
<p>I have expected the opposite:</p>
<p><span class="math-container">$d_1(x,y)<d(x,y)\Rightarrow \{x|d(x,y... | José Carlos Santos | 446,262 | <p>Because if <span class="math-container">$d(x,y)<\varepsilon$</span>, then <span class="math-container">$d_1(x,y)<d(x,y)<\varepsilon$</span>.</p>
|
2,454,455 | <p>I know this is a soft and opinion based question and I risk that this question get's closed/downvoted but I still wanted to know what other persons, who are interested in mathematics, think about my question.</p>
<p>Whenever people are talking about the most beautiful equation/identity Euler's identity is cited in... | CopyPasteIt | 432,081 | <p>Looking at the answers and comments I feel compelled to offer this as a compromise,</p>
<h1><span class="math-container">$\frac {e^{i \pi} + e^{-i \pi}}{2} = -1$</span></h1>
|
1,729,893 | <p>Let $\nu_1, \nu_2, \nu_3 \in \mathbb{R}$ not all be zero.</p>
<p>I wish to show
$$\nu_1^2 + \nu_1\nu_2 + \nu_2^2+\nu_2\nu_3 + \nu_3^2 > 0\text{.}$$</p>
<p><a href="http://www.wolframalpha.com/input/?i=x%5E2%2Bx*y%2By%5E2%2By*z%2Bz%5E2+%3E+0" rel="nofollow">Wolfram</a> seems to suggest splitting this into cases,... | Will Jagy | 10,400 | <p>$$
\frac{1}{2}
\left(
\begin{array}{ccc}
2 & 1 & 0 \\
1 & 2 & 1 \\
0 & 1 & 2
\end{array}
\right)
$$
is positive definite. If you leave off the $1/2$ factor, you have the Hessian matrix of second partial derivatives of the form. </p>
<p>Meanwhile, if we allow $x,y,z$ to take only integer valu... |
3,151,143 | <p>I want to find a symmetric matrix <span class="math-container">$A$</span>, whose eigenvalues are <span class="math-container">$4$</span> and <span class="math-container">$-1$</span>. One of the eigenvectors corresponding to the eigenvalue <span class="math-container">$4$</span> is <span class="math-container">$(2,3)... | gt6989b | 16,192 | <p><strong>HINT</strong></p>
<p>Pick your favorite eigenvector , say <span class="math-container">$(1, 0)^T$</span> and remember that the diagonalized form of <span class="math-container">$A$</span> looks like <span class="math-container">$A= V^{-1}DV$</span> for special <span class="math-container">$D$</span> and <sp... |
356,940 | <p>According to Wolfie:</p>
<p>$2^{-1} \bmod 5 = 3$</p>
<p><a href="http://www.wolframalpha.com/input/?i=2%5E-1+mod+5" rel="nofollow">http://www.wolframalpha.com/input/?i=2%5E-1+mod+5</a></p>
<p>Why is that?</p>
| Alfonso Fernandez | 54,227 | <p>Because $2\cdot3=1\pmod5$ - in this sense, $3$ is the multiplicative inverse of $2$ modulo $5$, and this is formally expressed as $2^{-1}=3\pmod5$</p>
|
147,338 | <p>I'm helping a student through a course in mathematics. In the course text, we came across the following problem concerning the Lagrange multiplier technique.
Given a differentiable function with continuous partial derivatives $f:\mathbb{R}\to\mathbb{R}:(x,y)\mapsto f(x,y)$ that has to be extremized and a constraint... | Stefan Smith | 55,689 | <p>In the Lagrange multiplier technique, what you really need is a point $(x^*,y^*)$ where $\nabla f(x^*,y^*)$ and $\nabla g(x^*,y^*)$ are linearly dependent. If the text says otherwise, they are wrong. This is slightly different from what you wrote down. Try $g(x,y)=x^2+y^2$ and $f(x,y) = (x-1)^2=0$, which has a co... |
1,243,750 | <p>$A \in \mathbb{R}^{n\times n}$, with $A^2 = 1$ and $A\ne\pm1$</p>
<p>Show that the only eigenvalues of $A$ are $1$ and $-1$.</p>
| Nescrio | 230,074 | <p>$$
Av = \lambda v \Rightarrow A^2v = \lambda Av = v
$$</p>
<p>So
$$
\lambda Av = v \Rightarrow \lambda \lambda v = \lambda ^2v = v
$$</p>
<p>This shows $\lambda = 1$ or $\lambda = -1$.</p>
|
587,484 | <p>Show that $\left(\displaystyle\frac{\partial (u,v)}{\partial (x,y)}\right)\left(\displaystyle\frac{\partial (x,y)}{\partial (r,s)}\right)=\displaystyle\frac{\partial (u,v)}{\partial (r,s)}$. Thus, prove that $\displaystyle\frac{\partial (u,v)}{\partial (x,y)}=\frac{1}{\displaystyle\frac{\partial (x,y)}{\partial (u,v... | aaaaa | 112,182 | <p>Let $$u_1=u, u_2=v, x_1=x, x_2=y, r_1=r, r_2=s$$ Here I'll use the symbol you use for the determinant for the matrix itself, that is:
$$\left(\frac{\partial (a_1, a_2)}{\partial (b_1,b_2)}\right)_{ij}:=\frac{\partial a_j}{\partial b_i}$$
Then
$$\left(\left(\frac{\partial (x,y)}{\partial (r,s)}\right)\left(\frac{\par... |
1,888,732 | <p>When I had the calculus class about the limit, one of my classmate felt confused about this limit:</p>
