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 |
|---|---|---|---|---|
422,196 | <p>After a long reflection, I've decided I won't go to graduate school and do a thesis, among other things. I personally can't cope with the pressure and uncertainty of an academic job.</p>
<p>I will therefore move towards a master's degree in engineering and probably work in industry. However, I'm still passionate abo... | Todd Trimble | 2,926 | <p>As someone who has gone through a in-some-ways similar experience, I don't think you would be ostracized, and yes, you absolutely could still publish work. You might need people to endorse your work, though, for example if you lack an academic affiliation but want to publish on the arXiv.</p>
<p>Generally speaking, ... |
422,196 | <p>After a long reflection, I've decided I won't go to graduate school and do a thesis, among other things. I personally can't cope with the pressure and uncertainty of an academic job.</p>
<p>I will therefore move towards a master's degree in engineering and probably work in industry. However, I'm still passionate abo... | Igor Belegradek | 1,573 | <p>Something that I don't see mentioned in other answers is that a number of research engineers in academia are making substantial contributions to mathematics. Simply put, many excellent applied mathematicians choose to be employed as engineers in academia. This may be the most natural route for someone interested in ... |
806,015 | <p>The question I'm working on is the following:
Let $C_R$ be a contour in the shape of a wedge starting at the origin, running along the real axis to $x=R$, then along the arc $0 \leq \theta \leq 2\pi/3$, then back down to the origin along the ray $\theta=2\pi/3$. </p>
<p>Evaluate the limit as $R$ approaches infinity... | Silynn | 152,754 | <p>It's a third order pole, so you need to use the higher order formula if you aren't doing a series expansion.</p>
<p>$$Res(f,c)=\frac{1}{(n-1)!}\lim_{z\rightarrow c}\frac{d^{n-1}}{dz^{n-1}}((z-c)^{n}f(z))$$</p>
<p>So in this case,
$$Res(f,e^{\frac{i\pi}{3}})=\frac{1}{2!}\lim_{z\rightarrow e^{\frac{i\pi}{3}}}\frac{... |
806,015 | <p>The question I'm working on is the following:
Let $C_R$ be a contour in the shape of a wedge starting at the origin, running along the real axis to $x=R$, then along the arc $0 \leq \theta \leq 2\pi/3$, then back down to the origin along the ray $\theta=2\pi/3$. </p>
<p>Evaluate the limit as $R$ approaches infinity... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\... |
2,692,112 | <p>Let $A$ be $m\times n$ matrix with full column rank where $m > n$. Let $P = A(A^TA)^{-1}A^T$. How do we show that $P$ is SPD (symmetric positive definite)? Proving that it is symmetric is trivial, but how can I show it is positive definite?</p>
| M. Winter | 415,941 | <p>Some facts about positive semi-definite (PSD) matrices:</p>
<blockquote>
<ol>
<li>If $X$ is any matrix, then $X^\top X$ is always PSD.</li>
<li>The inverse of an (invertible) PSD matrix is also PSD.</li>
<li>A matrix $X$ is PSD if and only of $x^\top Xx\ge 0$ for all vectors $x$.</li>
</ol>
</blockquote>
... |
3,473,911 | <blockquote>
<p>What is the derivative of <span class="math-container">$f: \mathbb C \to \mathbb R$</span> where <span class="math-container">$f(z)=z\bar z$</span>?</p>
</blockquote>
<p>Not sure how to go about differentiating this function.</p>
<p>Is it just <span class="math-container">$f'(z)=\bar z$</span>? Not ... | Community | -1 | <p>As you know, the given equation has extrema at <span class="math-container">$x=\pm1$</span>. These correspond to values of the polynomial</p>
<p><span class="math-container">$$1-5-c$$</span> and <span class="math-container">$$1+5-c$$</span> (the RHS was moved to the left).</p>
<p>Hence the polynomial will grow fro... |
3,473,911 | <blockquote>
<p>What is the derivative of <span class="math-container">$f: \mathbb C \to \mathbb R$</span> where <span class="math-container">$f(z)=z\bar z$</span>?</p>
</blockquote>
<p>Not sure how to go about differentiating this function.</p>
<p>Is it just <span class="math-container">$f'(z)=\bar z$</span>? Not ... | fleablood | 280,126 | <blockquote>
<p>I know that between consecutive real roots of f there is a real root of f′. Now f′ in this case is 5x4−5 which always has two real roots. So the claim should be true for all c.</p>
</blockquote>
<p><span class="math-container">$A \implies B$</span> does not mean <span class="math-container">$B \impli... |
2,148,484 | <p>Find the value of the series $$\sum_{n=0}^\infty \frac{n^{2}}{2^{n}}.$$ I tried the problem but not getting the sum. Please help.</p>
| Elementarium | 413,840 | <p>To give a rigorous answer, we first need to find the radius of convergence of the series $$\displaystyle\sum_{n=0}^{\infty}\dfrac{n^2}{2^n}=\displaystyle\sum_{n=0}^{\infty}n^2\left(\dfrac{1}{2}\right)^n$$</p>
<p>To do so, let's apply the Cauchy-Hadamard theorem. Consider the general term $a_n=n^2$. Then,</p>
<p>$$... |
4,632,471 | <p>I thought this was an interesting concept so I tried to make an impossible map using this.
I also tested it on a solving website and it ended up giving me 6 different colors...<a href="https://i.stack.imgur.com/Imt8I.png" rel="nofollow noreferrer">A picture of it</a></p>
<p><a href="https://i.stack.imgur.com/I8OEL.p... | Adrien I | 1,038,959 | <p>Machines are powerful, but humans are apparently better (joke) !</p>
<p>Here is my solution.</p>
<p><a href="https://i.stack.imgur.com/xMvdm.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xMvdm.png" alt="solution" /></a></p>
<p>EDIT : I think an interesting method would be to color the area that ... |
4,632,471 | <p>I thought this was an interesting concept so I tried to make an impossible map using this.
I also tested it on a solving website and it ended up giving me 6 different colors...<a href="https://i.stack.imgur.com/Imt8I.png" rel="nofollow noreferrer">A picture of it</a></p>
<p><a href="https://i.stack.imgur.com/I8OEL.p... | David G. Stork | 210,401 | <p>Adrien I writes "Machines are powerful but humans are apparently better!"</p>
<p><em>Really</em>?? For graph coloring??</p>
<p>Here's the graph coloring found in 0.000435 seconds by <em>Mathematica</em>:</p>
<p><a href="https://i.stack.imgur.com/DqlyD.png" rel="noreferrer"><img src="https://i.stack.imgur.... |
4,058,947 | <p>Have got into a pretty heated debate with a friend, and looking online there's lacking proof
<a href="https://i.stack.imgur.com/S30uM.png" rel="nofollow noreferrer">Line contained in a plane</a></p>
<p>Is a line that is contained within a plane, considered parallel to it? By my understanding it is parallel , if at a... | user2661923 | 464,411 | <p>I'll throw my hat into the ring.</p>
<p>I certainly agree that it is a matter of convention.</p>
<p>However, it is also reasonable to presume that 3 dimensional Geometry is (in effect) an extension of 2 dimensional Geometry.</p>
<p>In 2 dimensional Geometry, I have never heard of the specification that a line is par... |
3,614,178 | <p>I was going through this article on geometric series on Wikipedia and found this diagrammatic representation of an infinite geometric series with a said common factor of (1/2) but shouldn't the common factor be (1/4) based on the diagram?
