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 |
|---|---|---|---|---|
359,212 | <p>I mean, $\Bbb Z_p$ is an instance of $\Bbb F_p$, I wonder if there are other ways to construct a field with characteristic $p$?
Thanks a lot!</p>
| Vicfred | 85,162 | <p>There are two nice constructions I know, the first one is already mentioned. For every prime $p$ you can 'extend' it and construct fields with $p^n$ elements. (this depends on the fact that there are irreducible polynomials of every degree in $\mathbb{F}_p$, exercise: give some examples to show why this might be tru... |
2,663,537 | <p>Suppose G is a group with x and y as elements. Show that $(xy)^2 = x^2 y^2$ if and only if x and y commute.</p>
<p>My very basic thought is that we expand such that $xxyy = xxyy$, then multiply each side by $x^{-1}$ and $y^{-1}$, such that $x^{-1} y^{-1} xxyy = xxyy x^{-1}$ , and therefore $xy=xy$.</p>
<p>I realiz... | Xander Henderson | 468,350 | <p>The proof can be written fairly concisely as follows:</p>
<p>\begin{align}
(xy)^2 = x^2 y^2
&\iff (xy)(xy) = (xx)(yy) && (\text{expand both sides}) \\
&\iff xyxy = xxyy && (\text{association}) \\
&\iff x^{-1} (xyxy) y^{-1} = x^{-1} (xxyy) y^{-1} &&(\text{cancelation}) \\
&... |
610,672 | <p>Could anyone help me with homework or give me a hint? Any help would be highly appreciated.</p>
<p>Given a set of N distinct objects:</p>
<p>How many ways are there to pick any number of them to be in a pile while the rest are in anotherpile? If your answer is written in terms of binomial coecients, use the Binom... | gt6989b | 16,192 | <p><strong>Hint</strong></p>
<ol>
<li>There are the following choices for the number of objects in the first pile: 0, 1, ..., N</li>
<li>If the first pile has $k$, how much does the second pile have?</li>
<li>To reach such an arrangement, pick $k$ elements for the first pile from the entire group. Is order important? ... |
610,672 | <p>Could anyone help me with homework or give me a hint? Any help would be highly appreciated.</p>
<p>Given a set of N distinct objects:</p>
<p>How many ways are there to pick any number of them to be in a pile while the rest are in anotherpile? If your answer is written in terms of binomial coecients, use the Binom... | robjohn | 13,854 | <p><strong>Hint 1:</strong> There are $\binom{n}{k}$ ways to put $k$ items into the first pile and $n-k$ items into the second pile. Consider what this means for all $k$.</p>
<p><strong>Hint 2:</strong> Each item can either be in the first pile or the second pile ($2$ options for each item).</p>
|
87,963 | <p>Assume that $L/K$ is an extension of fields and $[L:K]=n$, with $n$ composite. Assume that $p\mid n$, can we always produce a subextension of degree $p$ and if not under what conditions can it be done? I would guess this is very false, but I couldn't come up with any trivial counterexamples.</p>
| Soka | 20,290 | <p>Suppose $L/K$ is Galois with group $G$. Then subextensions of degree $m$ over $K$ correspond to normal subgroups of $G$ of index $m$. Given an abelian group you can always find subgroups of order $p$, where $p$ is a prime divisor of $\vert G \vert =n$. So to find a counterexample, you should (with $L/K$ Galois) star... |
4,558,460 | <p>According to the implicit function theorem(on <span class="math-container">$\mathbb R^2$</span> for simplicity), if <span class="math-container">$\displaystyle\frac{\partial f}{\partial y}\ne 0$</span> at <span class="math-container">$(x_0, y_0)$</span>, then on a neighborhood of <span class="math-container">$(x_0, ... | Hans Lundmark | 1,242 | <p>Take <span class="math-container">$f(x,y)=(y-x^2)^2$</span> and any point <span class="math-container">$(x_0,y_0)$</span> on the curve <span class="math-container">$y=x^2$</span>. Then <span class="math-container">$\nabla f(x_0,y_0) = (0,0)$</span>, so the implicit function theorem doesn't apply, but still the equat... |
839,124 | <p>This is similar to an exercise I just posted. The necessary part is easy, but the sufficient condition I'm having trouble seeing.</p>
<p>$\Rightarrow$. Since $(x,y)=g,$ there exist integers $x_1, y_1$ such that $x=gx_1, y=gy_1$. Since $[x,y]=l$, there exist integers $x_2, y_2$ such that $l=xx_2=yy_2$. Then
$$l=... | André Nicolas | 6,312 | <p>Let $x=g$ and let $y=l$. (We do need to assume that $l\gt 0$ and $g\ge 0$.) </p>
|
27,089 | <p>Off hand, does anyone know of some useful conditions for checking if a ring (or more generally a semiring) has non-trivial derivations? (By non-trivial, I mean they do not squish everything down to the additive identity.) Part of the motivation for this is that I was thinking about it the other day, and had troubl... | Robin Chapman | 4,213 | <p>There is a notion of a universal derivation for an algebra. I'll assume
everything is commutative for simiplcitity. If $A$ is a $k$-algebra
($k$ a commutative ring) then there is an $A$-module $\Omega_{A/k}$,
the module of <em>Kahler differentials</em> of $A$ over $k$ and a $k$-derivation
$d:A\to\Omega_{A/k}$ which ... |
27,089 | <p>Off hand, does anyone know of some useful conditions for checking if a ring (or more generally a semiring) has non-trivial derivations? (By non-trivial, I mean they do not squish everything down to the additive identity.) Part of the motivation for this is that I was thinking about it the other day, and had troubl... | Steve Huntsman | 1,847 | <p>The Banach algebra of bounded functions on a finite set turns out to be semisimple, and therefore carries no nonzero derivations by the results of <a href="http://www.jstor.org/pss/2373262" rel="nofollow">Johnson, B. E. "Continuity of derivations on commutative algebras". <em>Amer. J. Math.</em> <strong>91</strong>, ... |
44,868 | <p><strong>Bug introduced in version 8 or earlier and fixed in 10.0</strong></p>
<hr>
<p>I have created a notebook with two cells. This is the content of the first:</p>
<pre><code>g = Graph[{1 \[UndirectedEdge] 2, 2 \[UndirectedEdge] 3, 1 \[UndirectedEdge] 3, 1 \[UndirectedEdge] 4, 4 \[UndirectedEdge] 5, 4 \[Undirec... | Jacob Akkerboom | 4,330 | <p>This looks like a bug to me. Here is a slightly more minimal example.</p>
<pre><code>ue = UndirectedEdge;
g = Graph[ue @@@ {{1, 2}, {2, 3}, {1, 3}, {1, 4}, {4, 5}, {4, 6}}];
vl = VertexList[g]
aM = AdjacencyMatrix[g];
vLM = aM[[VertexIndex[g, 1]]];
Pick[vl, vLM, 0];
VertexList[g]
</code></pre>
<p>Output</p>
<bloc... |
897,633 | <p><strong>First question:</strong></p>
<p>Let's say we have a hypothesis test:</p>
<p>${ H }_{ 0 }:u=100$
and ${ H }_{ 1 }:u\neq 100$.</p>
<p>The sample has a size of 10 and gives an average $u=103$ and a p-value = 0.08.
