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 |
|---|---|---|---|---|
2,631,733 | <blockquote>
<p>Let $E = \mathcal{C}^0([a,b],\mathbb{R})$, provided with the $||\cdot ||_{\infty}$ norm. Let $\phi: \mathbb{R} \rightarrow \mathbb{R}$ that is $\mathcal{C}^1$. Show that the function given by $\Psi:E \rightarrow \mathbb{R}$:
$$ \Psi(f) = \int^{b}_{a}\phi(f(x))dx $$ is differentiable.</p>
</blockquo... | Aloizio Macedo | 59,234 | <p>We have that
\begin{align*}
\Psi(f+h)&=\int_a^b\phi \circ (f+h) \\
&=\int_a^b\big(\phi(f(x))+\phi'(f(x))h(x)+\epsilon_{f(x)}(h(x))\big)dx \\
&=\Psi(f)+\int_a^b\phi'(f(x)) h(x)dx+\int_a^b\epsilon_{f(x)}(h(x))dx.
\end{align*}</p>
<p>So the obvious candidate for the derivative at $f$ is $h \mapsto \int_a^b... |
3,020,285 | <p>I was reading this <a href="https://math.stackexchange.com/questions/443418/calculating-int-fraceizz-dz-on-the-semi-circle-given-by-rei-theta">post</a> and in the comments someone said that the difficult in calculating the limit as <span class="math-container">$r$</span> goes to <span class="math-container">$0$</spa... | caverac | 384,830 | <p>That function is called an <a href="https://en.wikipedia.org/wiki/Activation_function" rel="nofollow noreferrer">Activation Function</a>, and it is actually what makes Neuronal Networks interesting.</p>
<p>To give you an example, imagine you have a network with one input <span class="math-container">$x$</span> and ... |
3,020,285 | <p>I was reading this <a href="https://math.stackexchange.com/questions/443418/calculating-int-fraceizz-dz-on-the-semi-circle-given-by-rei-theta">post</a> and in the comments someone said that the difficult in calculating the limit as <span class="math-container">$r$</span> goes to <span class="math-container">$0$</spa... | Bion Alex Howard | 689,685 | <p>"AND" gate implies both are True. The neural correlate of boolean true/false is the sigmoid function. Thus, we can multiply the sigmoid of two values to make a continuous correlate of "and"</p>
<pre><code>def neural_and(a, b):
return sigmoid(a) * sigmoid(b)
</code></pre>
<p>if both a and b ar... |
3,301,881 | <p>I'm reading the proof that <span class="math-container">$L_XY=[X, Y]$</span> on page 225 <a href="https://books.google.com/books?id=xQsTJJGsgs4C&q=225#v=snippet&q=225&f=false" rel="nofollow noreferrer">in this book</a> and I believe it is not quite correct to due an error in equation 20.6. The author wri... | Alex Provost | 59,556 | <p>I think you are correct: <span class="math-container">$\phi_{-t *}:T_{\phi_t(p)}M \to T_pM$</span> should satisfy <span class="math-container">$$ \phi_{-t *} \left(\frac{\partial}{\partial x^j}\rvert_{\phi_t(p)} \right) = \frac{\partial\phi^i_{-t}}{\partial x^j}(\phi_t(p)) \frac{\partial}{\partial x^i}\rvert_p.$$</s... |
11,618 | <p>I'm teaching a preparatory course on mathematics at a university. The content is mostly calculus, manipulating expressions and solving equations and inequalities. I show a couple of simple derivations/proofs and ask the students to occasionally prove some simple equality, so the course is by no means rigorous. Most ... | Rory Daulton | 1,905 | <p>Here is a method I use in all my calculus classes to show that it <em>may</em> be true that the derivative of $e^x$ is itself.</p>
<p>Sketch a graph of $y=e^x$ high up on the board and sketch coordinate axes below that on the board. Say that the lower graph will be the derivative of the upper graph.</p>
<p>Start a... |
1,786,514 | <p>Let $S$ be a set containing $n$ elements and we select two subsets: $A$ and $B$ at random then the probability that $A \cup B$ = S and $A \cap B = \varnothing $ is?</p>
<p>My attempt</p>
<p>Total number of cases= $3^n$ as each element in set $S$ has three option: Go to $A$ or $B$ or to neither of $A$ or $B$</p>
<... | André Nicolas | 6,312 | <p>The idea is good, but the total number of cases is not $3^n$. </p>
<p>Let us assume that subsets are chosen independently, with all pairs of subsets equally likely. Then in effect for every element of $S$ we flip a fair coin twice. If the result is head, head, the element ends up in both $A$ and $B$. If the result ... |
3,706,190 | <p>Well it is pretty wierd for me to see this question, the function already is power series isn't it?
Am I missing the purpose of the excersize?</p>
| quasi | 400,434 | <p>First consider <span class="math-container">$x^n-1$</span> . . .
<p>
If <span class="math-container">$n=1$</span> then <span class="math-container">$x^n-1=x-1$</span> which is irreducible in <span class="math-container">$\mathbb{Z}_p[x]$</span>.
<p>
If <span class="math-container">$n > 1$</span> then <span class=... |
3,463,970 | <p>I am trying to see if someone can help me understand the isomorphism between <span class="math-container">$V$</span> and <span class="math-container">$V''$</span> a bit more <strong>intuitively</strong>.</p>
<p>I understand that the dual space of <span class="math-container">$V$</span> is the set of linear maps fro... | celtschk | 34,930 | <p>Maybe it helps if we first widen our view, in order to then narrow it again and see the double-dual as special case.</p>
<p>So let's start with functions (<em>any</em> functions, for now) <span class="math-container">$f:X\to Y$</span>. Let's as concrete example, take <span class="math-container">$X=Y=\mathbb R$</sp... |
3,463,970 | <p>I am trying to see if someone can help me understand the isomorphism between <span class="math-container">$V$</span> and <span class="math-container">$V''$</span> a bit more <strong>intuitively</strong>.</p>
<p>I understand that the dual space of <span class="math-container">$V$</span> is the set of linear maps fro... | Tristan Duquesne | 678,209 | <p>This natural isomorphism only arises in finite-dimensional vector spaces. Do note that there exists isomorphisms between <span class="math-container">$V$</span> and <span class="math-container">$V^*$</span> as well, but these need coordinates (or rather, an inner product) to be properly defined, so they're never a "... |
1,273,353 | <p>Suppose the Roulette table has 37 numbers (European Roulette table). During 37 spins, I always do the same bet: 35 numbers straight (35 chips in 35 different numbers).
Then:</p>
<ol>
<li>the probability of winning the 37 consecutive spins is $(\frac{35}{37})^{37}\approx 0.1279$,</li>
<li>the probability of losing ... | Daniel Fischer | 83,702 | <p>To prove that $\lim\limits_{n\to \infty} p_n = 0$, we can, since $p_n > 0$ for all $n$, show that $\lim\limits_{n\to\infty} \frac{1}{p_n} = +\infty$.</p>
<p>Rewriting in a convenient manner, we obtain</p>
<p>\begin{align}
\frac{1}{p_n} &= \prod_{k=0}^n \frac{b+k}{a+k}\\
&= \prod_{k=0}^n \biggl(1 + \frac... |
175,661 | <blockquote>
<p>Prove that the series $\sum_{n=1}^{\infty}\left\Vert x\right\Vert ^{n} $, $x\in\mathbb{R}^{n} $, does not converge uniformly on the unit ball $\left\{ x\in\mathbb{R}^{n}\mid\left\Vert x\right\Vert <1\right\} $. </p>
</blockquote>
<p>I am not sure how to show this. What I got to is that the given s... | Mohamed | 33,307 | <p>It holds if we add : $X$ a Hausdorff space</p>
|
617,598 | <p>Does anyone know any examples of $f$'s for which $-\triangle u(x) = k f(u(x))$ has an explicit solution (i.e. a formula for the solution, not a numerical approximation scheme) in terms of $k$?</p>
<p>I am interested in examples where $f\geq 0$ is neither constant nor linear. Optimally I would be interested in a smo... | abiessu | 86,846 | <p>It looks like you are applying fraction addition rules to fraction multiplication:</p>
<p>$$\frac ab+\frac cd=\frac {a\cdot d+b\cdot c}{b\cdot d}$$</p>
<p>Instead you should use:</p>
<p>$$\frac ab\cdot \frac cd=\frac {a\cdot c}{b\cdot d}$$</p>
<p>So we have</p>
<p>$$\frac{x^2+16y^2}{x} \cdot \frac{x^2+4xy}{x-4y... |
617,598 | <p>Does anyone know any examples of $f$'s for which $-\triangle u(x) = k f(u(x))$ has an explicit solution (i.e. a formula for the solution, not a numerical approximation scheme) in terms of $k$?</p>
<p>I am interested in examples where $f\geq 0$ is neither constant nor linear. Optimally I would be interested in a smo... | Zhoe | 99,231 | <p>As mentioned in the comments, by factoring out $x$ in the second fraction, $x^2+4xy$ becomes $x(x+4y)$. Which leaves you with:</p>
<p>$$\frac{x^2+16y^2}{x} \cdot \frac{x(x+4y)}{x-4y}=\frac{(x^2+16y^2)(x+4y)}{x-4y}$$</p>
