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 |
|---|---|---|---|---|
95,242 | <p>Is it possible to use <code>ProbabilityScalePlot</code> to show different plot markers in a single dataset, such as in going from <code>plot2</code> to <code>plot3</code> below?</p>
<pre><code>nPoints = 10;
x = RandomVariate[NormalDistribution[1, 1], nPoints];
y = RandomVariate[LogNormalDistribution[1, 1], nPoints]... | george2079 | 2,079 | <p>More complicated, but I thought it interesting to see how to generate the plot from first principles:</p>
<pre><code>nPoints = 10;
x = RandomVariate[NormalDistribution[1, 1], nPoints];
y = RandomVariate[LogNormalDistribution[1, 1], nPoints];
z = RandomVariate[WeibullDistribution[1, 1], nPoints];
data = {x, y, z};
n... |
95,242 | <p>Is it possible to use <code>ProbabilityScalePlot</code> to show different plot markers in a single dataset, such as in going from <code>plot2</code> to <code>plot3</code> below?</p>
<pre><code>nPoints = 10;
x = RandomVariate[NormalDistribution[1, 1], nPoints];
y = RandomVariate[LogNormalDistribution[1, 1], nPoints]... | kglr | 125 | <pre><code>colorF = Piecewise[{{Red, MemberQ[x, #[[1]]]}, {Green, MemberQ[y, #[[1]]]}}, Blue] &;
Normal[ProbabilityScalePlot[Flatten[{x, y, z}]]] /.
Point[p_] :> ({colorF @ #, PointSize[.015], Point @ #} & /@ p)
</code></pre>
<p><a href="https://i.stack.imgur.com/SQGSh.png" rel="nofollow noreferrer"><i... |
7,025 | <p>Many commutative algebra textbooks establish that every ideal of a ring is contained in a maximal ideal by appealing to Zorn's lemma, which I dislike on grounds of non-constructivity. For Noetherian rings I'm told one can replace Zorn's lemma with countable choice, which is nice, but still not nice enough - I'd lik... | Guillermo Mantilla | 2,089 | <p>There are certain kinds of rings that came to my mind when I saw this question. $K[v]:=$ Coordinate ring of an affine variety $V$ over a field $K$ , and $C(X, F)$:= the ring of continuous $F$-valued ($F$ a topological field) functions on a compact space $X$. In both examples one can construct maximal ideals as zero ... |
7,025 | <p>Many commutative algebra textbooks establish that every ideal of a ring is contained in a maximal ideal by appealing to Zorn's lemma, which I dislike on grounds of non-constructivity. For Noetherian rings I'm told one can replace Zorn's lemma with countable choice, which is nice, but still not nice enough - I'd lik... | David E Speyer | 297 | <p>It seems to me like there are two points here. </p>
<p>(1) A ring is called noetherian if any ascending chain of ideals terminates. I think all the standard facts about noetherian rings can be proved without choice: $\mathbb{Z}$ is noetherian; fields are noetherian; $A$ noetherian implies $A[x]$ noetherian; that qu... |
7,025 | <p>Many commutative algebra textbooks establish that every ideal of a ring is contained in a maximal ideal by appealing to Zorn's lemma, which I dislike on grounds of non-constructivity. For Noetherian rings I'm told one can replace Zorn's lemma with countable choice, which is nice, but still not nice enough - I'd lik... | Neel Krishnaswami | 1,610 | <p>You should take a look at Coquand and Lombardi's "A Logical Approach to Abstract Algebra". </p>
<p>They observe that commutative rings have a purely equational description, and so there are very strong metatheorems that apply to this theory: Birkhoff's completeness theorem for equational logic, of course; and also ... |
7,025 | <p>Many commutative algebra textbooks establish that every ideal of a ring is contained in a maximal ideal by appealing to Zorn's lemma, which I dislike on grounds of non-constructivity. For Noetherian rings I'm told one can replace Zorn's lemma with countable choice, which is nice, but still not nice enough - I'd lik... | Ingo Blechschmidt | 31,233 | <p>If the ring is countable (or the image of a linear well-ordering), then no choice of any kind (not even countable choice) and in fact not even the law of excluded middle is required: There is an explicit construction, admissible by the standards of constructive mathematics.</p>
<p>This result is due to Krivine and w... |
4,066,601 | <p>The question is</p>
<blockquote>
<p>Find all solutions <span class="math-container">$z\in \mathbb C$</span> for the following equation: <span class="math-container">$z^2 +3\bar{z} -2=0$</span></p>
</blockquote>
<p>I have attempted numerous methods of approaching this question, from trying to substitute <span class="... | Lukas | 844,079 | <p>Plugging in <span class="math-container">$x+iy$</span> for <span class="math-container">$z$</span> seems to be a good idea actually. You get