link to the diagram:
<a href="https://commons.wikimedia.org/wiki/File:Geometr... | Robert Israel | 8,508 | <p>I claim <span class="math-container">$\log_2(3) < 5/3 < \log_5(17)$</span>.</p>
<p>The first is equivalent to <span class="math-container">$2^{5/3} > 3$</span>, i.e. <span class="math-container">$2^5 > 3^3$</span>. That's easy: <span class="math-container">$2^5 = 32$</span> and <span class="math-contai... |
25,778 | <p>Is there a simple numerical procedure for obtaining the derivative (with respect to $x$) of the <a href="http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse">pseudo-inverse</a> of a matrix $A(x)$, without approximations (except for the usual floating-point limitations)? The matrix $\frac{\mathrm{d}}{\m... | S. Carnahan | 121 | <p>This is not a complete answer.</p>
<p>According to the Wikipedia page you linked, the pseudoinverse $A^+$ is not a continuous function of $A$, as it jumps around when $A$ is ill-conditioned. Therefore, you can't expect $A^+(x)$ to always have a derivative in terms of the matrix derivative of $A(x)$.</p>
<p>I supp... |
25,778 | <p>Is there a simple numerical procedure for obtaining the derivative (with respect to $x$) of the <a href="http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse">pseudo-inverse</a> of a matrix $A(x)$, without approximations (except for the usual floating-point limitations)? The matrix $\frac{\mathrm{d}}{\m... | Gabriel Mitchell | 6,908 | <p>Since your goal is to take into account the effect of perturbations in the matrix elements on the least squares solution you may find the following useful:</p>
<p><a href="http://www.jstor.org/pss/2156807" rel="nofollow">http://www.jstor.org/pss/2156807</a></p>
<p><a href="http://en.wikipedia.org/wiki/Total_least_... |
2,906,314 | <p>If I have vectors a and b sharing a common point of intersection then I know how to calculate angle between them by using the formula for dot product. But whether b lies to the right or left of a if I am moving along a can not be gotten from this. </p>
<p>What would be the easiest way to find out whether b lies lef... | Jordi Cruzado | 96,872 | <p>use cross product. The sign of cross product of a and b will let you know if the angle is clockwise or counter clockwise.</p>
|
24,939 | <p>Sorry, I'm just starting to learn mathematica.</p>
<p>I have the following two-argument function:</p>
<pre><code>h[{x_, y_}] := x ^ y
</code></pre>
<p>When I do the following:</p>
<pre><code>Map[h, {{1, 2}, {2, 2}, {3, 2}}]
</code></pre>
<p>I get the expected output:</p>
<pre><code>{1, 4, 9}
</code></pre>
<p>... | SEngstrom | 6,453 | <p>If you want to use Map[] that is possible too:</p>
<pre><code>#[[1]]^#[[2]] & /@ {{1, 2}, {2, 2}, {3, 2}}
</code></pre>
|
1,946,144 | <p>I have a rough idea of how to solve this nonautonomous equation.</p>
<p>$x'=3x+sin(2t)$ </p>
<p>$\int 1\, dx=\int (3x+sin(2t))\, dt$</p>
<p>$x = 3xt - \frac{cos(2t)}{2} + constant$</p>
<p>$(1-3t)x = -\frac{1}{2}cos(2t) +constant$</p>
<p>$x = -\frac{cos(2t)}{2(1-3t)}+\frac{constant}{1-3t}$</p>
<p>Does this look... | Community | -1 | <p>Rewrite as</p>
<p>$$x'-3x=e^{3t}(e^{-3t}x)'=\sin2t,$$</p>
<p>then</p>
<p>$$x=e^{3t}\int e^{-3t}\sin2t\,dt.$$</p>
<p>To evaluate the integral, work by parts, twice, or switch to complex exponentials.</p>
|
3,964,172 | <blockquote>
<p>Does the following limit exist?
<span class="math-container">$$\lim_{x \to 0} x\sin\left(\frac{1}{x}\right)$$</span></p>
</blockquote>
<p>If yes, then I have to find it; if no, then I need to give reason why.</p>
<p>I couldn’t figure out how to prove this formally, can someone please help me out.</p>
| Community | -1 | <p>You can let <span class="math-container">$$z=\frac{1}{x}$$</span> so then the limit becomes <span class="math-container">$$\lim_{z \rightarrow \infty} \frac{1}{z} \sin z \:,$$</span> which clearly pulls down to zero.</p>
|
3,964,172 | <blockquote>
<p>Does the following limit exist?
<span class="math-container">$$\lim_{x \to 0} x\sin\left(\frac{1}{x}\right)$$</span></p>
</blockquote>
<p>If yes, then I have to find it; if no, then I need to give reason why.</p>
<p>I couldn’t figure out how to prove this formally, can someone please help me out.</p>
| Community | -1 | <p>Yes, the limit there exist. Note that the funcion <span class="math-container">$f(x)=\sin(x)$</span> is a <a href="https://en.wikipedia.org/wiki/Bounded_function" rel="nofollow noreferrer">bounded function</a>, with <span class="math-container">$$|\sin(x)|\leq 1 \iff -1\leq \sin(x)\leq 1$$</span>
Therefore, <span ... |
3,964,172 | <blockquote>
<p>Does the following limit exist?
<span class="math-container">$$\lim_{x \to 0} x\sin\left(\frac{1}{x}\right)$$</span></p>
</blockquote>
<p>If yes, then I have to find it; if no, then I need to give reason why.</p>
<p>I couldn’t figure out how to prove this formally, can someone please help me out.</p>
| mentallurg | 125,420 | <p><span class="math-container">$$-1 \le \sin {x} \le 1$$</span></p>
<p><span class="math-container">$$-|x| \le x \sin \frac{1}{x} \le |x|$$</span></p>
<p><span class="math-container">$$\lim_{x \to 0} -|x| \le \lim_{x \to 0} x \sin \frac{1}{x} \le \lim_{x \to 0} |x|$$</span></p>
<p><span class="math-container">$$0 \le ... |
2,993,166 | <p>Suppose I have an operator valued function, <span class="math-container">$\omega\mapsto A(\omega)$</span>; for each <span class="math-container">$\omega$</span>, <span class="math-container">$A(\omega):X\to Y$</span>, is a bounded linear operator with <span class="math-container">$X$</span> and <span class="math-con... | fleablood | 280,126 | <p>Well, you are lucky.</p>
<p><span class="math-container">$x^3 - x - 6 = 0\implies$</span></p>
<p><span class="math-container">$x^3 - x = 6\implies$</span></p>
<p><span class="math-container">$x(x^2 - 1) = x(x+1)(x-1)=6$</span> and it just happens that <span class="math-container">$6 = 1*2*3$</span> so if you set ... |
411,261 | <p>Question: Use double integral to find the volume of the solid enclosed by the spheres $x^2+y^2+z^2=1$ and $x^2+y^2+(z-1)^2=1$</p>
<p>Alright so I tried to doing this by myself and I'm not sure if this is right. Could someone check over my work?</p>
<p>Curve of intersection:
\begin{align*}
x^2 + y^2 + z^2 &= x^... | robjohn | 13,854 | <p>We will use the identities
$$
\begin{align}
\sum_{k=m}^n\binom{n}{k}\binom{k}{m}
&=\sum_{k=m}^n\binom{n}{m}\binom{n-m}{k-m}\\
&=\binom{n}{m}2^{n-m}\tag{1}
\end{align}
$$
and
$$
\binom{k-1}{2}=\binom{k}{2}-\binom{k}{1}+\binom{k}{0}\tag{2}
$$
noting that
$$
\binom{k-1}{2}=\left\{\begin{array}{}
0&\text{for... |
3,321,863 | <p>The eigenvalues of a positive-definite matrix are guaranteed to be <span class="math-container">$> 0$</span>; but does anyone know of sufficient conditions when they will also all be <span class="math-container">$\le 1$</span>?</p>
| dylan7 | 163,751 | <p>I just want to expand on @ Kavi Rama Murphy's comment, to show it's iff.</p>
<p>If <span class="math-container">$A$</span> is a Positive Definite matrix then let <span class="math-container">$T$</span> be the associated linear operator, and <span class="math-container">$V$</span> an inner product space, with an inn... |
3,593,702 | <p>I've discovered and am trying to understand power sets, specifically how to calculate the power sets of a set. I found the <a href="http://www.ecst.csuchico.edu/~akeuneke/foo/csci356/notes/ch1/solutions/recursionSol.html" rel="nofollow noreferrer">algorithm's description</a>, which concluded with this:</p>
<p><span... | Alain Remillard | 278,299 | <p>When doing probability, the important thing is to consider order (or not) in both numerator (favorable outcomes) and denominator (total outcomes).</p>
<p>Regarding the first problem. If order is important, then total outcomes are
<span class="math-container">$$A_7^{30}=\frac{30!}{23!}$$</span>
and favorable outcom... |
317,756 | <p>Let $L>1$. I am looking for the value, or the leading asymptotics for $L\to\infty$, of
$$\int_1^L\int_1^L\int_1^L\int_1^L \dfrac{\mathrm dx_1~\mathrm dx_2 ~ \mathrm dx_3 ~ \mathrm dx_4}{(x_1+x_2)(x_2+x_3)(x_3+x_4)(x_4+x_1)}$$
More generally, I'd like to know the leading asymptotics of an expression like this wit... | Martin | 507,947 | <p>The general asymptotics as <span class="math-container">$L \to\infty$</span> of
<span class="math-container">$$
I_m(L) = \int_1^L \cdots \int_1^L \frac {dx_1 \cdots dx_{m}} {(x_1+x_2)(x_2+x_3) \cdots (x_{m} +x_1)}
$$</span>
for <span class="math-container">$m\in\mathbb N$</span> can be computed in the following way:... |
1,460,012 | <p>Heres the problem in my textbook:<br>
Show that if B and C have two equal columns, and A is any matrix for which AB and AC are defined, then AB and AC also have two equal columns.</p>
<p>This wasn't hard to figure out. But the question also says "Find a similar result involving matrices with two equal rows." I've... | Mesmerized student | 221,927 | <p>Try $BA$ and $CA$.