The level of significance is 0.05.</p>
<p>I'm asked the following question (exam):</p>
<p>A) ... | Satish Ramanathan | 99,745 | <p>A P-value can be reported more formally in terms of a fixed level α test. Here α is a
number selected independently of the data, usually 0.05 or 0.01, more rarely 0.10. We
reject the null hypothesis at level α if the P-value is smaller than α , otherwise we fail to
reject the null hypothesis at level α.</p>
<p>Now ... |
2,290,395 | <p>What if in Graham’s Number every “3” was replaced by “tree(3)” instead? How big is this number? Greater than Rayo’s number? Greater than every current named number?</p>
| mgiroux | 586,861 | <p>The TREE function grows much much faster than any construction of knuth up arrows. Because of this, inserting the TREE function into Grahams number would yield a number still very close to TREE(3). It would be like trying to create a number larger than a googolplex by adding a 1 on the end. You would be better off i... |
3,073,361 | <p>I think I understood 1-forms fairly well with the help of these two sources. They are dual to vectors, so they measure them which can be visualized with planes the vectors pierce.</p>
<ul>
<li><a href="https://the-eye.eu/public/WorldTracker.org/Physics/Misner%20-%20Gravitation%20%28Freeman%2C%201973%29.pdf" rel="nof... | Michael Paris | 458,204 | <blockquote>
<ol>
<li>How can I visualize the wedge between two 1-forms <span class="math-container">$α∧β$</span>?</li>
</ol>
</blockquote>
<p>First we need to understand what the wedge actually does. In your case it creates a new fully anti-symmetric tensor (think determinant) of order 2. A 2-form is a thing that take... |
1,154,592 | <p>I was doing some basic Number Theory problems and came across this problem :</p>
<blockquote>
<p>Show that if $a$ and $n$ are positive integers with $n\gt 1$ and $a^{n} - 1$ is prime, then $a = 2$ and $n$ is prime</p>
</blockquote>
<p><strong>My Solution : (Sloppy)</strong></p>
<blockquote>
<ul>
<li>$a^{n}-... | quid | 85,306 | <p>The proof is alright there are two or three details though (the same issue twice actually), oe was already pointed out in comments: </p>
<ul>
<li><p>Likely you should exclude the case $a=1$ right away. Just by saying $1^n -1 = 0$ is not prime so assume $a>1$.</p></li>
<li><p>You cannot derive from $a^n -1$ being... |
1,652,165 | <p>On a empty shelf you have to arrange $3$ cans of soup, $4$ cans of beans, and $5$ cans of tomato sauce. What is the probability that none of the cans of soup are next to each other?</p>
<p>I tried working this out but get very stuck because I'm not sure that I'm including all the possible outcomes. </p>
| André Nicolas | 6,312 | <p>We have $12$ cans, soup (S) and other (O). An arrangement is a $12$-letter word in the alphabet S, O. We assume all arrangements are equally likely. There are $\binom{12}{3}$ of them. </p>
<p>Now we count the arrangements in which the soup cans are separated, the "favourables. Here we use a little trick. Line up th... |
177,209 | <p>I found the following problem while working through Richard Stanley's <a href="http://www-math.mit.edu/~rstan/bij.pdf">Bijective Proof Problems</a> (Page 5, Problem 16). It asks for a combinatorial proof of the following:
$$ \sum_{i+j+k=n} \binom{i+j}{i}\binom{j+k}{j}\binom{k+i}{k} = \sum_{r=0}^{n} \binom{2r}{r}$$
w... | Will Orrick | 3,736 | <p><strong>Short summary:</strong> On the right we are summing the number of words of $r$ $a$s and $r$ $b$s over $0\le r\le n.$ Denote the set of words with $r$ $a$s and $r$ $b$s by $U_r.$ On the left we are computing the number of triples of words, the first with $i$ $a$s and $j$ $b$s, the second with $j$ $a$s and $k... |
4,542,985 | <p>I want to fully understand the probabilistic interpretation. As in, I know once we have a probabilistic model, we differentiate for maximum likelihood and find the weights/regressors but what i really find difficult to grasp is how exactly are we developing a probabilistic model for linear regression.
I have see th... | Matija | 1,096,797 | <p>The statement is true if a <em>random variable</em> is a measurable function <span class="math-container">$V:\Omega\rightarrow E$</span> to <span class="math-container">$E\subseteq\mathbb R$</span> equipped with the canonical <span class="math-container">$\sigma$</span>-algebra (the Borel algebra induced by one of t... |
8 | <p>Contexts have backticks, which conflict with the normal way to enter inline code. How do I enter an inline context, since the initial approach:</p>
<pre><code>`System``
</code></pre>
<p>doesn't work ( `System`` ).</p>
| Verbeia | 8 | <p>You can also use the HTML markup <code><code>...</code></code>. This has the advantage that you can bold and italicise inside it, like so:</p>
<pre><code><code>f[x_*Pattern*]:= 50.`**watch out**</code>
</code></pre>
<p>Results in</p>
<p><code>f[x_<em>Pattern</em>]:= 50.` <strong>watch out<... |
2,972,950 | <p>Everything on this question is in complex plane.</p>
<p>As the book describes a property of a winding number, it says that:</p>
<blockquote>
<p>Outside of the [line segment from <span class="math-container">$a$</span> to <span class="math-container">$b$</span>] the function <span class="math-container">$(z-a) / ... | Eric Wofsey | 86,856 | <p>Note that <span class="math-container">$\frac{z-a}{z-b}$</span> is unchanged if we add the same number to each of <span class="math-container">$z$</span>, <span class="math-container">$a$</span>, and <span class="math-container">$b$</span>. So, we may translate all of our points by <span class="math-container">$-a$... |
25,488 | <p>I have noticed a common pattern followed by many students in crisis:</p>
<ul>
<li>They experience a crisis or setback (injury, illness, tragedy, etc)</li>
<li>This causes them to miss a lot of class.</li>
<li>They may stay away from class longer than they "need to" because of shame: they feel that since t... | TomKern | 15,671 | <p>Set realistic intermediate goals for students while they're trying to catch up. By default their goal might be along the lines of completely catching up within a week. That's unrealistic, and if they fail at it, they'll be demotivated and can easily get stuck in a failure/demotivation feedback spiral.</p>
<p>Setting... |
3,860,330 | <p>I am interested in proving what family of functions have the property
<span class="math-container">$$f'(x)=f^{-1}(x)$$</span>
I've never dealt with a differential equation of this form, hence I could only go as far as to gather a little data:</p>
<p><span class="math-container">$$f'(x)=f^{-1}(x)\implies f(f'(x))=x$$... | Jan Eerland | 226,665 | <p>Well, we know that:</p>
<p><span class="math-container">$$\text{y}\left(\text{y}^{-1}\left(x\right)\right)=x\tag1$$</span></p>
<p>Where <span class="math-container">$\text{y}^{-1}\left(x\right)$</span> is the inverse of the function <span class="math-container">$\text{y}\left(x\right)$</span>.</p>
<p>So, using your ... |
10,977 | <p>When I taught calculus, I posted my notes after the lecture. Then I had the students fill out a mid-quarter evaluation, and a lot of them wanted me to post my notes before class.</p>
<p>What I started doing was printing and handing out the notes to them, leaving the examples blank so they can fill those in. Many ... | Jessica B | 4,746 | <p>I think what works depends on the local culture to a fair extent, ie what the expectations of the students are. </p>
<p>As a student myself, I almost always had to copy down everything in lectures, and no-one seemed to have a problem with that. One lecturer gave us gappy notes, which we hated, because it made the e... |
10,977 | <p>When I taught calculus, I posted my notes after the lecture. Then I had the students fill out a mid-quarter evaluation, and a lot of them wanted me to post my notes before class.</p>
<p>What I started doing was printing and handing out the notes to them, leaving the examples blank so they can fill those in. Many ... | user14622 | 14,622 | <p>In my experience, guided notes seem to be correlated to poor learning. While preference for guided notes tends to split about 50/50, invariably the bottom half in terms of academic performance is dominated by people who use guided notes. Now is this causal? I'm not sure; it seems reasonable that "poor" stu... |
3,509,441 | <p>Given a Complex Matrix <span class="math-container">$A$</span> which is <span class="math-container">$n \times n$</span>. How would I go about showing that <span class="math-container">$A^*A$</span> is <span class="math-container">$$\sum_{i=1}^n \sum_{j = 1}^n | a_{ij} |^2$$</span></p>
<p>Here <span class="math-con... | lulu | 252,071 | <p>Think of there being <span class="math-container">$3$</span> sets: Set <span class="math-container">$A$</span>, set <span class="math-container">$B$</span>, and set <span class="math-container">$C$</span>. for each element <span class="math-container">$s\in \{1, \cdots, n\}$</span> you have three choices as to whe... |