<span class="math-container">$$(x^2-y^2+3x-2)+i(2xy-3y)=0$$</span>
For a complex number to be zero both the real and the imaginary part have to be zero, so we get <span class="... |
215,834 | <p>I have the following plot </p>
<pre><code>Show[Graphics[Axes -> True], ParametricPlot[al[t], {t, 0, 1}],
ParametricPlot[be[t], {t, 0, 1}]]
</code></pre>
<p>where <code>al[t]</code> and <code>be[t]</code> are parametric plots of a BezierFunction. </p>
<p>I would like to add arrows to the midpoint of the param... | kglr | 125 | <pre><code>SeedRandom[1]
al = BezierFunction[RandomReal[{-1, 1}, {14, 2}]];
be = BezierFunction[RandomReal[{-1, 1}, {20, 2}]];
</code></pre>
<p>You can temporarily redefine <code>Line</code> as <code>Arrow</code> using <code>Block</code> and use <code>ParametricPlot</code>:</p>
<pre><code>Block[{Line = Arrow},
Pa... |
215,834 | <p>I have the following plot </p>
<pre><code>Show[Graphics[Axes -> True], ParametricPlot[al[t], {t, 0, 1}],
ParametricPlot[be[t], {t, 0, 1}]]
</code></pre>
<p>where <code>al[t]</code> and <code>be[t]</code> are parametric plots of a BezierFunction. </p>
<p>I would like to add arrows to the midpoint of the param... | wmora2 | 40,277 | <p>To put arrows in a curve I usually use this code</p>
<pre><code>ParametricPlot[...] /. Line[fig___] :> {Arrowheads[ConstantArray[0.06, 4]], Arrow[fig]};
</code></pre>
<p><a href="https://i.stack.imgur.com/2B8SW.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/2B8SW.png" alt="enter image descript... |
153,448 | <p>On the complex plane, I have a transformation "T" such that :</p>
<p>$z' = (m+i)z + m - 1 - i$ ($z'$ is the image and $z$ the preimage, $z$ and $z'$ are both complex number)</p>
<p>and $m$ is a real number. </p>
<p>I'd need to determine "$m$" such that this transformation "T" is a rotation.</p>
<p>I know a r... | Gigili | 181,853 | <p>$$\int_{0}^{f(x)}t^2dt=\frac{{f(x)}^3}{3}=x^3(1+x)^2$$</p>
<p>$$\Downarrow$$</p>
<p>$${f(x)}^3=3x^3(1+x)^2$$
$$\Downarrow$$</p>
<p>$${f(2)}^3=3 \cdot2^3(1+2)^2=3^3 \cdot 2^3$$
$$\Downarrow$$</p>
<p>$$f(2)=3 \cdot 2 =6$$</p>
|
2,252,206 | <p>This question is related to <a href="https://math.stackexchange.com/questions/1574196/units-of-group-ring-mathbbqg-when-g-is-infinite-and-cyclic">this</a> one, in that I am asking about the same problem, but not necessarily about the same aspect of the problem.</p>
<p>I need to identify all units of the group ring ... | Travis Willse | 155,629 | <p><strong>Hint</strong> The Hermitian isometry condition $||T v|| = || v ||$ is equivalent to $\langle T v, T v \rangle = \langle v, v \rangle$, and one can use the <a href="https://en.wikipedia.org/wiki/Polarization_identity#For_vector_spaces_with_complex_scalars" rel="nofollow noreferrer">Hermitian polarization iden... |
10,468 | <p>I know that many graph problems can be solved very quickly on graphs of bounded degeneracy or arboricity. (It doesn't matter which one is bounded, since they're at most a factor of 2 apart.) </p>
<p>From Wikipedia's article on the clique problem I learnt that finding cliques of any constant size k takes linear time... | David Eppstein | 440 | <p>Bounded degeneracy or arboricity just means that the graph is sparse (number of edges is proportional to number of vertices in all subgraphs).</p>
<p>Some ideas that have been used for fast algorithms on these graphs:</p>
<ul>
<li><p>Order the vertices so that each vertex has only d neighbors that are later in the... |
10,468 | <p>I know that many graph problems can be solved very quickly on graphs of bounded degeneracy or arboricity. (It doesn't matter which one is bounded, since they're at most a factor of 2 apart.) </p>
<p>From Wikipedia's article on the clique problem I learnt that finding cliques of any constant size k takes linear time... | anonymous | 17,256 | <p>One article that provides algorithms for the MDS problem on graphs of bounded arboricity is "Minimum Dominating Set Approximation in Graphs of Bounded Arboricity" by Lenzen and Wattenhofer <a href="http://www.disco.ethz.ch/publications/disc10_LW_204.pdf" rel="nofollow">http://www.disco.ethz.ch/publications/disc10_LW... |
1,634,741 | <p>$22+22=4444$</p>
<p>$43+46=618191$</p>
<p>$77+77=?$</p>
<p>What should come in place of $?$</p>
<p>I cannot see any logic in $43+46=618191$. Is there any?</p>
| 2.71828-asy | 302,548 | <p>Let $S = a + ar + ar^2 + ar^3 ...$</p>
<p>Then $S-Sr = (a + ar + ar^2 + ar^3 ... ar^n) - (ar + ar^2 + ar^3 + ar^4 ... ar^{n+1}) = a - ar^{n+1}$</p>
<p>Factoring out an S we have $S(1-r) = a-ar^{n+1}$</p>