If we know that $AB$ and $AC$ have some equal column, then $(AB)^T=B^TA^T$ and $(AC)^T=C^T A^T$ have the equal row.</p>
|
1,714 | <p>Say I have a function $f(x)$ that is given explicitly in its functional form, and I want to find its Fourier transform[1]. If $f$ is too complicated to have an analytic expression for $\hat f(k)$, how do I obtain it numerically?</p>
<p>The naive and stupid way, which I currently use, is evaluating the integral for ... | ftiaronsem | 13,131 | <p>Since I recently had to deal with this issue too, I wanted to provide full working example, comparing the <code>FourierTransform</code> result to the <code>Fourier</code> result. This code is based on David Z's excellent answer. </p>
<p>First let's define our function (in this case the ground state of the quantum h... |
2,800,416 | <p>this is my first question and I don't quite understand how do I confront this equation:</p>
<p>$z^2+i\sqrt{32}z-6i=0$</p>
<p>I tried using the quadratic formula but it doesn't seem to give me a correct answer, any help will be much obliged.</p>
<p>Thank you! :)</p>
| poyea | 498,637 | <p>Note that $z^2+i\sqrt{32}z-6i=(z+i\sqrt{8})^2+(8-6i)$. The problem is essentially$$(z+i\sqrt{8})^2+(8-6i)=0$$</p>
|
411,309 | <p>I'm trying to understand the equation:</p>
<p>$$\frac{1}{2\pi i} \int_C \left( \frac{x}{n} \right)^s \frac{ds}{s} = \theta(x-n).$$</p>
<p>Here $x\in \mathbb{R}, x\geq 0$, and $C = \{s:\operatorname{Re}(s) = \sigma\}$ is a coutour, with fixed $\sigma > 0$ and $t = \operatorname{Im}(s)$ ranging.</p>
<p>This is ... | robjohn | 13,854 | <p>I assume that $x/n\gt0$ (real implied). If so, we can replace $x/n$ by $x$. So we need to look at
$$
\int_Cx^{\large s}\frac{\mathrm{d}s}{s}
=\int_Ce^{\large s\log(x)}\frac{\mathrm{d}s}{s}
$$
If $x\gt1$, then $\log(x)\gt0$. Then we use a contour that looks like a backwards "D"; going from $\sigma-i\infty$ to $\sigma... |
3,154,609 | <p>Let <span class="math-container">$L$</span> denote a linear operator and <span class="math-container">$v\in V$</span>. Does the expression <span class="math-container">$$c_0L^nv + c_1L^{n-1}v + \cdots + c_{n-1}L^1v + c_nL^0v = 0$$</span>
has a special name and what properties are known? For example, I know that if <... | Will | 652,741 | <p>Yes, if <span class="math-container">$(A,\omega,\phi)$</span> is mutually independent.</p>
<p>Indeed, if <span class="math-container">$(A,\omega,\phi)$</span> is mutually independent, then <span class="math-container">$A$</span> is independent of <span class="math-container">$(\omega,\phi)$</span> so you can write ... |
3,709,040 | <p>I'm reading Goldblatt's Topoi and trying to practice categorical reasoning, generalizing the example of <span class="math-container">$\mathbf{Set}^\rightarrow$</span> being a topos.</p>
<p>So, let <span class="math-container">$\mathcal{C}$</span> be a category with a subobject classifier <span class="math-container... | Daniel Schepler | 337,888 | <p>The first question to ask would be: what <em>are</em> the monomorphisms of <span class="math-container">$\mathcal{C}^{\rightarrow}$</span>? To answer this question, let us use a Yoneda lemma style of argument: first, note that <span class="math-container">$\operatorname{Hom}_{\mathcal{C}^{\rightarrow}}(0 \to U, X \... |
2,894,637 | <p>As stated the condition is:</p>
<ol>
<li>$\int_0^\infty f(x) dx=0$ </li>
<li>$f(x)$ continuous on $x\in[0,\infty)$</li>
</ol>
<p>What I would like to prove is $\lim_{x\to\infty}f(x)=0$</p>
<p>It is easy to prove that if $\lim_{x\to\infty}f(x)$ exists, or $\lim_{x\to\infty}f(x)=\infty$.</p>
<p>But I would like to... | user1551 | 1,551 | <p>A quick and dirty way to prove the theorem is to note that the matrix representation of $\operatorname{ad}(A)$ (i.e. $I\otimes A-A^T\otimes I$) is similar to $C=I\otimes A-A\otimes I$, because $A$ is similar to $A^T$. By extending the field to $\mathbb C$, you may further assume that $A$ is upper triangular. Now the... |
975,759 | <p>Is differentiation a continuous function from $C^1[a,b] \to C[a,b]$?</p>
<p>I think it is but I can't prove it... Would it be possible to prove it using theory about closed sets in $C[a,b]$ and their preimage? My problem here would be to check <strong>all</strong> closed sets and the closeness of their preimage so ... | Anthony Carapetis | 28,513 | <p>These questions are much easier to approach using the sequential definition of continuity: for metric spaces $X$ and $Y$, a map $L : X \to Y$ is continuous if and only if $L(x_n) \to L(x)$ for every convergent sequence $x_n \to x$ in $X$. </p>
<p>The answers to your questions of course depend on what topology/metri... |
2,050,426 | <p>My question is really simple. How can I show intuitively to my complex analysis students that the sine function is unbounded? What kind of behavior makes the complex sine function different from the real one in this sense?</p>
| Jean Marie | 305,862 | <p>Here is a picture displaying $|\sin(x+iy)|$ as a function of $x$ and $y$ that may be appealing to students: the $x$ variable conveys the <em>circular</em> trigonometry part ($-2 \pi < x < 2 \pi$) and the $y$ variable ($-1 < y < 1$) accounts for the <em>hyperbolic</em> trigonometry part:</p>
<p><img sr... |
278,486 | <p>I'm just curious because I was trying to come up with a weird function with weird discontinuities. Then I thought</p>
<p>$$f(x)=\prod_{n=1}^\infty \dfrac{1}{1-\frac{1}{nx}}$$</p>
<p>So what's this product?</p>
| Geoff Robinson | 13,147 | <p>For amended question. When $x >1,$ the log of the partial product to $k$ terms
is $\sum_{n=1}^{k} \log(1 - \frac{1}{nx}),$ Let $r$ be the smallest integer with
$rx >1.$ Then the partial product is greater than $\sum_{n=r}^{k} \frac{1}{nx}$ which tends to infinity as $k$ increases, since the harmonic series di... |
201,370 | <p>I would like to plot entropy on a 2-simplex, i.e., I want to plot a function for x,y,z s.t. x+y+z=1. My strategy is to take bounds {x,0,1}, {y,0,1-x}, and compute z=1-x-y. However, there is some problem with logarithm in this approach. The following yields an empty plot:</p>
<pre><code>Plot3D[x + Log[y], {x, 0, 1},... | Ulrich Neumann | 53,677 | <p>Try <code>RegionFunction</code></p>
<pre><code>Plot3D[x + Log[y], {x, 0, 1}, {y, 0, 1}, RegionFunction -> Function[{x, y}, 0 <= y <= 1 - x]]
</code></pre>
<p><a href="https://i.stack.imgur.com/Fu26M.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Fu26M.png" alt="enter image descript... |
3,945,610 | <p>If <span class="math-container">$a, b, c, d>0$</span>, such that <span class="math-container">$a+b+c=1$</span>, prove that: <span class="math-container">$$a^3+b^3+c^3+abcd\ge \min\left(\frac{1}{4}, \frac{1}{9}+\frac{d}{27}\right).$$</span></p>
<p>I tried solving it as follows:
<span class="math-container">$$a^3+b... | River Li | 584,414 | <p>My second solution:</p>
<p>By using Schur's inequality and the identity
<span class="math-container">$$a^3 + b^3 + c^3 + \frac{15}{4} abc =
\frac{3}{4}[a(a-b)(a-c) + b(b-c)(b-a) + c(c-a)(c-b)] + \frac{1}{4}(a+b+c)^3,$$</span>
we have
<span class="math-container">$$a^3 + b^3 + c^3 + \frac{15}{4} abc \ge \frac{1}{4}.... |
748,013 | <p>Definition: $f(X)=${$f(x)|x\in X$}, "$|A|$"represents the number of elements in the set A.
<br/>In the title, $f:S\to T$, "$iff$" means "if and only if".$S$and $T$ are finite sets.
Two definitions for being onto:
<br/>1.If for every element $ t$ in $T$, there exists some $s\in S$, such that $f(s)=t$, then it's onto... | daw | 136,544 | <p>There is a formal difference between the tupel space
$$
\mathbb R^n:=\mathbb R \times \dots \times \mathbb R
$$
and the space of row vectors $\mathbb R^{1,n}$, which is defined as the space 'numbers arranged in a row' (think arranged in a spreadsheet). In the book you linked, the difference is emphasized by using d... |
39,802 | <p>Suppose $f:X \to Y$ is a function of sets. Then we can take the quotient $X/\text{~}$ by identifying $x \text{~} y$ if and only if $f(x)=f(y)$. Now suppose instead that $f:X \to Y$ is a map of simplicial sets. I want to emulate this homotopically, by adding a 1-simplex between $x$ and $y$ if there is a 1-simplex fro... | David Roberts | 4,177 | <p>You are looking at the coequaliser of the kernel pair, so my guess would be to take the homotopy pullback of $f$ along itself, then look at the nerve of the groupoid $X\times_Y X \rightrightarrows X$ in $sSet$ this gives rise to, then form the diagonal (=hocolim) of this bisimplicial set. I guess this comes with a m... |
234,845 | <p>How do we prove that a function $f$ is measurable if and only if $\arctan(f)$ is measurable?</p>
<p>If I use the definition of measurable functions, that is, a function is measurable if and only if its inverse is measurable?</p>
| Martin Argerami | 22,857 | <p><em>(I will assume we are talking about real-valued functions)</em></p>
<p>To prove the first implication, assume that $f$ is measurable.</p>
<p>Take $V\subset\mathbb R$ open. Write $g=\arctan$. Then $(g\circ f)^{-1}(V)=f^{-1}(g^{-1}(V))$. As $g$ is continuous, $g^{-1}(V)$ is open. As $f$ is measurable, $f^{-1}(g^... |
2,720,188 | <p>If we take the inner product $\langle f, g \rangle = \displaystyle\int_{-\pi}^{\pi} f(t) \overline{g(t)} dt$ on $L^2 ((-\pi, \pi))$, which allows functions to $\mathbb{C}$, then it's not hard to check that $\Big(\frac{e^{int}}{\sqrt{2\pi}}\Big)_{n=1}^{\infty}$ is an orthonormal <em>sequence</em> in the space, but I'... | Disintegrating By Parts | 112,478 | <p>One of the more interesting proofs of completeness uses complex analysis. Consider the expression
$$
F(\lambda) = \frac{i}{1-e^{-2\pi i\lambda}}\int_{0}^{2\pi}e^{-i\lambda t}g(t)dt.