1,315,805 | <blockquote>
<p>Let the series $$\sum_{n=1}^\infty \frac{2^n \sin^n x}{n^2}$$. For $x\in (-\pi/2, \pi/2)$, when is the series converges?</p>
</blockquote>
<p>By the root-test:</p>
<p>$$\sqrt[n]{a_n} = \sqrt[n]{\frac{2^n\sin^n x}{n^2}} = \frac{2\sin x}{n^{2/n}} \to 2\sin x$$</p>
<p>Thus, the series converges $\iff ... | Barry | 90,638 | <p>I like the ratio test here:</p>
<p>$$\begin{split}
L &= \lim_{n\to\infty} \left|\frac{2^{n+1} \sin^{n+1} x}{(n+1)^2} \frac{n^2}{2^n\sin^n{x}} \right| \\
&=\lim_{n\to\infty} \left|\frac{2n^2\sin{x}}{(n+1)^2}\right| \\
&=\lim_{n\to\infty} \left|\frac{2\sin{x}\cdot n^2}{n^2}\right| \\
&= |2\sin{x}|
\en... |
754,583 | <p>Write <span class="math-container">$$\phi_n\stackrel{(1)}{=}n+\cfrac{n}{n+\cfrac{n}{\ddots}}$$</span> so that <span class="math-container">$\phi_n=n+\frac{n}{\phi_n},$</span> which gives <span class="math-container">$\phi_n=\frac{n\pm\sqrt{n^2+4n}}{2}.$</span> We know <span class="math-container">$\phi_1=\phi$</span... | achille hui | 59,379 | <p>For any given $n \ne 0, -4$, let $( a_{n,m} )_{m\in\mathbb{Z}_{+}}$ be the sequence
defined by
$$a_{n,m} = \begin{cases}n,& m = 1\\\displaystyle n + \frac{n}{a_{n,m-1}},&m > 1\end{cases}$$
Let $\displaystyle\;\mu_n = \frac{n+\sqrt{n(n+4)}}{2}\;$ and
$\displaystyle\;\nu_n = \frac{n-\sqrt{n(n+4)}}{2}\;$. It... |
39,762 | <p>Happy new year mathematica gurus of stack exchange!</p>
<p>As I see it one of the major obstacles in getting decent at programming mathematica is that, not only do you need to learn how certain commands work, but rather that you mainly need to understand how to write your syntax. This is a typical such situation, I... | Mr.Wizard | 121 | <p>To begin with you should try to avoid explicit <code>For</code> loops in most cases. See <a href="https://mathematica.stackexchange.com/q/7924/121">Alternatives to procedural loops and iterating over lists in Mathematica</a>. You will often also benefit from a functional rather than mutable style. Consider for exa... |
2,977,645 | <p>I'm trying to prove that <span class="math-container">$\sqrt[n]{\frac{s}{t}}$</span> is irrational unless both s and t are perfect nth powers. I have found plenty of proofs for nth root of an integer but cannot find anything for rationals. Also trying to work up from the proofs I have found is rather difficult.</p>
... | Arthur Hertz | 610,350 | <p>To build off Vasya and Calum's helpful comments:</p>
<p>Let <span class="math-container">$\frac{s}{t}$</span> be an irreducible fraction (to address Vasya's comment).</p>
<p>Continuing with the main proof, recall Vasya's comment:</p>
<blockquote>
<p><span class="math-container">$\sqrt[n]{\frac{s}{t}} = \frac{\s... |
997,463 | <p>For example, a complex number like $z=1$ can be written as $z=1+0i=|z|e^{i Arg z}=1e^{0i} = e^{i(0+2\pi k)}$.</p>
<p>$f(z) = \cos z$ has period $2\pi$ and $\cosh z$ has period $2\pi i$.</p>
<p>Given a complex function, how can we tell if it is periodic or not, and further, how would we calculate the period? For ex... | Aaron Maroja | 143,413 | <p><strong>Hint:</strong>
Make the substitution $x = \frac{3}{2}\tan\theta \Rightarrow dx = \frac{3}{2} \sec^2 \theta \ \ d\theta$.</p>
|
1,523,392 | <p>This is question 2.4 in Hartshorne. Let $A$ be a ring and $(X,\mathcal{O}_X)$ a scheme. We have the associated map of sheaves $f^\#: \mathcal{O}_{\text{Spec } A} \rightarrow f_* \mathcal{O}_X$. Taking global sections we obtain a homomorphism $A \rightarrow \Gamma(X,\mathcal{O}_X)$. Thus there is a natural map $\alp... | Babai | 36,789 | <p>Let <span class="math-container">$g\in\hom_{ring}(A,\Gamma(X,\mathcal{O}_X)$</span></p>
<p>Cover <span class="math-container">$X$</span> by affine open subsets <span class="math-container">$\{U_i=Spec(A_i)\}_{i\in I}$</span>.</p>
<p>Now, the inclusion <span class="math-container">$U_i\hookrightarrow X$</span> gives ... |
2,416,071 | <p>I have this integral: $\displaystyle \int^{\infty}_0 kx e^{-kx} dx$.</p>
<p>I tried integrating it by parts:</p>
<p>$\dfrac{1}{k}\displaystyle \int^{\infty}_0 kx e^{-kx} dx = ... $. But I'm stuck </p>
<p>now. Can you help me please?</p>
| Tony Delgado | 477,171 | <p>Just take $kx = y$ so you will have $$\int_{0}^\infty\frac{y}{e^y}dy$$</p>
<p>Take $u = y$ and $dv = \frac{1}{e^y}$, after integrate just remeber change $y = kx$ </p>
|
1,558,256 | <p>The standard Normal distribution probability density function is $$p(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2},\int_{-\infty}^{\infty}p(t)\,dt = 1$$ i.e., mean 0 and variance 1. The cumulative distribution function is given by the improper integral $$P(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^2/2}\,dt$$ Describe... | njuffa | 114,200 | <p>Before embarking on crafting a custom implementation, it seems advisable to check whether the CDF of the standard normal distribution is supported as a built-in function in the programming environment of your choice. For example, <a href="http://www.mathworks.com/help/stats/normcdf.html" rel="nofollow">MATLAB</a> of... |
5,528 | <p>Let H be a subgroup of G. (We can assume G finite if it helps.) A complement of H in G is a subgroup K of G such that HK = G and |H∩K|=1. Equivalently, a complement is a transversal of H (a set containing one representative from each coset of H) that happens to be a group.</p>
<p>Contrary to my initial naive... | Greg Kuperberg | 1,450 | <p>If $H$ is normal, then there is a well-known textbook answer. $H$ has a complement if and only if $G$ is a semidirect product of $H$ and its complement. The complement is isomorphic to the quotient $G/H$, and if you don't know whether there is one, you can do an extension calculation to see if it exists. For insta... |
5,528 | <p>Let H be a subgroup of G. (We can assume G finite if it helps.) A complement of H in G is a subgroup K of G such that HK = G and |H∩K|=1. Equivalently, a complement is a transversal of H (a set containing one representative from each coset of H) that happens to be a group.</p>
<p>Contrary to my initial naive... | Theo Johnson-Freyd | 78 | <p>This doesn't quite answer your question, but rather answers the question in the opposite direction. Thus, while I can't say precisely what subgroups have complements, I can give conditions that the complements must satisfy.</p>
<p>To present a group as $G = KH$ with $H \cap K = 1$ is equivalent to the following:</... |
5,528 | <p>Let H be a subgroup of G. (We can assume G finite if it helps.) A complement of H in G is a subgroup K of G such that HK = G and |H∩K|=1. Equivalently, a complement is a transversal of H (a set containing one representative from each coset of H) that happens to be a group.</p>
<p>Contrary to my initial naive... | Derek Holt | 35,840 | <p>Let me mention one other condition. Suppose the finite group $G$ has an abelian Sylow $p$-subgroup $P$ for some prime $p$. Then Burnside's Transfer Theorem says that $P$ has a normal complement in $G$ if and only if $P$ is in the centre of its normalizer in $G$; i.e. $P \le Z(N_G(P))$.</p>
<p>This often provides t... |
3,165,460 | <p>I am reading a survey on Frankl's Conjecture. It is stated without commentary that the set of complements of a union-closed family is intersection-closed. I need some clearer indication of why this is true, though I guess it is supposed to be obvious. </p>
| Henno Brandsma | 4,280 | <p>The argument in a comment is worth restating: if <span class="math-container">$X$</span> is a set without isolated points (a crowded space) that is <span class="math-container">$T_1$</span> then no scattered subset (countable or not) can be dense in <span class="math-container">$X$</span>:</p>
<p>Let <span class="m... |
2,118,931 | <p>If $A$ is a $4\times2$ matrix and $B$ is a $2\times 3$ matrix, what are the possible values of $\operatorname*{rank}(AB)$?</p>
<p>Construct examples of $A$ and $B$ exhibiting each possible value of $\operatorname*{rank}(AB)$ and explain your reasoning.</p>
| dz_killer | 410,800 | <p>rank = number of pivots in reduced form</p>
<p>Okay, so in terms of pivots per column, matrix A can have at most 2 pivots and matrix B also 2 pivots at most. AB which is 4*3 can have 3 pivots at most so I don't really understand why the formula above is true.</p>
|
636,089 | <p>Let the function
$$
f(x) = \begin{cases} ax^2 & \text{for } x\in [ 0, 1], \\0 & \text{for } x\notin [0,1].\end{cases}
$$ Find $a$, such that the function can describe a probability density function. Calculate the expected value, standard deviation and CDF of a random variable X of such distribution.</p>
<p... | Sergio Parreiras | 33,890 | <p>First notice the PDF is $3x^2$ <strong>only in the interval [0,1]</strong> it is zero outside.