<p>Finally, $$S = {(a - ar^{n+1})\over(1-r)}$$</p>
<p>In your case, you are trying to find $5^4 + 5^5 + 5^6 .... |
4,531,652 | <p>In my school book, I read this theorem</p>
<blockquote>
<p>Let <span class="math-container">$n>0$</span> is an odd natural number (or an odd positive integer), then the equation <span class="math-container">$$x^n=a$$</span> has exactly one real root.</p>
</blockquote>
<p>But, the book doesn't provide a proof, onl... | nonstudent | 1,089,358 | <p>The case <span class="math-container">$a=0$</span> is obviously trivial. Suppose that <span class="math-container">$a>0$</span>. This implies <span class="math-container">$x^n>0\implies x>0$</span>, where <span class="math-container">$n$</span> is an odd positive integer.</p>
<p>Thus, we can apply the <em>r... |
4,052,760 | <blockquote>
<p>Prove that <span class="math-container">$\int\limits^{1}_{0} \sqrt{x^2+x}\,\mathrm{d}x < 1$</span></p>
</blockquote>
<p>I'm guessing it would not be too difficult to solve by just calculating the integral, but I'm wondering if there is any other way to prove this, like comparing it with an easy-to-c... | Dr. Wolfgang Hintze | 198,592 | <p>Still simpler.</p>
<p>For <span class="math-container">$0<x<1$</span> we have <span class="math-container">$x^2<x$</span>.</p>
<p>Hence <span class="math-container">$\sqrt{x^2+x}\lt \sqrt{x+x}=\sqrt{2}\sqrt{x}$</span>, and the integral can be estimated as <span class="math-container">$\int_0^1 \sqrt{x^2+x}&... |
121,653 | <p>What is information about the existence of rational points on hyperelliptic curves over finite fields available?</p>
| Michael Zieve | 30,412 | <p>[Edited to remove material subsumed and improved by Felipe's answer.]</p>
<p>Here is some historical info. Dickson studied this question in his 1909 paper "Definite forms in a finite field". For Dickson, a "definite form" is a homogeneous $f(x,z)\in\mathbb{F}_q[x,z]$ which takes nonzero square values for all $(x,... |
1,068,103 | <p>The question is about approximating the continuous function in an interval $[a, b]$. If we consider the linear space of all such functions endowed with the norm</p>
<p>$$||f|| = \max_{x \in [a, b]}|f(x)|$$ </p>
<p>then the best error approximating a function $f$ using polynomials $p \in \pi_n$ (with $\pi_n$ is den... | Robert Israel | 8,508 | <p>Hint: if $p$ is a good approximation to $f$ and $q$ is a good approximation to $g$,
try approximating $f+g$ by $p + q$.</p>
|
1,068,103 | <p>The question is about approximating the continuous function in an interval $[a, b]$. If we consider the linear space of all such functions endowed with the norm</p>
<p>$$||f|| = \max_{x \in [a, b]}|f(x)|$$ </p>
<p>then the best error approximating a function $f$ using polynomials $p \in \pi_n$ (with $\pi_n$ is den... | Exodd | 161,426 | <p>Let
$$E_n(f)=a,\qquad E_n(g)=b$$
There exist two polynomials $p_1$ and $p_2$ such that
$$\|f-p_1\|<a+\epsilon,\qquad \|g-p_2\|<b+\epsilon$$
so
$$E_n(f+g)\le\|f+g-p_1-p_2\|\le a+b+2\epsilon$$
for all epsilon, so
$$E_n(f+g)\le a+b=E_n(f)+E_n(g)$$</p>
|
175,723 | <p>I am reading Goldstein's Classical Mechanics and I've noticed there is copious use of the $\sum$ notation. He even writes the chain rule as a sum! I am having a real hard time following his arguments where this notation is used, often with differentiation and multiple indices thrown in for good measure. How do I get... | Robert Israel | 8,508 | <p>The minimal polynomial of $(\theta^2-\theta)/2$ is ${z}^{3}+11\,{z}^{2}+36\,z+4$.</p>
<p>One way to get this is: if $t = (\theta^2-\theta)/2$, express $t^3 + b t^2 + c t + d$ as a rational linear combination of $1$, $\theta$ and $\theta^2$, and solve the system of equations that say that the coefficients of $1$, $... |
81,728 | <p>The question is to compute or estimate the following probabilty.</p>
<p>Suppose that you have $N$ (e.g. $30$) tasks, each of which repeats every $t$ min (e.g. $30$ min) and lasts $l$ min (e.g. $5$ min). If the tasks started at uniformly random point in time yesterday, what is the probability that there is a time to... | John Jiang | 4,923 | <p>Consider a circle of length $t+\ell$. Then I think your problem is asking if I drop $N$ points uniformly at random onto that circle, what is the probability that at least $m$ of them are in an interval of length $\ell$. When $m/(N-m) >> \ell / t$, so that the tail event that there are $m$ points in a fixed int... |
196,902 | <p>Hello fellow Ace Users.</p>
<p>Currently I'm working on a project to implement Peridynamics.
This is a discretization technique in the fashion of a meshless particle method.