$$
The function $F(\lambda)$ is holomorphic in $\mathbb{C}\setminus\mathbb{Z}$, with possible first order poles at $\lambd... |
10,948 | <p>This is a simple question, but its been bugging me. Define the function $\gamma$ on $\mathbb{R}\backslash \mathbb{Z}$ by
$$\gamma(x):=\sum_{i\in \mathbb{Z}}\frac{1}{(x+i)^2}$$
The sum converges absolutely because it behaves roughly like $\sum_{i>0}i^{-2}$.</p>
<p>Some quick facts:</p>
<ul>
<li>Pretty much by c... | Qiaochu Yuan | 290 | <p>As long as we're talking about the Weierstrass function, consider the parallels between the following:</p>
<p>1) Given a lattice $\Gamma$ in $\mathbb{R}$, the quotient $\mathbb{R}/\Gamma$ is topologically a circle. One way to describe it is to write down some functions on $\mathbb{R}$ which are invariant under $\... |
110,599 | <blockquote>
<p>Why isn't $M = \mathbb{C}[x,y,z]/(xz-y)$ a flat $R = \mathbb{C}[x,y]$-module?</p>
</blockquote>
<p>The reason given on the book is "the surface defined by $y-xz$ doesn't lie flat on the $(x,y)$-plane". But I don't understand why this can be a reason.</p>
<p>Since</p>
<blockquote>
<p>An $R$-module... | Matt E | 221 | <p>This answer is similar to the others; perhaps it will help to see the same
points made by yet another person.</p>
<p>First of all, it might help to note that $\mathbb C[x,y,z]/(xz-y)$ is
isomorphic to $\mathbb C[x,z]$. So you are looking at the map $\mathbb C[x,y]
\to \mathbb C[x,z]$ defined by $x \mapsto z, y \... |
102,976 | <p>Is there a way to evaluate a string containing RPN in Mathematica?</p>
<p>SE thinks this question is too short, so let me expand on it.
Do you know of any function, that provides the following functionality?</p>
<pre><code>EvalRPN["5 4 + 3 /"]
</code></pre>
<blockquote>
<p>3</p>
</blockquote>
<p>Or even symbol... | C. E. | 731 | <p>A rule based option:</p>
<pre><code>Clear[rpn]
$InfixOperators = Plus | Subtract | Times | Divide;
$PrefixOperators = Cos | Sin | Sqrt;
operand = Except@Join[$InfixOperators, $PrefixOperators];
rpn[{a : operand ..., b : operand, c : operand, op : $InfixOperators, d___}] := rpn[{a, op[b, c], d}]
rpn[{a : operand ..... |
3,004,210 | <p>A smilar question has been asked before <a href="https://math.stackexchange.com/questions/23503/create-unique-number-from-2-numbers">Create unique number from 2 numbers</a>.</p>
<blockquote>
<p>is there some way to create unique number from 2 positive integer numbers? Result must be unique even for these pairs: 2 an... | Ross Millikan | 1,827 | <p>I showed in <a href="https://math.stackexchange.com/questions/3003672/convert-infinite-2d-plane-integer-coords-to-1d-number/3003770#3003770">this answer</a> how to adapt the Cantor pairing function to accept negative as well as positive arguments. It is not a complex calculation, a few adds and one multiply. The p... |
56,337 | <p>Suppose I have the following list:</p>
<pre><code>list = {a, b, c, d}
</code></pre>
<p>I want to generate this result:</p>
<pre><code>{{f[a, a], f[a, b], f[a, c], f[a, d]}, {f[b, b], f[b, c],
f[b, d]}, {f[c, c], f[c, d]}, {f[d, d]}}
</code></pre>
<p>What could be the shortest way?</p>
<p>The list elements can... | alancalvitti | 801 | <pre><code>Outer[f, list, list] /.
f[x_, y_] /;
First@First@Position[list, x] > First@First@Position[list, y] :>
Sequence[]
</code></pre>
<p>What's annoying here is projecting <code>#[[1,1]]&</code>. How to make <code>{{1}} <= {{2}}</code> evaluate <code>True</code>? </p>
|
56,337 | <p>Suppose I have the following list:</p>
<pre><code>list = {a, b, c, d}
</code></pre>
<p>I want to generate this result:</p>
<pre><code>{{f[a, a], f[a, b], f[a, c], f[a, d]}, {f[b, b], f[b, c],
f[b, d]}, {f[c, c], f[c, d]}, {f[d, d]}}
</code></pre>
<p>What could be the shortest way?</p>
<p>The list elements can... | kglr | 125 | <pre><code>f1 = SplitBy[Tuples[f @@ #, 2] /. ( f[x__] /; Not[OrderedQ[{x}]] :> (## &[])), First]&
f1 @ list // Grid // TeXForm
</code></pre>
<blockquote>
<p>$\begin{array}{cccc}
f(a,a) & f(a,b) & f(a,c) & f(a,d) \\
f(b,b) & f(b,c) & f(b,d) & \text{} \\
f(c,c) & f(c,d) &... |
3,229,020 | <p>My lecturer set as a bonus exercise the following induction proof:</p>
<p>If <span class="math-container">$G$</span> is a finite abelian group <span class="math-container">$|G| = p_1^{n_1} \cdots p_s^{n_s}$</span> is the decomposition of <span class="math-container">$|G|$</span> into a product of distinct prime num... | Cardioid_Ass_22 | 631,681 | <p>We'll induct on the number of distinct prime factors in the group's order. As you've said, the base case is pretty easy. Now, suppose the statement has been proven for all abelian groups, <span class="math-container">$H$</span>, such that <span class="math-container">$|H|$</span> has <span class="math-container">$n$... |
119,981 | <p>Let $C/\mathbb Q$ be a smooth projective curve of genus $g\geq 2$ or a smooth affine curve of genus $g \geq 1$. The exact sequence</p>
<p>$1 \to \pi_1^{et}(C \otimes_\mathbb Q \bar{\mathbb Q}) \to \pi_1^{et}(C) \to \operatorname{Gal}(\bar{\mathbb Q}|\mathbb Q) \to 1$</p>
<p>gives a homomorphism from $\operatorname... | Michael Greinecker | 35,357 | <p>I think an answer that discusses the actual institutional details of how the Fed controls the money supply would be off-topic here. Also, the Fed works slighlty differently from the ECB in that regard and there is more than one method of influencing the money supply (take a look at the wikipedia page on <a href="htt... |
3,034,766 | <p>So is all it's saying that if there are two functions that have the same derivatives for every single <span class="math-container">$x$</span> in the interval, then <span class="math-container">$f(x) = g(x) + \alpha$</span>, means that the second function is just the exact same as <span class="math-container">$f(x)$<... | Dando18 | 274,085 | <p>Yes, <span class="math-container">$\int f(x)\ dx$</span> is a <em>family</em> of functions that only differ by a constant, which is typically denoted by <span class="math-container">$c$</span>. This constant <span class="math-container">$c$</span> can be viewed as a shift vertically of the function.</p>
<p>The idea... |
4,203,235 | <p>Let <span class="math-container">$G$</span> be a group and let <span class="math-container">$ a \in G$</span>. If <span class="math-container">$A = \{a \}$</span>, prove <span class="math-container">$C_G(a) = C_G(a^{-1})$</span></p>
<p>Proof:
Since <span class="math-container">$C_G(a) \le G$</span>, it is given that... | Dionel Jaime | 462,370 | <p>For any subset <span class="math-container">$A \subset G$</span>, we define <span class="math-container">$C_G(A)$</span> as</p>
<p><span class="math-container">$$C_G(A):= \{x\in G| xa = ax \ \text{for all} \ a \in A \} $$</span></p>
<p>This is set of all elements in your group which commute with <span class="math-co... |
3,652,205 | <p>I am trying to compute the following double integral:
<span class="math-container">$$I=\iint_S \frac{2-4xy}{(9-xy)(8+xy)}dxdy$$</span>
with <span class="math-container">$S=[0,1]\times[0,1].$</span></p>
<p><strong>What I have tried:</strong></p>
<ol>
<li>I have written the integral as follows:
<span class="math-con... | user97357329 | 630,243 | <p>In the following I'll address a solution in one line to the OP's question in comments. We may observe the simple facts that <span class="math-container">$\int_0^1 \frac{\log(t)}{a+1-t}\textrm{d}t=\int_0^1 \frac{1/(1+a)\log(t)}{1-(1/(1+a))t}\textrm{d}t=-\operatorname{Li}_2\left(\frac{1}{1+a}\right)$</span> and <span ... |
599,467 | <p>I would like to summarize my formula.