<p>Alternately if you had not seen the cancellations and had ended up with $x^3+16xy^2+4x^2y+64... |
2,728,317 | <p>As I know when you move to "bigger" number systems (such as from complex to quaternions) you lose some properties (e.g. moving from complex to quaternions requires loss of commutativity), but does it hold when you move for example from naturals to integers or from reals to complex and what properties do you lose?</p... | Patrick Stevens | 259,262 | <p>The most important ones as I see it:</p>
<ul>
<li>Naturals to integers: lose well-orderedness, gain "abelian group" (and, indeed, "ring").</li>
<li>Integers to rationals: lose discreteness, gain "field".</li>
<li>Rationals to reals: lose countability, gain "Cauchy-complete".</li>
<li>Reals to complexes: lose a comp... |
2,728,317 | <p>As I know when you move to "bigger" number systems (such as from complex to quaternions) you lose some properties (e.g. moving from complex to quaternions requires loss of commutativity), but does it hold when you move for example from naturals to integers or from reals to complex and what properties do you lose?</p... | fleablood | 280,126 | <p>Losing order is the most important but we also lose that if $b > 1$ then $b^z$ is injective so $\ln z$ (or $\log_b z$) is no longer a function but an equivalence class.</p>
|
2,728,317 | <p>As I know when you move to "bigger" number systems (such as from complex to quaternions) you lose some properties (e.g. moving from complex to quaternions requires loss of commutativity), but does it hold when you move for example from naturals to integers or from reals to complex and what properties do you lose?</p... | J.G. | 56,861 | <p>Many people have said the crucial thing is order. I'd defend another fact we lose: that a number is self-conjugate. It's not obvious this matters, since we don't bother defining a "conjugate" of a real. But there's a reason I bring it up. The Cayley-Dickson construction can be thought of as a dimension-dou... |
2,728,317 | <p>As I know when you move to "bigger" number systems (such as from complex to quaternions) you lose some properties (e.g. moving from complex to quaternions requires loss of commutativity), but does it hold when you move for example from naturals to integers or from reals to complex and what properties do you lose?</p... | djechlin | 79,767 | <p>$\mathbb C$ is more rigid. Holomorphic functions are analytics. This means complex manifolds are polynomial-like and more akin to algebraic varieties than to real manifolds. By contrast $\mathbb R$ manifolds can be glued together using functions like $\exp(-1/x^2)$ that can $C^\infty$-smoothly transition from one fu... |
217,483 | <p><strong>Prove that if $I \subset \mathbb{R}$ is an open interval and $f: I \to \mathbb{R}$ differentiable, and $f$ has only one critical point $x_0$ and this critical point is a local minimum, then $x_0$ is also the absolute min of $f$, using Rolle's and IVT.</strong></p>
<p>To me this seems "obvious", because if $... | Julian Kuelshammer | 15,416 | <p>I'm not sure if the following is allowed but:</p>
<p>Inserting the first equation in the second we get:
$\alpha+1=\alpha+\gamma+\delta$.</p>
<p>Now we can substract $\alpha$ at both sides to get $1=\gamma+\delta$, which is a contradiction.</p>
|
3,767,935 | <p>Prove or disprove that if <span class="math-container">$$\prod\limits_{x=2}^{\infty} f(x)=0$$</span> and <span class="math-container">$f(x)\neq0$</span> for any <span class="math-container">$x\geq0$</span> then <span class="math-container">$$\prod\limits_{x=2}^{\infty} f(x\varphi)=0$$</span> for any constant <span c... | Eric Towers | 123,905 | <p>Let is say we want to disprove it. Then we need a <span class="math-container">$\varphi$</span> for which this product is not zero. One way to obtain that is for <span class="math-container">$f$</span> to be <span class="math-container">$1$</span> at all integer multiples of <span class="math-container">$\varphi$<... |
3,252,737 | <p>Show that <span class="math-container">$$\int_{0}^{\pi/2} \sin^3x \cos^2 x \cos7x ~ dx= \frac{1}{60}$$</span>
I could solve this integral by making use of the special expansion <span class="math-container">$$\cos 7x=64 \cos^7 x- 112\cos^5 x + 56 \cos^3 x -7\cos x$$</span> and then using <span class="math-container"... | Z Ahmed | 671,540 | <p>Use <span class="math-container">$$\int_{0}^{a} f(x) dx=\int_{0}^a f(a-x) dx. ~~~(1)$$</span> Then <span class="math-container">$$I=\int_{0}^{\pi/2} \sin^3x \cos^2 x \cos7x ~ dx. ~~~(2)$$</span> becomes<span class="math-container">$$I=\int_{0}^{\pi/2} \cos^3x \sin^2 x (-\sin7x) ~ dx. ~~~(3)$$</span> By adding (2) an... |
2,760,982 | <p>Let $n\in\mathbb{Z}$. What are all solutions, for $z\in\mathbb{C}$, of $ z^n=\overline{z} $?</p>
<p>To solve it, I tryed to write the term in Polar-Form and than take the logarithm. Because of the "ln(r)" term I was unable to find a solution.</p>
<p>I also tryed to write $ z^n $ as a Binomial, but this was not hel... | MPW | 113,214 | <p><strong>Hint:</strong> Assuming $z=re^{it}\neq 0$ (which may or may not be a solution depending on the sign of $n$), your equation is equivalent to$$r^ne^{int}=re^{-it}$$
$$r^{n-1}e^{i(n+1)t} = 1\tag{uses $r\neq 0$}$$
$$\implies r=1,(n+1)t=2k\pi$$
This is enough for you to enumerate the solutions. You must consider ... |
4,087,134 | <p>I have the polynomial <span class="math-container">$ p(x) = ax^3 + bx^2 + cx + d $</span>. I have to show that:</p>
<p><span class="math-container">$ \int_{-1}^{1} p(x) = p(- \frac{1}{\sqrt{3}}) + p (\frac{1}{\sqrt{3}}) $</span></p>
<p>I'm kind of stuck. My idea so far is to use "proof by symmetry" and the... | J.G. | 56,861 | <p>Since <span class="math-container">$X^2=|X|^2$</span>, if <span class="math-container">$E[X^2]$</span> exists then <span class="math-container">$E[X^2]=E[|X|^2]$</span>, so if <span class="math-container">$\sigma_X$</span> is finite <span class="math-container">$\mu_X^2+\sigma_X^2=\mu_{|X|}^2+\sigma_{|X|}^2$</span>.... |
233,846 | <p>This may be related to <a href="https://mathematica.stackexchange.com/q/98147">How to discretize a BezierCurve?</a>, but this question deals with <code>BSplineCurve</code>s with specific <code>SplineWeights</code>, so I don't think the answers there will help here.</p>
<hr />
<p><strong>Background</strong></p>
<p>I ... | kglr | 125 | <p><strong>1. To get <em>"the black curve (to) completely obscure the red curve"</em></strong></p>
<p>You can replace <code>BSplineCurve</code>s with <code>Line</code>s using <a href="https://reference.wolfram.com/language/ref/BSplineFunction.html" rel="noreferrer"><code>BSplineFunction</code></a>:</p>
<pre><... |
1,184,501 | <p>I need help putting this in $0/0$ or $\infty/\infty$:</p>
<p>$$\lim_{x\to 0}{1\over xe^{x^{-2}}}$$</p>
<p>I've tried every possible combination, and I don't get what I'm missing. Using a graphic calculator, you easily see that the $\lim_{x\to 0}$ of this function is $0$.</p>
| egreg | 62,967 | <p>Substitute $t=1/x$ and compute separately the limits from the right and from the left:
$$
\lim_{x\to0^+}\frac{1}{x\exp(x^{-2})}=
\lim_{t\to\infty}\frac{t}{\exp(t^2)}\overset{\mathrm{(H)}}{=}
\lim_{t\to\infty}\frac{1}{2t\exp(t^2)}=0
$$
and
$$
\lim_{x\to0^-}\frac{1}{x\exp(x^{-2})}=
\lim_{t\to-\infty}\frac{t}{\exp(t^2)... |
361,212 | <blockquote>
<p>Suppose $X_1$ is a standard normal random variable. Define
$$X_2=\begin{cases} -X_1, &\text{if} \,\, |X_1|<1 \\ \,\,\,\,X_1, & \text{otherwise}\end{cases}$$ Obtain the cumulative distribution function of $X_1+X_2$ in terms of the cumulative distribution function of a standard normal rand... | Did | 6,179 | <p>It seems that $F_Y(y)=F_X(u(y))$ with $u(y)=\frac12y$ if $|y|\geqslant2$, $u(y)=-1$ if $-2\leqslant y\lt0$ and $u(y)=+1$ if $0\leqslant y\lt2$.</p>
|
749,090 | <p>Prove $\ a_{n}<2^{n} $ for every natural number n, where $\ a_{n} $ is defined recursively by $$ a_{1}=1, a_{2}=2, a_{3}=3, a_{n}=a_{n-3}+a_{n-2}+a_{n-1},\ for\ n>=4$$
Once I get the explicit equation, proving this would be easy with induction, however I'm having trouble finding it. I can't find the connecti... | marty cohen | 13,079 | <p>Consider coordinates
where $0 \le x, y \le 3$.
This has area $9$.</p>
<p>The points where
$y \ge x+1$
make up the part of this square
on or above the line
$y = x+1$.