AceGen/AceFEM provides the feature of arbitrary nodes per element which suits my need perfectly as such a peridynamic particle interacts with ... | BHudobivnik | 47,826 | <p>I can mostly help with the implementation aspect of meshfree methods and answer point two. Although Mathematica can plot anything you want, but you might need to program it yourself.</p>
<ol start="2">
<li>In the new version of AceGen/FEM 6.923, prof Korelc introduced a new SMSIO functions and there is improvement ... |
3,489,345 | <p>My goal is to find the values of <span class="math-container">$N$</span> such that <span class="math-container">$10N \log N > 2N^2$</span></p>
<p>I know for a fact this question requires discrete math. </p>
<p>I think the problem revolves around manipulating the logarithm. The thing is, I forgot how to manipula... | Claude Leibovici | 82,404 | <p>In the real domain, consider the function
<span class="math-container">$$f(x)=5\log(x)-x$$</span> The first derivative cancels at <span class="math-container">$x=5$</span> and by the second derivative test, this is a maximum. So, there is a limited range of <span class="math-container">$x$</span> where <span class="... |
53,185 | <p>Let us consider a noncompact Kähler manifold with vanishing scalar curvature but nonzero Ricci tensor. I'm wondering what can it tell us about the manifold. The example (coming from physics) has the following Kähler form</p>
<p><span class="math-container">$$K = \bar{X} X + \bar{Y} Y + \log(\bar{X} X + \bar{Y} Y)$$<... | Zatrapilla | 4,129 | <p>The Ricci scalar is the average gaussian curvature in all the two-dimensional subspaces passing through the point, I believe. Whence you can derive the 'meaning'.</p>
|
53,185 | <p>Let us consider a noncompact Kähler manifold with vanishing scalar curvature but nonzero Ricci tensor. I'm wondering what can it tell us about the manifold. The example (coming from physics) has the following Kähler form</p>
<p><span class="math-container">$$K = \bar{X} X + \bar{Y} Y + \log(\bar{X} X + \bar{Y} Y)$$<... | diverietti | 9,871 | <p>On a $n$-dimensional Kähler manifold $(X,\omega)$, the Ricci form is (minus) the curvature of the canonical bundle $K_X$ endowed with the induced metric. Thus, if $X$ has zero Ricci curvature then its canonical bundle is flat. Thus, the structure group can be reduced to a subgroup of the special linear group $SL(n,\... |
53,185 | <p>Let us consider a noncompact Kähler manifold with vanishing scalar curvature but nonzero Ricci tensor. I'm wondering what can it tell us about the manifold. The example (coming from physics) has the following Kähler form</p>
<p><span class="math-container">$$K = \bar{X} X + \bar{Y} Y + \log(\bar{X} X + \bar{Y} Y)$$<... | Gunnar Þór Magnússon | 4,054 | <p>Dear Peter, I don't think one can say anything about such manifolds because the scalar curvature is too weak an invariant to be of use. Here is an infinite family of non-diffeomorphic compact examples to support my claim; for non-compact ones, remove a subvariety.</p>
<p>Let $Y$ be a projective manifold of dimensio... |
2,211,075 | <p>I don't understand the following example from Math book.</p>
<p>Solve for the equation <code>sin(theta) = -0.428</code> for <code>theta</code> in <code>radians</code> to 2 decimal places. where <code>0<= theta<= 2PI</code>.</p>
<p>And this is the answer:</p>
<p><code>theta=-0.44 + 2PI = 5.84rad and theta = ... | Matrefeytontias | 392,482 | <p>You cannot just replace the tangent formula by the derivative ; doing so would mean that you are actually taking the limit when x -> 0, but in that case, you must also divide by $sin(0)$, which you cannot do obviously.</p>
<p>What you should do here is use equivalent functions, namely the following :</p>
<p>$sin(x... |
956,680 | <p>$\displaystyle\lim_{x\to0}\frac{x^2+1}{\cos x-1}$</p>
<p>My solution is:</p>
<p>$\displaystyle\lim_{x\to0}\frac{x^2+1}{\cos x-}\frac{\cos x+1}{\cos x+1}$</p>
<p>$\displaystyle\lim_{x\to0}\frac{(x^2+1)(\cos x+1)}{\cos^2 x-1}$</p>
<p>$\displaystyle\lim_{x\to0}\frac{(x^2+1)(\cos x+1)}{-(1-\cos^2 x)}$</p>
<p>Since... | Paul | 17,980 | <p>This Limit doesn't exist! Notice that as $x\to 0$, $x^2+1 \to 1$ and $\cos x-1 \to 0$; So the Limit $\to -\infty.$</p>
|
3,905,629 | <p>I need to compute a limit:</p>
<p><span class="math-container">$$\lim_{x \to 0+}(2\sin \sqrt x + \sqrt x \sin \frac{1}{x})^x$$</span></p>
<p>I tried to apply the L'Hôpital rule, but the emerging terms become too complicated and doesn't seem to simplify.</p>
<p><span class="math-container">$$
\lim_{x \to 0+}(2\sin \s... | robjohn | 13,854 | <p>For <span class="math-container">$x\in\left(0,\frac\pi2\right]$</span>, the concavity of <span class="math-container">$\sin(x)$</span> says
<span class="math-container">$$