<blockquote>
<p><span class="math-container">$$\lim_{x\to \infty} \sqrt{x^2+x+1}-\sqrt{x^2+3x+1} = -1$$</span></p>
</blockquote>
<p>What he thought that since <span class="math-container">$x^2 > x$</span> and <s... | Sarvesh Ravichandran Iyer | 316,409 | <p>Let our problem be of the form $\displaystyle\lim_{x \to \infty} \sqrt[n]{p(x)} - \sqrt[n]{q(x)}$, where $p$ and $q$ are $n$th degree monic(coefficient of $x^n$ is $1$) polynomials. First of all, without loss of generality, we will rewrite $p$ and $q$ in the forms:
$$
p = (x + a)^n + \sum_{i=0}^{n-1} a_ix^i, q = (x ... |
1,888,732 | <p>When I had the calculus class about the limit, one of my classmate felt confused about this limit:</p>
<blockquote>
<p><span class="math-container">$$\lim_{x\to \infty} \sqrt{x^2+x+1}-\sqrt{x^2+3x+1} = -1$$</span></p>
</blockquote>
<p>What he thought that since <span class="math-container">$x^2 > x$</span> and <s... | Claude Leibovici | 82,404 | <p>Consider the general case $$A=\sqrt{x^2+ax+b}-\sqrt{x^2+cx+d}=x\left(\sqrt{1+\frac a x+\frac b {x^2}}-\sqrt{1+\frac c x+\frac d {x^2}}\right)$$ and use the fact that, for small $y$, using the generalized binomial theorem or Taylor series, $$\sqrt{1+y}=1+\frac{y}{2}-\frac{y^2}{8}+O\left(y^3\right)$$ For the first rad... |
2,665,539 | <p>How to solve the following system of equations</p>
<blockquote>
<p>$$
\begin{cases}
a+c=12\\
b+ac+d=86\\
bc+ad=300\\
bd=625\\
\end{cases}
$$</p>
</blockquote>
| Community | -1 | <p>Well, $b=\cfrac {625}{d}$ from your last equation, and $a=-c+12$ from your first equation.</p>
<p>Substitute $b$ and $a$ into your second equation to get $\cfrac{625}{d}+(-c+12)(c)+d=86$</p>
<p>Do the same for your third equation.</p>
|
2,348,290 | <p>The question and its answer is given in the following pictures:<a href="https://i.stack.imgur.com/xYhqD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xYhqD.png" alt="enter image description here"></a></p>
<p><a href="https://i.stack.imgur.com/Gv3IA.png" rel="nofollow noreferrer"><img src="https... | J.G. | 56,861 | <p>In answer to your first question, on that interval the integrand vanishes, so there is no contribution to the integral. In answer to your second question, on each of the length-$1$ intervals considered the floor factor is constant but $e^{-x}$ is not.</p>
|
2,348,290 | <p>The question and its answer is given in the following pictures:<a href="https://i.stack.imgur.com/xYhqD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xYhqD.png" alt="enter image description here"></a></p>
<p><a href="https://i.stack.imgur.com/Gv3IA.png" rel="nofollow noreferrer"><img src="https... | stity | 285,341 | <p>$$ \forall x \in [n, n+1[, \lfloor x \rfloor = n$$</p>
<p>and $$\int_0^1{\lfloor x \rfloor e^{-x}dx} = \int_0^1{0 e^{-x}dx} =0$$</p>
|
1,241,639 | <p>How do I prove that $\lim_{(x,y)\to (0,0)} \frac{x^2 y }{x^2 + y^2} = 0$?</p>
<p>I can prove this by notifying $x=rcos\theta$ and $y=rsin\theta$, but I remember that it could also be proven by squeeze theorem.</p>
<p>How do I prove this using squeeze theorem?</p>
| agha | 118,032 | <p>By AM-GM inequality we have:</p>
<p>$$2|xy|\leq x^2+y^2$$</p>
<p>So:</p>
<p>$$0<\frac{|xy|}{x^2+y^2}\leq\frac{1}{2}$$</p>
<p>Multiply both sides by $|x|$:</p>
<p>$$0\leq \frac{|x^2y|}{x^2+y^2} \leq \frac{|x|}{2}$$</p>
<p>Now $\frac{|x|}{2}$ goes to zero when $(x,y) \to 0$, so $\frac{|x^2y|}{x^2+y^2}$ also g... |
1,090,974 | <p>We have two functions of time $f(t)$ and $g(t)$, for which convolution and correlation are defined as following:</p>
<p>Convolution: $(f(t)\ast g(t))(\tau) = \int_{-\infty}^\infty{f(t)g(\tau-t)dt}$</p>
<p>Correlation: $(f(t)\star g(t))(\tau) = \int_{-\infty}^\infty{f^\ast(t)g(\tau+t)dt}$</p>
<p>In the english wik... | Timitry | 157,781 | <p>I figured out the answer while writing down the question. Here it is:</p>
<p>$(f^\ast(-t)\ast g(t))(\tau) = \int_{-\infty}^\infty{f^\ast(-t)g(\tau-t)dt} = -\int_{\infty}^{-\infty}{f^\ast(t)g(\tau+t)dt} = \int_{-\infty}^{\infty}{f^\ast(t)g(\tau+t)dt} = (f(t)\star g(t))(\tau)$</p>
<p>In the second step, the substitu... |
341,621 | <p>Is there an inequality such as
$$(a+b)^2 \leq 2(a^2 + b^2)$$
for higher powers of $k$
$$(a+b)^k \leq C(a^k + b^k)?$$</p>
| Samuel | 405 | <p>The <a href="http://en.wikipedia.org/wiki/Generalized_mean#Generalized_mean_inequality">generalized mean inequality</a> states that