$p$ and $y$ are constant value, $10000$ and $0.65$.</p>
<p>When $n = 3$, my formula recalculate the result of $n = 2$. I don't want to recalculate. Is there way to summarize or other formula for that equivalent?</p>
<p>$$x_n=(p+x_{n-1})y$$</p>
<p><strong>Update :</strong></p>... | dato datuashvili | 3,196 | <p>so we have like this yes</p>
<p>$x_n=(10000+x_{n-1})*0.65$</p>
<p>so we have</p>
<p>$x_n=6500+0.65*x_{n-1}$</p>
<p>rewrite it as</p>
<p>$x_n-0.65*x_{n-1}=6500$</p>
<p><a href="http://www.economics.utoronto.ca/osborne/MathTutorialSF/FODF.HTM" rel="nofollow">http://www.economics.utoronto.ca/osborne/MathTutorialS... |
1,517,189 | <p>My first question here..sorry if I'm not very specific but I try to be.</p>
<p>A T-tetromino has three connected blocks in a line and another one above the middle block. How many ways can one be painted on the grid if orientation matters? What about if it doesn't?</p>
| robjohn | 13,854 | <p>In <a href="https://en.wikipedia.org/wiki/Asymptotic_analysis" rel="nofollow">Asymptotic Analysis</a>, $f\sim g$ means $\lim\limits_{n\to\infty}\frac{f(n)}{g(n)}=1$, and $f\lesssim g$ means $\limsup\limits_{n\to\infty}\frac{f(n)}{g(n)}\le1$. Thus, asymptotically less than would mean
$$
\limsup_{n\to\infty}\frac{f(n)... |
3,097,590 | <p>I am reading my textbook and I see this:</p>
<p><a href="https://i.stack.imgur.com/Na9oQ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Na9oQ.png" alt="enter image description here"></a></p>
<p>So why are we allowed to just restrict the domain like this? I can see that we're trying to make it s... | eyeballfrog | 395,748 | <p>The book solution is glossing over an important point--the function <span class="math-container">$1/\sqrt{x^2-a^2}$</span>, and thus its antiderivative, is only defined on the set <span class="math-container">$(-\infty,-a)\cup(a,\infty)$</span>, which is not a connected set. So to find an antiderivative properly, ea... |
2,192,314 | <p>Once upon a time, I was looking at interesting properties of prime numbers. One thing I noticed was that if we take the <strong>absolute values of the differences</strong> between each prime, and repeat this process on the differences recursively, the first column turns out to always be $1$ (With the exception of th... | Klangen | 186,296 | <p>You have re-discovered <a href="https://en.wikipedia.org/wiki/Gilbreath%27s_conjecture" rel="noreferrer">Gilbreath's conjecture</a>, namely that creating a sequence in which each term $s_n$ is the difference between the consecutive primes $p_{n+1}-p_n$, and then repeating this process for the newly created sequence,... |
4,593,584 | <p>A R.V. <span class="math-container">$X$</span> has a pdf of the form <span class="math-container">$f_{X}(x) = e^{-x} u(x)$</span> and an independent R.V. <span class="math-container">$Y$</span> has a pdf of <span class="math-container">$f_{Y}(y) = 3e^{-3Y} u(y)$</span> using characteristic functions, find the pdf of... | Mark Saving | 798,694 | <p>It is in the line</p>
<blockquote>
<p>Next, we can represent <span class="math-container">$S$</span> as
<span class="math-container">$$S = (S \setminus S_1) \cup (S_1 \setminus S_2) \cup \cdots \cup (S_k \setminus S_{k + 1}) \cup \cdots \cup D$$</span></p>
</blockquote>
<p>Every other line of the proof works without... |
1,177,871 | <p>I need to find a bijective map from $A=[0,1)$ to $B=(0,1).$ Is there a standard method for coming up with such a function, or does one just try different functions until one fits the requirements?</p>
<p>I've considered some "variation" of the floor function, but not sure if $\left \lfloor{x}\right \rfloor$ is bije... | Archy Will He 何魏奇 | 65,082 | <p>To find the bijection from $[0,1)$ to $(0,1)$, remember that you are basically looking for a function that maps a subset of $\mathbb{R}$ to another subset of $\mathbb{R}$. </p>
<p>(An example would be $f:R\mapsto (0,1)$ where $f(x)=1/(1+e^x)$)</p>
<blockquote>
<p>.. assuming one
exists, is this equivalent to s... |
2,283,705 | <p>In <a href="https://arxiv.org/pdf/0910.5004.pdf" rel="nofollow noreferrer">F.M.S. Lima's paper</a> on Riemann zeta-type functions, he conjectures the following formula:
<span class="math-container">$$\sum_{n=1}^\infty{\frac{\zeta(2n)}{2n(2n+1)\dots(2n+N)}\frac{1}{4^n}}=\frac{1}{2}\left[\frac{\ln{\pi}}{N!}-\frac{H_N}... | Dr. Fabio M. S. Lima | 99,872 | <p>Let me just tell you that my conjecture, mentioned above, has been proved true (by myself) in Theorem 2 of: Annali di Matematica Pura ed Applicata vol.194, p.1015 (2015). There in Theorem 3 of this paper, I prove a generalization of Theorem 2 which may be useful for answering the question by tyobrien.</p>
<p>Prof... |
2,751,831 | <p>Are there any results along the following lines: </p>
<p>Let $\Gamma_1$ and $\Gamma_2$ be groups with respective finite index subgroups $\Gamma_0^i$ for $i=1,2$. If $\Gamma_1 \cap \Gamma_2 \leq \Gamma_0^i$ for $i=1,2$ can we conclude that $[\Gamma_1 \ast_{\Gamma_1 \cap \Gamma_2} \Gamma_2: \Gamma_0^1 \ast_{\Gamma_1 ... | Lee Mosher | 26,501 | <p>This is not true, and in fact the exact opposite holds in most cases. </p>
<p>Just as an example, if $\Gamma_1$ and $\Gamma_2$ are infinite cyclic groups, and if just one of $\Gamma^1_0$ or $\Gamma^2_0$ is a proper subgroup, then the index is infinite. </p>
|
2,397,077 | <p>The "Power of a Point" or "Intersecting Chords" theorem states that for any point in a plane, if a line is drawn that intersects a circle, the distance from the point to one of the intersections multiplied by the distance from the point to the other intersection is a constant for that point, no matter what line you ... | Intelligenti pauca | 255,730 | <p>Here's a neat example: $OA\cdot OB=2$.</p>
<p><a href="https://i.stack.imgur.com/vnGCJ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/vnGCJ.png" alt="enter image description here"></a></p>
|
442,043 | <p>Assume I have a non-empty finite set $S$ with $x=|S|$. I want to divide the set $S$ into subsets $S_1, S_2, .., S_n$ (<em>Edit:</em> Yes, $S = \cup S_i$, and I'm embarrassed that I forgot to include that) such that: </p>
<ul>
<li>$ |S_i| = y, \forall 1 \le i \le n$ (The cardinality of each subset is fixed) </li>... | hardmath | 3,111 | <p><strong>Update:</strong> After filling up several sheets of scratch notes, and much Internet browsing, it dawned on me that the dual of the designs considered here are <a href="http://designtheory.org/library/encyc/topics/pbd.pdf" rel="noreferrer">pairwise balanced designs</a> on $n$ "points" (namely the sets $S_i$ ... |
2,639,304 | <p>I'm looking for a monotonic increasing function $f(x)$ defined on $[0,\infty)$ so that $f(0)=0$ and with $f'(x)$ approaching $0$ as $x$ approaches $0$, and which approaches 1 as $x \rightarrow \infty$. </p>
<p>The function might look roughly like the figure below, but that particular function is just a shifted and... | dxiv | 291,201 | <p>$f(x) = 1 - e^{-a x^2}$ with $a \gt 0$ satisfies the conditions $f(0)=f'(0)=0$, $\lim_{x \to \infty} f(x) = 1$.</p>
|
66,485 | <p>How do I evaluate the mean end-to-end squared distance of a FENE ideal chain at fixed inverse temperature $\beta$ in the canonical ensemble?</p>
<p>This quantity is defined as the mean value
$$\left<\left(\sum_{i=1}^{N-1} \vec r_i\right)\cdot\left(\sum_{j=1}^{N-1} \vec r_j\right)\right>$$</p>
<p>Under the c... | yarchik | 9,469 | <p>I believe it is not possible to solve this problem with Mathematica in the presented form. Partly because there inconsistencies in the formulation:</p>
<pre><code>Plot[-Log[1 - 16 (Abs[x] - 1)^2], {x, 1 - 1/4, 1 + 1/4}]
</code></pre>
<p>We see that the potential is, in fact, defined on a larger domain [3/4,5/4]</p... |
2,101,241 | <p>Find the remainder when $$140^{67}+153^{51}$$ is divided by $17$.</p>
<p>$$140\equiv 4 \pmod {17}$$
$$67\equiv 16 \pmod{17}$$