\frac2\pi\le\frac{\sin(x)}x\le1
$$</span>
Therefore,
<span class="math-container">$$
\underbrace{\left(\frac4\pi\sqrt{x}-\sqrt{x}\right)^x}_{\lef... |
2,823,758 | <p>I was learning the definition of continuous as:</p>
<blockquote>
<p>$f\colon X\to Y$ is continuous if $f^{-1}(U)$ is open for every open $U\subseteq Y$</p>
</blockquote>
<p>For me this translates to the following implication:</p>
<blockquote>
<p>IF $U \subseteq Y$ is open THEN $f^{-1}(U)$ is open</p>
</blockq... | Evan Wilson | 570,598 | <p>The two definitions are equivalent to each other for metric spaces. To see that the first definition implies the second, let $\epsilon>0$ and $y=f(x)$. The open ball $B_\epsilon(y)$ is open in $Y$. Therefore $f^{(-1)}(B_\epsilon(y))$ must be open in $X$. Therefore, it contains the open ball $B_\delta(x)$ for smal... |
205,671 | <p>How would one go about showing the polar version of the Cauchy Riemann Equations are sufficient to get differentiability of a complex valued function which has continuous partial derivatives? </p>
<p>I haven't found any proof of this online.</p>
<p>One of my ideas was writing out $r$ and $\theta$ in terms of $x$ a... | James S. Cook | 36,530 | <p>A less standard approach: Take $\{e_{r},e_{\theta}\}$ as a basis in polar coordinates then the Jacobian matrix for any function on $\mathbb{R}^2$ has the form:
$$ [df] = \left[ \begin{array}{cc} U_r & U_{\theta} \\ V_r & V_{\theta} \end{array} \right] $$
<strong>Define</strong> complex-differentiability via ... |
2,615,185 | <p>The title is not complete, since it would be too long. Consider the following statement:</p>
<blockquote>
<p>Let $U \subset \mathbb{R}^n$ be open, connected and such that its one-point compactification is a manifold. Then, this compactification must be (homeomorphic to) the sphere $S^n$.</p>
</blockquote>
<p>Is ... | Moishe Kohan | 84,907 | <p>I do not expect any simple proofs of this result. A good exercise would be to prove this for domains in $R^2$ without using anything about the classification of surfaces or the Schoenflies theorem in $R^2$. </p>
<p>Here is a proof of your statement (in the topological category). I will assume that $n\ge 2$ since th... |
2,402,410 | <p>I defined the "function":</p>
<p>$$f(t)=t \delta(t)$$</p>
<p>I know that Dirac "function" is undefined at $t=0$ (see <a href="http://web.mit.edu/2.14/www/Handouts/Convolution.pdf" rel="nofollow noreferrer">http://web.mit.edu/2.14/www/Handouts/Convolution.pdf</a>).</p>
<p>In Wolfram I get $0 \delta(0)=0$ (<a href=... | Ethan Bolker | 72,858 | <p>You do understand that the Dirac delta "function" isn't a function, since you too put the word in quotes. To justify assertions about it you have to see how those assertions behave in the integrals that involve the delta function. ("Behavior inside integrals" is the idea behind distributions.) That's the essence of ... |
1,554,285 | <p>Here's my problem:</p>
<blockquote>
<p>In Ohio, 55% of the population support the republican candidate in an
upcoming election. 200 people are polled at random. If we suppose that
each person’s vote (for or against) is a Bernoulli random variable
with probability p, and votes are independent,</p>
<p>(a... | Rowan | 229,922 | <p>Hints:</p>
<p>Let $X=\sum_{i=1}^{200}X_i$, where $X_i=\{\text{No.i person votes for the democratic candidate}\}$, so that $$X_i\sim
\begin{pmatrix}
1 & 0\\
p & 1-p\\
\end{pmatrix}$$</p>
<p>You have already calculated $E(X)$ and $D(X)$. So according to CLT, $X\sim?$</p>
|
2,490,128 | <p>Over the domain of integers, if $(a-c)|(ab+cd)$ then $(a-c)|(ad+bc)$.</p>
<p>Note: $x|y$ means "$x$ divides $y$," i.e. $\exists k\in \mathbb{Z}. y=x\cdot k$</p>
<p>This is part of an assignment on GCD, Euclidean algorithm, and modular arithmetic.</p>
<p>My approach:</p>
<p>If $a-c$, divides a linear combination ... | Nosrati | 108,128 | <p>Let $x=\dfrac1u$ then
\begin{align}
I
&= \int\dfrac{-u}{\sqrt{u^2+u+1}}du \\
&= -\int\dfrac{2u+1}{2\sqrt{u^2+u+1}}du+\dfrac12\int\dfrac{1}{\sqrt{(u+\frac12)^2+\frac34}}du \\
&= -\sqrt{u^2+u+1}+\dfrac12\operatorname{arcsinh}\dfrac{2u+1}{\sqrt{3}}+C \\
&= -\dfrac{\sqrt{x^2+x+1}}{x}+\dfrac12\operatorna... |
1,879,395 | <p>I am trying to learn generating functions so I am trying this recurrence:</p>
<p>$$F(n) = 1 + \frac{n-1}{n}F(n-1)$$</p>
<p>But I am struggling with it. Luckily the base case can be anything since $F(1)$ will multiply it by $0$ anyway, so let's say $F(0) = 0$. Then I tried this:</p>
<p>$$G(x) = \sum_{n=0}^{\infty}... | alans | 80,264 | <p>If you are interested in an easier proof:</p>
<p>Observe: </p>
<p>$F(1)=1$, $F(2)=\frac{3}{2}$, $F(3)=2$, $F(4)=\frac{5}{2}$ $\dots$ </p>
<p>Now it is easy to get the pattern $F(n)=\frac{n+1}{2}$ and prove it by induction. </p>