This is a right triangle
with sides 2 and 2,
so its area is 2.</p>
<p>The probability wanted is
the ratio of these
or 2/9.</p>
<p>More generally,
su... |
3,845,520 | <blockquote>
<p><strong>The length approximately equals width. The length is three times the height. The volume of the building is about <span class="math-container">$0.009 km^3$</span>.</strong></p>
</blockquote>
<hr />
<h2><em>The answer is 100 m by 300 m by 300 m.</em></h2>
<p>This question is supposed to be solved ... | user2661923 | 464,411 | <p>I wasn't going to answer, because I agree with Ananyapam De's comment that this is the wrong site. However, from all of the reactions...</p>
<p><span class="math-container">$\left(13 - \lfloor \sqrt{13} \rfloor \right)^{\lfloor \sqrt{13} \rfloor}.$</span></p>
|
1,832,512 | <p>How can you find the inverse of $f(x) = 2x^2+8x+13?$ This is what I've tried so far:</p>
<ol>
<li>$y = 2x^2+8x+13$</li>
<li>$x = 2y^2+8y+13$</li>
<li>$x-13 = 2y^2+8y$</li>
<li>$x-13=y(y+8)$</li>
</ol>
<p>This is where I got stuck. To be clear, I want to write $x$ in terms of $y.$ <strong>Credit to Jenna</strong> f... | Mary | 315,109 | <p>I think you meant to write $x$ in terms of $y$?
$$y=2x^2+8x+13$$$$y-13=2x^2+8x$$$$\frac{y-13}{2}=x^2+4x=(x+2)^2-4$$$$\frac{y-13}{2}+4=(x+2)^2$$$$\pm\sqrt{\frac{y-13}{2}+4}=x+2$$$$x=\pm\sqrt{\frac{y-13}{2}+4}-2$$</p>
|
1,896,546 | <p>I have a set of 3D points (I'll call them "outer" points) and a 2D point ("inner" point).</p>
<p>I need to quickly calculate a "good" third coordinate for the inner point so that it would place the constructed 3D point as "close" to the outer points as possible. "Close" may be defined as a minimum sum of distances ... | John Wayland Bales | 246,513 | <p>Well, without knowing much more about the general problem the best I can do is probably what you already know, but I will show it anyway.</p>
<p>Given three entry or exit points $(x_1,y_1,z_1),\,(x_2,y_2,z_2),\,(x_3,y_3,z_3)$ first find the $\textbf{normal vector}$ of the plane containing them by taking the cross p... |
1,896,546 | <p>I have a set of 3D points (I'll call them "outer" points) and a 2D point ("inner" point).</p>
<p>I need to quickly calculate a "good" third coordinate for the inner point so that it would place the constructed 3D point as "close" to the outer points as possible. "Close" may be defined as a minimum sum of distances ... | David K | 139,123 | <p>I think a problem like this is better solved by making as good a model
as you can of the way the real-life object is constructed,
not by taking some mathematical abstraction.</p>
<p>The elevation of most points in a tunnel follows the elevation of a
line painted down the center of the roadway within that tunnel.
(T... |
3,462,507 | <p>Before diving into the Sherman-Morrison formula, Meyer in <em>Matrix Analysis and Applied Linear Algebra</em>, pg. 124, starts with</p>
<p><span class="math-container">$$
(I+cd^T)^{-1} = I - \frac{cd^T}{1+dc^T}
$$</span></p>
<p>where <span class="math-container">$c,d$</span> are vectors, and says "it's straightfor... | Christopher Gadzinski | 380,117 | <p>Well, <span class="math-container">$I + \alpha cd^T$</span> with <span class="math-container">$\alpha \in \mathbb{R}$</span> is the general form of a matrix that fixes vectors orthogonal to <span class="math-container">$d$</span> by multiplication on the left and fixes vectors orthogonal to <span class="math-contain... |
1,883,443 | <p>I need to compare (asymptoticly) between $\left ( \frac{\ln n +4 \ln\ln n}{\ln n} \right )^{\ln n}$ and $16^{\ln\ln n}$. The options are $\Theta , \omega, o$.
My work so far:</p>
<p>I denoted $t_n=\ln n$ to make things cleaner.</p>
<p>The first sequence is
$\left ( 1+\frac{4 \ln t_n}{t_n} \right )^{t_n}$. It looks... | samerivertwice | 334,732 | <p>The negation of "You know nothing." is "You do not know nothing." Which implies that your knowledge is non-zero; not that it is necessarily absolute.</p>
|
193,053 | <p>Assume that </p>
<p>$$T(n) = 2T\left(\frac{n}{2}\right) + \Theta(n \log n)$$</p>
<p>By <a href="http://en.wikipedia.org/wiki/Master_theorem#Generic_form_2" rel="nofollow">Generic form of master theorem</a> with $a = 2$, $b = 2$ and $f(n) = c \, n \log n$, it can easily be proved that $T(n) = \Theta(n \log^2 n)$.<... | Did | 6,179 | <p>This was already answered multiple times on the site but here we go. Let $S(k)=2^{-k}T(2^k)$, then $S(k)=S(k-1)+\Theta(k)$, hence $S(k-1)+Ak\leqslant S(k)\leqslant S(k-1)+Bk$ for some finite $A$ and $B$. Iterating this yields $S(0)+A\sum\limits_{i=1}^ki\leqslant S(k)\leqslant S(0)+B\sum\limits_{i=1}^ki$, hence $S(k)... |
1,144,044 | <p>I need to find values of $x_1, x_2, x_3, $ with $0 \leq (x_1+x_2+x_3) \leq 1$ from the two follow equations:</p>
<p>$10 = 12x_1+3.6x_3 $</p>
<p>$39=14x_2+1.4x_3$</p>
<p>With the restriction
$0 \leq{x_1, x_2, x_3} \leq1 $</p>
<p>I've tried every way I could think of, but can't find a way to keep $x_1+x_2+x_3 &... | Mankind | 207,432 | <p>I would rewrite your equations as</p>
<p>$x_1 = \frac{10-3.6x_3}{12},$</p>
<p>$x_2 = \frac{39-1.4x_3}{14}.$</p>
<p>Then I would plug these two expressions (where the right-hand-side contains <em>only</em> $x_3$) for $x_1$ and $x_2$ into $x_1+x_2+x_3$. Then you get something that depends only on $x_3$, and you the... |
1,144,044 | <p>I need to find values of $x_1, x_2, x_3, $ with $0 \leq (x_1+x_2+x_3) \leq 1$ from the two follow equations:</p>
<p>$10 = 12x_1+3.6x_3 $</p>
<p>$39=14x_2+1.4x_3$</p>
<p>With the restriction
$0 \leq{x_1, x_2, x_3} \leq1 $</p>
<p>I've tried every way I could think of, but can't find a way to keep $x_1+x_2+x_3 &... | Khosrotash | 104,171 | <p>you have 3 variable ,and two equation . so : find all of them by one variable ,and put in first relation for example
$$x_3=\alpha$$ $$x_1=\frac{10-3.6\alpha}{12}\\x_2=\frac{39-1.4\alpha}{14}\\ 0\leq \frac{10-3.6\alpha}{12}+\frac{39-1.4\alpha}{14}+\alpha \leq 1$$</p>
|
1,107,013 | <p>Suppose that $f$ is a differentiable real function in an open set $E \subset \mathbb{R^n}$, and that $f$ has a local maximum at a point $x \in E$. Prove that $f'(x)=0$</p>
| symplectomorphic | 23,611 | <p>Hint: what happens to $$\frac{x+1}{\sqrt{x^2+1}}$$ as $x\to\infty$? And what is the amplitude of $2\cos x$?</p>
|
1,752,577 | <p>Let X be a topological space. All that I know is Borel $\sigma$ algebra on X is the smallest $\sigma$ algebra generated by $T_X$ i.e. set of all open sets in X. Is there any other characterization of Borel $\sigma$ algebra on X? or we can show the above assertion using the definition only.</p>
| Daniel Valenzuela | 156,302 | <p>Hint: for finite products, the product topology is generated by products of open sets. Hence you only need to use the definition to show what you need.</p>
|
3,109,300 | <p>For the case that <span class="math-container">$m\geq0$</span> I don't need to apply L'Hospital.</p>
<p>Let <span class="math-container">$m<0$</span></p>
<p>We have <span class="math-container">$x^m=\frac{1}{x^{-m}}$</span></p>
<p>We also know that <span class="math-container">$x^{-m}\rightarrow 0$</span> as <... | Mark Viola | 218,419 | <blockquote>
<p><strong>Direct application of L'Hospital's Rule does not provide a tractable way forward as mentioned in the OP.</strong> </p>
</blockquote>
<hr>
<p>To see this, we begin by writing (for <span class="math-container">$m<0$</span>, <span class="math-container">$|m|\in\mathbb{N}$</span>)</p>
<p><s... |
194,123 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/150482/probability-of-a-random-binary-string-containing-a-long-run-of-1s">Probability of a random binary string containing a long run of 1s?</a> </p>
</blockquote>
<p><strong><em>EDIT</strong>: Cocopuffs b... | mercio | 17,445 | <p>Define $Q(L) = P(L)/2^{L+1}$,
so that your recurrence relation becomes : $Q(L) = 4^{-L} + \sum_{i=1}^{L-1} Q(L+i)$.</p>
<p>Then you get $Q(L+1) - Q(L) = (1-4).4^{-L-1} + Q(2L) + Q(2L+1) - Q(L+1)$, thus $2Q(L+1) - Q(L) = -3.4^{-L-1} + Q(2L) + Q(2L+1)$.
