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 |
|---|---|---|---|---|
12,922 | <p>European universities are running the so called "Bologna plan" which includes the recommendation of keeping some percentage of the final grades to be filled with various activities during the lessons period. Like everything in life, it has pros and cons.</p>
<p>I will tell you the cons I have seen (in the particula... | Nick C | 470 | <p>I will use the term "formative assessment" to refer to the "activity" points you describe.</p>
<p>Here are some positives to awarding points in this way:</p>
<ol>
<li><b>You can give students the opportunity to practice math “under pressure”. </b>Not every “formative assessment” has to be called out as such before... |
2,443,640 | <p>Give a vector function $r(t)$ representing the intersection of the surfaces:</p>
<p>$x^2 + y^2 = z^2$ and $y = xz$</p>
| Will Jagy | 10,400 | <p>The beginning is some version of the Extended Euclidean Algorithm...</p>
<p>$$ 1729 \cdot 82 - 801 \cdot 177 = 1 $$ </p>
<p>$$ 1729 \cdot 82 \equiv 1 \pmod{801} $$
$$ 1729 \cdot 82 \equiv 0 \pmod{1729} $$
$$ 801 \cdot (-177) \equiv 1 \pmod{1729} $$
$$ 801 \cdot (-177) \equiv 0 \pmod{801} $$</p>
<p>What ... |
2,443,640 | <p>Give a vector function $r(t)$ representing the intersection of the surfaces:</p>
<p>$x^2 + y^2 = z^2$ and $y = xz$</p>
| Donald Splutterwit | 404,247 | <p>Note that $801=3^2 \times 89 $ and $1729 =7 \times 13 \times 19 $ and so they are coprime.
Use Euclid's Algorithm to calulate
\begin{eqnarray*}
1729 \times 82 \equiv 1 \pmod{801} \\
801 \times 1552 \equiv 1 \pmod {1729}
\end{eqnarray*}</p>
<p>A solution can be constructed easily by $x=676 \times 1729 \times 82 +14... |
2,443,640 | <p>Give a vector function $r(t)$ representing the intersection of the surfaces:</p>
<p>$x^2 + y^2 = z^2$ and $y = xz$</p>
| Piquito | 219,998 | <p>The better I know for this sort of problems with large numbers is to go to a linear diophantine equation in two variables which is translation of the two given modular equalities and solving it completely finding out first a particular solution. This way we have
$$x=676+801X=1447+1729Y.................(*)$$
When the... |
3,548,866 | <blockquote>
<p>In how many ways can the word "kosmos" be arranged so that the same
letters do not come together? <span class="math-container">$(84)$</span></p>
</blockquote>
<p>We have the word <span class="math-container">$k_1o_1s_1m_1o_2s_2$</span> - <span class="math-container">$6$</span> letters but two lett... | Mick | 42,351 | <p>Initially, we have 6 objects with 2 sets of objects with 2 equal members each. The total number of ways = <span class="math-container">$\dfrac {6!}{2!2!}$</span></p>
<p>Next, we count those with OO and SS. We first “tie” the objects with the same nature together to form one unit, and there are two such units. Ther... |
1,387,170 | <p>Let $V$ be a vector space with dimension $n$ such that $\{v_1,\cdots,v_n\}$ is its basis. Take $A\equiv\{a_1,\cdots,a_p\}\subset V$ with $p>n$.</p>
<p>How do can I show that $A$ is linearly dependent <strong>without having to do all those summations</strong>. Is there an easier more direct proof than that?</p>
... | egreg | 62,967 | <p>There are a constructive and a non constructive proof. The constructive proof is known as “Steinitz Lemma”:</p>
<blockquote>
<p><em>If $B$ is a finite spanning set (in particular, a basis) of a finitely generated vector space $V$ and $A$ is a (finite) linearly independent subset of $V$, then $|A|\le|B|$.</em></p>... |
932,711 | <p>A simple question, but I like to be clinical with my choice of words:</p>
<p>I have a complex number, $z=-i$. </p>
<p>If I were to calculate the argument of this complex number, $arg(z) = tan^{-1}( \frac{-1}{0}) \approx -\frac{π}{2}rad$.</p>
<p>I know that <em>something</em> is tending towards infinity. What is ... | Nagase | 117,698 | <p>I'll develop my hint a bit more.</p>
<p>Suppose you knew that four softwares have a bug, say $A$, $B$, $C$, and $D$. Let $P(x)$ mean "x has a bug". You could then symbolize the situation with the following formula:</p>
<blockquote>
<p>$P(A) \wedge P(B) \wedge P(C) \wedge P(D)$</p>
</blockquote>
<p>But that's no... |
5,751 | <p>I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general public. The best I could think of was Goodstein sequences: if you take a particular instance of Goodstein's theorem,... | Pablo Zadunaisky | 17,353 | <p>I am surprised no one has mentioned any examples from physics. Fluid mechanics for example obtains some beautiful results by assuming that a fluid is continuous, while at the molecular level it is a discrete object. An example from my physics highschool textbook:</p>
<blockquote>
<p>Suppose you have a circular gr... |
5,751 | <p>I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general public. The best I could think of was Goodstein sequences: if you take a particular instance of Goodstein's theorem,... | Piotr Hajlasz | 121,665 | <p>A good example is a solution to Hilbert's third problem: <strong><em>it is not possible to cut the unit cube into finitely many polyhedral pieces and reassemble then as the regular tetrahedron of unit volume.</em></strong> The shortest proof requires construction of a Dehn's invariant which, in turn, requires the ex... |
3,166,159 | <p>If <span class="math-container">$f: \mathbb{R} \to\mathbb{R} $</span>, <span class="math-container">$x \ne 0,1$</span></p>
<p>Find all functions <span class="math-container">$f(x)$</span> such that <span class="math-container">$f(x)+f\left(\frac{1-x}{x}\right)=1-x$</span></p>
<p>My try:</p>
<p>Letting <span class... | jmerry | 619,637 | <p>Chasing that idea in the comments: <span class="math-container">$f(x)+f\left(\frac{1-x}{x}\right) = 1-x$</span>, then <span class="math-container">$f\left(\frac{1-x}{x}\right) + f\left(\frac{2x-1}{1-x}\right) = \frac{2x-1}{x}$</span>, then <span class="math-container">$f\left(\frac{2x-1}{1-x}\right) + f\left(\frac{... |
3,762,322 | <p>I'm looking for the name of the following expression:</p>
<p><span class="math-container">$$
\frac{\sum_{i} x_i^2}{\sum_i x_i}
$$</span></p>
<p>This can be understood as a type of average in which each value is weighted by itself – in other words, a type of average in which near-zero values are ignored. Here are som... | sk29910 | 28,434 | <p>We can think of the expression</p>
<p><span class="math-container">$$
\frac{x^n}{\sum_i x_i^n}
$$</span></p>
<p>as a polynomial approximation of the softmax weighting</p>
<p><span class="math-container">$$
\frac{e^{x/T}}{\sum_i e^{x_i/T}}.