$$\dfrac{a+b}2\leq \left(\dfrac{a^k+b^k}2\right)^{1/k},$$
with equality if and only if $a=b$, from which it follows that
$$(a+b)^k\leq 2^{k-1}(a^k+b^k),$$
with equality if and only if $... |
2,941,579 | <p>In Taylor's series, to determine the number of terms needed to obtain the desired accuracy, sometimes one needs to solve inequalities of the form <span class="math-container">$$\frac{a^n}{n!}<b,$$</span>
where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are fixed positive n... | Andrei | 331,661 | <p>Because you started with <span class="math-container">$$0\lt|x-3|\le\delta$$</span>
Multiply <strong>every</strong> term in this inequality by <span class="math-container">$4$</span> you get <span class="math-container">$$4\cdot 0\lt4\cdot |x-3|\le4\cdot\delta$$</span>
or, just using the last two terms <span class=... |
866,921 | <p>On the <a href="http://chat.stackexchange.com/rooms/36/mathematics">Mathematics chat</a> we were recently talking about the following problem <a href="https://math.stackexchange.com/users/32016/chriss-sis">@Chris'ssis</a> had to solve during an interview :</p>
<p>$$3\times 4=8$$
$$4\times 5=50$$
$$5\times 6=30$$ ... | Community | -1 | <p>The answer is $42$.</p>
<p>$69$ is also the answer.</p>
<p>"Purple feelings" is also an answer.</p>
<p>The truth of each of these is, of course, vacuous. :)</p>
<hr>
<p>If the question is posed as something other than multiplication, then it is the fault of the questioner for miscommunicating.</p>
<p>Although,... |
866,921 | <p>On the <a href="http://chat.stackexchange.com/rooms/36/mathematics">Mathematics chat</a> we were recently talking about the following problem <a href="https://math.stackexchange.com/users/32016/chriss-sis">@Chris'ssis</a> had to solve during an interview :</p>
<p>$$3\times 4=8$$
$$4\times 5=50$$
$$5\times 6=30$$ ... | hrkrshnn | 63,095 | <p>$$p(x)=$$</p>
<p>$$-\frac{1486263915627335609976345925580307452480}{198824918770116952269605821139049374259}-\frac{23535858736574459335924875719051524464677 x}{1789424268931052570426452390251444368331}+\frac{1532186339457747628597246965489647712097745599 x^2}{742539494635629574624160683858739355082631760}-\frac{530... |
866,921 | <p>On the <a href="http://chat.stackexchange.com/rooms/36/mathematics">Mathematics chat</a> we were recently talking about the following problem <a href="https://math.stackexchange.com/users/32016/chriss-sis">@Chris'ssis</a> had to solve during an interview :</p>
<p>$$3\times 4=8$$
$$4\times 5=50$$
$$5\times 6=30$$ ... | gnasher729 | 137,175 | <p>The answer stares you right in the face.</p>
<p>7 x 8 is a question mark.</p>
<p>Now I should add that one moderator apparently believed this was a joke answer. It should be obvious that it isn't. If I wanted to make a joke, I would have added a comment. Mathematics is about the manipulation of symbols, and this i... |
866,921 | <p>On the <a href="http://chat.stackexchange.com/rooms/36/mathematics">Mathematics chat</a> we were recently talking about the following problem <a href="https://math.stackexchange.com/users/32016/chriss-sis">@Chris'ssis</a> had to solve during an interview :</p>
<p>$$3\times 4=8$$
$$4\times 5=50$$
$$5\times 6=30$$ ... | Kal S. | 117,757 | <p>We rewrite the riddle using prime decomposition.</p>
<p><strong>Notation:</strong> <span class="math-container">$2^{x_1}3^{x_2}5^{x_3}7^{x_4}\leftrightarrow (x_1,x_2,x_3,x_4)$</span>
We have:
<span class="math-container">$$
(a_1,a_2,a_3,a_4) \odot (b_1,b_2,b_3,b_4) = (c_1,c_2,c_3,c_4)\\
(0,1,0,0) \odot (2,0,0,0) = (... |
2,125,136 | <p>I understand the usual motivation behind the truth table for the logical connective <span class="math-container">$\to$</span>.</p>
<p>However, I would like to know if there is a more fundamental reason for that truth table. Something that would have to do with arguments and validity.</p>
<p>A.G.Hamilton writes in <e... | Asinomás | 33,907 | <p>assuming you counted correctly:</p>
<p>The probability that he misses the first attempt is $\dfrac{4319}{4320}$, the probability he misses the second is $ \dfrac{4318}{4319}$ and so on.</p>
<p>The probability he doesn't guess is the product of all these fractions, almost everything cancels and you get $\dfrac{4310... |
163,043 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/160847/polynomials-irreducible-over-mathbbq-but-reducible-over-mathbbf-p-for">Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$</a> </p>