Together with $Q(2) = 4^{-2} + Q(3)$, we have an equivalent syst... |
1,955,225 | <p>In working with a particular gene for fruit flies, geneticists classify an individual fruit fly as $\small \text{dominant, hybrid or recessive}$. In running an experiment, an individual fruit fly is crossed with a hybrid, then the offspring is crossed with a hybrid and so forth. The offspring in each generation are ... | Antoni Parellada | 152,225 | <p>I hope you don't mind it if I dumb it down (mainly to make sure I get it)...</p>
<p>So we have </p>
<p>$$P=\overset{\begin{matrix}\color{blue}{\text{Dom}}&\color{blue}{\text{Hyb}}&\color{blue}{\text{Rec}}\end{matrix}}{\begin{bmatrix}
&.5 & .5 & 0 & \\
&.25 & .5 & .25 & \\
... |
189,119 | <p>How to calculate my age given date of birth? I have read other questions and answers but is not satisfied with answers. My question is how do I calculate my age if my date of birth is 18.03.2000?</p>
| Michael E2 | 4,999 | <p>Let's start with a system with exact coefficients:</p>
<pre><code>frexact = fr /. Equal -> Subtract /.
x_Real :> Rationalize[10^16 (Rationalize@x)]/10^16
(*
{-(29/(10000000000000000 x^4)) + 1/x^2 - 1/(-x + y)^2 - 1/(-x + z)^2,
-(29/(10000000000000000 y^4)) + 1/y^2 + 1/(-x + y)^2 - 1/(-y + z)^2,
-(29/(100... |
1,478,103 | <p>I'm looking for examples of subtle errors in reasoning in a mathematical proof. An example of a 'false' proof would be</p>
<blockquote>
<p>Let $a=b>0$. Then $a^2 - b^2 = ab - b^2$. Factoring, we have $(a-b)(a+b) = b(a-b)$, which after cancellation yields $b = a = 2b$ and thus $1=2$. </p>
</blockquote>
<p>Howe... | NoChance | 15,180 | <p>The following deduction is not correct:</p>
<p>$i^2 = i*i = {\sqrt{-1}} * {\sqrt{-1}} = {\sqrt{(-1) * (-1)}} = {\sqrt{1}} = 1$</p>
|
299,304 | <p>I had posted the following problem on <a href="https://math.stackexchange.com/questions/2722893/the-modulus-of-a-polynomial-are-the-same-is-1">stack exchange</a> before.</p>
<blockquote>
<p>Suppose $\lambda$ is a real number in $\left( 0,1\right)$, and let $n$ be a positive integer. Prove that all the roots of t... | Noam D. Elkies | 14,830 | <p>The claim is true for $\lambda = 0$, when $f(x) = x^n + 1$,
and for $\lambda = 1$, when $f(x) = (x+1)^n$. We show that this
implies the claim for all intermediate $\lambda$, by proving that
the number of zeros on the unit circle is a non-decreasing
function of $\lambda$.</p>
<p>Let $\lambda = e^t$ with $-\infty &l... |
2,010,158 | <p>Let $b >0$ , let $B= \{ f \in C^r([-b,b]) : f(x) = f(-x) for \ \ 0\leq x\leq b\}$, and let $A$ be the set of all polynomials that contain only terms of even degree (with domains restricted to $[-b,b]$). Then the uniform closure of $A$ is $B$.</p>
<p>I am not getting any clue how to solve the problem. Help Neede... | Bob Jones | 370,252 | <p>First you need to show that all uniform limits $f$ of sums of even-degree polynomials satisfy $f(x)=f(-x)$ and are continuous. The first is because even-degree polynomials and their sums all satisfy this relation, so any limit should too. The second part, that uniform limits of continuous functions are continuous, i... |
392,580 | <p>How to evaluate the following
$$\int_0^{\infty} \frac{\sin (ax)}{e^{\pi x} \sinh(\pi x)} dx $$
Given hints says to construct a rectangle $0\to R\to R+i\to i \to 0$ and consider $\displaystyle f(z):=\frac{e^{iaz}}{e^{2\pi z}-1} $ and evaluate around it but that does not help.</p>
<p><strong>ADDED::</strong> I need ... | Caran-d'Ache | 66,418 | <p>Well, it's not the direct answer on you questions (because you were given a direct hint in the link above), but an alternative way.
You can look at it like the Laplace transform:
$$\int_0^{\infty} \frac{\sin (ax)}{e^{\pi x} \sinh(\pi x)} dx=2\int_0^{\infty} \frac{\sin (ax)}{1-e^{-2\pi x}}e^{-2\pi x} dx=2\int_0^{\inf... |
1,563,686 | <p>Let $\sim$ be define so that $a\sim b$ exactly when $a \times b$ is divisible by $3$. Is this an equivalence relation? If not, which of the three properties (reflexive, symmetric, transitive) does not hold?</p>
<p>Solution:</p>
<p>We need to test each of the following cases to see if they hold.</p>
<p>Here are my... | BrianO | 277,043 | <p>$3$ is prime, so $a\sim b \iff 3|ab \iff 3|a \text{ or } 3|b$. </p>
<p>$\sim$ is not transitive: $1\sim 3$ and $3\sim 1$, but not $1\sim 1$.</p>
<p>As just seen, $\sim$ is not reflexive.</p>
<p>It is symmetric, obviously.</p>
|
2,164,465 | <p>Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.</p>
<p>Where would a counting concept like nCr and nPr fall into this mix?</p>
| hmakholm left over Monica | 14,366 | <p>There is no strong convention for these notations.</p>
<p>In most cases the $C$ and $P$ notations are self-delimiting, because the arguments are typeset as subscripts or superscripts. They can be written as ${}_nC_r$ or ${}^nC_r$ or $C^n_r$, where in each case it is unambiguous that the argument expression is whate... |
3,117,260 | <p>So according to the commutative property for multiplication:</p>
<p><span class="math-container">$a \times b = b \times a$</span> </p>
<p>However this does not hold for division</p>
<p><span class="math-container">$a \div b \neq b \div a$</span> </p>
<p>Why is it that in the following case:</p>
<p><span class="... | Jennifer | 506,739 | <p>Start with 56 × 100 ÷ 8.</p>
<p>Swap the first two factors which is definitely allowed: 100 × 56 ÷ 8.</p>
<p>Now put parentheses around part of it: 100 × (56 ÷ 8).</p>
<p>Again, swap: (56 ÷ 8) × 100.</p>
<p>Take off the parentheses: 56 ÷ 8 × 100.</p>
<p>And there you go! transforming one into the other using o... |
50,209 | <p>Over the past few days I have been pondering about this: I enjoy technical things (like programming and stuff) and try to find the patterns and algorithms in everything. My life is number oriented. I'll spend all day working on a programmatic problem. I'll spend however much time is needed to think of an elegant/eff... | Paul Z | 8,311 | <p>If you truly enjoy numbers and programming, yet you're having trouble in a high school math course, it is a safe bet that <em>there is nothing wrong with you</em>. The problem is that your math teachers for the past 4 years (at least) have almost certainly been blithering idiots. This is unfortunately quite common (... |
2,270,346 | <p>I have a pretty simple question here it looks like but I just can't seem to do it. I'd like to be able to do it the easiest way possible. </p>
<blockquote>
<p>Solve $\dot{x}=y$ and $\dot{y}=x$ for $x(t)$ and $y(t)$.</p>
</blockquote>
<p>I need solve these two equations so I can draw a phase plane portrait and s... | Lutz Lehmann | 115,115 | <p>You get
$$
\frac{d}{dt}(y^2-x^2)=2(y\dot y-x\dot x)=0
$$
so that all solutions lie on the hyperbolic curves $y^2-x^2=C$.</p>
|
2,894,807 | <p>Please, does anyone know of a algorithm to compute the integer part $n$ of natural logarithm of an integer $x$?</p>
<p>$$n = \lfloor \ln(x) \rfloor$$</p>
<p>Preferably using integer arithmetic only (akin to <a href="https://en.wikipedia.org/wiki/Integer_square_root#Algorithm_using_Newton's_method" rel="nofollo... | Community | -1 | <p>I guess that the most efficient is to tabulate all the integer powers of $e$ and use a dichotomic search.</p>
<p>For $64$ bits integers, $44$ tables entries will suffice and the answer is found in $6$ steps.</p>
<p>For a million bits, $693148$ entries and $20$ comparisons only. But this is impractical because the ... |
1,025,642 | <p>Let $X$ be a non-empty set. Suppose that $d_1$ and $d_2$ are two possibly different metrics on $X$. Let $\tau_i$ denote the topology generated by the metric $d_i$ ($i\in\{1,2\}$).</p>
<p>The following are known:</p>
<ul>
<li>$\tau_1=\tau_2\equiv\tau$;</li>
<li>$(X,d_1)$ is a totally bounded metric space;</li>