$$</span></p>
<p>One difference is that <span class="math-container">$e^0 = 1... |
2,477 | <p>When talking with students it frequently happens that they misunderstand what you meant. The common example is the amount of rigor that one would consider "a proof", but there are other things, like</p>
<ul>
<li>smooth meaning <span class="math-container">$C^2$</span> or <span class="math-container">$C^\in... | vonbrand | 123 | <p>Define your terms, if needed point out there are alternative definitions/notations (and perhaps point some of them out). When using them, be careful to go back to first definitions whenever it makes sense.</p>
|
959,628 | <p>So the series is $$P_k: 2^2 + 4^2 + 6^2 + ... + (2k)^2 = \frac{2k(k+1)(2k+1)}3$$</p>
<p>and i have to replace $P_k$ with $P_{k+1}$ to prove the series.</p>
<p>I have to show that $$\frac{2k(k+1)(2k+1)}3 + [2(k+1)]^2 = \frac{2(k+1)(k+2)(2k+3)}3$$ but I don't know how.</p>
| Mathronaut | 53,265 | <p>This equals $$\displaystyle\lim_{h\to 0}\frac{9-h}{9-h-9}=\lim_{h\to 0}\frac{9-h}{-h}=\lim_{h\to0}\frac{9}{-h}+1=-\infty$$</p>
|
959,628 | <p>So the series is $$P_k: 2^2 + 4^2 + 6^2 + ... + (2k)^2 = \frac{2k(k+1)(2k+1)}3$$</p>
<p>and i have to replace $P_k$ with $P_{k+1}$ to prove the series.</p>
<p>I have to show that $$\frac{2k(k+1)(2k+1)}3 + [2(k+1)]^2 = \frac{2(k+1)(k+2)(2k+3)}3$$ but I don't know how.</p>
| Arian | 172,588 | <p>$$\lim_{x\to 9^-}\frac{x^2-9x}{x^2-18x+81}=\lim_{x\to 9^-}\frac{x(x-9)}{(x-9)^2}=\lim_{x\to 9^-}\frac{x}{x-9}=\lim_{\epsilon\to 0}\frac{9-\epsilon}{9-\epsilon-9}=\lim_{\epsilon\to 0}\frac{9-\epsilon}{-\epsilon}=-\infty$$</p>
|
83,900 | <p>Given a convex surface $S$ in $\mathbb{R}^3,$ you can associate to each point the length of the inward pointing normal contained inside $S.$ Call this quantity $s(x).$ Now, given a positive function $f$ on the sphere $S^2,$ are necessary and sufficient conditions known so that $f$ can be realized as $s$ for some con... | alvarezpaiva | 21,123 | <p>Dear Igor,</p>
<p>Since the answer to your problem is "no" I'm not going to dwell on that, but I feel there is something interesting behind your question that does not admit such a clear cut answer. </p>
<p>Two known facts:</p>
<p><strong>1.</strong> Consider the manifold of oriented lines in three-space and, in ... |
2,138,165 | <blockquote>
<p>Let <span class="math-container">$a,b,c,d$</span> be positive real numbers such that <span class="math-container">$a+b+c+d=1$</span>, then prove that <span class="math-container">$$S=\sqrt{4a+1}+\sqrt{4b+1}+\sqrt{4c+1}+\sqrt{4d+1} < 6$$</span></p>
<p>Source : <a href="http://imomath.com/pcpdf/f1/f41.... | 5xum | 112,884 | <p>1) We choose a sequence of rational numbers because the function is $1$ at all rational numbers, thus $\lim_{n\to\infty} f(x_n)=1$ (more on that below).</p>
<p>It would also be possible to prove discontinuity using the $\epsilon-\delta$ definition, but this particular textbook chose the convergent sequences way of ... |
810,032 | <p>A consequence of the Lagrange theorem:</p>
<blockquote>
<p>Let $G$ a finite group and $H$ a subgroup of G. Then $|H| \mid |G|$.</p>
</blockquote>
<p>is that each subgroup $\neq <i_d>$ of $D_4$, which has $8$ elements , has either $2$ or $4$ elements..</p>
<p>But.... $1 \text{ divides also }8$..Isn't it p... | Robert Israel | 8,508 | <p>Two of the roots will be near the roots $1$ and $-1$ of $-2x^2 + 2$, the other two will be near $\infty$. Solving numerically for $\epsilon = .01$,
those roots are approximately $-100$ and $200$.
So it looks like you want $x \sim c \epsilon^{-1}$. Indeed with $x = y/\epsilon$ the equation becomes $y^4 - y^3 - 2 y^... |
3,368,130 | <p>Hi I'm having a lot of trouble with the following problem, I tried using the equation dQ/dt = 3Q/2006-3t but it doesn't work.</p>
<blockquote>
<p>A tank contains 90 kg of salt and 2000 L of water. Pure water enters a tank at the rate 6 L/min. The solution is mixed and drains from the tank at the rate 3 L/min. How... | Conrad | 298,272 | <p>This is the identity theorem for analytic functions in a slight abuse of notation. </p>
<p>The setting is that a given (usually by a formula like a Dirichlet series or integral but could be something else too) analytic <span class="math-container">$F(s)$</span> is defined on a domain <span class="math-container">$D... |
3,632,502 | <p>I have line whose end points p1 and p2 coordinates I know.
The line has a width w (basically it forms a rectangle)</p>
<p>How to check if a point P(x,y) lies inside this line.</p>
<p>What I tried was taking perpendicular distance from P to the line and check if its less than half the width. </p>
<p>But this will ... | Apass.Jack | 580,448 | <p>Let us show the following more general proposition.</p>
<p>Let <span class="math-container">$a_{1},a_{2},\cdots,a_{n}$</span> be positive real numbers. Then there exist <span class="math-container">$\mu_{i}\in\{-1,1\},i=1,2,\cdots,n$</span> such
<span class="math-container">$$0\le(\mu_{1}a_{1}+\mu_{2}a_{2}+\cdots+\... |
2,747,826 | <p>If I entered the lottery, and the chance of winning was 5%, what are the chances I win at least once in 10 draws?
I've searched everywhere, but Google Calculator did not help me :(</p>
| Rócherz | 451,007 | <p>Sounds like a <a href="https://en.wikipedia.org/wiki/Binomial_distribution" rel="nofollow noreferrer">binomial distribution</a> to me.</p>
|
1,962 | <p>I am only a beginner in math.se, (and to math as well in some technical ways, not in terms of enthusiasm though) in my view i feel that math.se has a very good amount of wealth of knowledge in math and is rapidly increasing. I think in my opinion, at some point in future it could give math class room courses a run f... | Jeff Atwood | 153 | <p>This is certainly a laudable goal; I would <em>absolutely</em> say the #1 purpose of any Stack Exchange site is <strong>learning</strong> within an expert community on a specific topic.</p>
<p>However, I think the goals you outline are more akin to something like</p>
<p><a href="http://www.khanacademy.org/" rel="n... |
1,864,161 | <p>The main question is :</p>
<p>$z^x=x$, $z^y=y$, $y^y=x$</p>
<p>Find $z$, $y$, $x$.</p>
<p>My method : </p>
<p>I first attempted to get two equation for the unknowns $x$ and $y$.</p>
<p>We can happily write :</p>
<p>$z=x^{1/x}$ and $z=y^{1/y}$</p>
<p>Thus we get,
$x^{1/x}=y^{1/y}$
Which is,
$x^y=y^x$.</p>
<p>... | Lukas Cerny | 354,982 | <p>One solution just by looking at it is $z=x=y=1$.</p>
|
13,276 | <p>I have been doing volunteer high-school math tutoring in person. But nowadays, I think there should also be an online way of connecting a student in need of help, with someone willing to help them. This could be in real time, or via message.</p>
<p>Are there any phone apps or websites that will do this? They sho... | yathish | 2,191 | <p>Why not use Google Hangouts? It enables screen sharing both ways - effectively turning your screen into a blackboard. As others pointed out there's probably no free and yet good site/app that aligns with what you propose. However, functionality-wise hangouts would suffice. Having a dedicated website would certainly... |
13,276 | <p>I have been doing volunteer high-school math tutoring in person. But nowadays, I think there should also be an online way of connecting a student in need of help, with someone willing to help them. This could be in real time, or via message.</p>
<p>Are there any phone apps or websites that will do this? They sho... | jfkoehler | 7,003 | <p>There are a few options, and they are free. First, google hangouts and google classroom are fine for this. If you want, you can just use drawing in a google doc or something for the pictures.</p>
<p>If you want a little higher level functionality, you can checkout some kind of courseware like Moodle, Blackboard, o... |
510,750 | <p>I have a molecule with one coordinate system ( denote as x,y,z ) where the origin is center of mass of the molecule. I have to define another coordinate system (p,q,r) for a local motion. (shown in the figure.) I know the unit vectors of p,q,r axises respective to x,y,z coordinates and origin of the p,q,r coordinat... | rschwieb | 29,335 | <p>Personally, I remember my linear algebra experience well because it taught me about rings and modules. Unfortunately, this is a dangerous strategy because not so many people are attracted by this abstraction :)</p>
<p>However, I do remember enjoying some of the projects in <em>Numerical Linear Algebra</em> by Trefe... |
267,507 | <p>Figure shows a rectangle $ABCD$ and an isosceles triangle $\triangle DEC$.</p>