<p>First you can check that the induction base $F(1)=1$ holds. </p>
<p>In the induct... |
2,214,030 | <p>$\mathbb{R}^{13}$ has two subspaces such that dim(S)=7 and dim(T)=8 <br/></p>
<p>⒜ max dim (S∩T)=?<br/>
⒝ min dim (S∩T)=?<br/>
⒞ max dim (S+T)=?<br/>
⒟ min dim (S+T)=?<br/>
⒠ dim(S∩T) + dim (S+T)=?</p>
| Jaideep Khare | 421,580 | <p>It is '$0 <|x-a| $' , not '$0 \le |x-a|$'. This suggests that :
$$|x-a| \neq 0 \implies x \neq a$$</p>
<p>Id est, $x$ tends to $a$ but is never ever exactly equal to $a$.</p>
<p>Also, absolute values aren't always positive, they are always <strong>NON-NEGATIVE</strong>.</p>
|
733,553 | <p>It's been a long time since high school, and I guess I forgot my rules of exponents. I did a web search for this rule but I could not find a rule that helps me explain this case:</p>
<p>$ 2^n + 2^n = 2^{n+1} $</p>
<p>Which rule of exponents is this?</p>
| Abraham Zhang | 112,045 | <p>If you realise that there are $2$ of $2^n$, then we have
$$2^1\times2^n$$
If we are multiplying $2$ by itself <strong>n</strong> times and then multiplying the result by another $2$, we get $2$ multiplied by itself <strong>n+1</strong> times, which is $$2^{n+1}$$</p>
|
2,469,720 | <p>Math problem:</p>
<blockquote>
<p>Find $x$, given that $ \, 2^2 \times 2^4 \times 2^6 \times 2^8 \times \ldots \times 2^{2x} = \left( 0.25 \right)^{-36}$</p>
</blockquote>
<p>To solve this question, I changed the left side of the equation to $2^{2+4+6+ \ldots + 2x}$ and the right side to: $\frac{2^{74}}{3^{36}}$... | Dr. Sonnhard Graubner | 175,066 | <p>use that $$2+4+6+8+...+2x=2^{x(x+1)}$$ and $$\left(\frac{1}{4}\right)^{-36}=2^{72}$$ and you will have $$2^{2^{x(x+1)}}=2^{72}$$</p>
|
4,146,081 | <p>How can I demonstrate the Jacobi identity:</p>
<p><span class="math-container">\begin{equation}
[S_{i}, [S_{j},S_{k}]] + [S_{j}, [S_{k},S_{i}]] + [S_{k}, [S_{i},S_{j}]] = 0 ~,
\end{equation}</span></p>
<p>using the infinitesimal generators <span class="math-container">$S_{\kappa}$</span> for a continuous group, wher... | Dietrich Burde | 83,966 | <p>The Jacobi identity can be derives formally by expanding <span class="math-container">$[S_i,S_j]=S_iS_j-S_jS_i$</span>. Define the <em>associator</em> of <span class="math-container">$S_i,S_j,S_k$</span> by
<span class="math-container">$$
(S_i,S_j,S_k)=(S_iS_j)S_k-S_i(S_jS_k)
$$</span>
This is zero if the product is... |
370,007 | <p>A river boat can travel a 20km per hour in still water. The boat travels 30km upstream against the current then turns around and travels the same distance back with the current. IF the total trip took 7.5 hours, what is the speed of the current? Solve this question algebraically as well as graphically..</p>
<p>I st... | lab bhattacharjee | 33,337 | <p>$ord_pa=m\iff a^m\equiv1\pmod p$ and $ord_pb=n\iff b^n\equiv1\pmod p$ where $n$ is any integer</p>
<p>$\implies a^{lcm(m,n)}\equiv1\pmod p,b^{lcm(m,n)}\equiv1\pmod p$</p>
<p>$\implies (ab)^{lcm(m,n)}\equiv1\pmod p\implies ord_p(ab)$ divides lcm$(m,n)$</p>
<p>Conversely, let $ord_p(ab)=h$ and $(m,n)=d$ and $\fra... |
269,665 | <p>Is Klein bottle an algebraic variety? I guess no, but how to prove. How about other unorientable mainfolds? </p>
<p>If we change to Zariski topology, which mainfold can be an algebraic variety? </p>
| Zhen Lin | 5,191 | <p>Any complex manifold is not merely an orientable manifold but an <em>oriented</em> manifold. Hence the Klein bottle cannot be a complex manifold (and so not complex algebraic).</p>
<p>Indeed, consider the holomorphic tangent bundle $T M$ of a complex manifold $M$. We define an orientation as follows: take a complex... |
3,416,895 | <p>here's the relevant question: <a href="https://math.stackexchange.com/q/193157/716946">If $\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$ then $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$</a></p>
<p>In the accepted answer, <strong>doesn't the last inequality only work if <span class="math-container">$... | Martin R | 42,969 | <p>You have that
<span class="math-container">$$ \tag{*}
\sigma_n\geqslant \frac 1n\sum_{j=1}^ks_j+\frac{n-k}n\inf_{l\geqslant k}s_l
$$</span>
and you are right that this is <span class="math-container">$\ge \frac 1n\sum_{j=1}^ks_j+\inf_{l\geqslant k}s_l$</span> only if <span class="math-container">$\inf_{l\geqslant k... |
2,649,283 | <p>There are these two questions that my professor posted, and they absolutely stumped me:</p>
<p>$ \vdash (\exists x. \bot) \implies P $
and
$(\exists x. \top) \vdash (\forall x. \bot ) \implies P$.</p>
<p>What do I even do with the $(\exists x. \bot)$ part? It got me stuck for quite some time. Any help will be ap... | Bram28 | 256,001 | <p>Here are some proofs in the Fitch system:</p>
<p>$\def\fitch#1#2{\begin{array}{|l}#1 \\ \hline #2\end{array}}$ </p>
<p>$\fitch{
1.