<li>... | Arian | 172,588 | <p>Your sequence can be rewritten as
$$a_n=\frac{1\cdot3\cdot...\cdot(2n-1)}{2\cdot4\cdot...\cdot2n}=\frac{1\cdot2\cdot3\cdot...\cdot(2n-1)\cdot2n}{(2\cdot4\cdot...\cdot2n)^2}=\frac{(2n)!}{2^{2n}(n!)^2}$$
Using Stirling's approximation we get
$$a_n=\frac{(2n)!}{2^{2n}(n!)^2}\sim\frac{1}{2^{2n}}\cdot\frac{\sqrt{2\pi 2n}... |
1,025,642 | <p>Let $X$ be a non-empty set. Suppose that $d_1$ and $d_2$ are two possibly different metrics on $X$. Let $\tau_i$ denote the topology generated by the metric $d_i$ ($i\in\{1,2\}$).</p>
<p>The following are known:</p>
<ul>
<li>$\tau_1=\tau_2\equiv\tau$;</li>
<li>$(X,d_1)$ is a totally bounded metric space;</li>
<li>... | robjohn | 13,854 | <p><strong>Using $\boldsymbol{1+x\le e^x}$</strong></p>
<p>Since $1+x\le e^x$ for all $x\in\mathbb{R}$,
$$
\begin{align}
a_n
&=\prod_{k=1}^n\frac{2k-1}{2k}\\
&=\prod_{k=1}^n\left(1-\frac1{2k}\right)\\
&\le\prod_{k=1}^ne^{-\large\frac1{2k}}\\[3pt]
&=e^{-\frac12H_n}\tag{1}
\end{align}
$$
where $H_n$ are ... |
979,299 | <p>Assuming that I have $\{x_1,\ldots, x_N\}$ - an iid (independent identically distributed) sample size $N$ of observations of random variable $\xi$ with unknown mean $m_1$, variance (second central moment) $m_{c_2}$ and second raw moment $m_2$. I try to use sample mean $\overline{x}=\frac{1}{N}\sum_{i=1}^Nx_i$ as an ... | UserX | 148,432 | <p>You are calculating different areas. If you want to split the areas it should be like this;</p>
<p>$(1)$ The box bounded by $x=[0,1], y=[0,1]$ has area $1$. $(2)$ The area under $y=\frac{1}{x^2}$ from $1$ to $\infty$ is $\int_1^{\infty} \frac{1}{x^2} \mathrm{d}x=1$. Now the area $(3)$ you want to calculate isn't $\... |
979,299 | <p>Assuming that I have $\{x_1,\ldots, x_N\}$ - an iid (independent identically distributed) sample size $N$ of observations of random variable $\xi$ with unknown mean $m_1$, variance (second central moment) $m_{c_2}$ and second raw moment $m_2$. I try to use sample mean $\overline{x}=\frac{1}{N}\sum_{i=1}^Nx_i$ as an ... | Jack M | 30,481 | <p>The graph of $\frac 1 {x^2}$ does not have that form of symmetry. In fact, if you flip the graph around the $y=x$ axis as you're trying to do, you get the function's inverse, $\frac 1 {x^{1/2}}$. It should now be obvious that the two functions aren't the same:</p>
<p><img src="https://i.stack.imgur.com/4613v.png" a... |
1,447,089 | <p>I am trying to prove that (0.1) is uncountable given that R is uncountable.</p>
<p>I start by assuming that (0.1) is countable. </p>
<p>Then there exists a bijective map between (0.1) and N.</p>
<p>I guess then we can construct bijective map for (1.2) also.</p>
<p>this shows that each (i-1,i) for i=intergers is ... | marty cohen | 13,079 | <p>To show
$\sqrt{ c} − \sqrt{c − 1}
\geq \sqrt{c + 1} −\sqrt{c}
$.</p>
<p>This is the same as
$\sqrt{c}
\ge \frac{\sqrt{c + 1}+\sqrt{c - 1}}{2}
$.</p>
<p>If
$f(x) = x^{1/2}
$,
then
$f'(x) = \frac12x^{-1/2}
$
and
$f''(x) = -\frac14x^{-3/2}
< 0
$
so
$f(x)$
is concave
which means that
$f(\frac{a+b}{2})
\ge \frac{f... |
443,099 | <p>I remember hearing someone say "almost infinite" on one of the science-esque youtube channels. I can't remember which video exactly, but if I do, I'll include it for reference.</p>
<p>As someone who hasn't studied very much math, "almost infinite" sounds like nonsense. Either something ends or it doesn't, there rea... | Jonathan Livingstone | 179,060 | <p>Saying that something massively big is 'almost infinite' is no different from saying that 1 is almost infinite, since the difference between infinity and 1 and between infinity and massively big is exactly the same - namely, infinite. </p>
|
156,585 | <p>I am struggling to evaluate the following integral:<br>
$$\int \frac{1}{(1-x^2)^{3/2}} dx$$<br>
I tried a lot to factorize the expression but I didn't reach the solution.
Please someone help me.</p>
| rajai 7 | 157,650 | <p>$\displaystyle (1-x^2)^{\frac{3}{2}} =x^3(\frac{1}{x^2}-1)^{\frac{3}{2}}$</p>
<p>From it you can substitute:</p>
<p>$\displaystyle [\frac{1}{x^2}-1] =z$</p>
<p>By differentiating both sides
we will get $\displaystyle \frac{-2}{x^3}dx=dz$. </p>
<p>In this way we can also solve the integral. </p>
|
612,468 | <p>I've been using "conditional random variables" as a notation aid with some good success in problem solving. But I've heard people claim that one shouldn't define conditional random variables.</p>
<p>By a conditional random variable for $X$ given $Y$, a "pseudo" random variable $(X|Y)$ with the density function $f_... | Adam Williams | 567,416 | <blockquote>
<p>But is this abuse of notation sound?</p>
</blockquote>
<p>As others have noted in the comments, the answer is <strong>not quite</strong>. But to inform your understanding of why not, it may be helpful for you to read about the concept of <a href="https://en.wikipedia.org/wiki/Conditional_expectation"... |
16,080 | <p>I'm having trouble solving problem 12 from Section 1.2 in Hatcher's "Algebraic Topology".</p>
<p>Here's the relevant image for the problem:
<img src="https://i.stack.imgur.com/QNb5W.png" alt="enter image description here"></p>
<p>I'm trying to find $\pi_1(R^3-Z)$, where $Z$ is the graph shown in the first figure. ... | Ronnie Brown | 28,586 | <p>The general kind of graph in <span class="math-container">$\mathbb R^3$</span> that might be involved is as follows:</p>
<p><a href="https://i.stack.imgur.com/4zxFz.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/4zxFz.jpg" alt="Graph in <span class="math-container">$mathbb R ^3$</span>"></a> </... |
2,906,832 | <p>I want to write $\csc$ and $\tan$ and terms of classical trigonometric functions like $\sin$ and $\cos$. I know about the identity $\sin(x)^2+\cos(x)^2=1$. But I am not sure where to go from here. </p>
| blub | 369,816 | <p>You have the following identity relating the cosecant to the sine:
$$\csc(x)=\frac{1}{\sin(x)}$$</p>
<p>Similarly, you have</p>
<p>$$\tan(x)=\frac{\sin(x)}{\cos(x)}$$</p>
<p>relating the tangent to sine and cosine. These are the classics, but since the trigonometric functions all have interesting relationships am... |
1,789,077 | <blockquote>
<p>There is a square <span class="math-container">$Q$</span> consisting of <span class="math-container">$(0,0), (2,0), (0,2), (2,2)$</span>.</p>
<p>A point <span class="math-container">$P$</span> satisfies following condition:</p>
<p>The straight line passing through <span class="math-container">$P$</span>... | Rodrigo de Azevedo | 339,790 | <p>Since $\sin (n x) = \dfrac{\mathrm{e}^{i n x} - \mathrm{e}^{-i n x}}{2 i}$ and $\cos (n x) = \dfrac{\mathrm{e}^{i n x} + \mathrm{e}^{-i n x}}{2}$,</p>
<p>$$\begin{array}{rl} \sin(2x)-2\sin(3x)+\sin(4x) &= \left(\dfrac{\mathrm{e}^{i 2 x} - \mathrm{e}^{-i 2 x}}{2 i}\right) - 2 \left(\dfrac{\mathrm{e}^{i 3 x} - \m... |
1,789,077 | <blockquote>
<p>There is a square <span class="math-container">$Q$</span> consisting of <span class="math-container">$(0,0), (2,0), (0,2), (2,2)$</span>.</p>
<p>A point <span class="math-container">$P$</span> satisfies following condition:</p>
<p>The straight line passing through <span class="math-container">$P$</span>... | mathreadler | 213,607 | <p>A rather systematic approach would be to use FFT together with the convolution theorem to create linear equation system to find and then solve a polynomial in the complex exponential.</p>
<p><strong>Strength</strong> could solve ANY of these (to the extent we can solve polynomial equations).</p>
<p>$$\sum_{k\in \m... |
1,789,077 | <blockquote>
<p>There is a square <span class="math-container">$Q$</span> consisting of <span class="math-container">$(0,0), (2,0), (0,2), (2,2)$</span>.</p>
<p>A point <span class="math-container">$P$</span> satisfies following condition:</p>