To get the CDF you just have to use $F(x)=\int_{-\infty}^{x}f(z)dz$ and for the expected value: $E[X]=\int_{-\infty}^{+\infty}z\cdot f(z)dz$.</p>
|
13,835 | <p>Given that $$X = \{(x,y,z) \in \mathbb{R}^3 |\, x^2 + y^2 + z^2 - 2(xy + xz + yz) = k\}\,,$$ where $k$ is a constant. Also given that a group $G$ is represented by $$\langle g_1,g_2,g_3|\, g_1^2 = g_2^2 = g_3^2 = 1_G\rangle\,.$$ $G$ acts on $X$ such that $$g_1 \cdot (x,y,z) = (2(y+z) - x,y,z)\,,$$ $$g_2 \cdot (x,y,z... | Markeur | 4,448 | <p>This answer is actually to reply to whuber, who offered the solution. I'd like to reply in the comment section, however, as this reply is going to be very long and it might take more than 5 comments, I think it'd be better to reply here instead. I'd like to apologise for violating the rules.</p>
<p>Anyway, I've tri... |
3,285,036 | <p>Obviously this cannot happen in a right rectangle, but otherwise - as Sin(0) or 180 or 360 equals 0, I guess there is no way to find out what the original angle was?</p>
| Paras Khosla | 478,779 | <p>The general solution of the equation is as follows. Note that <span class="math-container">$\pi \text{ rad}=180^{\circ}$</span>. <span class="math-container">$$\sin \theta=0\implies \theta=n\pi, n\in \mathbb{Z}=\{..., -\pi, 0,\pi,2\pi,...\}\equiv \theta=n\cdot180^{\circ}, n\in\mathbb{Z}$$</span></p>
|
3,285,036 | <p>Obviously this cannot happen in a right rectangle, but otherwise - as Sin(0) or 180 or 360 equals 0, I guess there is no way to find out what the original angle was?</p>
| Community | -1 | <p><strong>Caution:</strong></p>
<p>The equation </p>
<p><span class="math-container">$$\sin(x)=0$$</span></p>
<p>has infinitely many solutions, each an integer multiple of a half turn.</p>
<p>But</p>
<p><span class="math-container">$$\arcsin(0)=0$$</span></p>
<p>and nothing else, because the arc sine is defined ... |
2,593,627 | <p>I struggle to find the language to express what I am trying to do. So I made a diagram.</p>
<p><a href="https://i.stack.imgur.com/faHgE.png" rel="noreferrer"><img src="https://i.stack.imgur.com/faHgE.png" alt="Graph3parallelLines"></a></p>
<p>So my original line is the red line. From (2.5,2.5) to (7.5,7.5).</p>
<... | user | 505,767 | <p>The line equation between $(2.5,2.5)$ to $(7.5,7.5)$ is:</p>
<p>$$y=x$$</p>
<p>For a <strong>vertical shifting</strong> of $\pm1$, the equation for blu line and yellow line are</p>
<p>$$y=x+1$$</p>
<p>$$y=x-1$$</p>
<p>For a <strong>parallel shifting</strong> of $\pm1$, the equation for blu line and yellow line ... |
257,567 | <p>Assuming <a href="http://www.springer.com/gp/book/9783642649059" rel="nofollow noreferrer">Bishop's</a> constructive mathematics, is it true that any real-valued square matrix with <strong>distinct</strong> roots of the characteristic polynomial can be diagonalized? By distinct, I mean <strong>apart</strong>: $x \ne... | Denis Serre | 8,799 | <p><strong>Edit</strong>. I think now that your question concerns Gårding's theory of <em>hyperbolic polynomials</em>. </p>
<p>A homogeneous polynomial of degree $d$ in $N$ real variables is hyperbolic in the direction $\bf e$ if for every vector $X$, the roots of the polynomial $t\mapsto p(X+t{\bf e})$ are real. We m... |
1,873,180 | <p>The final result should be $C(n) = \frac{1}{n+1}\binom{2n}{n}$, for reference.</p>
<p>I've worked my way down to this expression in my derivation:</p>
<p>$$C(n) = \frac{(1)(3)(5)(7)...(2n-1)}{(n+1)!} 2^n$$</p>
<p>And I can see that if I multiply the numerator by $2n!$ I can convert that product chain into $(2n)!$... | Michael Hardy | 11,667 | <p>Neither is strictly correct: the expression $\displaystyle\int f(x)\,dx$ is not really unambiguously defined as identifying some particular mathematical object except when the context makes it clear. When it has a precisely defined meaning, it's because of the context that it's precisely defined.</p>
<p>If you mea... |
630,838 | <p>I was woundering if anyone knows any good references about Kähler and complex manifolds? I'm studying supergravity theories and for the simpelest N=1 supergravity we'll get these. Now in the course-notes the're quite short about these complex manifolds. I was hoping someone of you guys might know a good (quite compl... | Wintermute | 67,388 | <p>Geometry, Topology, and Physics by Nakahara is a great reference with some good examples but it does not have a ton of Kahler material in it, still I have found it very helpful in my research. Something very complete is Lectures on Kahler Geometry by Moroianu. You can also find the Ballman lectures on Kahler geometr... |
1,970,458 | <p>Consider a stock that will pay out a dividends over the next 3 years of $1.15, $1.8, and 2.35 respectively. The price of the stock will be $48.42 at time 3. The interest rate is 9%. What is the current price of the stock?</p>
| Wolfy | 217,910 | <p>We have $D_1 = 1.15, D_2 = 1.8, D_3 = 2.35, P_3 = 48.42$ and $r = .09$. We know from the time value of money that
$$P_0 = \frac{D_1 + P_1}{1+r} = \frac{D_1}{1+r} + \frac{D_2 + P_2}{(1+r)^2} = \frac{D_1}{1+r} + \frac{D_2}{(1+r)^2} + \frac{D_3 + P_3}{(1+r)^3}$$ Now just plug in what we have and you are done :)</p>
|
14,847 | <p>I thought a simple Mathematica kerning machine (for adjusting the space between characters) would be interesting, but I'm having trouble with the locators. (There are a number of other questions related to this, and I've read the answers, but as yet without finding a solution, or understanding them that well.)</p>
... | jVincent | 1,194 | <p>First off, <code>Dynamic[exp]]</code> redraws whenever anything that appears in expression changes. Think of Manipulate as being just a <code>Dynamic[code]</code> with some nice shortcuts to build controllers that can change things that appear in <code>code</code>. In your case, you have a <code>Dynamic[LocatorPane[... |
14,847 | <p>I thought a simple Mathematica kerning machine (for adjusting the space between characters) would be interesting, but I'm having trouble with the locators. (There are a number of other questions related to this, and I've read the answers, but as yet without finding a solution, or understanding them that well.)</p>
... | Michael E2 | 4,999 | <p>I'm sure the OP has figured the problem out already, so I don't know why I'm bothering except that I think future users might appreciate a good <code>Manipulate</code>-based solution as an alternative, especially as an example of how to work with <code>Locators</code>.</p>
<p>A few changes: The most important thin... |
424,209 | <p>I am a Computer Science student. While going through some random maths topics I came across Chaos Theory. I wanted to know if there are any applications of it in CS.