}{
\fitch{
2.\exists x. \bot}{
\fitch{
3.\bot
}{
4.P \quad \bot \text{ Elim } 3}
\\
5.P \quad \exists \text{ Elim } 2, 3-4} \\
6. \exists x. \bot \rightarrow P \quad \rightarrow \tex... |
1,970,235 | <p>If I remember right, $f(x)$ is continuous at $x=a$ if</p>
<ol>
<li><p>$\lim_{x \to a} f(x)$ exists</p></li>
<li><p>$f(a)$ exists</p></li>
<li><p>$f(a) = \lim_{x \to a} f(x)$</p></li>
</ol>
<p>So $\lim_{x \to 0^{-}} \sqrt{x}$ exists? Thus $\lim_{x \to 0^{-}} \sin(\sqrt{x})$ <a href="https://math.stackexchange.com/q... | operatorerror | 210,391 | <p>Alright, I'll bite with the classic proof for the limit from the right, where the square root function has real value. </p>
<p>Fix $\epsilon>0$. Then taking $\delta=\epsilon^2$ we get
$$
|x|<\epsilon^2\Rightarrow |\sqrt{x}|<\epsilon
$$</p>
|
1,970,235 | <p>If I remember right, $f(x)$ is continuous at $x=a$ if</p>
<ol>
<li><p>$\lim_{x \to a} f(x)$ exists</p></li>
<li><p>$f(a)$ exists</p></li>
<li><p>$f(a) = \lim_{x \to a} f(x)$</p></li>
</ol>
<p>So $\lim_{x \to 0^{-}} \sqrt{x}$ exists? Thus $\lim_{x \to 0^{-}} \sin(\sqrt{x})$ <a href="https://math.stackexchange.com/q... | Alex M. | 164,025 | <p>You make a single mistake in your question: you forget to specify the domain of definition of $\sqrt \cdot$. You know that this is $[0, \infty)$, so "limit in $0$" here means, necessarily, "limit from the right" - simply because there is nothing to the left of $0$ in $[0, \infty)$.</p>
<p>If you prefer to be more p... |
1,618,373 | <p>Prove that $S_4$ cannot be generated by $(1 3),(1234)$</p>
<p>I have checked some combinations between $(13),(1234)$ and found out that those combinations cannot generated 3-cycles.</p>
<p>Updated idea:<br>
Let $A=\{\{1,3\},\{2,4\}\}$<br>
Note that $(13)A=A,(1234)A=A$<br>
Hence, $\sigma A=A,\forall\sigma\in \langl... | CopyPasteIt | 432,081 | <p>Here we copy/paste/modify another <a href="https://math.stackexchange.com/a/3843575/432081">answer</a>:</p>
<p>Define</p>
<p><span class="math-container">$\; \tau = (13)$</span></p>
<p><span class="math-container">$\;\sigma = (1234)$</span>
<br>
<br>
<span class="math-container">$1^{st} \text {group of calculations:... |
139,575 | <p>I use Magma to calculate the L-value, yields</p>
<p>E:=EllipticCurve([1, -1, 1, -1, 0]);
E;
Evaluate(LSeries(E),1),RealPeriod(E),Evaluate(LSeries(E),1)/RealPeriod(E);</p>
<p>Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - x over Rational Field
0.386769938387780043302394751243 3.09415950710224034641915800995
0.... | Tim Dokchitser | 3,132 | <p>To expand my comment, there are at least 3 subtle ways to get BSD wrong:</p>
<p>1) The BSD period over ${\mathbb Q}$ is the real period $\Omega_\infty$ when $E({\mathbb R})$ is connected ($\Delta(E)<0$) and $2\Omega_\infty$ when it has two connected components ($\Delta(E)>0$). The same thing happens over numb... |
111,795 | <p>I need to add a small graphics on top of a larger one, and the small graphics should stick very close to the large one, with their axis aligned. Here's a minimal code to work with, using some elements from this question/answer :</p>
<p><a href="https://mathematica.stackexchange.com/questions/22521/how-to-make-a-pl... | Cham | 6,260 | <p>This is a partial solution, using <code>Epilog</code> and <code>Inset</code>. It has an alignment problem, especially after we resize the picture by hand inside the <code>Manipulate</code> box. Also, without resizing the whole, playing with the parameters may give an alignment problem after a while. How to fix th... |
3,553,644 | <p>I am taking a Introduction to Calculus course and am struggling to understand how derivatives can represent tangent lines.</p>
<p>I learned that derivatives are the rate of change of a function but they can also represent the slope of the tangent to a point. I also learned that a derivative will always be an order... | Tryst with Freedom | 688,539 | <p>The derivative represents the slope of the tangent, not the equation of a tangent line. </p>
<p>For understanding why it is so, we delve into the question of 'what is the derivative?', the fundamental idea of finding the derivative is taking a point on the curve and another point, which is extremely close to it, an... |
3,223,705 | <p>I have a task for school and we need to plot a polar function with MATLAB. The function is <span class="math-container">$r = 1-2\cos(6\theta)$</span>.</p>
<p>I did this and I'm getting exactly the same as on Wolfram Alpha: <a href="https://www.wolframalpha.com/input/?i=polar+plot+r%3D1-2" rel="nofollow noreferrer">... | the_candyman | 51,370 | <p>Matlab code:</p>
<pre><code> clear all
close all
nPoints = 500; % Number of points for the plot
theta = linspace(0, 2*pi, nPoints); % Define the theta points
r = 1 - 2*cos(6*theta); % Evaluate the radius for each theta
x = r.*cos(theta); % Evaluate x for each theta
y = r.*sin(theta); % Evaluate y for ea... |
2,062,706 | <p>I have the following function:</p>
<p>\begin{equation}
f(q,p) = q \sqrt{p} + (1-q) \sqrt{1 - p}
\end{equation}</p>
<p>Here, $q \in [0,1]$ and $p \in [0,1]$.</p>
<p>Now, given some value $q \in [0,1]$ what value should I select for $p$ in order to maximize $f(q,p)$? That is, I need to define some function $g(q)$ s... | yurnero | 178,464 | <p>By Cauchy-Schwarz,
$$
[q \sqrt{p} + (1-q) \sqrt{1 - p}]^2\leq(q^2+(1-q)^2)(p+1-p)=q^2+(1-q)^2.
$$
To have equality, we require
$$
\frac{\sqrt{p}}{\sqrt{1-p}}=\frac{q}{1-q}\iff \boxed{p=\frac{q^2}{1-2q+2q^2}}\in[0,1].
$$</p>
|
1,186,825 | <p>Prove $$\lim_{n\to\infty}\int_0^1 \left(\cos{\frac{1}{x}} \right)^n\mathrm dx=0$$</p>
<p>I tried, but failed. Any help will be appreciated.</p>
<p>At most points $(\cos 1/x)^n\to 0$, but how can I prove that the integral tends to zero clearly and convincingly?</p>
| Siminore | 29,672 | <p>Your integral coincides with
$$
\int_1^{+\infty} \frac{(\cos u)^n}{u^2}\mathrm{d}u.