<p>The straight line passing through <span class="math-container">$P$</span>... | Jack D'Aurizio | 44,121 | <p>By dividing both sides of our equation by $\sin x$ and letting $t=\cos x$ we get:</p>
<p>$$ U_1(t)-2U_2(t)+U_3(t) = 0 \tag{1}$$
where $U_i$ are <a href="http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html" rel="nofollow">Chebyshev polynomials of the second kind</a>. $(1)$ is equivalent to:
$$ 0 = 2... |
1,789,077 | <blockquote>
<p>There is a square <span class="math-container">$Q$</span> consisting of <span class="math-container">$(0,0), (2,0), (0,2), (2,2)$</span>.</p>
<p>A point <span class="math-container">$P$</span> satisfies following condition:</p>
<p>The straight line passing through <span class="math-container">$P$</span>... | colormegone | 71,645 | <p>Even if you don't have the "product-to-sum formulas" handy, the "angle-addition formulas" will get you there fast enough:</p>
<p>$$ \sin(2x) \ + \ \sin(4x) \ - \ 2\sin(3x) \ \ = \ \ \sin(3x - x) \ + \ \sin(3x + x) \ - \ 2\sin(3x) $$</p>
<p>$$ = \ \ [ \sin(3x) \cos x \ - \ \cos(3x) \sin x] \ + \ [ \sin(3x) \cos x \... |
3,997,992 | <p>Taken from <a href="https://artofproblemsolving.com/wiki/index.php/1970_Canadian_MO_Problems/Problem_10" rel="nofollow noreferrer">https://artofproblemsolving.com/wiki/index.php/1970_Canadian_MO_Problems/Problem_10</a></p>
<p>Problem <br>
Given the polynomial <span class="math-container">$f(x)=x^n+a_{1}x^{n-1}+a_{2}... | Juan Castaño | 854,352 | <p>The area <span class="math-container">$\mathcal{A}$</span> of a triangle <span class="math-container">$ABC$</span> can be obtained with
<span class="math-container">$$\mathcal{A}=\frac{1}{2}\left|\overrightarrow{AB}\times\overrightarrow{AC}\right|.$$</span>
Then you should write
<span class="math-container">$$\frac{... |
3,168,381 | <p>Given <span class="math-container">$f(x) = \frac{{4x}}{\sqrt{x}-3}$</span>, what's the domain of <span class="math-container">$g(x) = \frac{{1}}{f(x)}$</span> ?</p>
<p>My textbook includes in the answers <span class="math-container">$x \neq 9$</span>, which I think is erroneous.</p>
| Marvin | 544,299 | <p>The domain of <span class="math-container">$f(x)$</span> includes all the values of <span class="math-container">$x$</span>, except for <span class="math-container">$x = 9$</span> (which makes the denominator of <span class="math-container">$f(x)$</span> equal to <span class="math-container">$0$</span>) and <span cl... |
4,418,091 | <p>Is <span class="math-container">$\mathbb Q-\mathbb N$</span> dense in <span class="math-container">$\mathbb R$</span>? I believe that the solution is about two relatively prime integers. However, I do not know how to proceed.</p>
| egreg | 62,967 | <p>Let <span class="math-container">$r\in\mathbb{R}$</span> and <span class="math-container">$\varepsilon>0$</span>. Then</p>
<ol>
<li><span class="math-container">$(r-\varepsilon,r+\varepsilon)\cap\mathbb{Q}$</span> is infinite, because <span class="math-container">$\mathbb{Q}$</span> is dense in <span class="math-... |
157,301 | <p>Here is the limit I'm trying to find out:</p>
<p><span class="math-container">$$\lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)}$$</span></p>
<p>Since it is an indeterminate form, I simply applied l'Hopital's Rule and I ended up with:</p>
<p><span class="math-container">$$\lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)} = \... | Santosh Linkha | 2,199 | <p>$$ \lim_{x \rightarrow 0 }\frac{x^3}{\tan^3 (2x) } =
\lim_{x \rightarrow 0 } \left ( \frac{2x}{\tan (2x) } \right )^3 \frac{1}{2^3} = \frac{1}{2^3}$$
$$ $$</p>
|
157,301 | <p>Here is the limit I'm trying to find out:</p>
<p><span class="math-container">$$\lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)}$$</span></p>
<p>Since it is an indeterminate form, I simply applied l'Hopital's Rule and I ended up with:</p>
<p><span class="math-container">$$\lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)} = \... | Madrit Zhaku | 34,867 | <p>$\lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)}$=$\lim_{x\rightarrow 0} \frac{x^3}{\frac{\sin^3(2x)}{\cos^3(2x)}}$=$\lim_{x\rightarrow 0} \frac{x^3\cos^3(2x)}{\sin^3(2x)}$=$\lim_{x\rightarrow 0}\frac{x\cdot x\cdot x \cdot\cos^3(2x)}{\sin(2x)\cdot
\sin(2x)\cdot\sin(2x)}$=$\lim_{x\rightarrow 0}\frac{2x\cdot 2x\cdot 2x \... |
106,265 | <p>I am a beginner in <em>Mathematica</em>, so take this into account. I could not find anything similar in the internet so far.</p>
<p>I am trying to find a closed form solution for <code>A</code> in the equation below as a function of <code>n</code>, where <code>n</code> is countable and grows large. Potentially is ... | Dr. belisarius | 193 | <p>$\sum _{t=1}^n l(t) \left(\frac{\text{A}+\text{B}}{2}+b(t)-\text{D}-\text{E}\right)^2-\sum _{t=1}^n l(t)
\left(b(t)+\frac{\text{B}+\text{C}}{2}-\text{D}-\text{E}\right)^2=0$</p>
<p>$\sum _{t=1}^n (\text{A}-\text{C}) l(t) (\text{A}+4 b(t)+2 \text{B}+\text{C}-4 \text{D}-4 \text{E})=0$</p>
<p>Then</p>
<p>$\text{A... |
921,644 | <p>Is this series convergent $1+\dfrac{1}{4}-\dfrac{1}{9}-\dfrac{1}{16}+\dfrac{1}{25}+\dfrac{1}{36}-\dfrac{1}{49}-\dfrac{1}{64}+\cdots\ ?$</p>
<p>Can we write this series as function of $n?$</p>
| André Nicolas | 6,312 | <p>The series is absolutely convergent, so we may rearrange it as we please.</p>
<p>The sum $\frac{1}{4}-\frac{1}{16} + \frac{1}{36}-\frac{1}{64}+\cdots$ is just $\frac{1}{4}$ of the well-known series $1-\frac{1}{4}+\frac{1}{9}-\frac{1}{16}+\cdots$, which has sum $\frac{\pi^2}{12}$. This can be obtained quickly from ... |
4,364,370 | <p>For <span class="math-container">$$\Phi(x) = (2\pi)^{-\frac 12}\int_{-\infty}^x \mathrm{e}^{-t^2/2}\,\mathrm{d}t,$$</span> it is claimed in the proof of Lemma 8.12 of <a href="https://www.hse.ru/data/2016/11/24/1113029206/Concentration%20inequalities.pdf" rel="nofollow noreferrer">this book</a> that we have the asym... | Oliver Díaz | 121,671 | <p>Let <span class="math-container">$\phi(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$</span>. Notice that
<span class="math-container">$$\begin{align}
-\frac{d}{dx}\Big(\Big(\frac1x-\frac1{x^3}\Big)\phi(x)\Big)&=(1-3x^{-4})\phi(x)\\
&\leq \phi(x)\leq (1+x^{-2})\phi(x)=-\frac{d}{dx}\Big(\frac{\phi(x)}{x}\Big)
\end{alig... |
542,639 | <blockquote>
<h3>Problem:</h3>
<p>The collection $\mathcal{B}$ of subsets of the form $V = \{ x + yk : k
> \in \mathbb{Z} \} $ for $x,y \in \mathbb{Z}$ is a basis for some
topology of $\mathbb{Z}$</p>
</blockquote>
<h3>Solution attempt:</h3>
<p>Pick $n \in \mathbb{Z}$. and Put $N = \{ nk : k \in \mathbb{Z... | Brian M. Scott | 12,042 | <p>I’m going to assume that the definition of $\mathscr{B}$ given in the question is slightly incorrect, and that $\mathscr{B}$ is actually the collection of all sets of the form $a+b\Bbb Z=\{a+bk:k\in\Bbb Z\}$ such that $a,b\in\Bbb Z$ and $b\ne 0$; without that last restriction $\{a\}\in\mathscr{B}$ for each $a\in\Bbb... |
2,346,987 | <p>Let $G_{w}$ be the set of all wighted graphs $(G,f)$ with $G = (V,E)$, $V = \{1,...,n\}$ for an $n \in \mathbb N$ and $f: E \to \mathbb N$ the wight function.
I need to prove or disprove that the set is countable. </p>
<p>This is how I proceeded:</p>
<p>assume that $G_{w}$ is countable, then we can make a list of ... | 5xum | 112,884 | <p>Your proof is incorrect because you did not prove that $V'$ is not in the original list.</p>
<hr>
<p>To actually solve the problem, here's two facts to think about:</p>
<ol>
<li>A countable union of countable sets is countable.</li>
<li>For each $n$, the set of all weighted graphs on $\{1,2,\dots, n\}$ is countab... |
2,346,987 | <p>Let $G_{w}$ be the set of all wighted graphs $(G,f)$ with $G = (V,E)$, $V = \{1,...,n\}$ for an $n \in \mathbb N$ and $f: E \to \mathbb N$ the wight function.