I tried searching on the internet about this but ended up only with <a href="https://security.stackexchange.com/questions/31000/does-chaos-theory-have-... | nonlinearism | 59,567 | <p>I think the most fascinating account of chaos in computer science I have read comes when you analyze the attempted solutions to NP problems. </p>
<p>Here's a link that explains how the "hardness" of a Sudoku is related to chaotic behavior of solutions
<a href="http://www.nature.com/srep/2012/121011/srep00725/full/s... |
119 | <p>One thing I could never wrap my head around is how <code>Flatten</code> works when provided with a matrix as the second argument, and the <em>Mathematica</em> help isn't particularly good on this one.</p>
<p>Taken from the <a href="http://reference.wolfram.com/mathematica/ref/Flatten.html"><code>Flatten</code></a> ... | Leonid Shifrin | 81 | <p>One convenient way to think of <code>Flatten</code> with the second argument is that it performs something like <code>Transpose</code> for ragged (irregular) lists. Here is a simple example:</p>
<pre><code>In[63]:= Flatten[{{1,2,3},{4,5},{6,7},{8,9,10}},{{2},{1}}]
Out[63]= {{1,4,6,8},{2,5,7,9},{3,10}}
</code></pre... |
119 | <p>One thing I could never wrap my head around is how <code>Flatten</code> works when provided with a matrix as the second argument, and the <em>Mathematica</em> help isn't particularly good on this one.</p>
<p>Taken from the <a href="http://reference.wolfram.com/mathematica/ref/Flatten.html"><code>Flatten</code></a> ... | ConvexMartian | 10,492 | <p>I learned a lot from WReach's and Leonid's answers and I'd like to make a small contribution: </p>
<p>It seems worth emphasizing that the primary intention of the list-valued second argument of <code>Flatten</code> is merely to flatten certain levels of lists (as WReach mentions in his <strong>List Flattening</stro... |
119 | <p>One thing I could never wrap my head around is how <code>Flatten</code> works when provided with a matrix as the second argument, and the <em>Mathematica</em> help isn't particularly good on this one.</p>
<p>Taken from the <a href="http://reference.wolfram.com/mathematica/ref/Flatten.html"><code>Flatten</code></a> ... | Wjx | 6,084 | <p>This is a old question, but frequently asked by a <strong>lot</strong> of people. Today when I was trying to explain how this works, I came across a quite clear explanation, so I think sharing it here would be helpful for further audience.</p>
<h2>What do index means?</h2>
<p>First let's make clear what <strong>in... |
864,212 | <p>While trying to look up examples of PIDs that are not Euclidean domains, I found a statement (without reference) on the <a href="http://en.wikipedia.org/wiki/Euclidean_domain">Euclidean domain</a> page of Wikipedia that</p>
<p>$$\mathbb{R}[X,Y]/(X^2+Y^2+1)$$</p>
<p>is such a ring. After a good deal of searching, I... | user26857 | 121,097 | <blockquote>
<p>In P. Samuel, <em>Anneaux factoriels</em>, pages 36-37, it's proved that $A=\mathbb R[X,Y]/(X^2+Y^2+1)$ is a UFD. </p>
</blockquote>
<p>In the following we denote by $x,y$ the residue classes of $X,Y$ modulo $(X^2+Y^2+1)$. Thus $A=\mathbb R[x,y]$ with $x^2+y^2+1=0$.</p>
<blockquote>
<p><em>Lemma.<... |
3,520,354 | <p>In the problem <span class="math-container">$\frac{8.01-7.50}{3.002}$</span></p>
<p>Why would the answer be <span class="math-container">$0.17$</span> and not <span class="math-container">$0.170$</span>? My least amount of <em>sig figs</em> is <span class="math-container">$3$</span> in the original equation. The o... | Community | -1 | <p>It is often regarded as good practice to give <span class="math-container">$1$</span> fewer sig. figs. than in the given numbers. So you are right, the least number of sig figs in the original equation is <span class="math-container">$3$</span> therefore give <span class="math-container">$2$</span> in the final ans... |
3,197,046 | <p>I'm interested in <span class="math-container">${\bf integer}$</span> solutions of </p>
<p><span class="math-container">$$abcd+1=(ecd-c-d)(fab-a-b)$$</span></p>
<p>subject to <span class="math-container">${\bf a,b,c,d \geq 2}$</span>, and <span class="math-container">${\bf e,f \geq 1}$</span>. </p>
<p><span class... | Community | -1 | <p><span class="math-container">$1=(f-1)abcd-fabc-fabd-acd+ac+ad-bcd+bc+bd$</span></p>
<p>Only if an odd number of odd terms exists, can this work. That takes at least one, even variable (not just term). </p>
<p>mod 3, we have that when turned into addition of possible negative equivalents, the 2 mod 3's either need... |
226,097 | <p>I am having a problem with the following exercise. Can someone help me please.</p>
<p>Find all functions $f$ for which $f'(x)=f(x)+\int_{0}^1 f(t)dt$</p>
<p>Thank you in advance</p>
| Michael Hardy | 11,667 | <p>$$
\frac{df}{dx} = f + c
$$
$$
\frac{df}{f+c} = dx
$$
(provided $f+c\ne 0$).
$$
\log_e|f+c| = x + \text{another constant}
$$
$$
|f+c| = e^{x+\text{other constant}} = (e^x\cdot\text{positive constant})
$$
$$
f+c = (e^x\cdot\text{some nonzero constant})
$$
But since this was "provided $f+c\ne0$, we need to check whet... |
396,085 | <p>The length of three medians of a triangle are $9$,$12$ and $15$cm.The area (in sq. cm) of the triangle is</p>
<p>a) $48$</p>
<p>b) $144$</p>
<p>c) $24$</p>
<p>d) $72$</p>
<p>I don't want whole solution just give me the hint how can I solve it.Thanks.</p>
| newzad | 76,526 | <p>You know that medians divide a triangle to 6 equal areas. If you find one of them, multiplying with 6 give you the area of whole triangle. Let's denote one area as $S$, now see the figure:
<img src="https://i.stack.imgur.com/SqjxT.png" alt="enter image description here"></p>
<p>I guess you saw the right triangle.</... |
396,085 | <p>The length of three medians of a triangle are $9$,$12$ and $15$cm.The area (in sq. cm) of the triangle is</p>
<p>a) $48$</p>
<p>b) $144$</p>
<p>c) $24$</p>
<p>d) $72$</p>
<p>I don't want whole solution just give me the hint how can I solve it.Thanks.</p>
| Miikash Rainag | 470,654 | <p>In this type of questions, given medians always make triplet (a right triangle). From these given triplet area of triangle can be find easily
A=4/3{area of right triangle form by triplet}</p>
<p>As according to your question:
A=4/3{0.5×(9×12)}
=72</p>
|
396,085 | <p>The length of three medians of a triangle are $9$,$12$ and $15$cm.The area (in sq. cm) of the triangle is</p>
<p>a) $48$</p>
<p>b) $144$</p>
<p>c) $24$</p>
<p>d) $72$</p>
<p>I don't want whole solution just give me the hint how can I solve it.Thanks.</p>
| Piquito | 219,998 | <p>There exists a formulae giving the area of a triangle in function of its medians. It is
<span class="math-container">$$A=\frac13\sqrt{2\alpha^2\beta^2+2\beta^2\gamma^2+2\gamma^2\alpha^2-\alpha^4-\beta^4-\gamma^4}$$</span> where it is clear what are <span class="math-container">$\alpha,\beta$</span> and <span class="... |
2,275,785 | <p>I asked a similar question last night asking for an explanation of the statement, however I was unable to find how to prove such a statement, so I have a proof, however I think it is wrong, so I'm just asking for it to be checked and if it is, for it to be corrected, thanks! </p>
<p><strong>Question</strong></p>
<... | helloworld112358 | 300,021 | <p>It looks like you have all the right ideas here. Your notation is somewhat confusing in some spots, and you perhaps mention more than is necessary. Here is a short summary:</p>
<p>Suppose $x\ge 0$. Then for some $i\in\mathbb{Z}_{\ge0}$, we have $x\in [i,i+2]$, so $x\in\bigcup_{i=0}^\infty[i,i+2]$. Suppose $x<0$.... |
660,899 | <p>find the unit normal $\bf \hat{N}$ of</p>
<p>$${\bf r}=6 \mathrm{e}^{-14 t}\cos(t){\bf i}+6 \mathrm{e}^{-14 t}\sin(t){\bf j}$$</p>
<p>The answer should be in vector form. Use t as parameter. Write $e^x$ for exponentials.</p>
<p>Have been working with this a long time now but cant get the right answer.