<p>$AD=BC=z$;$AB=DC=y$;$DE=CE=x$</p>
<p>One solution is as follows.</p>
<p>We know that the pythagoras theorem holds for a right triangle $\triangle EBC$.</p>
<p>In the right triangle, $\triangle EBC$.</p>
<p>$z^2+(\frac{y}{2})^{2}=x... | Calvin Lin | 54,563 | <p>If I understood you correctly, what you want is a proof of pythagorean theorem for the isosceles right triangle. A proof is as follows, and is similar to a proof of the generalized pythagorean theorem.</p>
<p>Let $ABCD$ be a square of side length $2l$. Let $E, F, G, H$ be the midpoints of $AB, BC, CD, DA$ respecti... |
1,623,003 | <p>Say you take a dice and you roll it twice so that you have a pair (X,Y) where X represents the first roll, Y represents the second. </p>
<p>When you have a distribution like max(X,Y), what type of distribution is that? It's not uniform, obviously, because the outcomes of max(X,Y) are clearly more favorable towards ... | Em. | 290,196 | <p>"Describe" is a vague term in this case. There are two possibilities that I assume they mean. They either will accept a distribution table, which would amount to writing down all possibilities and probabilities like you say. The other way is to give the pmf. To get the pmf of $Z = \max\{X,Y\}$, you want to do the fo... |
527,747 | <p>Prove that for all positive integers k $\le$ n, the inequality $p_k(n)$ $\le$ $(n-k+1)^{(k-1)}$ holds.</p>
<p>Is it true that $p_k(n)$ is a polynomial function of n?</p>
<p>(Number 29 chapter 5)
I don't even know where to start with the first part of this inequality proof.</p>
| Smylic | 100,361 | <p>Each summand is at most $n - k + 1$, 'cause other $k-1$ summands are positive. So you have at most $n - k + 1$ variants for each summand. But if you know first $k-1$ of them you definitely know the last one. So the number of different not greater than $(n - k + 1)^{k-1}$.</p>
|
1,579,077 | <p>This expression has appeared in my work:</p>
<p>$$P_n(x)=\sum _{j=0}^n\binom{n}{j}\, j^n x^j $$</p>
<p>It seems too simple not to be already somewhere (perhaps in a different form). Has anyone seen it before?</p>
| Marko Riedel | 44,883 | <p>We can add to the collection of properties of the polynomials</p>
<p>$$P_n(x) = \sum_{j=0}^n {n\choose j} j^n x^j$$</p>
<p>by observing that</p>
<p>$$j^n = \frac{n!}{2\pi i}
\int_{|z|=\epsilon} \frac{1}{z^{n+1}} \exp(jz) \; dz$$</p>
<p>which yields for the sum
$$\frac{n!}{2\pi i}
\int_{|z|=\epsilon} \frac{1}{z^{... |
1,366,737 | <p>Demonstrate by induction: $3^{2n} + 7 = 4k$ is true, for any $n\in \mathbb N$. I need to demonstrate this using the induction principle.</p>
<p>So far I have:</p>
<p><strong>$n = 1$</strong></p>
<p>$$3^{2\cdot 1} + 7 = 4\cdot k $$
$$9 + 7 = 4k$$
$$16 = 4k$$
$$k = 4$$</p>
<p>So it checks for $n=1$.</p>
<p><stron... | Bernard | 202,857 | <p>I'll give a proof with congruences: as $3^2\equiv 1\mod8$,
$$3^{2^n}+7=(3^2)^n+7\equiv 1^n+7\equiv 0\mod 8.$$
<em>A fortiori</em>, $\;3^{2^n}+7\equiv 0\mod 4$.</p>
|
1,366,737 | <p>Demonstrate by induction: $3^{2n} + 7 = 4k$ is true, for any $n\in \mathbb N$. I need to demonstrate this using the induction principle.</p>
<p>So far I have:</p>
<p><strong>$n = 1$</strong></p>
<p>$$3^{2\cdot 1} + 7 = 4\cdot k $$
$$9 + 7 = 4k$$
$$16 = 4k$$
$$k = 4$$</p>
<p>So it checks for $n=1$.</p>
<p><stron... | marty cohen | 13,079 | <p>Just to be contrary,
here is a non-induction proof.</p>
<p>$\begin{array}\\
3^{2n}+7
&=3^{2n}-1+8\\
&= (3^n+1)(3^n-1)+8
\end{array}
$</p>
<p>Since $3^n$ is odd
(for $n \ge 1$),
$3^n-1$ and
$3^n+1$
are consecutive even numbers,
so one is exactly divisible by $2$
and the other is divisible by
(at least) $4$.... |
2,162,676 | <p><a href="https://i.stack.imgur.com/nafvt.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nafvt.png" alt="enter image description here"></a></p>
<p>I am currently rereading the proof of Lemma 136. given above and I understand almost every part of the proof, except for the next one. </p>
<p>Namely... | Emo | 127,234 | <p>When talking about topological properties you need to know that different topologies (or different metrics) give different consequences even if we are working on the same set.</p>
<p>Example, if you take the set of real numbers and {0} as a subset, {0} is not open on the Euclidean metric, but it is so in the discre... |
2,162,676 | <p><a href="https://i.stack.imgur.com/nafvt.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nafvt.png" alt="enter image description here"></a></p>
<p>I am currently rereading the proof of Lemma 136. given above and I understand almost every part of the proof, except for the next one. </p>
<p>Namely... | zygzzy | 420,281 | <p>let for a given distance "d" B(x,r) be open ball center x for r > 0</p>
<p>x is in the closure of C if $B(x,r) \cap C \ne \emptyset$ for all r > 0</p>
<p>x is in the interior of C if there exists r > 0 such that $B(x,r) \subset C$</p>
<p>x is in the frontier of C if $B(x,r) \cap C \ne \emptyset$ for all r > 0, an... |
1,558,850 | <p>I'm solving the 1D wave equation
\begin{equation}
\frac{\partial^2 \eta}{\partial t ^2} - \frac{\partial^2 \eta}{\partial x ^2} = 0
\end{equation}
with boundary conditions
\begin{equation}
\frac{\partial \eta}{\partial x} = 0 \qquad \qquad \text{on} \qquad \qquad x = 0, 1
\end{equation}
using Finite Element Method.... | Fryderyk | 186,371 | <p>I think I found a way...</p>
<p>Rearranging the Crank-Nicolson formulation as follows:
\begin{align*}
\eta_j^{n+1} - \frac{\Delta t }{2} p_j^{n+1} &= \eta_j^{n} + \frac{\Delta t }{2} p_j^{n} \\
M_{ij} p_j^{n+1} +\frac{\Delta t}{2} S_{ij} \eta_j^{n+1} &= M_{ij} p_j^{n} -\frac{\Delta t}{2} S_{ij} \eta_j^... |
1,930,866 | <p>For a homework assignment, one of the questions requires finding the points where a parametric line in vector form intersects with a sphere whose radius changes. </p>
<p>The line $r(s)$ is denoted by $\textbf{OP} + s\textbf{v}$, where $P = (x,y,z)$, and $\textbf{v} = \langle a,b,c\rangle$ (In the problem, there are... | G Cab | 317,234 | <p>Find the plane that contains the center of the sphere (origin ?) and the line, i.e. two points of it, that you can easily determine (one is $\mathbf P$, and the other could be $\mathbf {P+v}$). Then find the distance of the line from the center (cross product of $\mathbf P$ with $\mathbf v /|\mathbf v|$).
Now everyt... |
1,979,666 | <p>Let be $I=\{1,2,\ldots, n\}$ is the set of indices.
I have an increasing sequence of positive real numbers: $$0<a_1<a_2<\ldots <a_n.$$
<strong>Question.</strong> How to denote the set $A$ correctly?</p>
<p>My attemps are:</p>
<ol>
<li>$A=\{a_i \in \mathbb{R}^+: a_i < a_{i+1}, i=1,2, \ldots,n-1\}.$<... | Joe Temple | 1,032,403 | <p>Here is another way: We can write Little's Law as : X = Work/T. Scaling laws are most often in this form: Relative X =Scale/Relative Response Time. These Laws are just normalized versions of Little's law where Work = the work done at scale = 1 and T is the response time for that work.</p>
|
195,413 | <p>I have tried to write a code to check whether a line is parallel to the plane in Mathematica.
Using that a plane has the normal vector of <span class="math-container">$\vec{n}=(a,b,c)$</span> and a line has a direction vector <span class="math-container">$\vec{L}=(l,m,n),$</span> the condition is <span class="math-c... | Roman | 26,598 | <p>You can give vector-values to variables and use scalar products:</p>
<pre><code>planenormal = {1, 2, 3};
linedirection = {3, 1, 9};
planenormal.linedirection == 0
</code></pre>
<blockquote>
<p>False</p>
</blockquote>
|
2,078,947 | <p>We all know of other languages, Spanish, English, Frensh, German...</p>
<p>...<strong>but</strong> <strong>is there</strong> <strong>another</strong> way to quantify and Analyse the world than <strong>Mathematics</strong> (in our age) ?</p>
| Nowibananatzki | 385,751 | <p>Suppose a matrix A and one of its columns <span class="math-container">$v_k$</span> is a linear combination of the others, we can form an equation as such:</p>
<p><span class="math-container">$$a_1v_1 + \ldots + a_{k-1}v_{k-1} + a_{k+1}v_{k+1} + \ldots + a_nv_n = a_kv_k$$</span></p>
<p>If the column vectors of A are... |
3,748,284 | <p>I have already tried Wolfram Alpha (not pro) and I don't know whether I can access MATLAB for free, any software than downloaded for free and is easy to use will work. I need it for computing complex Hermite polynomials and solving differential equations of order more than 3.</p>
| Lysistrata | 246,677 | <p>I suggest looking at Octave, a free Matlab-like replacement.
Also consider SageMath.