</blockquote>
<p>Anyon... | KReiser | 21,412 | <p>Yes. The polynomial $x^4+1$ does the trick. </p>
|
163,043 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/160847/polynomials-irreducible-over-mathbbq-but-reducible-over-mathbbf-p-for">Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$</a> </p>
</blockquote>
<p>Anyon... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\rm\ \ x^2 +\, 2\cdot3\cdot 5 $</p>
|
63,641 | <p>Here is the problem: Two mathematicians meet at a bar. They like each other and tend to collaborate. But it is not so clear what problems could be of common interest to both of them. Of course, the traditional way is they keep describing they work or their field in general so that hopefully they catch something at t... | Tim Dokchitser | 3,132 | <p>No, such an isomorphism does not always exist, and the smallest counterexample is $G=C_5\rtimes C_4$ with $C_4$ acting faithfully. It is not hard to see that the only automorphisms of $G$ are inner, and that they cannot map an element of order 4 to its inverse.</p>
|
859,493 | <p>Given that any fixed integer $n>0$, let $S=\{1,2,3,4,...,n\}$. Now a Red-Blue subset of $S$ is called $T$. Every element of $T$ is given a colour (either red or blue). For instance $\{17 (\text{red})\}, \{1 (\text{red}), 5 (\text{red})\}$ and $\{1(\text{red}),5(\text{blue})\}$ are $3$ different subsets of $S$.</p... | kingW3 | 130,953 | <p>By the binomial theorem $$(2x^2+(x-3))^8=\sum_{k=0}^8{8\choose k}2^{8-k}x^{16-2k}(x-3)^{k}$$
for $k<8$ term power of $x$ is bigger than $1$,so $k=8$ than it simplifies to find coefficient of $x$ in $(x-3)^8$ which is $8\cdot(-3)^7=-17496$</p>
|
1,920,781 | <p>Let's say I have a basic equation (here a linear combination):
$$ax_1 + bx_2=y$$</p>
<p>Is there any way to get the value of the sum $x_1+x_2$ if $a$, $b$ and $y$ is known?</p>
| Bobson Dugnutt | 259,085 | <p>If there are no further restrictions on $x_1$ and $x_2$, then we cannot <em>in general</em> (see <em>Lacklub</em>s comment below) say anything about $x_1+x_2$, as the system is <a href="https://en.wikipedia.org/wiki/Underdetermined_system" rel="nofollow">underdetermined</a>, i.e., there are more variables than equat... |
3,609,791 | <p><span class="math-container">$f(x) = \frac{x}{(1+x^2)}$</span></p>
<p>I need to find the range of function.</p>
<p>My method-</p>
<p>Let <span class="math-container">$f(x)=y$</span> </p>
<p>Then <span class="math-container">$x²y-x+y=0$</span></p>
<p><span class="math-container">$$x=1\pm\frac{\sqrt{(1-4y²)}}{2y}... | joeb | 362,915 | <p>Short answer ...</p>
<p>In your efforts to describe the solution set of the statement <span class="math-container">$$y = x/(1+x^2),$$</span> you manipulated algebraically, and arrived at the statement <span class="math-container">$$x = (1 + \sqrt{1-4y^2})/2y \qquad \text{or} \qquad x = (1 + \sqrt{1-4y^2})/2y,$$</sp... |
2,407,700 | <p>If $x$ satisfies $x^2 + 3x + \frac{3}x + \frac{1}{x^2} = 26$ and $x$ can be written as $a + \sqrt{b}$ where $a$ and $b$ are positive integers, then find $a + b$.</p>
<p>This equation becomes $x^4+3x^3-26x^2+3x+1=0$ which has four solutions. One of the solutions is $2+\sqrt3$ which has the form $a+\sqrt b$. So, $a+b... | GAVD | 255,061 | <p>Let $t = x + \frac{1}{x}$, then the equation is $$t^2 - 2 + 3t = 26,$$ or $$t^2 + 3t -28 = 0.$$</p>
<p>Then, we have $t = 4$ or $t = -7$. </p>
<p>If $t=4$, then $x + \frac{1}{x} = 4$, then $x = 2 \pm \sqrt{3}$.</p>
<p>If $t=-7$, then $x + \frac{1}{x} = -7$, then $x = \frac{1}{2}(-7\pm \sqrt{45})$.</p>
<p>Because... |
2,407,700 | <p>If $x$ satisfies $x^2 + 3x + \frac{3}x + \frac{1}{x^2} = 26$ and $x$ can be written as $a + \sqrt{b}$ where $a$ and $b$ are positive integers, then find $a + b$.</p>
<p>This equation becomes $x^4+3x^3-26x^2+3x+1=0$ which has four solutions. One of the solutions is $2+\sqrt3$ which has the form $a+\sqrt b$. So, $a+b... | Davood | 477,916 | <p>We can factor this equation as follows:
$$x^4+3x^3-26x^2+3x+1 = (x^2-4x+1) (x^2+7x+1).$$</p>
<hr>
<p>One can check that $2\pm \sqrt3$ are solutions to the $(x^2-4x+1)$,<br>
alse we can check that $\dfrac{-7\pm 3\sqrt5}{2}$ are solutions to the $(x^2+7x+1)$.</p>
|
65,892 | <p>Hello!</p>
<p>Let $M$ be an almost complex manifold. Let $TM$ denote its tangent bundle. Then we have the decomposition $TM\otimes\mathbb{C}=T^{1,0}M\oplus T^{0,1}M$ corresponding to the eigenvalues of the almost complex structure. This decomposition yields the decomposition:
$$
\Lambda^r(T^\star M\otimes\mathbb{C}... | diverietti | 9,871 | <p>Call $C^{\infty}_{p,q}(M)$ the space of smooth complex sections of the bundle $\Lambda^{p,q}T^*_M$ and let $2n$ be the real dimension of $M$. </p>
<p>The fact that
$$
dC^{\infty}_{p,q}(M)\subset C^{\infty}_{p+2,q-1}(M)+C^{\infty}_{p+1,q}(M)+C^{\infty}_{p,q+1}(M)+C^{\infty}_{p-1,q+2}(M)
$$
follows immediately from t... |
3,636,040 | <p>I am reading a paper titled "Cross-Calibration for Data Fusion of EO-1/Hyperion and Terra/ASTER" which mentions a particular minimization problem
<img src="https://i.stack.imgur.com/UgZrB.png" alt="Frobenius norm of y(I)-r(I)x"></p>
<p>I am not able to comprehend how this equation is converted (during implementatio... | dohmatob | 168,758 | <p><strong>Note.</strong> For ease of writing, everything is transposed in my answer, but should change anything in the implementation...</p>
<hr>
<p>So you have <span class="math-container">$L_m$</span> constrained problems of minimization of Frobenius norm. The objective function of the <span class="math-container"... |
2,706,061 | <p>This can be rewritten as $A\vec{x}=0$, where $A=(1, -2, 3), \vec{x}=(x, y, z)^T$.
I understand that one can find vectors perpendicular to this place by finding the basis for the null space, as the null space is perpendicular to the row space.</p>
<p>This basis is $(2, 1, 0), (-3, 0, 1)$. I believe that the space de... | Tal-Botvinnik | 331,471 | <p>This is because of linearity of the dot product. First, write your $A$ matrix as a vector $\vec y$, it will be simpler in a second, and note that what you mean by $A\vec x$ is nothing more than $\vec y\cdot \vec x$. I.e. your plane consists in all vectors perpendicular to the vector $y$. I guess you know this but I'... |
1,621,989 | <p>If we have any two co-prime positive integers <em>x</em> and <em>y</em>, does there always exist a positive integer <em>C</em> such that all integers greater than <em>C</em> can be expressed as <em>Ax</em>+<em>By</em> where <em>A</em> and <em>B</em> are also non-negative integers?</p>
<p>Do we have a formula to cal... | David | 119,775 | <p>I'm going to use more conventional notation otherwise my head will explode ;-)</p>
<p><strong>Theorem</strong>. Let $a,b$ be coprime positive integers, let $c$ be an integer, and consider the equation
$$ax+by=c\ .$$</p>
<ul>
<li>If $c=ab$ then the equation has no solutions in positive integers $x,y$.</li>
<li>If ... |
470,081 | <p>There is a question from an old topology prelim that is somewhat giving me a hard time. Consider the cylinder $X= S^1 \times [-1,1]$. Now we define an equivalence relation $\sim$ as follows: For points $v,v' \in S^1$, we have $(v,-1) \sim (v',-1)$ and $(v,1) \sim (v',1)$. I am asked to show that the quotient space $... | Stefan Hamcke | 41,672 | <p>Try to find for a point $((a,b),z)\in S^1\times [-1,1]$ a point $(x,y,z)$ in $S^2$ with the same $z$-coordinate and $x=\lambda a,\ y=\lambda b$ for a $\lambda$ which is a function in $z$.</p>
<p>Then show that the map $f:((a,b),z)\mapsto (x,y,z)$ is surjective and continuous, and that it induces a map $\tilde f:X^*... |
4,453,356 | <p>I had been having trouble understanding a proof of the irrational nature of √2.</p>
<p>I found this proof in the first page of the foreward to <a href="https://www.cs.ru.nl/%7Efreek/comparison/comparison.pdf" rel="nofollow noreferrer"><em>17 theorem provers of the world</em></a> where a 'geometrical proof' (is that ... | TonyK | 1,508 | <p>If <span class="math-container">$\sqrt 2$</span> is rational, it can be expressed as <span class="math-container">$p/q$</span> where <span class="math-container">$p,q$</span> are <em>integers</em>. (Is this what you were missing?) The proof then goes on to construct a rational representation of <span class="math-con... |
688,577 | <p>A beautiful polyhedron with 20 hexagons and 60 pentagons can be seen here: <a href="http://robertlovespi.wordpress.com/2013/11/03/a-polyhedron-with-80-faces/" rel="nofollow">http://robertlovespi.wordpress.com/2013/11/03/a-polyhedron-with-80-faces/</a> . Euler formula and the corresponding Diophantine equation give a... | vonbrand | 43,946 | <p>There are 13 <a href="http://en.wikipedia.org/wiki/Archimedean_solid" rel="nofollow">Archimedean solids</a>, none of them with your recipe.</p>
|