I need to prove or disprove that the set is countable. </p>
<p>This is how I proceeded:</p>
<p>assume that $G_{w}$ is countable, then we can make a list of ... | Josh | 7,090 | <p>As far as I understand, you're simply adding a new weighted edge between a new vertex and some other vertex in the $G_n$. There is no reason this graph cannot be in your list.</p>
<p>In any case, the set is countable, assuming your graphs are simple. There are finitely many simple graphs of $n$ vertices for every $... |
1,376,392 | <p>Let $(\Omega,\mathcal{F},\mathbf{P})$ denote a probability space, $(S,\mathcal{M})$ denote a measurable space, and $X : (\Omega,\mathcal{F},\mathbf{P}) \rightarrow (S,\mathcal{M})$ denote a measurable function (thought of as a random variable). Then there is a pushforward measure induced on $(S,\mathcal{M})$ (though... | Nate Eldredge | 822 | <p>Just so that everyone knows what we are talking about here, let me rephrase in more familiar notation.</p>
<p>Suppose $(\Omega, \mathcal{F}, P)$ is a probability space, and $(M, \mathcal{M})$ is a measurable space. If $X : \Omega \to M$ is a random variable (i.e. a $(\mathcal{F}, \mathcal{M})$-measurable function)... |
3,464,295 | <p>I thought that QR algorithm decomposes a matrix into an orthogonal matrix Q and a upper triangular matrix R using GramSchmidth process for singular matrices but, what is meant by Explicit and Implicit QR algorithms? and how will they help in decomposing a non-singular matrix?</p>
| CallMeStag | 659,266 | <p>The explict QR algorithm for computing eigenvalues of a matrix <span class="math-container">$A$</span> works like this:</p>
<ol>
<li>Compute <span class="math-container">$QR$</span> decomposition <span class="math-container">$A_i = Q_i R_i$</span></li>
<li>Set <span class="math-container">$A_{i+1} = R_i Q_i (=Q^{\m... |
5,119 | <p>If $\mathcal{F}_1 \subset \mathcal{F}_2 \subset \dotsb$ are sigma algebras, what is wrong with claiming that $\cup_i\mathcal{F}_i$ is a sigma algebra?</p>
<p>It seems closed under complement since for all $x$ in the union, $x$ has to belong to some $\mathcal{F}_i$, and so must its complement.</p>
<p>It seems close... | chenwins | 120,002 | <p>Let <span class="math-container">$\Omega=[0,1]$</span>, <span class="math-container">$A_{0}= \{\emptyset, \Omega \}$</span> and <span class="math-container">$A_{k}=\sigma \{[0,\frac{1}{2^k}],[\frac{1}{2^k},\frac{2}{2^k}],[\frac{2}{2^k},\frac{3}{2^k}],.....,[\frac{2^k-1}{2^k},1]\}$</span></p>
<p>pick irrational numbe... |
151,192 | <p>By Cantor's normal form theorem, any ordinal $r$ can be expressed as:
$r=\omega^{k_1}a_1 + \omega^{k_2}a_2 + \ldots$. ($k_1>k_2>\ldots$)</p>
<p>I want to know whether the class of all $a_i$'s is countable or not.</p>
<p>If it is not countable how do i prove a problem such as:
$\omega^{k_1}a_1 + \omega^{k_2}a... | Asaf Karagila | 622 | <p>If you require that $k_i$ and $a_i$ are all strictly less than $\omega$ then this simply corresponds to finite sequences of natural numbers, which is a countable collection.</p>
<p>Now remember that <strong>every</strong> ordinal has a unique Cantor normal form, so every ordinal can be expressed like this. If, howe... |
818,015 | <p>I try to get the variables for this equation:</p>
<p>$$\begin{cases}
6x_1 + 4x_2 + 8x_3 + 17x_4 &= -20\\
3x_1 + 2x_2 + 5x_3 + 8x_4 &= -8\\
3x_1 + 2x_2 + 7x_3 + 7x_4 &= -4\\
0x_1 + 0x_2 + 2x_3 -1x_4 &= 4
\end{cases}$$</p>
<p>So i started with:</p>
<p>$$ \begin{pmatrix}
6 & 4 &am... | Tunk-Fey | 123,277 | <p><strong>HINT :</strong>
\begin{align}
\lim_{x\to 0}\frac{1−\cos x}{x^2}&=\lim_{x\to 0}\frac{1−\cos x}{x^2}\cdot\frac{1+\cos x}{1+\cos x}\\
&=\lim_{x\to 0}\frac{1−\cos^2 x}{x^2(1+\cos x)}\\
&=\lim_{x\to 0}\frac{\sin^2 x}{x^2(1+\cos x)}\\
&=\lim_{x\to 0}\frac{\sin x\cdot\sin x}{x\cdot x\cdot(1+\cos x)}... |
60,750 | <p>If we have two finitely generated residually finite groups $G$ and $H$, is there are relation between</p>
<p>the profinite completions $\hat{G},\hat{H}$ and the profinite completion of a semidirect
product $\hat{G \rtimes H}$</p>
<p>(and analogous question for pro-p completions)</p>
| Community | -1 | <p>Take a finite non-abelian simple group $A$ and consider the wreath product $G=A\wr \mathbb Z$. Let $N$ be any subgroup of finite index of $G$. Then $N\cap A^{\mathbb Z}\ne 1$. Let $g$ be a non-trivial element in the intersection. Suppose that the $i$-th coordinate $g_i$ of $g$ is not $1$. Since $A$ has trivial cente... |
60,750 | <p>If we have two finitely generated residually finite groups $G$ and $H$, is there are relation between</p>
<p>the profinite completions $\hat{G},\hat{H}$ and the profinite completion of a semidirect
product $\hat{G \rtimes H}$</p>
<p>(and analogous question for pro-p completions)</p>
| Ahmed Elsawy | 21,286 | <p>Let $\mathscr{P}$ be any property
such that whenever a group has $\mathscr{P}$ then all its subgroups also have $\mathscr{P}$.
In [1] Theorem 3.1, K. W. Gruenberg has proved that if the wreath product
$W= A \wr B$, is residually $\mathscr{P}$,
then either $B$ is $\mathscr{P}$ or $A$ is abelian. </p>
<p>Consider $W=... |
1,653,934 | <p>I'm a software engineering and mathematics student, I was searching for disciplines of mathematics that would go well with my engineering degree, and found a lot of people recommended that software engineers should learn at least a bit of linear algebra, giving book recomendations and else, but I couldn't find any r... | bubba | 31,744 | <p>"Software engineering" is an enormously broad term. Let's assume you end up writing software in industry, somewhere (like me). The kinds of mathematics that are useful will depend very much on the applications/functionality of the software.</p>
<p>Some examples:</p>
<ol>
<li><p>Graphics/games: People will tell you... |
2,448,082 | <p>Trying to solve $$\int 27x^3(9x^2+1)^{12} dx$$
I know the process and formula of integration by parts. When I set $u = 9x^2 + 1$, $du = 18x dx$. I am stuck on the next step as 18x does not line up with the $27x^3$. </p>
| xpaul | 66,420 | <p>I think you don't need to use integration by parts. Substitution is enough. Let $u=9x^2+1$. Then $du=18xdx$ or $xdx=\frac1{18}du$, $x^2=\frac19(u-1)$ and hence
\begin{eqnarray}
\int 27x^3(9x^2+1)^{12} dx&=&27\int x^2(9x^2+1)^{12} xdx\\
&=&27\cdot\frac19\cdot\frac{1}{18}\int(u-1)u^{12}du\\
&=&... |
119,443 | <p>My buddy and I are arguing over something that cropped up in this past weekend's Texas Hold'em tournament.</p>
<p>A player got "knocked out" (lost all their chips) early on in the game. The person hosting the tournament told them they could buy back in, and my buddy got upset. He argued that it gave the player buyi... | Ross Millikan | 1,827 | <p>You can just argue that all players start with the same options and therefore (assuming equal skill) have the same chance of winning. What has changed is the value to each player of having somebody knocked out. If there are $n$ players originally, if there is no buy back in, your probability of winning if you are ... |
119,443 | <p>My buddy and I are arguing over something that cropped up in this past weekend's Texas Hold'em tournament.</p>
<p>A player got "knocked out" (lost all their chips) early on in the game. The person hosting the tournament told them they could buy back in, and my buddy got upset. He argued that it gave the player buyi... | Michael Joyce | 17,673 | <p>(1) If it's a game among friends, then I don't think that worrying about who has a theoretical advantage is more important than having a good time and letting a buddy buy back into the game. But the buy-in structures should be designed so that all players can comfortably rebuy if it is a friendly game.</p>
<p>(2) ... |
203,995 | <p>Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$.</p>
<p>Let $f,g\in \mathbb{C}[x,y]$. </p>
<p>Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. </p>
<p>Given $h\in \mathbb{C}[x,y]$, how to determine whether $h\in (f,g)$ or not? </p>
<p>I have trie... | joro | 12,481 | <p>Too long for a comment.</p>
<p>All CASes have bugs, so if you are using CAS solution, better
run on as many CASes as you can.</p>
<p>Comment suggests to use Groebner basis, but this leads to
the question "How do you compute Groebner basis without CAS?"</p>
<p>If your ideal is complicated enough, computing Groebne... |
2,992,411 | <p><span class="math-container">\begin{cases}
\frac {dP}{dt} = rP(t)(1-\frac {P(t)}{K}) ,t \geq 0 \\
P(0) = P_o
\end{cases}</span></p>
<p><span class="math-container">$r, K$</span> and <span class="math-container">$P_o$</span> are positive constants.</p>
<p>We say that <span class="math-container">$P(t), t \geq0... | Robert Z | 299,698 | <p>Hint. Assume that <span class="math-container">$\{f_n\}_{n\in\mathbb{N}}$</span> is a countable list of ALL such functions from <span class="math-container">$\mathbb{Z}^2\to \{-1,1\}$</span>. Now define a new function <span class="math-container">$g:\mathbb{Z}^2\to \{-1,1\}$</span> such that for any <span class="mat... |
248,900 | <p>Let $\mathfrak{n}$ be a $2k$ dimensional $2$-step nilpotent Lie algebra and suppose that its center is $k$ dimensional. Does $\mathfrak{n}$ admit symplectic structure?</p>
<p>Let $\{f_1,\dots,f_k\}$ be a basis of the center of $\mathfrak{n}$ and complete it to a basis of $\mathfrak{n}$ $\{e_1,\dots,e_k,f_1,\dots,f_... | Dietrich Burde | 32,332 | <p>It may be interesting to note that the minimal dimension for a $2$-step nilpotent Lie algebra without symplectic structure is $6$.