<a href="https://www.sagemath.org/" rel="nofollow noreferrer">https://www.sagemath.org/</a></p>
|
2,837,230 | <p>When using vector notation in coordinate systems (Cartesian coordinates) we see that the magnitude of a vector in two dimensions is equal to the square root of its Y component squared added to its X component squared (Pythagorean theorem).<br>
But the same calculation is done for a three dimensional vector that has ... | Josh B. | 4,308 | <p>A different variation of the Pythagorean theorem in 3D:
This one relates areas!</p>
<p>The black lines are perpendicular to each other.</p>
<p><a href="https://i.stack.imgur.com/Fh3Kn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Fh3Kn.png" alt="Triangle"></a></p>
<p>$Area(\Delta ABC)^2 = Are... |
2,837,230 | <p>When using vector notation in coordinate systems (Cartesian coordinates) we see that the magnitude of a vector in two dimensions is equal to the square root of its Y component squared added to its X component squared (Pythagorean theorem).<br>
But the same calculation is done for a three dimensional vector that has ... | user361424 | 361,424 | <p>Especially for the purposes of physics, let's not make this more complicated than it has to be. Start with the Pythagorean theorem:</p>
<p>$$h = \sqrt{x^2 + y^2}$$</p>
<p>Now consider the same in three dimensions. Choose at random two of the three vectors, which are by hypothesis both at right angles to the thir... |
158,907 | <p>Here is a simple demo code, with which I thought I could rotate a 3D point set around the different axes:</p>
<pre><code>SeedRandom[1];
data = RandomReal[10, {20000, 3}];
Manipulate[
e = EulerMatrix[{alpha Degree, beta Degree, gamma Degree}];
datanew = e.# & /@ data;
Graphics3D[{PointSize[Small], Point[da... | Nasser | 70 | <p>This answers the slowness part.</p>
<p>It is slow because you are using exact numbers for degrees and the matrix was not packed then. Compare the timing:</p>
<p>Before:</p>
<p><img src="https://i.stack.imgur.com/NkDan.png" alt="Mathematica graphics"></p>
<p>After</p>
<p><img src="https://i.stack.imgur.com/kAQs... |
158,907 | <p>Here is a simple demo code, with which I thought I could rotate a 3D point set around the different axes:</p>
<pre><code>SeedRandom[1];
data = RandomReal[10, {20000, 3}];
Manipulate[
e = EulerMatrix[{alpha Degree, beta Degree, gamma Degree}];
datanew = e.# & /@ data;
Graphics3D[{PointSize[Small], Point[da... | José Antonio Díaz Navas | 1,309 | <p>After my previous comments. I have tried with this (I have used the package Developer in order to pack the array, thus, making increasing the speed of the operations with the huge data):</p>
<pre><code><< Developer`;
SeedRandom[1];
data = ToPackedArray[RandomReal[10, {20000, 3}]];
Manipulate[
Graphics3D[
... |
2,324,718 | <p>To find the inverse of a function, you switch the spots of $x$ and $y$ in the function and solve for the new $y$ output. For example:</p>
<p>$y = 3 + \sin\ x$</p>
<ul>
<li>Switch positions of $x$ and $y$</li>
</ul>
<p>$x = 3 + \sin\ y$</p>
<ul>
<li>Solve for this new $y$</li>
</ul>
<p>$y = \sin^{-1}(x-3)$</p>
... | JJacquelin | 108,514 | <p>Elementary explanation with the example $$y(x)=3+\sin(x)$$</p>
<p>This means that putting a value $x$ into the above equation returns a value $y$.</p>
<p>To make things more simple, we ignore that this function is not bijective and should be restricted to a limited range.</p>
<p>Then, we look for a function $g$ s... |
2,389,247 | <p>I need to solve $$\int|x^n| \, dx$$ <strong>if n is an odd positive integer</strong>.</p>
<p>I think it should be $\dfrac{|x|^{n+1}}{n+1}+c$, but this is wrong. I am new to integration so I don’t know what to do when there is modulus function.</p>
<p>The answer given at back of book is $\dfrac{|x^n|x}{n+1} +c$.</p... | Community | -1 | <p><strong>Short answer:</strong></p>
<p>$|x^n|$ is an even function so that its antiderivative must be odd (to a constant), i.e. have the sign of $x$. This explains</p>
<p>$$\frac{|x^n|x}{n+1}.$$</p>
|
3,429,601 | <blockquote>
<p>If a variable line <span class="math-container">$x\cos\alpha+y\sin\alpha=p$</span> which is a chord of the hyperbola <span class="math-container">$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$</span>, <span class="math-container">$(b>a)$</span> subtend a right angle at the centre of hyperbola,then prove tha... | Blue | 409 | <p>The line equation seems a bit of a distraction. Here's how I'd solve the problem.</p>
<hr>
<p><a href="https://i.stack.imgur.com/UbzFo.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/UbzFo.png" alt="enter image description here"></a></p>
<p>Suppose <span class="math-container">$H$</span> and <s... |
34,664 | <p>Here is the post for context: <a href="https://math.stackexchange.com/questions/4415677/is-today-is-presidents-day-a-proposition-propositional-function-or-neither#4415684">Is "today is Presidents' Day" a proposition, propositional function, or neither?</a></p>
<p>Both the answers are really great and t... | Paul D. P. Blum | 371,438 | <p>As I cannot yet comment I shall add to the answer proposed by @John Bentin.</p>
<p><a href="https://en.wikipedia.org/wiki/Occam%27s_razor" rel="nofollow noreferrer">Occam's Razor (OR)</a> is a well known and arguably well established philosophical principle. Essentially, OR states that if both answers are equal in c... |
1,453,336 | <p>Let us define half-space as
$$ C = \{x\mid a^Tx\leq b\} $$
Intuitively (or geometrically), I understand why halfspace is not affine. But while I prove that half-space is convex, it seems to hold for affine case.</p>
<hr>
<p>Let us choose any $x_1,x_2\in C$ and $x=\theta x_1 + (1-\theta)x_2$, then
$$ a^Tx = a^T(\th... | Avijit Dasgupta | 284,830 | <p>The inequality $a^Tx=a^T(\theta x_1+(1-\theta)x_2)\leq \theta b+(1-\theta)b=b$ holds if and only if when $0\leq \theta \leq 1$. If suppose, $\theta <0 $ the value of $(1-\theta)$ becomes negative. We cannot really draw any conclusion about the inequality between L.H.S and R.H.S, i.e., $a^T(\theta x_1+(1-\theta)x... |
3,940,974 | <p><span class="math-container">$((A \land (A \rightarrow B)) \rightarrow B$</span></p>
<p><span class="math-container">$((A \land (A \rightarrow B))$</span> is premise
and <span class="math-container">$B$</span> is goal</p>
<p><span class="math-container">$((A \land (\neg A \lor B))$</span></p>
<p><span class="math-c... | Saeed | 858,459 | <p>Hint: replace <span class="math-container">$A \to B$</span> with its equivalent: <span class="math-container">$B \lor \lnot A$</span> .</p>
|
1,131,959 | <p>Let $G$ be a finite group and $K$ a field with $\mathrm{char}(K) \nmid |G|$, and let $M$ be a cyclic $KG$-module.</p>
<p>For a cyclic generator $v \in M$ we can pick group elements $g_1, \dots, g_d \in G$ such that $\{ g_1v, \dots, g_d v \}$ is a $K$-basis of $M$. Does that automatically mean that $\{ g_1w, \dots, ... | H.Kakuhama | 211,966 | <p>I think your trial will be not success. Because of the point of view "decomposition to the irreducible representations";</p>
<p>$K[G] \cong \oplus_{i=1}^h V_i$ as $K[G]$-module for some sub $K[G]$-modules $\{V_i\}_i$. (Giving $K[G]$-module and giving representation of $G$ are equivalent. And it is possible to take ... |
1,942,599 | <p>How can I calculate sums of the form:
$$\sum_{i=-1}^{5}\sum_{j = -5}^{1}(1 + 2i + 3j + 4ij)?$$</p>
<p>Shall I calculate it like first for $i = -1, j = \{-5,\dots, 1\}$, then for $i = 2, j = \{-5, \dots, 1\}$? Is there some shortcut for calculating those kinds of sums, i.e. something akin to summation of arithmetic ... | Brian M. Scott | 12,042 | <p>The method that you propose is certainly the most straightforward, but it is a bit tedious. You can speed things up by breaking it up into four sums and dealing with each of them separately:</p>
<p>$$\sum_{i=-1}^5\sum_{j=-5}^11+2\sum_{i=-1}^5\sum_{j=-5}^1i+3\sum_{i=-1}^5\sum_{j=-5}^1j+4\sum_{i=-1}^5\sum_{j=-5}^1ij\... |
744,412 | <p>If I take the $k \in \mathbb{N}$ power of 10 and mod it by a large prime, I notice that the remainders repeat at some point.</p>
<p>For instance $10^k mod~7$ seems to repeat every $8$th value of $k$.</p>
<p>Given $k$ (the power of ten), and a large prime number $p$, how can I compute the size of the cycle?</p>
<p... | Sandeep Silwal | 138,892 | <p>For $a^k \pmod p$, this cycle that you refer to has the name "order". We generally write this as $\text{ord}_p(a)$, that is the minimum $k$ such that $a^k \equiv 1 \pmod p$. (Then the cycle starts back at $a$). There is not really a good way to estimate this well. A weak estimate comes from Fermat's Little Theorem,... |
3,382,130 | <p>Let <span class="math-container">$h(y) = y\sqrt y$</span>.</p>
<p>The derivative of it is <span class="math-container">$\frac {3y} {2 \sqrt y}$</span> <em>or</em> <span class="math-container">$\frac {3 \sqrt y} 2$</span>, when more simplified. Although both graphs of the derivative are the same, when simplified fur... | k.stm | 42,242 | <p>You haven’t even given a domain for <span class="math-container">$h$</span>.</p>
<p>Here’s an answer that probably addresses your confusion:</p>
<ul>
<li>The maximal domain of the formal assignment <span class="math-container">$y ↦ \frac {3y} {2 \sqrt y}$</span> within the real numbers is <span class="math-contain... |
24,904 | <p>So I am having a lot of trouble proving this. It was for an assignment due tuesday, but the prof said I can have a bit of extra time with this question. </p>
<p>Please do not give me the answer because I am sure my prof is on this website :P</p>
<p>okay so the question is as follows:</p>
<blockquote>
<p>Let $R$... | Andrea Mori | 688 | <p>So, I guess that for $m$ and $n$ positive you need to prove that
$$
\left(\sum_{i=1}^m\,1_R\right)\left(\sum_{i=1}^n\,1_R\right)=
\left(\sum_{i=1}^{mn}\,1_R\right).