My answer i... | leonbloy | 312 | <p>You can apply the <a href="http://en.wikipedia.org/wiki/Law_of_total_expectation" rel="nofollow">law of total expectation</a> (or "tower rule"), which says $E(Y) = E(E(Y\mid X))$</p>
<p>In your case, you know that $E(Y\mid X)= a X +b$. Hence $E(Y)=E(a X+b) = a \mu_x +b$</p>
|
2,745,884 | <p>If random variables $X$ and $Y$ are independent and $X$ and $Z$ are independent, are $X$ and $Y \cup Z$ independent?</p>
| Sally G | 156,064 | <p>No. The event $A_ i,_j$ that $i,j$ have the same birthdays is pairwise independent, so $A_1,_2$ and $A_2,_3$ are independent, as well as $A_1,_2$ and $A_1,_3$, but $A_1,_2, A_2,_3$ and $A_3,_1$ are clearly not independent. As person 1 having the same birthday as person 2 and person 3 having the same birthday as pers... |
34,049 | <p>A person has a sheets of metal of a fixed size.</p>
<p>They are required to cut parts from the sheets of metal. </p>
<p>It's desireable to waste as little metal as possible. </p>
<p>Assume they have sufficient requirements before making the first cut to more than use one sheet of metal</p>
<p>What is the name of... | Henry | 6,460 | <p>Depending on the precise details of the question, this looks like a 2-dimensional <a href="http://en.wikipedia.org/wiki/Cutting_stock_problem" rel="nofollow">cutting stock problem</a> or <a href="http://en.wikipedia.org/wiki/Packing_problem" rel="nofollow">packing problem</a> </p>
|
4,058,600 | <p>Please pardon the elementary question, for some reason I'm not grocking why all possible poker hand combinations are equally probable, as all textbooks and websites say. Just intuitively I would think getting 4 of a number is much more improbably than getting 1 of each number, if I were to draw 4 cards. For example,... | saulspatz | 235,128 | <p>Yes, that's true, but they mean that any particular hand of 5 cards has the same probability as any other hand of 5 cards. Once you start talking about the probability of a pair or four of a kind, you're talking about the probability of getting one of a number of hands. To put it another way, the probability of draw... |
4,058,600 | <p>Please pardon the elementary question, for some reason I'm not grocking why all possible poker hand combinations are equally probable, as all textbooks and websites say. Just intuitively I would think getting 4 of a number is much more improbably than getting 1 of each number, if I were to draw 4 cards. For example,... | D. G. | 581,400 | <p>This is just like flipping a coin. You are just as likely to get exactly
<span class="math-container">$$HHTHTTH$$</span>
as you are to get
<span class="math-container">$$HHHHHHH$$</span>
You are intuitively grouping poker hands into their categories, and you are right that for example four of a kind is less likely t... |
698,743 | <blockquote>
<p>Let the real coefficient polynomials
$$f(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}$$
$$g(x)=b_{m}x^m+b_{m-1}x^{m-1}+\cdots+b_{1}x+b_{0}$$
where $a_{n}b_{m}\neq 0,n\ge 1,m\ge 1$, and let
$$g_{t}(x)=b_{m}x^m+(b_{m-1}+t)x^{m-1}+\cdots+(b_{1}+t^{m-1})x+(b_{0}+t^m).$$
Show that</p>
<p>... | Sylvain Biehler | 132,773 | <p>let $x_i$ be the roots of $f$.</p>
<p>$g_t$ and $f$ are not primes iif $g_t(x_i) \neq 0$ for every $x_i$.</p>
<p>So $g_t$ and $f$ are primes iif $t$ no in the finite set of roots of $$b_{m}x_i^m+(b_{m-1}+t)x_i^{m-1}+\cdots+(b_{1}+t^{m-1})x_i+(b_{0}+t^m)$$</p>
|
48,629 | <p>Recently I began to consider algebraic surfaces, that is, the zero set of a polynomial in 3 (or more variables). My algebraic geometry background is poor, and I'm more used to differential and Riemannian geometry. Therefore, I'm looking for the relations between the two areas. I should also mention, that I'm interes... | Sándor Kovács | 10,076 | <p>It seems to me that your interest is not in algebraic geometry, but in the differential geometry of spaces defined by algebraic equations. An algebraic variety defined over $\mathbb R$ or $\mathbb C$ is a manifold away from the singularities. The singular set is is a proper closed subset, where closed means defined ... |
88,199 | <p>Is there a function that would satisfy the following conditions?:</p>
<p>$\forall x \in X, x = f(f(x))$ and $x \not= f(x)$,</p>
<p>where the set $X$ is the set of all triplets $(x_1,x_2,x_3)$ with $x_i \in \{0,1,\ldots,255\}$.</p>
<p>I would like to find a function that will have as an input RGB color values (tri... | dls | 1,761 | <p>If $f(x)=1-x$, then $f(f(x))=1-(1-x)=x$ (applied to each component).</p>
|
88,199 | <p>Is there a function that would satisfy the following conditions?:</p>
<p>$\forall x \in X, x = f(f(x))$ and $x \not= f(x)$,</p>
<p>where the set $X$ is the set of all triplets $(x_1,x_2,x_3)$ with $x_i \in \{0,1,\ldots,255\}$.</p>
<p>I would like to find a function that will have as an input RGB color values (tri... | Paul | 16,158 | <p>What about this: Define
$$g(x)=x-1\mbox{ if }x\mbox{ is odd; and }x+1\mbox{ if } x \mbox{ is even.}$$
Then $g$ maps from $\{0,1,\ldots,255\}$ to $\{0,1,\ldots,255\}$. Note also that $g(x)\neq x$ for all $x\in\{0,1,\ldots,255\}$, and $g(g(x))=x$. Now we can define $f$ as the following:
$$f(x_1,x_2,x_3)=(g(x_1),x_2,x... |
88,199 | <p>Is there a function that would satisfy the following conditions?:</p>
<p>$\forall x \in X, x = f(f(x))$ and $x \not= f(x)$,</p>
<p>where the set $X$ is the set of all triplets $(x_1,x_2,x_3)$ with $x_i \in \{0,1,\ldots,255\}$.</p>
<p>I would like to find a function that will have as an input RGB color values (tri... | ofer | 17,209 | <p>$f(x_1,x_2,x_3)=(255−x_1,255−x_2,255−x_3)$ should do the work.</p>
|
88,199 | <p>Is there a function that would satisfy the following conditions?:</p>
<p>$\forall x \in X, x = f(f(x))$ and $x \not= f(x)$,</p>
<p>where the set $X$ is the set of all triplets $(x_1,x_2,x_3)$ with $x_i \in \{0,1,\ldots,255\}$.</p>
<p>I would like to find a function that will have as an input RGB color values (tri... | Phira | 9,325 | <p>If you prefer the large color change, you should rather take $$f(x,y,z)= (x+128 \bmod 256,y+128 \bmod 256,z+128 \bmod 256).$$</p>
|
1,677,035 | <p>I'm new to this website so I apologize in advance if what I'm going to ask isn't meant to be posted here.</p>
<p>A bit of background though: I haven't been to school in 6 years and the last level I've graduated was Grade 7 due to financial problems, as well as my mom frequently being in and out of the hospital. I a... | Community | -1 | <p>Books by Bernard Child, SL Loney are awesome.
Do try problems in calculus of one variable by IA Maron.
Thomas's Calculus is alse good for in-depth knowledge.
Problems in mathematics is good for it's problems (there are too many of them).</p>
<p>Best of luck for your studies.</p>
|
2,540,992 | <blockquote>
<p>An infinite sequence of increasing positive integers is given with bounded first differences.</p>
<p>Prove that there are elements <span class="math-container">$a$</span> and <span class="math-container">$b$</span> in the sequence such that <span class="math-container">$\dfrac{a}{b}$</span> is a positiv... | Erick Wong | 30,402 | <p><a href="https://www.renyi.hu/%7Ep_erdos/1935-04.pdf" rel="nofollow noreferrer">Here is a 1935 paper</a> of a relatively young Erdős proving in a few lines that a sequence of positive integers which don't divide one another must have lower density zero, as a consequence of the fact that <span class="math-container">... |
4,521,661 | <p>Calls arrive according to Poisson process with parameter <span class="math-container">$$</span>. Lengths of the calls are iid with cdf <span class="math-container">$F_x(x)$</span>. What is the probability distribution of the number of calls in progress at any given time?</p>
<p>I am confused, is the answer then just... | Matthew H. | 801,306 | <p>Here is my approach, without queuing theory.</p>
<p>Let <span class="math-container">$T>0$</span> be any given time, and take <span class="math-container">$\Big\{[t_{i-1},t_i),s_i\Big\}_{i=1}^n$</span> as any uniform tagged partition of <span class="math-container">$[0,T)$</span> with <span class="math-container"... |
631,163 | <p>As a student in high school, I never bothered to memorize equations or methods of solving, rather I would try to identify the logic behind the operations and apply them. However, now that I've begun to teach Algebra in high school, I find it rather frustrating when students either a) memorize methods of solving the ... | String | 94,971 | <p>From my perspective, it is not just a simple "pro et contra memorization". Actually it is of great value to memorize formulas in general so that they are readily available. What may instead be the issues is rather</p>
<ol>
<li>How students memorize (reflected/un-reflected)</li>
<li>What they do with the memorized s... |
2,317,625 | <p>How do you compare $6-2\sqrt{3}$ and $3\sqrt{2}-2$? (no calculator)</p>
<p>Look simple but I have tried many ways and fail miserably.
Both are positive, so we cannot find which one is bigger than $0$ and the other smaller than $0$.