$$153 \equiv 0 \pmod{17}$$
$$51\equiv 0 \pmod{17}$$</p>
<p>$$\Rightarrow 140^{67}+153^{51}\equiv 4^{16}+0 \equiv 1\pmod{17}$$</p>
<p>Solution should be $13$. What's wrong?</p>
| Miz | 205,000 | <p>One of the rules for you are using is correct the other one is wrong.</p>
<p>Suppose $p$ is a prime then</p>
<blockquote>
<p>If $a \equiv b \ (\bmod p)$ then $a^n\equiv b^n (\bmod p)$</p>
</blockquote>
<p>However </p>
<blockquote>
<p>If $m \equiv n (\bmod p) \not\Rightarrow a^m \equiv a^n (\bmod p)$. </p>
</... |
419,678 | <p>I need your help. I have to prove absolute convergence and conditional convergence to this two series. I have no idea how to do this.</p>
<blockquote>
<p>a) $$\sum_{n=1}^\infty \frac{n (x-1)^{n}}{2^{n} (3n - 1)}$$</p>
<p>b) $$\sum_{n=1}^\infty \frac{1}{2n -1}\left(\frac{x+2}{x-1}\right)^{n}
$$</p>
</bloc... | Mhenni Benghorbal | 35,472 | <p><strong>Hint:</strong> You can use the ratio test and note that, if the series converges absolutely, then it converges conditionally. </p>
|
971,457 | <p>It is asked that I find a function such that $$10-f(x)=2\int_0^xf(t)dt.$$ I tried giving a new function F(x) such that ${dF(x)\over dx}=f(x)$, but all I got was a new equation $$F(x)=10x-2\int_0^xF(t)dt.$$ So how do we find such function. Thanks in advance! (I am new to differential equations, so I do not know much ... | Travis Willse | 155,629 | <p>You have the right idea: Differentiating both sides gives
$$-f'(x) = 2f(x),$$
which has general solution $$f(x) = C e^{-2x}.$$ Evaluating both sides at a convenient value of $x$ (say, $x = 0$) determines $C$ uniquely.</p>
|
382,549 | <p>Let $A = diag \left (\lambda_1, ..., \lambda_n \right ) \in \mathbb{R}^{n \times n}$, with $\lambda_1 < \lambda_2 < ... < \lambda_n$.<br>
Let $u = \left (u_1, ..., u_n \right ) ^T \in \mathbb{R}^n$, with $u_i \neq 0 \ \ \forall i$.<br>
How can be shown that: </p>
<ol>
<li>For any $\alpha \in \mathbb{R}, \... | Berci | 41,488 | <p><strong>Hint:</strong> Let
$$g(\tau):=\prod_i(\lambda_i-\tau)\,\cdot f(\tau)=\prod_i(\lambda_i-\tau)+\alpha\,\sum_i\,\left(u_i^2\cdot\prod_{j\ne i}(\lambda_j-\tau) \right)\,,$$
which is a polynomial of $\tau$ (of degree $n$). By (a big amount of) hypothesis, we have that $g(\lambda_i)\ne 0$, and if we prove that th... |
2,418,384 | <p>So I made a 10 x 10 board. I assumed that the board is a checker board and found that each tile would have three of the same color (either black or white) and 1 different color. However, I do not know what else to do with this information.</p>
| quasi | 400,434 | <p>Hints:
<p>
Assume an exact cover is possible.
<p>
How many $T$-tiles are needed?
<p>
Some will be majority white, others majority black.
<p>
But there must be the same number of each.</p>
|
2,418,384 | <p>So I made a 10 x 10 board. I assumed that the board is a checker board and found that each tile would have three of the same color (either black or white) and 1 different color. However, I do not know what else to do with this information.</p>
| Michael L. | 153,693 | <p>We let $a$ be the number of tiles covering three white squares and one black square and $b$ be the number of tiles covering one white square and three black squares. Then, we must have $3a+b = a+3b = 50$, since $50$ white squares and $50$ black squares must each be covered. This is only satisfied for $$a = b = \frac... |
3,643,773 | <p>I have a circle with center (0,0) and radius 1. I have calculated a random point inside a circle by generating a random angle <span class="math-container">$a=random()\cdot 2\pi $</span> and a random distance smaller than or equal to the radius <span class="math-container">$b=random()\cdot r$</span>. The center of th... | Brian Tung | 224,454 | <p>The point is at angle <span class="math-container">$a$</span>, so the line must be at angle <span class="math-container">$a+\frac\pi2$</span>. (Do you see why?) Provided that <span class="math-container">$a$</span> is not a multiple of <span class="math-container">$\frac\pi2$</span>, one can then write</p>
<p><sp... |
4,582,099 | <p>Find limit of the given function:</p>
<p><span class="math-container">$$\lim_{x\rightarrow0} \frac{(4^{\arcsin(x^2)} - 1)(\sqrt[10]{1 - \arctan(3x^2)} - 1)}{(1-\cos\tan6x)\ln(1-\sqrt{\sin x^2})}
$$</span></p>
<p>I tried putting 0 instead of x
<span class="math-container">$$\lim_{x\rightarrow0} \frac{(4^{\arcsin(x^2)... | Sine of the Time | 1,118,406 | <p><span class="math-container">$$\lim_{x\to 0} \frac{(4^{\arcsin(x^2)} - 1)(\sqrt[10]{1 - \arctan(3x^2)} - 1)}{(1-\cos\tan6x)\ln(1-\sqrt{\sin x^2})}
$$</span>
<span class="math-container">$4^{\arcsin(x^2)} - 1=e^{\arcsin(x^2) \log 4}-1\sim x^2\log 4 $</span></p>
<p><span class="math-container">$\sqrt[10]{1 - \arctan(3... |
2,783,922 | <p>I will start by the usual definition of a path connected set:</p>
<blockquote>
<p>A subset $A$ of a topological space $X$ is path connected when, for every pair of points $a,b\in A$, there exists a continuous function $f:[0,1]\rightarrow A$ such that $f(0)=a$ and $f(1)=b$.</p>
</blockquote>
<p>This question may ... | emma | 229,471 | <p>According to the <a href="https://en.wikipedia.org/wiki/Inverse_function_theorem" rel="nofollow noreferrer">inverse function theorem</a>, we have
$$(f^{-1})'(a)=\frac{1}{f'(f^{-1}(a))}$$
with $a=-2.$
With $f'(x)=3x^2+2x+2$ and $f^{-1}(-2)=\frac{1}{(-2)^3+(-2)^2+2\cdot(-2)}=-\frac{1}{8}$ we get
$$(f^{-1})'(-2)=\frac{... |
160,355 | <p>If i import a .xyz file in Mathematica that will show me the moelcule in the default way. But i want to visualise the .xyz file in the following manner.
<a href="https://i.stack.imgur.com/7SwPp.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7SwPp.jpg" alt="enter image description here"></a></p>
| Sumit | 8,070 | <p>Sticks and balls</p>
<pre><code>file = "ben.xyz";
atm = Import[file, "VertexTypes"];
xyz = Import[file];
coord = xyz[[1, 4, 1]];
balls = Cases[xyz[[1, 4, 2]], Sphere[x_, y_] -> {x, y}];
sticks= Cases[xyz[[1, 4, 2]], Cylinder[x_, y_] -> {x, y}];
Graphics[{EdgeForm[Black], FaceForm[White],
{Thickness[#[[2]]/1... |
1,123,979 | <p>I am totally confused with the substitution method of evaluating indefinite integrals, especially those with rational functions and polynomials. I have 2 cases, which if I made to understand, would break ice for my future cases.</p>
<p>case 1: Evaluate: $\int x\sqrt{x+2}dx $</p>
<p>case 2: Evaluate: $\int \frac{x^... | Upax | 157,068 | <p>when you replace x you've to replace x everywhere. If $u=x+2$ then $x=u-2$ and $dx= du$. This means that your integral become
\begin{equation}
\int (u-2) \sqrt u du = \int u \sqrt{u} du -2 \int \sqrt{u} du=\frac{2}{15}(3u-10) u^{\frac{3}{2}}
\end{equation}
Now you go back by replacing $u$ by $x+2$. For the second in... |
1,907,182 | <p>I have got short question: How to draw Arc in 2 demension know only start and end point in OpenGL?</p>
<p>Actually, something already wrote, however, this algorithm does not accomplish its purpose. Here we go:</p>
<pre><code> 1. Get startXY
2. Get endXY
3. Const num_segments = numeric below than 0
4. Calc midPo... | Mick | 42,351 | <p>The formula for the equation of the circle using $(x_{start}, y_{start})$ and $(x_{end}, y_{end})$ as two endpoints of its diameter is:-</p>
<p>$$(x - x_{start})(x - x_{end}) + (y - y_{start})(y - y_{end}) = 0$$</p>
<p>It saves some time than going through the process of finding the midpoint. </p>
|
1,907,182 | <p>I have got short question: How to draw Arc in 2 demension know only start and end point in OpenGL?</p>
<p>Actually, something already wrote, however, this algorithm does not accomplish its purpose. Here we go:</p>
<pre><code> 1. Get startXY
2. Get endXY
3. Const num_segments = numeric below than 0
4. Calc midPo... | Martien | 380,345 | <p>If you mean by arc any part of a circle there are an infinite number of circles going through two points.