$$
How many addends you have when you apply the distributive property on the LHS ?</p>
<p>Then you have to come up with something for when $m$ and $... |
958,486 | <p>I'm trying to prove that</p>
<p>$$x^2 + x + 1 = y^2$$</p>
<p>has no integer solution. So far I've tried proof by contradiction, but all of that seems to rely on me being able to factor this expression into some neat form where the solution is obvious. I've not been able to do so.</p>
| Mike | 17,976 | <p>Start by multiplying both sides by $4$.</p>
<p>$$4y^2=4x^2+4x+4$$
$$(2y)^2=(2x+1)^2+3$$
$$(2y)^2-(2x+1)^2=3$$</p>
<p>The only $2$ squares whose difference is $3$ are $4$ and $1$. So $2x+1=\pm1$, which has no positive solution. </p>
|
958,486 | <p>I'm trying to prove that</p>
<p>$$x^2 + x + 1 = y^2$$</p>
<p>has no integer solution. So far I've tried proof by contradiction, but all of that seems to rely on me being able to factor this expression into some neat form where the solution is obvious. I've not been able to do so.</p>
| said salah2020 | 763,257 | <blockquote>
<p><span class="math-container">$x^2 + x + 1 = y^2$</span> has no positive integer solutions.</p>
</blockquote>
<p>Assume that <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are positive: then<br>
<span class="math-container">$$x^2 - y^2 = -(x+1)$$</span>
Therefore... |
8,760 | <p>What are the three cube roots of -1?</p>
<p>Not sure if this is a trick question, But I have been asked this. one of the ansers is -1, what are the other 2?</p>
| Zarrax | 3,035 | <p>Write $-1$ in polar form as $e^{i\pi}$. In general, the cube roots of $r e^{i\theta}$ are given by $r^{1/3}e^{i\theta/3}$, $r^{1/3}e^{i(\theta/3 + 2\pi /3)}$ and $r^{1/3}e^{i(\theta/3 + 4\pi /3)}$. In your case $r = 1$ and $\theta = \pi$, so your cube roots are $e^{i\pi / 3}$, $e^{i\pi}$, and $e^{i 5\pi/ 3}$. Put ba... |
545,978 | <p>Can it be done?
We have the real field $(\Bbb R,+,-,\times,0,1,<)$, of course $(0,1,-,<)$ are definable using the rest.
We take an elementary non-archimedean extension. Can we define the original set of reals in it?</p>
| Andrés E. Caicedo | 462 | <p>No, you cannot: You have the supremum property for bounded <em>definable</em> sets, since this is first-order and your extension is elementary. If $\mathbb R$ is definable, it would be bounded (by any infinite element of the field). So there would be a "least" infinite number.</p>
|
2,158,115 | <p>I am trying to write down the set of all ordered pairs $(n,m)$ such that $n=2k+1:k\in\mathbb{Z}^{+}$ and $m=2l+1:l\in\mathbb{Z}^{+}$. How would that be written in set notation?</p>
| B. Mehta | 418,148 | <p>I would use $$\left\{(n,m)\in \mathbb{Z}^2 \mid \exists k, l \in \mathbb{Z}^+:n=2k+1,m=2l+1\right\}$$</p>
|
745,922 | <p>How did Leibniz know to write derivatives as $$\frac{dy}{dx}$$ so that everything would work out? For example, the chain rule: $$\frac{dy}{dz}=\frac{dy}{dx}\frac{dx}{dz}$$ Integration by Parts: $$xy=\int d(xy)=\int x\,dy+y\,dx \implies \int x\,dy =xy-\int y\,dx$$ Separable differential equations: $$\frac{dy}{dx}=\fr... | Michael Hardy | 11,667 | <p>Leibniz regarded $dx$ and $dy$ respectively as infinitely small increments of $x$ and $y$, so that $dy/dx$ is the ratio of the infinitely small change in $y$ corresponding to the infinitely small change $dx$ in $x$. Thus when $dy$ is $7$ times as big as $dx$ at a particular, point, then at that point $y$ is changin... |
745,922 | <p>How did Leibniz know to write derivatives as $$\frac{dy}{dx}$$ so that everything would work out? For example, the chain rule: $$\frac{dy}{dz}=\frac{dy}{dx}\frac{dx}{dz}$$ Integration by Parts: $$xy=\int d(xy)=\int x\,dy+y\,dx \implies \int x\,dy =xy-\int y\,dx$$ Separable differential equations: $$\frac{dy}{dx}=\fr... | Marcel Besixdouze | 29,892 | <p>As Michael points out, Leibniz thought of $\frac{dy}{dx}$ as a ratio between an infinitesimal quantity $dy$ and an infinitesimal quantity $dx$, and the integral as a sum of infinitesimal quantities $f(x)dx$. </p>
<p>You asked if Newton had a different perspective. Indeed, Newton never considered $\frac{dy}{dx}$ to ... |
542,582 | <p>I have a question which I'm unable to provide much working for as I'm not sure how to start it. </p>
<p>A metal sphere is placed in seawater to study the corrosive effect of seawater. If the surface area decreases at a rate of $35cm^2/year$ due to corrosion, how fast is the radius changing when the radius is $7cm$?... | Aaron | 9,863 | <p>You noted that $\frac{d}{dt}s=35$cm${}^2$/year, which is very close to correct. While the surface area is changing by that much, the derivative doesn't just measure change, but the direction of change. Since surface area is decreasing, $\frac{d}{dt}s=-35$cm${}^2$/year.</p>
<p>The next step is to differentiate the... |
381,771 | <p>Let <span class="math-container">$p$</span> be a prime. How many solutions <span class="math-container">$(x, y)$</span> are there to the equation <span class="math-container">$x^2 + 3y^2 \equiv 1 \pmod{p}$</span>? Here <span class="math-container">$x, y \in \{0, 1, \ldots p-1\}$</span>. This paper (<a href="https://... | Noam D. Elkies | 14,830 | <p>The count is <span class="math-container">$2$</span> and <span class="math-container">$6$</span> for <span class="math-container">$p=2$</span> and <span class="math-container">$p=3$</span> respectively,
and otherwise <span class="math-container">$p-1$</span> or <span class="math-container">$p+1$</span> according as ... |
381,771 | <p>Let <span class="math-container">$p$</span> be a prime. How many solutions <span class="math-container">$(x, y)$</span> are there to the equation <span class="math-container">$x^2 + 3y^2 \equiv 1 \pmod{p}$</span>? Here <span class="math-container">$x, y \in \{0, 1, \ldots p-1\}$</span>. This paper (<a href="https://... | Noam D. Elkies | 14,830 | <p>We give yet another approach to proving the result,
using nothing more advanced than properties of the Legendre symbol.</p>
<blockquote>
<p><strong>Proposition</strong>.