Taking the first minus the second in order to see the result positive or negative ge... | PM 2Ring | 207,316 | <p>Here's yet another way, for those who aren't comfortable with the $\gtrless$ or $\sim$ notation.</p>
<p>We can use crude rational approximations to $\sqrt 2$ and $\sqrt 3$.</p>
<p>$$\begin{align}
\left(\frac{3}{2}\right)^2 = \frac{9}{4} & \gt 2\\
\frac{3}{2} & \gt \sqrt 2\\
\frac{9}{2} & \gt 3\sqrt 2
\... |
1,114,007 | <p>How to simplify $$\arctan \left(\frac{1}{2}\tan (2A)\right) + \arctan (\cot (A)) + \arctan (\cot ^{3}(A)) $$ for $0< A< \pi /4$?</p>
<p>This is one of the problems in a book I'm using. It is actually an objective question , with 4 options given , so i just put $A=\pi /4$ (even though technically its disallo... | lab bhattacharjee | 33,337 | <p>As $0<A<\dfrac\pi4\implies\cot A>1\implies\cot^3A>1$</p>
<p>Like <a href="https://math.stackexchange.com/questions/523625/showing-arctan-frac23-frac12-arctan-frac125/523626#523626">showing $\arctan(\frac{2}{3}) = \frac{1}{2} \arctan(\frac{12}{5})$</a>,</p>
<p>$\arctan(\cot A)+\arctan(\cot^3A)=\pi+\arct... |
583,030 | <p>I have to show that the following series convergences:</p>
<p>$$\sum_{n=0}^{\infty}(-1)^n \frac{2+(-1)^n}{n+1}$$</p>
<p>I have tried the following:</p>
<ul>
<li>The alternating series test cannot be applied, since $\frac{2+(-1)^n}{n+1}$ is not monotonically decreasing.</li>
<li>I tried splitting up the series in ... | Beni Bogosel | 7,327 | <p>You could look at the partial sums:</p>
<p>$$\sum_{n=1}^{N}(-1)^n \frac{2+(-1)^n}{n+1}=\frac{2}{1}-\frac{2}{2}+\frac{2}{3}-\frac{2}{4}+...+\frac{1}{1}+\frac{1}{2}+...=2\sum_{k \leq N+1 \text{ odd}} \frac{1}{k}$$
and this diverges.</p>
|
4,285,143 | <p>Let's suppose I've got a function <span class="math-container">$f(x)$</span> where I'd like to differentiate with respect to <span class="math-container">$t$</span>, but <span class="math-container">$t$</span> depends on <span class="math-container">$x$</span>: <span class="math-container">$t(x)$</span>. Thus the wh... | amsmath | 487,169 | <p><span class="math-container">$$
\frac{df}{dt} = \frac{df}{dx}\frac{dx}{dt} = \frac{df}{dx}\left(\frac{dt}{dx}\right)^{-1} = \frac{f'(x)}{t'(x)}.
$$</span></p>
|
1,203,179 | <p>The problem I have is this:</p>
<p>Use suitable linear approximation to find the approximate values for given functions
at the points indicated:</p>
<p>$f(x, y) = xe^{y+x^2}$ at $(2.05, -3.92)$</p>
<p>I know how to do linear approximation with just one variable (take the derivative and such), but with two variabl... | Sloan | 217,391 | <p>Let $L(x,y)$=$f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$. Then $L(x,y) \approx f(x,y)$. Consider $(x_0,y_0)=(2,-4)$. Then, \begin{equation}
L(x,y)=2+9(x-2)+2(y+4) \implies f(2.05,-3.92) \approx L(2.05, -3.92)=2.61 \end{equation}
Notice, from a calculator, $f(2.05,-3.92)=2.7192$</p>
|
56,481 | <p>I don't know if the question trivial or not (if so I will delete it), but here is what I have:</p>
<p>I know that </p>
<pre><code>x x
(* x^2 *)
</code></pre>
<p>Looking at the full form gives</p>
<pre><code>FullForm[Times[x, x]]
(* Power[x, 2] *)
</code></pre>
<p>It is clear that the <code>Times</code> head h... | m_goldberg | 3,066 | <p><em>Mathematica</em> embraces the notion of canonical form. Internally it wants all expressions to be reduced to their canonical forms.</p>
<p><code>Times[x, x]</code> is not a canonical form, so it gets reduced to <code>Power[x, 2]</code>, which is.</p>
<p>I share Mr.Wizard's occasional frustration over the choic... |
3,038,965 | <p>Here's the question I'm puzzling over:</p>
<p><span class="math-container">$\textbf{Find the perpendicular distance of the point } (p, q, r) \textbf{ from the plane } \\ax + by + cz = d.$</span></p>
<p>I tried bringing in the idea of a dot product and attempted to get going with solving the problem, but I'm headin... | Chris Culter | 87,023 | <p>Try redoing the derivative of <span class="math-container">$1/(1-x)$</span> more carefully.</p>
|
670,781 | <p>Given $y=x\sqrt{a+bx^2}$. the tangent to $y$ at point $x=\sqrt5$ is also passing at point </p>
<p>$(3\sqrt5,\sqrt5).$ the area between $y=x\sqrt{a+bx^2}$ and the $x$-axis is equal to $18$.</p>
<p>Need to find $a,b$.</p>
<p>I have tried to differentiate and eliminate but failed...</p>
| DonAntonio | 31,254 | <p>$$f(x):=x\sqrt{a+bx^2}\implies f(\sqrt5)=\sqrt5\sqrt{a+5b}=\sqrt{5a+25b}\implies$$</p>
<p>since</p>
<p>$$f'(x)=\sqrt{a+bx^2}+\frac{bx^2}{\sqrt{a+bx^2}}=\frac{a+2bx^2}{\sqrt{a+bx^2}}\implies f'(\sqrt5)=\frac{a+10b}{\sqrt{a+5b}}$$</p>
<p>Thus, the tangent line at the tangency point $\;(\sqrt5\,,\,\sqrt{5a+25b})\;$ ... |
2,674,217 | <p>Let $\{ a_{n}\}_{n}$ be a sequence and let $a\in \mathbb{R}$. Define $\{ c_{n}\}_{n}$ as:</p>
<p>$$c_{n}=\frac{a_{1}+...+a_{n}}{n}.$$</p>
<p>I want to prove the following claim: if $\lim\limits_{n\to +\infty}a_{n}=+\infty$ then $\lim\limits_{n\to +\infty}c_{n}=+\infty$</p>
<p>Approach: Suppose $\lim\limits_{n\to ... | zhw. | 228,045 | <p>You were off to a good start. You showed</p>
<p>$$c_{n}>\frac{a_{1}+...+a_{n_{0}-1}}{n}+\frac{(n-(n_{0}-1))M}{n} $$</p>
<p>for $n\ge n_0.$ Thus $\liminf c_n$ is at least the $\liminf$ of the expression on the right. But the expression on the right has a limit, namely $0+M=M,$ and thus its $\liminf$ is the same.... |
2,336,535 | <p>I have a limit:</p>
<p>$$\lim_{(x,y)\rightarrow(0,0)} \frac{x^3+y^3}{x^4+y^2}$$</p>
<p>I need to show that it doesn't equal 0.</p>
<p>Since the power of $x$ is 3 and 4 down it seems like that part could go to $0$ but the power of $y$ is 3 and 2 down so that seems like it's going to $\infty$.</p>
<p>I wonder if t... | Fred | 380,717 | <p>Let $f(x,y)=\frac{x^3+y^3}{x^4+y^2}$. Then</p>
<p>$|f(x,x^2)| \to \infty$ for $x \to 0$.</p>
<p>Hence $\lim_{(x,y)\rightarrow(0,0)} \frac{x^3+y^3}{x^4+y^2}$ does not exist.</p>
|
1,387,184 | <p>Can someone show how to compute the residue of this function:
$$\frac{z}{e^z - 1}$$</p>
<p>I think can represent the Taylor series of $e^z$ as
$$e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots$$
Then, we have
$$\frac{z}{e^z - 1} = \frac{z}{(1 +z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots) -1}$$
$$ = 1 + 1... | Community | -1 | <p>Let $K$ be an algebraically closed field of characteristic $0$, $L$ be the Lie algebra $M_n(K),[.,.]$, $ad(L)=\{ad_x:y\rightarrow xy-yx=[x,y]|x\in L\}$, $der(L)$ be the set of derivations over $L$. </p>
<p>Prop 0. $L$ has no proper ideal. (well-known)</p>
<p>Prop 1. $ad(L)$ is an ideal of $der(L)$. </p>
<p>Proof.... |
1,386,683 | <p>I posted early but got a very tough response.</p>
<p>Point $A = 2 + 0i$ and point $B = 2 + i2\sqrt{3}$ find the point $C$ $60$ degrees ($\pm$) such that Triangle $ABC$ is equilateral. </p>
<p>Okay, so I'll begin by converting into polar form:</p>
<p>$A = 2e^{2\pi i}$ and $B = 4e^{\frac{\pi}{3}i}$</p>
<p>$\overli... | Eclipse Sun | 119,490 | <p>Since the modulus of $\dfrac{\bar z^2}{z}$ is $|z|$, the limit is $0$.</p>
|
2,558,267 | <p>Let $M$ be a finite dimensional von-Neumann algebra. We know this algebra is generated by its projections. My question maybe simple. Can one computing these projections? What about its minimal or central projections? </p>
<p>If possible please give me a reference for this. </p>
<p>Thanks </p>
| Ivan Burbano | 232,542 | <p>By the structure theorem of finite-dimensional <span class="math-container">$C^*$</span>-algebras (see, Takesaki's, Theory of Operator Algebras, Theorem 11.2), <span class="math-container">$M\cong\bigoplus_{r=1}^N M_{n_r}(\mathbb{C})$</span> for some unique <span class="math-container">$n_1,\dots,n_N\in\mathbb{N}^+$... |
4,495,044 | <p>Edit: There is an answer at the bottom by me explaining what is going on in this post.</p>