If you mean half of a circle there are TWO (for example upper half and lower half or left half and right half of a circle). So two points are still ambiguous (unless any of the two satisfies) defining a half cir... |
334,325 | <p>The function $u(x,y)$ satisfies the partial differential equation</p>
<p>$$\nabla^{2}u=\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0\text{ in }0<y<a, -\infty<x<\infty$$</p>
<p>and the boundary conditions $u \to 0$ as $x \to \pm\infty$, $\frac{\partial u}{\partial y}=0$ o... | Community | -1 | <p>For the first part you probably just need to transform the boundary conditions. For $y = a$:</p>
<p>$$
\bar{u}(k) = \int e^{-|x|}e^{-ikx}dx
$$</p>
<p>which you can easily evaluate by splitting the range of integration in half; i.e., $(-\infty,+\infty) = (-\infty,0] \cup [0,+\infty)$.</p>
<p>For $y=0$:</p>
<p>$$
... |
3,535,725 | <p>I have started working on this <span class="math-container">$\int_0^1 \frac{x^2}{\sqrt{1+x^5}} \, dx \leq \frac{1}{3}$</span> by using the fact that <span class="math-container">$\int_0^1 \frac{x^2}{\sqrt{1+x^5}} \, dx < \int_0^1 \frac{x^2}{\sqrt{x^5}} \, dx$</span>
But that didn't work.</p>
| marty cohen | 13,079 | <p>If <span class="math-container">$f(x) \ge 0$</span> and
<span class="math-container">$n, m \ge 0$</span>
then
<span class="math-container">$\int_0^1 \dfrac{x^n\,dx}{(1+f(x))^m}
\le \int_0^1 x^n\,dx
=\dfrac1{n+1}
$</span>.</p>
|
2,412,705 | <blockquote>
<p>Prove $\{a\}=\{b,c\} \iff a=b=c$</p>
</blockquote>
<p>When I try to demonstrate this I only get as far as $a=b \lor a=c \space$ and I can´t yet get to the $a=b \land a=c \space$ or $a=b \land b=c \space$ part of the demonstration </p>
| Andrew Bacon | 173,753 | <p>By the axiom of extensionality, if $\{b,c\}=\{a\}$, then $x \in \{b,c\}$ if and only if $x\in \{a\}$. So if $x$ is either $b$ or $c$, then $x$ is $a$. I.e. both $b$ and $c$ are identical to $a$.</p>
|
2,412,705 | <blockquote>
<p>Prove $\{a\}=\{b,c\} \iff a=b=c$</p>
</blockquote>
<p>When I try to demonstrate this I only get as far as $a=b \lor a=c \space$ and I can´t yet get to the $a=b \land a=c \space$ or $a=b \land b=c \space$ part of the demonstration </p>
| High GPA | 438,867 | <p>You said: $a\in\{b,c\} \Rightarrow a=b\lor a=c$.</p>
<p>ALSO: $b\in\{a\}\land c\in\{a\} \Rightarrow b=a\land c=a$</p>
|
2,412,705 | <blockquote>
<p>Prove $\{a\}=\{b,c\} \iff a=b=c$</p>
</blockquote>
<p>When I try to demonstrate this I only get as far as $a=b \lor a=c \space$ and I can´t yet get to the $a=b \land a=c \space$ or $a=b \land b=c \space$ part of the demonstration </p>
| Dave | 334,366 | <p>$(\Rightarrow)$ Suppose $\{a\}=\{b,c\}$. Then $\{b,c\}\subseteq\{a\}\implies b,c\in\{a\}$, so we must have $b=a$ and $c=a$ since $\{a\}$ has one element: $a$.</p>
<p>$(\Leftarrow)$ Suppose $a=b=c$. Then $\{a\}=\{a,a\}=\{b,c\}$. More explicitly, take $x\in\{a\}$, then $x=a=b$ since $a=b$, so $x\in\{b,c\}\implies \{a... |
2,068,643 | <p>I'm solving a problem about a <a href="https://en.wikipedia.org/wiki/Chemical_reactor#PFR_.28Plug_Flow_Reactor.29" rel="nofollow noreferrer">plug flow reactor</a> and I have this limit to compute. Just to control my result I asked <a href="http://www.wolframalpha.com/input/?i=lim_%7BR%20%5Cto%20%2B%5Cinfty%7D%20(1-e... | Duchamp Gérard H. E. | 177,447 | <p>One has
$$
\exp(h)=1+h+h\epsilon(h)
$$
with $\lim_{h\rightarrow 0} \epsilon(h)=0$ then, as $R\rightarrow +\infty$ one has
$$
\exp\left(\frac{x}{R+1}\right)=1+\frac{x}{R+1}+\frac{x}{R+1}\epsilon_1(R)
$$
with $\lim_{R\rightarrow +\infty} \epsilon_1(R)=0$. Substituting this (exact !) expression in your quotient gives ... |
87,869 | <p>Given the function $Z(s,N)= \sum \limits_{n=1}^{N}n^{-s}$.</p>
<p>In the limit $N \to \infty$ the function $Z(s,N) \to \zeta (s)$ Riemann Zeta function.</p>
<p>My question is: Is there a Functional equation for this function? I mean a relationship of the form $ Z(s,N)=G(s) Z(1-s,N)$.</p>
| anon | 11,763 | <p>Well, here's an answer - no, because then $G(s)=Z(s,N)Z(1-s,N)^{-1}$ would be invariant with respect to $N$, but it clearly isn't (check the graphs for various $N$ on W|A if you want to, or just plug in the trivial $N=1$ case). If you let $G$ vary with $N$ then of course there are such functions, but I doubt there i... |
51,596 | <p>Some time ago, I asked <a href="https://math.stackexchange.com/q/42276/8271">this</a> here. A restricted form of the second question could be this:</p>
<blockquote>
<p>If $f$ is a function with continuous first derivative in $\mathbb{R}$ and such that $$\lim_{x\to \infty} f'(x) =a,$$ with $a\gt 0$, then $$\lim_{x... | Pete L. Clark | 299 | <p>After reading some of the other answers, I thought it might be helpful to rephrase Shai Covo's answer in more physical language: suppose that instead of $y = f(x)$ we have $x = x(t)$, the position function of a particle at time $t$. (Of course it makes no mathematical difference what we call the variables...)</p>
... |
265,728 | <p>Consider the polynomial:</p>
<p>$$p(x) = \sum_{k=0}^{r}(-1)^{r-k} {r \choose k} x^{k(k-1) / 2}$$</p>
<p>I want to show that</p>
<p>$$p(x) = (x - 1)^{\lceil r/2 \rceil} \, q(x)$$</p>
<p>That is, $(x - 1)^{\lceil r/2 \rceil}$ is a factor of $p(x)$. Even better, find a formula for the quotient polynomial $q(x)$.</p... | Uri Bader | 89,334 | <p><strong>Required Claim:</strong> In a compact Lie group there are at most countably many conjugacy classes of closed subgroups.</p>
<p>I am the one who made the comment that the required claim could follow from an algebraic-groups-reasoning. I did not intend to say that this is obvious, rather that there is a conce... |
2,898,125 | <blockquote>
<p>I have the following initial value problem:
$$\tag{IVP}\label{IVP}
\begin{cases}
x^\prime(t) = -\sqrt[3]{x}\\
x(0) = x_0
\end{cases}
$$
and I have to show that for every $x_0 \in \mathbb R$ a solution to \eqref{IVP} exists and it is unique. </p>
</blockquote>
<p>Existence is not an issue, as the... | Rigel | 11,776 | <p>For $x_0 = 0$ there is multiplicity of solutions.
Indeed, one solution is $x(t) \equiv 0$, and another one is
$$
y(t) :=
\begin{cases}
(-\frac{2}{3} t)^{3/2}, & \text{if}\ t < 0,\\
0, & \text{if}\ t \geq 0.