In fact, only the direct product $\mathfrak{h}_5(\mathbb{R})\times \mathbb{R}$ is not symplectic in dimension $6$, $2$-step nilpotent. Here $\mathfrak{h}_5(\mathbb{R})$ denotes the $5$-... |
324,503 | <blockquote>
<p>Evaluate the limit
$$ \lim_{n\rightarrow\infty}{\frac{n!}{n^{n}}\left(\sum_{k=0}^{n}{\frac{n^{k}}{k!}}-\sum_{k=n+1}^{\infty}{\frac{n^{k}}{k!}} \right)} $$</p>
</blockquote>
<p>I use $$e^{n}=1+n+\frac{n^{2}}{2!}+\cdots+\frac{n^{n}}{n!}+\frac{1}{n!}\int_{0}^{n}{e^{x}(n-x)^{n}dx}$$
but I don't know h... | robjohn | 13,854 | <p>In <a href="https://math.stackexchange.com/a/160352">this answer</a>, it is shown, using integration by parts, that
$$
\sum_{k=0}^n\frac{n^k}{k!}=\frac{e^n}{n!}\int_n^\infty e^{-t}\,t^n\,\mathrm{d}t\tag{1}
$$
Subtracting both sides from $e^n$ gives
$$
\sum_{k=n+1}^\infty\frac{n^k}{k!}=\frac{e^n}{n!}\int_0^n e^{-t}\,... |
2,155,755 | <p>What is the difference (or connection) between the dimension of a vector space and the dimension in terms of bases?</p>
<p>For instance, when we talk about the vector space $\mathbb{R}^3$, we are talking about a 3-dimensional vector space. This vector space contains vectors with three elements: $(x_1, x_2, x_3)$.</... | user49640 | 417,455 | <p>Consider the vectors $e_1 =(1,0,0)$, $e_2 = (0,1,0)$ and $e_3 = (0,0,1)$ in $\mathbf{R}^3$. Then the vectors $e_1, e_2, e_3$ form a basis for the whole space, which has dimension $3$. This case is not a problem for you.</p>
<p>But what about the vectors $e_1$ and $e_2$? This is more like the case that is not clear ... |
17,423 | <p>In most of books on elementary algebra, intermediate algebra and college algebra, the degree of the non-zero polynomial <span class="math-container">$$f(x)=a_nx^n+\cdots a_1x+a_0$$</span> with <span class="math-container">$a_n\neq 0$</span> is defined to be <span class="math-container">$n$</span>. </p>
<p>But I am ... | user52817 | 1,680 | <p>Soon after introducing polynomials, students learn to add and subtract polynomials. Notice that if <span class="math-container">$f(x)$</span> and <span class="math-container">$g(x)$</span> are polynomials with degrees <span class="math-container">$n$</span> and <span class="math-container">$m$</span> respectively, a... |
3,634,864 | <p>Say we have a proposition If P then Q, and let Q = A <span class="math-container">$\land$</span> B</p>
<p>If we do a proof by contradiction, we assume the negation of Q: <span class="math-container">$\lnot (A \land B) = \lnot A \lor \lnot B$</span></p>
<p>From here, can we continue our contradiction proof by star... | Bram28 | 256,001 | <p>You need to show that <span class="math-container">$\neg A \lor \neg B$</span> implies a contradiction. So, showing that <span class="math-container">$\neg A$</span> by itself implies a contradiction is not enough, since <span class="math-container">$\neg A$</span> is not implied by <span class="math-container">$\ne... |
3,634,864 | <p>Say we have a proposition If P then Q, and let Q = A <span class="math-container">$\land$</span> B</p>
<p>If we do a proof by contradiction, we assume the negation of Q: <span class="math-container">$\lnot (A \land B) = \lnot A \lor \lnot B$</span></p>
<p>From here, can we continue our contradiction proof by star... | fleablood | 280,126 | <p>To do a contrapositive proof of <span class="math-container">$P \implies (A\land B)$</span> you must prove <span class="math-container">$(\lnot A \lor \lnot B) \implies \lnot P$</span> or in other words and assuming <span class="math-container">$(\lnot A\lor \lnot B)$</span> causes a contradiction to <span class="ma... |
1,303,263 | <p>Given 2 vectors,$u=(3,5)$,$v=(s,s^2)$,in what situations do u and v parallel?$(s≠0)$</p>
<p>In order to be parallel,$u$ must be proportional to $v$,vice verse.Let $k$ be a scalar $neq 0$,then $ku=(3k,5k)=v=(s,s^2)$,which gives $3k=s$;$5k=s^2 \rightarrow 5k=9k^2 (k \neq 0) \rightarrow 5=9k→k=\frac{5}{9}$,plug $k=\fr... | TravisJ | 212,738 | <p>This is a fun observation... but I think you have a mistake. I wrote a quick python script to generate the fibonacci numbers and primes and make the counts and this is what I get:</p>
<p>Between 5 and 8 there is 1 prime: 7</p>
<p>Between 8 and 13 there is 1 prime: 11</p>
<p>Between 13 and 21 there are 2 primes: ... |
1,303,263 | <p>Given 2 vectors,$u=(3,5)$,$v=(s,s^2)$,in what situations do u and v parallel?$(s≠0)$</p>
<p>In order to be parallel,$u$ must be proportional to $v$,vice verse.Let $k$ be a scalar $neq 0$,then $ku=(3k,5k)=v=(s,s^2)$,which gives $3k=s$;$5k=s^2 \rightarrow 5k=9k^2 (k \neq 0) \rightarrow 5=9k→k=\frac{5}{9}$,plug $k=\fr... | Mario Carneiro | 50,776 | <p>You can make sense of this pattern (which as yoann points out does not go on very far) as saying that the density of primes is a constant $\phi^{-3}\approx0.236$, since the gap length is $F_{n+1}-F_n=F_{n-1}$ and the conjectured prime count in this gap is $F_{n-4}\approx F_{n-1}\phi^{-3}$.</p>
<p>Unfortunately, thi... |
1,926,382 | <p>Let $T:\Bbb R^n\longrightarrow\Bbb R^n$ be a linear transformation, where $n\geq 2$. For $k\leq n$,
let $E=\{v_1,v_2,\dots,v_k\}$ contained in, equal to $R^n$ and $F=\{Tv_1,Tv_2,\dots,Tv_k\}$.
Then</p>
<p>a). If $E$ is linearly independent, then $F$ is linearly independent.</p>
<p>b). If $F$ is linearly independe... | Arnaud D. | 245,577 | <p>You are right when you say that a) is true if $T$ is one-to-one. However, you seem to assume that because $T$ is a transformation between two spaces of the same dimension it has to be one-to-one. This is <strong>not</strong> true, and you even give a counter-example yourself in your next paragraph!</p>
<p>Option b)... |
2,323,189 | <p>I've been asked to find the minimum and maximum of the following function:</p>
<p>$f(x,y) = x^2+y^2-x+1/4$ </p>
<p>On the region or restriction defined as:</p>
<p>$D$={${(x,y)\in\mathbb{R}^2:x^2+y^2\leq1; x+y\leq0}$}</p>
<p>First, I observed that $f$ is continuos, and after I did the graph of the region where I... | Ross Millikan | 1,827 | <p>A much simpler approach is to note that $f(x,y)=(x-\frac 12)^2+y^2$, the squared distance from $(\frac 12,0)$. Your critical point is the global minimum, but outside the region of interest. The minimum in your region will be the point closest to $(\frac 12,0)$. You can find that by taking the perpendicular to $x+... |
2,323,189 | <p>I've been asked to find the minimum and maximum of the following function:</p>
<p>$f(x,y) = x^2+y^2-x+1/4$ </p>
<p>On the region or restriction defined as:</p>
<p>$D$={${(x,y)\in\mathbb{R}^2:x^2+y^2\leq1; x+y\leq0}$}</p>
<p>First, I observed that $f$ is continuos, and after I did the graph of the region where I... | Doug M | 317,162 | <p>As Ross Millikan points out, you don not need calculus.</p>
<p>But since you are taking multivariate calculus, and you want to use your new tools.</p>
<p>You have found that there is no local minimum in the interior of the region.</p>
<p>You have found the point $(\frac 14,\frac 14)$ on the line $x+y = 0$</p>
<p... |
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