<em>Suppose <span class="math-container">$p$</span> is an odd prime and <span class="math-container">$D$</span> is not a multiple of <span class="m... |
2,143,658 | <blockquote>
<p>Show that no two of the spaces $(0, 1), (0, 1], [0, 1]$ are homeomorphic</p>
</blockquote>
<p><em>My Attempted Proof</em></p>
<p>We have $(0, 1) \subset (0, 1] \subset [0, 1]$.</p>
<p>We also have $(0, 1] = (0, 1) \cup \{1\}$ and thus $|(0, 1)| \neq |(0, 1]|$ and thus no bijective mapping $f : (0,... | 5xum | 112,884 | <blockquote>
<p>We also have $(0, 1] = (0, 1) \cup \{1\}$ and thus $|(0, 1)| \neq |(0, 1]|$ and thus no bijective mapping $f : (0, 1) \to (0, 1]$ exists</p>
</blockquote>
<p><strong>False</strong>. There exists a bijective mapping from $(0,1]$ to $(0,1)$. For example, one of them is</p>
<p>$$f(x) = \begin{cases}\fr... |
225,128 | <p>Six points are connected in pairs by lines each of which is either red or blue. Every pair of points is joined. Determine whether there must be a closed path having four sides all of the same colour. A path is closed if it begins and ends at the same point.</p>
| Bart Michels | 43,288 | <p>Let $f(x)=ax^3+bx^2+cx+d$. That means the sign of $f''(x)$ changes when $6ax+2b=0$, so when $x=\frac{-b}{3a}$.</p>
<p>Since the sum of the roots of $f$ equals $\frac{-b}{a}$, we have that $x=\frac{-b}{3a}=\frac{x_1+x_2+x_3}{3}$.</p>
|
225,128 | <p>Six points are connected in pairs by lines each of which is either red or blue. Every pair of points is joined. Determine whether there must be a closed path having four sides all of the same colour. A path is closed if it begins and ends at the same point.</p>
| wj32 | 35,914 | <p>You know that $f(x)=ax^3+bx^2+cx+d$, so $f''(x)=6ax+2b=0$ implies that $x=$ $___$. What do you know about the relationship between the coefficients $a,b,c,d$ and the roots $x_1,x_2,x_3$?</p>
|
1,046,864 | <p>Some time ago, I came across an anecdotal story about the "Big Book of Proofs" that God always keeps up in the heaven, which records valid proofs of all theorems in the world. A noted mathematician was quoted as saying that if he was to die tomorrow, the first thing he would do is to take a look at the book, since h... | daOnlyBG | 173,397 | <p>I think the gentleman you're thinking of is Paul Erdős: <a href="http://en.wikipedia.org/wiki/Proofs_from_THE_BOOK">http://en.wikipedia.org/wiki/Proofs_from_THE_BOOK</a></p>
|
2,257,878 | <p>We have to prove that the equation $\displaystyle \frac{x^5}{5!} +\frac{x^4}{4!} +\frac{x^3}{3!}+\frac{x^2}{2!} +x+1=0$ have exactly one real root .</p>
<p>My sir told me it is just an application of derivative .</p>
<p>But I could not understand what he mean by that . Can anybody please explain me .</p>
| Sri-Amirthan Theivendran | 302,692 | <p>You can prove a bound like $F_n<2^n$ using induction so that the power series conveges at least for $|x|<1/2$ and the manipulations are valid. More generally, for a linear recurrence with constant coefficients, the radius of convergence of the resulting power series will be the minimum of the reciprocals of th... |
2,257,878 | <p>We have to prove that the equation $\displaystyle \frac{x^5}{5!} +\frac{x^4}{4!} +\frac{x^3}{3!}+\frac{x^2}{2!} +x+1=0$ have exactly one real root .</p>
<p>My sir told me it is just an application of derivative .</p>
<p>But I could not understand what he mean by that . Can anybody please explain me .</p>
| Jair Taylor | 28,545 | <p>One way to do this is to first prove that the series converges and then do these calculations, as Foobaz John outlines. But in fact, all these manipulations are perfectly valid without even taking convergence into consideration if you frame them in the right way - that is, in terms of <em>formal power series</em>. ... |
1,009,169 | <p>I have to examine limit of following sequence $\{a_n\}_{ n \ge 1}$ $a_n = \sqrt[n]{\sum_{k=1}^{n}{(2 -\frac{1}{k})^k}}$. We know that $\lim_{n \to \infty} \sqrt[n]{a} = 1$ for $a > 0$ and we know that $\sqrt[n]{\sum_{k=1}^{n}{(2 -\frac{1}{k})^k}} > 0$ but sequence $a_n$ is increasing (I cannot prove it but I c... | Tacet | 186,012 | <p>We can apply <a href="https://en.wikipedia.org/wiki/Squeeze_theorem" rel="nofollow">squeeze theorem</a> to solve it.
$$\left(\lim_{n\to\infty} b_n = \lim_{n\to\infty}c_n = m \wedge \left(\exists \delta \in \mathbb{R}\right)\left(\forall n \in \mathbb{N}\right)\left( n \geq \delta \Rightarrow b_n \leq a_n \leq c_n \r... |
313,064 | <p>I have a problem with an exercise:
Find the probability that when throwing three dice, the six will occur on one of them (no matter which one), if on the facets of the remaining two dice, two different numbers will outcome.</p>
<p>The answer given to this exercise is 1/2. However I think it is $\frac{60}{6^3}=\frac... | polkjh | 54,811 | <p>With your edit, the question now is - 'given that the outcomes are all distinct, what is the probability that one of them is $6$'. Number of ways in which we can have distinct numbers on the three dice is $6*5*4=120$. And the number of ways in which none of them is $6$ is $5*4*3=60$. So the number of ways in which o... |
76,147 | <p>Is it possible to define a Borel $\sigma$-algebra on a topological space which is not second countable, i.e. one which does not have a countable base?</p>
<p>I am trying to learn measure theory and my intuition says that this is NOT possible, because if the space is not second countable, then there are "too many" ... | Community | -1 | <p>I found this post because I wondered something related, and I thought I'd post an observation here:</p>
<p>Yes, defining the Borel $\sigma$-algebra goes through trivially. But I think certain fundamental, "nice" theorems don't. For example, theorem 1.8 of big Rudin, page 11, says that if $u,v: X\to\mathbb{R}$ are m... |
313,723 | <p>Given a vector <span class="math-container">$\mathbf {x} =(x_{1},\ldots ,x_{n})$</span> the p-norm is defined as
<span class="math-container">$$
\left\|\mathbf {x} \right\|_{p}:={\bigg (}\sum _{i=1}^{n}\left|x_{i}\right|^{p}{\bigg )}^{1/p}
$$</span>
for <span class="math-container">$p \geq 1$</span>.</p>
<p>For <s... | Gerald Edgar | 454 | <p>A simple relation between <span class="math-container">$\|\cdot\|_p$</span> and <span class="math-container">$\|\cdot\|_{-p}$</span> is
<span class="math-container">$$
\displaystyle\left\|\frac{1}{\mathbf x}\right\|_{-p} = \frac{1}{\|\mathbf x\|_{p}}
$$</span>
where <span class="math-container">$\frac{1}{\mathbf x}$... |
612,058 | <blockquote>
<p>Let $g:\mathbb{R}\to \mathbb{R}$ and $f:\mathbb{R}\to \mathbb{R}$ be functions, such that $f(x)\leq g(x)$ for each $x \in \mathbb{R}$. Tell if $\sup(f)\leq \sup(g)$ is true or false. If true, prove it. If false give a counterexample.</p>
</blockquote>
<p>Well, $\sup(f)$ stands for the supremum of the... | drhab | 75,923 | <p>If $\sup f>\sup g$ then $\sup g$ must be finite and some $y\in\mathbb{R}$
must exist with $f\left(y\right)>\sup g\geq g\left(y\right)$. This
contradicts that $f\left(x\right)\leq g\left(x\right)$ for each $x$.</p>
|
3,866,495 | <p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be finite sets, prove that <span class="math-container">$|A\times B|=|A||B|$</span>.</p>