688,577 | <p>A beautiful polyhedron with 20 hexagons and 60 pentagons can be seen here: <a href="http://robertlovespi.wordpress.com/2013/11/03/a-polyhedron-with-80-faces/" rel="nofollow">http://robertlovespi.wordpress.com/2013/11/03/a-polyhedron-with-80-faces/</a> . Euler formula and the corresponding Diophantine equation give a... | John Joy | 140,156 | <p>I'm not sure if this leads anywhere, but I would ignore the hexagons and just deal with the region bounded by the planes that the pentagons lie on. This should result in a polyhedron with 60 triangles. Try to show that if these 60 triangles were edge-connected, that they would all lie on the same plane (and thus the... |
2,003,660 | <p>I got this exercise from the textbook Book of Proof, CH4 E12. I've tackle this problem in the following manner:</p>
<p>Suppose x is a real and $0 < x < 4$, it follows that,</p>
<p>\begin{align*}
&\Rightarrow 0 - 2 < x - 2 < 4 - 2 \\
&\Rightarrow 4 < (x - 2)^2 < 4\\
&\Rightarrow ... | Darío G | 27,454 | <p>We can do this proof by contradiction: We want to prove $P\rightarrow Q$ and we prove it by assuming $P$ and $\neg Q$, and getting a contradiction.</p>
<p>Suppose $0<x<4$ with $x\in\mathbb{R}$, and the inequality $\dfrac{4}{x(x-4)}\geq 1$ does not hold. </p>
<p>Then we have $0<x,0<4-x$ (both follow fro... |
1,493,783 | <p>We have $4$ pockets each containing $4$ placement positions, we want to obtain an estimate on the number of possible arrangements, when the exact set of allowed elements is given:
$$ls=[a,a,a,a,b,b,c,c,d,d,x,y] \tag{1}$$
and numbers [1:4] are used for empty slots. For example if one pocket is of the form -,a,-,- t... | Ian Miller | 278,461 | <p>16! ways to place all the items (include 4 null objects).
The four $a$'s are identical so 4! ways to interchange those.
The four $-$'s are identical so 4! ways to interchange those.
The two $b$'s are identical so 2! ways to interchange those.
The two $c$'s are identical so 2! ways to interchange those.</p>
<p>Answe... |
483,533 | <p>How to prove that the following linear map :</p>
<p>$T : \Bbb R^2 \to \Bbb R^2$ defined by $$T(e_1)=e_2$$
$$T(e_2)=0$$ </p>
<p>can't be diagonalized with respect to any basis. Here $e_i$s are standard basis.</p>
| Vishal Gupta | 60,810 | <p>Any map is diagonalizable if and only if it has the same number of linearly independent eigenvectors as the dimension of the space.</p>
<p>In your case, there is only one eigenvalue i.e. $0$ and the corresponding eigenspace is one dimensional. Hence, it does not have "enough" number of linearly independent eigenvec... |
483,533 | <p>How to prove that the following linear map :</p>
<p>$T : \Bbb R^2 \to \Bbb R^2$ defined by $$T(e_1)=e_2$$
$$T(e_2)=0$$ </p>
<p>can't be diagonalized with respect to any basis. Here $e_i$s are standard basis.</p>
| Robert Lewis | 67,071 | <p>One explanation by Yours Truly can be found in my answer to this question:</p>
<p><a href="https://math.stackexchange.com/questions/482688/proof-of-a-matrix-is-positive-semidefinite-iff-it-can-be-written-in-the-form-x">Proof of a matrix is positive semidefinite iff it can be written in the form $X'X$</a></p>
<... |
160,765 | <p>I have a functor $F:C \to D$ between poset-enriched categories, and I'd like to show that the induced map on classifying spaces is a homotopy-equivalence. To this end, I am trying to establish the presence of initial objects in all the fibers $d\setminus F$ and use the 2-categorical version of Quillen's Theorem A du... | Jonathan Chiche | 5,587 | <p>Dimitri Ara has just brought this question to my attention. Perhaps you will find the following useful. </p>
<p>Let us say that an object $z$ of a $2$-category $\mathcal{A}$ has a terminal object if, for every object $x$ of $\mathcal{A}$, the category $Hom_{\mathcal{A}}(x,z)$ has a terminal object. (In particular, ... |
3,563,419 | <p>What I did was writing <span class="math-container">$(1\, 2)(3\, 4)(1\, 3)(2\, 4)$</span> as <span class="math-container">$\bigl(\begin{smallmatrix}
1& 2 &3 & 4\\
2& 1 &4 & 3
\end{smallmatrix}\bigr)\bigl(\begin{smallmatrix}
1& 2 &3 &4 \\
3& 4& 1 & 2
\end{sma... | Luca Goldoni Ph.D. | 264,269 | <p>A rigorous proof could be the following:</p>
<ol>
<li><span class="math-container">$$
\begin{gathered}
\cos \left( {\frac{{\pi n^2 }}
{{n + 1}}} \right) = \cos \left[ {\pi \left( {n - 1} \right) + \frac{\pi }
{{n + 1}}} \right] = \hfill \\
\hfill \\