\end{cases}
$$</p>
|
1,303,868 | <p>Theorem:
$$\left\langle a^k \right\rangle = \left\langle a^{\gcd(n,k)}\right\rangle$$
Let G be a group and $$ a \in G$$ such that $$|a|=n$$
Then:</p>
<p>$$\left\langle a^k \right\rangle = \left\langle a^{\gcd(n,k)}\right\rangle$$</p>
<p>The proof begins by letting d = gcd(n,k) such that d is a divisor of k so ther... | absolute friend | 244,073 | <p>for the converse part</p>
<p>since $d=gcd(n,k)$ and therefore from $euclidian~ algorithem$ there exists </p>
<p>integers $s~\&~t$ such that</p>
<p>$ns+kt=d$ </p>
<p>This will imply that $a^d=a^{ns+kt}$</p>
<p>$\implies a^d=(a^n)^s (a^k)^t$</p>
<p>$\implies a^d=(a^k)^t$ since $|a|=n$</p>
<p>$\implies a^d ... |
11,294 | <p>Let $f(x)= \displaystyle \sum \limits_{n=1}^\infty \frac{\sin(nx)}{n^3}.$ Show that $f(x)$ is differentiable and that the derivative $f'(x)$ is continuous.</p>
<p>In class we solved a similar problem, and I think we had to show that both $f(x)$ and $f'(x)$ converge uniformly, but I am not really sure <em>wh... | Community | -1 | <p>Well, to see that $f(x)$ *converges uniformly* please apply the <em><a href="http://en.wikipedia.org/wiki/Weierstrass_M-test" rel="nofollow">Weierstrass M-Test</a></em>.</p>
<p>Now, i am aware of the fact that term by term differentiation and integration is possible if the series is uniformly convergent. But i don'... |
1,510,626 | <p><a href="http://i949.photobucket.com/albums/ad332/Fractur65/DSCN0311.jpg" rel="nofollow">This is my work on the problem,</a> not sure if I did this wrong or I'm missing some way to simplify this and continue from here.</p>
| Don | 283,786 | <p>Let $u=6x^2-8x+3$. Then $du=(12x-8)dx$=$4(3x-2)dx$. Then $\frac{du}{4}$=$(3x-2)dx$ So substituting $u$ and $du$ you get$\int \frac{u^3}{4}\,du$=$\frac{u^4}{16}$+$C$=$\frac{(6x^2-8x+3)^4}{16}$+$C$ Just wrote it all. No credit, as the answer was given above.</p>
|
1,216,227 | <p>I must compute $\int_{C}\frac{1}{z}dz$ where he branch of $\sqrt{z}$ satisfies $\sqrt{1}=1$.</p>
<p>Where $C$ is a positively oritented semi-cericle $|z|=1$, $0 \leq Arg(z) \leq \pi$.</p>
<p>I am confused on how to satisfy the branch requirement of $\sqrt{z}$. Usually, without branch requirements, I parametrize t... | Ron Gordon | 53,268 | <p>By "branch", we make a choice as to whether by "1", we mean $e^{i 0}$ or $e^{i 2 \pi}$, for example. In this case, $\sqrt{e^{i 0}} = e^{i 0} = 1$ while $\sqrt{e^{i 2 \pi}} = e^{i \pi} = -1$. So, by stating that $\sqrt{1}=1$, the problem means that $1=e^{i 0}$. </p>
<p>Thus, being on a single branch, we may then ... |
1,216,227 | <p>I must compute $\int_{C}\frac{1}{z}dz$ where he branch of $\sqrt{z}$ satisfies $\sqrt{1}=1$.</p>
<p>Where $C$ is a positively oritented semi-cericle $|z|=1$, $0 \leq Arg(z) \leq \pi$.</p>
<p>I am confused on how to satisfy the branch requirement of $\sqrt{z}$. Usually, without branch requirements, I parametrize t... | Xuqiang QIN | 137,389 | <p>A branch of $\sqrt z$ is defined whenever you take a slit off the complex plane. In your case, you can consider taking away the slit corresponding to $arg(z)=3\pi/2$. Then your function is defined on a simply connected domain and you can applies the fundamental theorem of calculus.
\begin{equation}
\int_C \frac{1}{... |
1,198,627 | <p>I have the following approximation: </p>
<p>$$f(x) \simeq f(a) + f^{'}(a)(x - a)$$</p>
<p>Letting $a = \mu_{x}$, the mean of $X$, a Taylor seties expansion of $y=f(x)$ about $\mu_{x}$ gives the approximation:
$$y=f(x) \simeq f(\mu_{x}) +f^{'}(\mu_{x})(x - \mu_{x})$$
Taking the variance of both sides yields:</p>
... | Cain | 28,606 | <p>Let X be a random variable with variance $Var(X)$ and let $a,b$ be two scalers, it then holds that $Var(aX + b) = a^{2}Var(X)$.</p>
<p>Apply this to $f(\mu_{x}) + f'(\mu_{x})(x-\mu_{x})$ to get the desired result.</p>
<p>An important thing to note is that $f(\mu_{x}),f'(\mu_{x}),\mu_{x} $ are all constants.</p>
|
1,640,740 | <p>I have the following topology over $\mathbb R$
$$ T = \{\emptyset\} \cup \{G\subseteq \mathbb R: \mathbb Q \setminus G \text{ is finite}\} $$
How could I study the closure of $\mathbb Q$ and $\mathbb R\setminus \mathbb Q$? Thanks in advance</p>
| Vincenzo Zaccaro | 269,380 | <p>Note that if $x\in \mathbb { R } $ everey open set $ A$ that contains x contains also infinite points of $\mathbb{Q}$ . The set of irrational numbers is closed, all rational points have some open sets in $\mathbb { Q}.$</p>
|
1,158,292 | <p>I'm totally lost. I've been trying to figure this out.
This is what I've figured out:</p>
<p>$dy/dx = 1/x$</p>
<p>$y$-intercept $= 1$</p>
<p>So I try to do $y-y_1 = m(x-x_1)+b,$ which I get as $y-1 = 1/x(x-0)+1,$ simplified to $y = 3.$</p>
<p>But I feel like that is totally wrong and well, obviously it isn't eve... | user84413 | 84,413 | <p>Let $(x,\ln x)$ be the point of tangency. </p>
<p>Then the slope of the tangent line is given by $m=\frac{1}{x}$, as you have, and the slope is also given by </p>
<p>$\displaystyle m=\frac{\ln x-1}{x-0}=\frac{\ln x-1}{x}$.</p>
<p>Setting these two expressions equal gives $\ln x-1=1$, so $\ln x=2$, $x=e^2$, and s... |
1,320,727 | <p>Construct an example in which a field $F$ is of degree 2 over two distinct subfields $A$ and $B$, but so that $F$ is not algebraic over $A\cap B$. </p>
<p>I'm having trouble thinking of an explicit example here.</p>
| Zev Chonoles | 264 | <p>Let $F=\mathbb{Q}(\pi,i)$, with $A=\mathbb{Q}(\pi)$ and $B=\mathbb{Q}(i\pi)$. Then $F$ is not algebraic over $A\cap B=\mathbb{Q}$.</p>
|
2,427,747 | <p>I want to prove that $2^{n+2} +3^{2n+1}$ is divisible by $7$ for all $n \in \mathbb{N}$ using proof by induction.</p>
<p>Attempt</p>
<p>Prove true for $n = 1$</p>
<p>$2^{1+2} + 3^{2(1) +1} = 35$</p>
<p>35 is divisible by 7 so true for $n =1$</p>
<p><em>Induction step</em>: Assume true for $n = k$ and prove true... | Donald Splutterwit | 404,247 | <p>\begin{eqnarray*}
2^{k+3}+3^{2k+3} =(9-7)2^{k+2} +9 \times 3^{2k+1}= 9(\color{blue}{2^{k+2}+3^{2k+1}})-\color{blue}{7} \times2^{k+2}
\end{eqnarray*}</p>
|
2,427,747 | <p>I want to prove that $2^{n+2} +3^{2n+1}$ is divisible by $7$ for all $n \in \mathbb{N}$ using proof by induction.</p>
<p>Attempt</p>
<p>Prove true for $n = 1$</p>
<p>$2^{1+2} + 3^{2(1) +1} = 35$</p>
<p>35 is divisible by 7 so true for $n =1$</p>
<p><em>Induction step</em>: Assume true for $n = k$ and prove true... | Piquito | 219,998 | <p>HINT.-$x+y\equiv0\pmod7$ this implies $2x+9y\equiv2x+2y\equiv2(x+y)\equiv0\pmod7$</p>
|
3,382,305 | <p>I know the two definitions for continuity, (sequential and epsilon-delta)</p>
<p>Given <span class="math-container">$x_0 \in D, \forall \epsilon > 0, \exists \delta > 0, |x - x_0| < \delta \rightarrow |f(x) - f(x_0)| < \epsilon $</span> </p>
<p>and</p>
<p>f is continuous if <span class="math-container... | azif00 | 680,927 | <p><span class="math-container">$\displaystyle \lim_{x \to x_0} f(x) = L$</span> implies <span class="math-container">$f$</span> is continuous at <span class="math-container">$x_0$</span> <strong>if and only if</strong> <span class="math-container">$L=f(x_0)$</span>.</p>
|
1,104,728 | <p>I want to determine the splitting field, galois group and intermediate fields of the polynomial $f(X)=(X^2+12)(X^3-5)\in\mathbb Q[X]$.</p>
<p>I want to obtain the splitting field by adjoining the roots of the polynomial to $\mathbb Q$ which is $\sqrt[3]{5}$ for $(X^3-5)$ but I don't understand how to determine the ... | mollyerin | 29,809 | <p>$\mathbb{R}^{n+1} \setminus \lbrace 0 \rbrace$ is homotopy equivalent to $\mathbb{S}^n$, and so in particular the $(n-1)$-st cohomology group
$$
H^{n-1}(\mathbb{R}^{n+1} \setminus \lbrace 0 \rbrace) = H^{n-1}(\mathbb{S}^n)
$$
vanishes. In particular, your closed form $\omega$ is exact; that is, there is an $(n-2... |
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