<p>Since <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are finite sets, then it can be represented as ... | Robert Shore | 640,080 | <p>For any <span class="math-container">$0\lt k \leq mn~ \exists q, r~(k=qn+r)$</span> with <span class="math-container">$0 \leq r \lt n$</span> and <span class="math-container">$q \in \Bbb N$</span>. Note that <span class="math-container">$q, r$</span> are unique with those properties. If <span class="math-container... |
177,181 | <p>I have a object which is similar to the shape of irregular rock like this <img src="https://i.stack.imgur.com/Zz55K.gif" alt="enter image description here"></p>
<p>I would like to find the volume of this. How to do it?</p>
<p>If I have to find the volume, what are the things I would need. eg., If it is cylindrical... | Emily | 31,475 | <p>As your comment indicates, you're not interested in rocks so much as tumors.</p>
<p>One possible approach is to use a tomographic technique. Many medical imaging tools image the body using tomography: that is, examining the body one "slice" at a time.</p>
<p>If you have access to such tools, or can derive such an ... |
1,361,766 | <p>I am stucked at this combinatorics/probability problem:</p>
<hr>
<p>There are 10 urns, each contains 8 balls numbered 1,2,...,8</p>
<p>If we randomly choose 1 ball from <strong>each</strong> urn, what is the probability that the <strong>total</strong> number of 1's will be <strong>exactly</strong> 3 and the <stro... | Brian Cheung | 248,555 | <p>First you choose $10-3-2=5$ urns out which are neither $1$ nor $4$ $(C^{10}_5\times6^5)$<br>
Among the remaining $5$ choose $3$ out, which are the $1$'s $(C^5_3)$<br>
The remaining 2 will be the $4$'s<br>
Number of ways satisfying the conditions $=C^{10}_5\times6^5\times C^5_3$<br>
Total number of ways $=8^{10}$<br>... |
1,361,766 | <p>I am stucked at this combinatorics/probability problem:</p>
<hr>
<p>There are 10 urns, each contains 8 balls numbered 1,2,...,8</p>
<p>If we randomly choose 1 ball from <strong>each</strong> urn, what is the probability that the <strong>total</strong> number of 1's will be <strong>exactly</strong> 3 and the <stro... | Community | -1 | <p>Select 3 out of 10 urns for 1's. Then 2 out of remaining 7 urns for 4's:</p>
<p>$$\binom{10}{3}\binom{7}{2}\left(\frac{1}{8}\right)^3\left(\frac{1}{8}\right)^2\left(1-\frac{2}{8}\right)^5=\frac{76545}{4194304}\approx 0.0183$$</p>
|
2,611,860 | <p>I am stuck in this exercise. it asks me to:</p>
<p>Proof that the only ring homomorphism $\phi:\Bbb Z_{n} \rightarrow \Bbb Z$ is the trivial one $\phi(n)=0$</p>
<p>I believe that I have to show that the kernel $K=\{n \in R|\phi(n)=0_{s} \}$</p>
<p>But I am not sure how to start, any hints will be appreciated</p>
| 1ENİGMA1 | 255,913 | <p>$\phi :\Bbb Z\to \Bbb Z_n$ such that $\phi(x)=\bar{x}$. Then
$$\rm{ker}\phi=\{x\in \Bbb Z:\phi(x)=\bar{0}\}$$
$$\rm{ker}\phi=\{x\in \Bbb Z:\phi(x)=\bar{x}=\bar{0}\}$$
$$\rm{ker}\phi=\{x\in \Bbb Z:x\sim 0\}$$
$$\rm{ker}\phi=\{x\in \Bbb Z:x-0=nk, k\in \Bbb Z\}$$
$$\rm{ker}\phi=\{x\in \Bbb Z:x=nk, k\in \Bbb Z\}=\langl... |
1,271,119 | <p>Say our world is actually 4-dimensional, and I have a tesseract sitting on my table. To a being only capable of perceiving 3 dimensions, what would it actually look like?</p>
<p>Would it just look like a cube?</p>
<p>I understand that the <em>projection</em> of a tesseract into 3D space looks like the image shown ... | Claude Leibovici | 82,404 | <p>You did fine. Repalicng your values in the term inside the radical, you have
$$a^5+b^2c^2+a^3c^2=d^5 k^{15}+d^5 k^{11}+d^4 k^6$$ $$b^4c+d^4+b^2cd^2=d^5 k^9+d^5 k^5+d^4$$ thet is to say that the ratio is $k^6$, its square root is $k^3$ which is $\frac ad$.</p>
|
1,313,670 | <p>The standard birthday question is, how many people have to be in a math class for the probability of two having a shared birthday is > 1/2. With standard assumptions about birthday distributions and independence of birthdays.</p>
<p>A slightly different question.</p>
<p>How many people need to be in a stadium for ... | gsmafra | 176,567 | <p>An estimate for a) would be</p>
<p>$p \approx ( \sum_{k=2 .. n} \mathcal{B}_{n, 1/365}(k) )^{365}$</p>
<p>$\mathcal{B}_{n,p}$ is the binomial distribution</p>
<p>The solution would have to be found numerically</p>
<p>What makes this equation an approximation is that the distributions $p(n_{people})$ which are ma... |
259,382 | <p>it seems first easy to me, but now i am tossing my head against wall not being able to solve the problem. i need to check for convergence of this sequence below. i dont know how to start although it seems to be very easy one</p>
<p>$\lim_{n \to \infty} i^{3n} = help = ?$ </p>
<p>i need help here. do i have to work... | Community | -1 | <p>Note that $i^3 = -i$. Hence, $i^{3n} = (-i)^n$.
$$i^{3n} = (-i)^n = \begin{cases} 1 & n \equiv 0 \pmod{4}\\ -i & n \equiv 1 \pmod{4}\\ -1 & n \equiv 2 \pmod{4}\\ i & n \equiv 3 \pmod{4} \end{cases}$$</p>
|
2,557,395 | <blockquote>
<p>Let $R$ be a ring and $M,N$ be $R-$modules. Let $\pi: M \to N$ be a
module homomorphism. Let $K$ be a $R$-submodule of $N$. Then
$\pi^{-1}(K)$ is a submodule of $M$</p>
</blockquote>
<p>Surprisingly, I couldn't find this result on google. </p>
<p>Is my proof correct?</p>
<p><strong>Proof:</str... | Fabio Lucchini | 54,738 | <p>You can also note that $\pi^{-1}(K)$ is the kernel of the $R$-module homomorphism $M\xrightarrow{\pi} N\longrightarrow N/K$, where $N\longrightarrow N/K$ is the canonical projection onto the factor module.
Since kernels are submodules this proves again that $\pi^{-1}(K)$ is an $R$-module.</p>
|
1,163,609 | <p>Find
$$\lim_{x \to \frac{3\pi}{4}}\frac{1+(\tan x)^{1/3}}{1-2\cos^2x}$$</p>
<p>I don't think series expansion will help, then how do I solve it?</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$1-2\cos^2x=-(2\cos^2x-1)=-\cos2x$$</p>
<p>and</p>
<p>$$\cos2x=\frac{1-\tan^2x}{1+\tan^2x}$$</p>
<p>and using $a^3-b^3=(a-b)(a^2+ab+b^2),$</p>
<p>$$1+(\tan x)^{1/3}=\frac{1+\tan x}{1+(\tan x)^{1/3}+(\tan x)^{2/3}}$$</p>
<p>For $x\to\dfrac{3\pi}4,\tan x\to\tan\dfrac{3\pi}4=\tan\left(\pi-\dfrac{\pi... |
1,163,609 | <p>Find
$$\lim_{x \to \frac{3\pi}{4}}\frac{1+(\tan x)^{1/3}}{1-2\cos^2x}$$</p>
<p>I don't think series expansion will help, then how do I solve it?</p>
| imranfat | 64,546 | <p>Alternatively, the limit can be transformed into a rational expression through the substitution $tanx=t^3$ From here you can deduce (through right triangle trigonometry) the relation $cos^2x=\frac{1}{t^6+1}$ The expression becomes after simplification $\frac{(1+t)(t^6+1)}{t^6-1}$ which can be reduced through factori... |
4,072,240 | <p>I'm stuck trying to solve this ODE and I wonder if someone can help, thanks in advance. The ODE is the following:</p>
<p><span class="math-container">$$(y+2)^2 y' + (x + 2y + 3)^2 = 0 $$</span></p>
<p>I first tried expanding the squared terms and trying to separate the variables but it was not very helpful. Then I n... | Robert Israel | 8,508 | <p>As given, it is not homogeneous, but you can make a substitution to make it homogeneous: <span class="math-container">$Y = y+2$</span>, <span class="math-container">$X = x - 1$</span></p>
|
2,604,548 | <p>I am trying to solve some <strong>ratio tests</strong>. I came across to my notes and i can't figure out how is that equation simplified.