= \cos \left[ {\pi \left( {n - 1} \right)} \right]\cos \l... |
3,432,861 | <p><span class="math-container">$y'$</span> and <span class="math-container">$y$</span> occur linearly in the ODE
<span class="math-container">$$y'=\frac{2x+y}{y-x}.$$</span>
yet it is a first order non-linear ODE. I can find a family solutions of this homogeneous ODE by using <span class="math-container">$y=vx \impli... | vic165 | 723,908 | <p><span class="math-container">$$ y'=\frac{x(2+\frac{y}{x})}{x(\frac{y}{x}-1)} $$</span>
then you can make a replacement <span class="math-container">$$ t=\frac{y}{x} \Rightarrow \frac{dy}{dx}=\frac{dt}{dx}x+t \Rightarrow \frac{dt}{dx}x+t=\frac{2+t}{t-1} $$</span>
It remains only to divide the variables and take... |
3,432,861 | <p><span class="math-container">$y'$</span> and <span class="math-container">$y$</span> occur linearly in the ODE
<span class="math-container">$$y'=\frac{2x+y}{y-x}.$$</span>
yet it is a first order non-linear ODE. I can find a family solutions of this homogeneous ODE by using <span class="math-container">$y=vx \impli... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: Write <span class="math-container">$$y'=\frac{2+\frac{y}{x}}{-1+\frac{y}{x}}$$</span> and substitute <span class="math-container">$$u=\frac{y}{x}$$</span></p>
|
3,432,861 | <p><span class="math-container">$y'$</span> and <span class="math-container">$y$</span> occur linearly in the ODE
<span class="math-container">$$y'=\frac{2x+y}{y-x}.$$</span>
yet it is a first order non-linear ODE. I can find a family solutions of this homogeneous ODE by using <span class="math-container">$y=vx \impli... | Z Ahmed | 671,540 | <p>You can get the singular solution of this equation by puttin <span class="math-container">$y=vx$</span>, but treat <span class="math-container">$v$</span> as constant for a change. then
<span class="math-container">$$v=\frac{2+v}{v-1} \implies v=1\pm \sqrt{3} ~~real~~.$$</span> so the singular solutions are
<span cl... |
307,822 | <p>Excuse my incorrect use of terminology, I hope my question is clear:</p>
<p>I am coding a Python module which tests whether a given number is a member of the Fibonacci series. No problem with that. Additionally, should a number not be a member of the series, I would like to test whether it is significantly close to... | venzen | 62,930 | <p>A suitable solution (suggested by <a href="https://math.stackexchange.com/users/856/rahul-narain">Rahul Narain</a>) is to simply calculate the margin as a fixed percentage of each successive Fibonacci number in the series:</p>
<pre><code>margin = 1 + 0.01n
</code></pre>
<p>This gives the desired outcome of allowin... |
2,060,633 | <p>Let $R=(\Bbb Z/5\Bbb Z$)[X]/ $\langle X^n \rangle $ with $n\in\Bbb N_+$. Show that $f\in R$ is a zero divisor if and only if $f_0=0$. And how many zero dividers exist?</p>
<p>I know that zero divispr means there exists non-zero $m$ such that $fm=0$ in $R$ and that $\Bbb Z/5\Bbb Z$ has no zero divisors, meaning $f_... | sami ullah | 976,897 | <p>Yeah, this question gave me a headache for a week but then it was easy once I realized how to look at the question. The first catch is that the probability is being calculated for each group. Do not confuse with a sequence of events.
The book has skipped a step as well. let's assume the initial state. Four groups an... |
1,574,290 | <p>How do I prove this? </p>
<p>For the Fibonacci numbers defined by $f_1=1$, $f_2=1$, and $f_n = f_{n-1} + f_{n-2}$ for $n ≥ 3$, prove that $f^2_{n+1} - f_{n+1}f_n - f^2_n = (-1)^n$ for all $n≥ 1$.</p>
| David Holden | 79,543 | <p>the inductive step:
$$
f^2_{n+1} - f_{n+1}f_n - f^2_n = (-1)^n \Leftrightarrow \\
f_{n+1}f_{n-1} - f_n^2 =(-1)^n \Leftrightarrow \\
(f_{n+2}-f_n)(f_{n+1}-f_n) -f_n^2 = (-1)^n \Leftrightarrow \\
f_{n+2}f_{n+1}-f_nf_{n+3} = (-1)^n \Leftrightarrow \\
f_{n+1}^2+f_nf_{n+1}-f_nf_{n+1}-f_nf_{n+2}=(-1)^n \Leftrightarrow \\
... |
2,423,086 | <p>Show, by listing the elements or how we can list the elements, that </p>
<p>$\mathbb{N}^3$ = $\mathbb{N×N×N}$ is a countable set.</p>
<p><strong>Attempt at a solution:</strong></p>
<p>I was thinking about making use of Cantor's diagonal argument. If this were $\mathbb{N×N}$ I could just do:</p>
<p>(1,1), (1,2), ... | Hagen von Eitzen | 39,174 | <p>For an explicit listing of the elements of $\Bbb N^3$, you might start with all $(x,y,z)$ with $x+y+z=3$, then those with $x+y+z=4$, etc.