<p>Define a function <span class="math-container">$f : R \to R$</span> by <span class="math-container">$f(x) = 1$</span> if <span class="math-container">$x = 0$</span> and <span class="math-container">$f(x) = 0$</span> if <spa... | Sourav Ghosh | 977,780 | <p>Taking absolute value of positive number os nothing but wasting time!</p>
<p>How you get contradiction from <span class="math-container">$0<\frac {1} {n} < \delta$</span> and <span class="math-container">$f(0) =1$</span> ?</p>
<p><span class="math-container">$1=f(0) <f(\frac{1}{n})=0$</span> is not true unl... |
3,703,981 | <p>If we consider an equation <span class="math-container">$x=2x^2,$</span> we find that the values of <span class="math-container">$x$</span> that solve this equation are <span class="math-container">$0$</span> and <span class="math-container">$1/2$</span>. Now, if we differentiate this equation on both sides with res... | Misha Lavrov | 383,078 | <p>Let <span class="math-container">$d = x - y$</span>. Then we want <span class="math-container">$x^3 - (x-d)^3 = 2020$</span>, which is a quadratic equation in <span class="math-container">$x$</span>. The discriminant is <span class="math-container">$24240d - 3d^4 = 3d (8080 - d^3)$</span>, which is nonnegative only ... |
2,668,616 | <p>I am fairly new at MAPLE and I'm having some trouble solving this ODE. </p>
<p>$$(t+1)\frac{dy}{dt}-2(t^2+t)y=\frac{e^{t^2}}{t+1}$$</p>
<p>My initial value problem is $$t>-1, y(0)=5$$</p>
<p>I put the equation in standard form and typed into maple</p>
<p><a href="https://i.stack.imgur.com/rctFl.png" rel="nofo... | user577215664 | 475,762 | <p><strong><em>Hint</em></strong>
$$(t+1)\frac{dy}{dt}-2(t^2+t)y=\frac{e^{t^2}}{t+1}$$
Since $t >-1$ then $t \neq -1$
$$y'-2ty=\frac{e^{t^2}}{(t+1)^2}$$
$e^{-t^2}$as integrating factor
$$(ye^{-t^2})'=\frac{1}{(t+1)^2}$$
$$(ye^{-t^2})=\int \frac{dt}{(t+1)^2}$$
$$y=e^{t^2}\int \frac{dt}{(t+1)^2}$$
Substitute $u=t+1$
$... |
194,547 | <p>I know the definition of a linear transformation, but I am not sure how to turn this word problem into a matrix to solve:</p>
<p>$T(x_1, x_2) = (x_1-4x_2, 2x_1+x_2, x_1+2x_2)$</p>
<p><strong>Find the image of the line that passes through the origin and point $(1, -1)$.</strong></p>
| Hagen von Eitzen | 39,174 | <p>The line passing through the origin and $(1, -1)$ is the set of points of the form $(t, -t)$ with $t\in\mathbb R$ (I suppose you are implicitly working over the reals).
We compute $$T(t,-t)=(t-4t,2t+t,t+2t)=(-3t,3t,3t).$$ That describes the line in 3D space through the origin and $(3, -3, -3)$ (or equivalently one c... |
3,274,172 | <p>Let <span class="math-container">$X$</span> a compact set. Prove that if every connected component is open then the number of components is finite.</p>
<p>Ok, <span class="math-container">$X = \bigcup C(x)$</span> where <span class="math-container">$C(x)$</span> is the connected component of <span class="math-conta... | J.-E. Pin | 89,374 | <p><strong>Hint</strong>. Since <span class="math-container">$X$</span> is compact and since, as you observed, the connected component form an open cover of <span class="math-container">$X$</span>, one can extract from them a finite subcover. On the other hand, the connected component form a partition of <span class="m... |
2,879,883 | <p>Suppose that $f$ and $g$ are differentiable functions on $(a,b)$ and suppose that $g'(x)=f'(x)$ for all $x \in (a,b)$. Prove that there is some $c \in \mathbb{R}$ such that $g(x) = f(x)+c$.</p>
<p>So far, I started with this:</p>
<p>Let $h'(x)=f'(x)-g'(x)=0$, then MVT implies $\exists$ c $\in \mathbb{R}$ such that... | Titus Moody | 583,516 | <p>Assume f-g is non-constant. Then f'-g' is not identically zero. But f'=g' by hypothesis. Contradiction. Hence f-g is constant. </p>
|
2,736,426 | <p>Let's imagine a point in 3D coordinate such that its distance to the origin is <span class="math-container">$1 \text{ unit}$</span>.</p>
<p>The coordinates of that point have been given as <span class="math-container">$x = a$</span>, <span class="math-container">$y = b$</span>, and <span class="math-container">$z = ... | user | 505,767 | <p>The vector point coordinates are $OP=(a,b,c)$ then the angles with $x,y,z$ with unitary vectors $e_1=(1,0,0),e_2=(0,1,0),e_3(0,0,1)$ are given by the dot product</p>
<ul>
<li>$\cos \alpha = \frac{OP\cdot e_1}{|OP||e_1|}=OP\cdot e_1=a$</li>
<li>$\cos \beta = \frac{OP\cdot e_2}{|OP||e_2|}=OP\cdot e_2=b$</li>
<li>$\co... |
283,183 | <p>I am looking for references that discuss Hecke operators $T_n$ acting on modular forms
for the principal congruence subgroup $\Gamma(N)$ of the modular group $SL(2,Z)$ and am happy to restrict to the case that $(n,N)=1$. Most textbooks (Diamond and Shurman, Koblitz etc.) that discuss Hecke operators for congruence s... | David Loeffler | 2,481 | <p>The reason why Hecke theory for $\Gamma(N)$ doesn't get much treatment in the literature is because you can easily reduce it to the $\Gamma_1(N)$ case. More precisely, you can conjugate $\Gamma(N)$ by $\begin{pmatrix} N & 0 \\ 0 & 1\end{pmatrix}$ to get a group intermediate between $\Gamma_0(N^2)$ and $\Gamm... |
283,183 | <p>I am looking for references that discuss Hecke operators $T_n$ acting on modular forms
for the principal congruence subgroup $\Gamma(N)$ of the modular group $SL(2,Z)$ and am happy to restrict to the case that $(n,N)=1$. Most textbooks (Diamond and Shurman, Koblitz etc.) that discuss Hecke operators for congruence s... | François Brunault | 6,506 | <p>The Hecke operators $T(n)$ and the dual Hecke operators $T'(n)$ acting as correspondences on the modular curve $Y(N)$ are defined by Kato in <em>$p$-adic Hodge theory and values of zeta functions of modular forms</em>, section 2.9 (in Kato's notation $Y(N)=Y(N,N)$). The action of $T(p)$ on Fourier expansions is give... |
1,146,824 | <p>The Russel's Paradox, showing $X=\{x|x\notin x\}$ can't exist is not very hard.
If $X \in X$, then $X \notin X$ by definiition, in the other case, $X \notin X$, then $X \in X$ by definition. Both cases are impossible.</p>
<p>But how about whole things $X=\{x|x=x\}$? $X \in X$ probably cause the problem, but I don't... | Mauro ALLEGRANZA | 108,274 | <p>Because in presence of the <a href="http://en.wikipedia.org/wiki/Axiom_schema_of_specification">Axion of Separation</a> (or <em>Axiom of Specification</em>), if the "universal set" $V = \{ x \mid x=x \}$ exists, we can have :</p>
<blockquote>
<p>$R = \{ x \mid x \in V \land x \notin x \}$</p>
</blockquote>
<p>an... |
1,015,498 | <p>I am merely looking for the result of the convolution of a function and a delta function.
I know there is some sort of identity but I can't seem to find it. </p>
<p>$\int_{-\infty}^{\infty} f(u-x)\delta(u-a)du=?$</p>
| Kevin Arlin | 31,228 | <p>The delta "function" is the multiplicative identity of the convolution algebra. That is, $$\int f(\tau)\delta(t-\tau)d\tau=\int f(t-\tau)\delta(\tau)d\tau=f(t)$$
This is essentially the definition of $\delta$: the distribution with integral $1$ supported only at $0$.</p>
|
3,013,355 | <p>I have been asked to prove that </p>
<p>(<span class="math-container">$a \to $</span>b) <span class="math-container">$\vee$</span> (<span class="math-container">$a \to $</span>c) = <span class="math-container">$a \to ($</span>b <span class="math-container">$\vee$</span> c).</p>
<p>I believe it is just the simple c... | Arrow | 223,002 | <p>Suppose <span class="math-container">$a,b,c,d\in R$</span> are powers of prime elements satisfying <span class="math-container">$ab=cd$</span>. Let us write <span class="math-container">$a\sim_R c$</span> when <span class="math-container">$a,c$</span> are associates in <span class="math-container">$R$</span>. Failur... |
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