Well, how is $\frac{\left(2\left(n+1\right)\right)!}{\left(n+1\right)!}\cdot \frac{n!}{\left(2n\right)!}$ simplified to $\frac{\left(2n+1\right)\left(2n+2\right)}{n+1}$</p>
| Mehrdad Zandigohar | 521,055 | <p>$[1]:\frac{(2(n+1))!}{(2n)!}$=$\frac{(2n+2)!}{(2n)!}$=$\frac{(2n+2)(2n+1)(2n)!}{(2n)!}=(2n+2)(2n+1)=2(n+1)(2n+1)$<br>
$[2]:\frac{(n)!}{(n+1)!}=\frac{(n)!}{(n+1)(n)!}=\frac{1}{n+1}$<br></p>
<p>Multiply 1 by 2:<br></p>
<p>$\frac{(2(n+1))!}{(2n)!}.\frac{(n)!}{(n+1)!}=\frac{(2n+2)(2n+1)}{n+1}=2(n+1)(2n+1)(\frac{1}{n+1... |
4,695 | <p>Let S be a finite set of primes in Q. What, if anything, do we know about K3 surfaces over Q with good reduction away from S? (To be more precise, I suppose I mean schemes over Spec Z[1/S] whose geometric fibers are (smooth) K3 surfaces, endowed with polarization of some fixed degree.) Are there only finitely man... | Community | -1 | <p>Some thoughts.</p>
<p>There are no such varieties when S = 1. This is a consequence of a theorem of Fontaine, MR1274493 (Schémas propres et lisses sur Z).</p>
<p>I think that one should only expect finitely many such varieties for any fixed S. Let me give an argument that uses every possible conjecture I know. The... |
148,466 | <p>Is it true that for every <span class="math-container">$t$</span> there is an <span class="math-container">$n$</span> and there exists a finite function
family, <span class="math-container">$\cal F$</span>, whose members are from <span class="math-container">$[n] \to \mathbb N$</span> (taking all different
values) a... | David E Speyer | 297 | <p>I can achieve $t=2$, $n=21$, $|\mathcal{F}|=7$. Every pair from $\mathcal{F}$ is defeated by some other element on $11$ of the $21$ coordinates. For future answer's sake, I propose that we report these parameters as $(t,n,c,p) = (2,21,7,11/21)$. If I am not mistaken, this is the first $(2,n,c,p)$ anyone has reported... |
1,246,694 | <p>This is the solution provided:
$$C(k+n-1,n) \times n!$$</p>
<p>I know that $C(k+n-1,n)$ denotes the number of ways n indistinguishable books can be placed on k distinguishable shelves. But I'm not too sure how multiplying that by $n!$ will give us the solution to the question :/ </p>
<p>On a side note, when we ar... | Marko Riedel | 44,883 | <p>As an incentive to further study here is an answer using species. The species under consideration is
$$\mathfrak{S}_{=k}(\mathfrak{S}(\mathcal{Z}))$$ which yields the exponential generating function $$\frac{1}{(1-z)^k}$$ for an answer of
$$n! [z^n] \frac{1}{(1-z)^k}$$
which is
$$n!\times {n+k-1\choose n}.$$</p>
|
407,570 | <p>The canonical one-form is defined here:
<a href="http://books.google.nl/books?id=uycWAu1yY2gC&lpg=PA128&dq=canonical%20one%20form%20hamiltonian&pg=PA128#v=onepage&q&f=false" rel="nofollow">http://books.google.nl/books?id=uycWAu1yY2gC&lpg=PA128&dq=canonical%20one%20form%20hamiltonian&p... | Rhys | 47,565 | <p>This sort of thing can be a bit confusing!</p>
<p>A local one-form on $N$ is given by $\sum_j y_j(x) dx_j$; this gives coordinates $(y_1,\ldots,y_n)$ on the fibres of $T^*N$, so we have local coordinates $(x_1,\ldots,x_n,y_1,\ldots,y_n)$ on the total space of $T^*N$. Therefore a local one-form on $T^*N$ is $\sum_j... |
407,570 | <p>The canonical one-form is defined here:
<a href="http://books.google.nl/books?id=uycWAu1yY2gC&lpg=PA128&dq=canonical%20one%20form%20hamiltonian&pg=PA128#v=onepage&q&f=false" rel="nofollow">http://books.google.nl/books?id=uycWAu1yY2gC&lpg=PA128&dq=canonical%20one%20form%20hamiltonian&p... | Alex Ortiz | 305,215 | <p>To make it even more transparent, I would do the following. Suppose <span class="math-container">$N$</span> is our manifold, <span class="math-container">$T^*N$</span> is the total space of the cotangent bundle, <span class="math-container">$\pi\colon T^*N\to N$</span> is the projection, and <span class="math-contai... |
1,833,470 | <p>I am having a problem with this calculus problem:</p>
<blockquote>
<p>The rate of change of the number of coyotes $N(t)$ in a population is
directly proportional to $650−N(t)$, where $t$ is the time in years. That
is,</p>
<p>$dN/dt=k(650−N)$.</p>
<p>When $t=0$, the population is $270$, and when $t=2... | mvw | 86,776 | <p>This can be integrated by multiplying both sides with a suitable integrating factor:
$$
650 k = k N + dN/dt \Rightarrow \\
650k e^{kt} = k e^{kt} N + e^{kt} dN/dt = (d/dt)(e^{kt} N) \Rightarrow \\
650 \int\limits_0^t k e^{k\tau} d\tau =
650 \left[ e^{k\tau} \right]_{\tau=0}^{\tau=t} =
650 \left( e^{kt} - 1 \right)... |
596,710 | <p>Help me? please How to solve this integral?
$$\int\frac{1+x^2}{1+x^4}\,dx$$</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$\displaystyle \int\frac{1+x^2}{1+x^4}dx=\int\frac{\frac1{x^2}+1}{x^2+\frac1{x^2}}dx$$</p>
<p>As $\displaystyle\int\left(\frac1{x^2}+1\right)dx=x-\frac1x$</p>
<p>write $\displaystyle x^2+\frac1{x^2}=\left(x-\frac1x\right)^2+2 $ and set $\displaystyle x-\frac1x=u$</p>
<p>Then use <a href="http://en.... |
2,303,803 | <p>It is given the $A,B$ and $C$ are the angles of a $\triangle ABC$.
$\angle A >\angle B$ and both the angles follow the equation:</p>
<p>$3 \sin x - 4 \sin ^3x-k=0 $ ; $k \in (0,1)$
and we have to figure out $\angle C$ out of the following options:</p>
<p>1.$\pi / 3$ </p>
<p>2.$\pi /2$</p>
<p>... | lab bhattacharjee | 33,337 | <p>If $0<\sin3x<1,$</p>
<p>$m360^\circ<3x<m360^\circ+180^\circ\iff m120^\circ<x<m120^\circ+60^\circ$ where $m$ is any integer</p>
<p>If $\sin3A=\sin3B$</p>
<p>$3A=n180^\circ+()^n3B$ where $n$ is any integer</p>
<p>$\implies A=n60^\circ+(-1)^n B$</p>
<p>We have $0<A+B<180^\circ\ \ \ \ (1),... |
115,345 | <p>If $a,b,c$ are primitive Pythagorean triples such that $c^2=a^2+b^2$, prove that this is unique primitive Pythagorean triple using $c$ as hypotenuse.</p>
<p>I'm not sure if I phrased this correctly. I want to ask, if $c^2=a^2+b^2$ where $a,b,c$ are pairwise coprime, prove that there are no $a_1$ and $b_1$ such that... | Robert Israel | 8,508 | <p>$63^2 + 16^2 = 33^2 + 56^2 = 65^2$</p>
|
2,771,404 | <p>This question is a trigonometric one from what I can tell. I understand that you need to differentiate the given function and the unknown variable on top ($dx$) will correspond to the $x$ unknowns, for example $5x^2$ will give us $10x\frac{dx}{dt}$.</p>
<p>With this specific question I can only get to $-\frac{\sec^... | Bernard | 202,857 | <p>Rewrite the relation as
$$t\,\tan x=4,\quad\text{so }\tan x=\frac 4t,$$
and differentiate this relation:
\begin{align}
&&&\mathrm d t\,\tan x+t(1+\tan^2x)\,\mathrm d x=0 \\[1ex]
&\text{whence}\qquad&\frac{\mathrm d x}{\mathrm d t}&=-\frac{\tan x}{t(1+\tan^2x)}=-\frac{t\tan x}{t^2+(t\tan x)^2}... |
3,426,659 | <p>Find the domain and range of <span class="math-container">$f(x)=\sin^{-1}(\sqrt{x^2+x+1})$</span></p>
<p><strong>Finding the domain:</strong></p>
<p><span class="math-container">$$x^2+x+1>=0 \text { it is always true }$$</span></p>
<p><span class="math-container">$$-1<=\sqrt{x^2+x+1}<=1$$</span>
<span cl... | Axion004 | 258,202 | <blockquote>
<p><span class="math-container">$$f(x)\in \left[2m\pi+\dfrac{\pi}{3},2m\pi+\dfrac{2\pi}{3}\right] \text { where m is integer }$$</span></p>
</blockquote>
<p>This isn't true because the range of the inverse sine function is from <span class="math-container">$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$</sp... |
4,440,622 | <p>Solve set of equations:
<span class="math-container">\begin{array}{rcl}
\sqrt{a^{2}+b^{2}}(a-1)&=&3a-\frac{b}{2}\\
\sqrt{a^{2}+b^{2}}(b-1)+6&=&3b+\frac{a}{2}\\
\end{array}</span></p>
<p>I've try to add, sub 2 equation. I also try multiple the first equation with <span class="math-container">$(b-1)$</... | Yuval Peres | 360,408 | <p>Call the required limit <span class="math-container">$L$</span>. Use the Taylor expansion
<span class="math-container">$$\cos t=1-t^2/2+O(t^4)$$</span> to deduce that
<span class="math-container">$$L=\lim_n \sum_{k=1}^n [2(k/n)^2 n^{-1}+O(n^{-2})] =\int_0^1 2x^2 \,dx=2/3 \,,$$</span>
since the sum is a Riemann sum.<... |
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