For each sum $s$, there are only finitely many choices of $x,y,z$ (namely $s-1\choose 2$), and these can be ordered lexically, i.e., sort by $x$, then $y$ (then $z$, but that's ir... |
2,642,547 | <p>If a subset A of $\mathbb{R}^n$ has no interior, must it be closed?</p>
<p>Can I prove this using the example of a subset A that consists of a single point, so A has no interior yet it is closed?</p>
| user87690 | 87,690 | <p>In general, every countable subset has empty interior, but not every countable subset is closed.</p>
|
2,642,547 | <p>If a subset A of $\mathbb{R}^n$ has no interior, must it be closed?</p>
<p>Can I prove this using the example of a subset A that consists of a single point, so A has no interior yet it is closed?</p>
| Asaf Karagila | 622 | <p>You have received numerous examples of countable sets which are with no interior, but not closed either. Let me note that the irrational numbers are not closed and have an empty interior. So it is not the countability which hinders your statement.</p>
<p>Let me address your secondary question. No, you cannot prove ... |
11,652 | <p>Last August, so-called "review audits" <a href="https://math.meta.stackexchange.com/q/10740/43351">were introduced</a> on MSE.</p>
<p>These are fake review samples added to the review queues, designed to catch those users who (in a dash to get a badge) practice mindless button mashing instead of careful considerati... | Lord_Farin | 43,351 | <p>No.</p>
<p>Please keep the review audits.</p>
|
1,809,871 | <p>I am trying to solve Poisson equation using FFT. The issue appears at wavenumber $k = 0$ when I want to get inverse Laplacian which means division by zero.</p>
<p>We have </p>
<p>${\nabla ^2}\phi = f$ </p>
<p>Taking FFT from both side we get:</p>
<p>$-k^2\hat\phi = \hat f $</p>
<p>or </p>
<p>$\hat\phi = \fra... | Vitality | 55,395 | <p><strong>THE PROBLEM</strong></p>
<p>Let me first restate the problem in 2 dimensions:</p>
<p>Solve the Poisson equation</p>
<p><span class="math-container">$\nabla^2\phi(x,y)=f(x,y)\;\;\;(x,y)\in[0, L_x]\times[0, L_y]$</span></p>
<p>subject to periodic boundary conditions, namely</p>
<p><span class="math-container">... |
2,276,429 | <p>I've tried multiple variations of polygons but can't find any that work. Do they exist?</p>
<p>Is it possible to draw a polygon on a grid paper and divide it into two equal parts by a cut of the shape shown on the Figure (a)? </p>
<p>Solve the same problem for a cut shown on Figure (b).</p>
<p>Solve the same prob... | tomi | 215,986 | <p><a href="https://i.stack.imgur.com/plHyU.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/plHyU.jpg" alt="enter image description here"></a></p>
<p>So your polygon can be a rectangle?</p>
<p>You need $a^2=(a+b)(a+c)-a^2$</p>
<p>$a^2=a^2+ac+ab+bc-a^2$</p>
<p>$a^2=ac+ab+bc$</p>
|
2,276,429 | <p>I've tried multiple variations of polygons but can't find any that work. Do they exist?</p>
<p>Is it possible to draw a polygon on a grid paper and divide it into two equal parts by a cut of the shape shown on the Figure (a)? </p>
<p>Solve the same problem for a cut shown on Figure (b).</p>
<p>Solve the same prob... | Andrei | 331,661 | <p>Here is something for (a) and (b)</p>
<p><a href="https://i.stack.imgur.com/722dH.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/722dH.png" alt="enter image description here"></a></p>
|
92,613 | <p>This is a question in Pinter's <em>A Book of Abstract Algebra</em>.</p>
<blockquote>
<p>Let $S=\{g\in G\mid \operatorname{ord}(g)=p\}$. Prove the order of $S$ is a multiple of $p-1$.</p>
</blockquote>
<p>In his solution Pinter says $a \in S$ implies that $a$ generates a subgroup with $p-1$ elements. Shouldn't th... | Justin Campbell | 4,842 | <p>You are correct, an element of order $n$ generates a subgroup with $n$ elements. Perhaps Pinter means that the subgroup generated by an element of order $p$ contains $p-1$ elements of order $p$ (namely, the nontrivial ones).</p>
|
1,405,809 | <p>So we have to find $\lim\limits_{x\to 0} \frac{a^x-1}{x}$ without using any series expansions or the L'Hopital's rule.<br>
I did it using both but I have no idea how to do it. I tried many substitutions but nothing worked. Please point me in the right direction.</p>
| Patrick Stevens | 259,262 | <p>It is precisely $f'(0)$, where $f(r) = a^r$.</p>
|
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