$$
For almost every $u>1$, $\lim_n (\cos u)^n =0$, and $$ \frac{|\cos u|^n}{u^2} \leq \frac{1}{u^2}.$$
Since $(u \mapsto u^{-2} ) \in L^1(1,+\infty)$, by the Dominated Convergence Theorem the integral converges to zero. Actually t... |
833,376 | <p>I know very little in the way of math history, but I question that was bothering me recently is where the terms open and closed came from in topology. I know that it's easy to ascribe a sense of openness/closedness to said sets, but I feel like there are a lot of other, more appropriate words that could have been us... | hugo mancera | 247,838 | <p>the followings archives corresponds to the doctoral thesis of Henri Lebesgue:</p>
<p>You can search for Analysis situs, old name for the actual topology.Lebesgue and Rham (french mathematicians) wrote about that subject.</p>
<p>sincerely</p>
<p>Hugo Mancera
Colombia</p>
|
1,410,163 | <p>Show that the limit of the function, $f(x,y)=\frac{xy^2}{x^2+y^4}$, does not exist when $(x,y) \to (0,0)$.</p>
<p>I had attempted to prove this by approaching $(0, 0)$ from $y = mx$, assuming $m = -1$ and $m = 1$. The result was $f(y, -y) = \frac{y}{1+y^2}$ and $f(y, y) = \frac{y}{1+y^2}$ as the limits which are ob... | Siminore | 29,672 | <p>This is elementary logic. Assume that $p \in B^c$ <em>and</em> $p \in A$. Then, by assumption, $p \in B$, a contradiction. Hence $B^c \subset A^c$. To conclude, just reverse the argument.</p>
<p>As you see, everything reduces to the properties of logical negation.</p>
|
3,320,193 | <blockquote>
<p>If given <span class="math-container">$P(B\mid A) =4/5$</span>, <span class="math-container">$P(B\mid A^\complement)= 2/5$</span> and <span class="math-container">$P(B)= 1/2$</span>, what is the probability of <span class="math-container">$A$</span>?</p>
</blockquote>
<p>I know I need to apply Bayes ... | mathsdiscussion.com | 694,428 | <p>Using venn diagram one of easy way to find solution(<a href="https://i.stack.imgur.com/ZiYVm.jpg" rel="nofollow noreferrer">https://i.stack.imgur.com/ZiYVm.jpg</a>)</p>
|
374,909 | <p>If <span class="math-container">$A\subseteq\mathbb{N}$</span> is a subset of the positive integers, we let <span class="math-container">$$\mu^+(A) = \lim\sup_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}$$</span> be the <em>upper density</em> of <span class="math-container">$A$</span>.</p>
<p>For <span class="math-con... | reuns | 84,768 | <p><span class="math-container">$\tau(n) \le k$</span> implies that <span class="math-container">$n=\prod_{i=1}^j p_i$</span> with <span class="math-container">$j\le k$</span>, thus <span class="math-container">$$\sum_{n=1,\tau(n)\le k}^\infty n^{-s} \le (1+\sum_{p \ prime} p^{-s})^k=\sum_n a_k(n) n^{-s}$$</span></p>
<... |
4,176,646 | <p>I need to find the directional derivatives for all vectors <span class="math-container">$u=[u_1\ \ u_2]\in \mathbb R^2$</span> with <span class="math-container">$\|u\|=1$</span> at <span class="math-container">$P_0=(0,0)$</span>, and determine whether <span class="math-container">$f$</span> is differentiable at <spa... | BillyJoe | 573,047 | <p>What you are trying to compute is <span class="math-container">$\nu_2(x)$</span>, the <span class="math-container">$2$</span>-adic valuation of <span class="math-container">$x$</span>, i.e. the highest exponent <span class="math-container">$\nu_2(x)$</span> such that <span class="math-container">$2^{\nu_2(x)}$</span... |
244,433 | <p>I have a list:</p>
<pre><code>data = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {6.*10^-9, 0.0025}, {8.*10^-9, 0.0025}, {1.*10^-8, 0.0025}, {7.*10^-9, 0.0023}, {3.*10^-9, 0.0025},...}
</code></pre>
<p>And I wanted to remove every third pair and get</p>
<pre><code> newdata = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {8.... | Rohit Namjoshi | 58,370 | <p>Another way</p>
<pre><code>MapIndexed[If[Divisible[First@#2, 3], Nothing, #1] &, data]
</code></pre>
<p><strong>Update</strong></p>
<p>One way to iterate is to use <code>Nest</code>.</p>
<pre><code>filter = MapIndexed[If[Divisible[First@#2, 3], Nothing, #1] &, #] &;
data = Range[20]; (* Easy to see what... |
244,433 | <p>I have a list:</p>
<pre><code>data = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {6.*10^-9, 0.0025}, {8.*10^-9, 0.0025}, {1.*10^-8, 0.0025}, {7.*10^-9, 0.0023}, {3.*10^-9, 0.0025},...}
</code></pre>
<p>And I wanted to remove every third pair and get</p>
<pre><code> newdata = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {8.... | AsukaMinato | 68,689 | <pre><code>Riffle[data[[;; ;; 3]], data[[2 ;; ;; 3]]]
</code></pre>
|
244,433 | <p>I have a list:</p>
<pre><code>data = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {6.*10^-9, 0.0025}, {8.*10^-9, 0.0025}, {1.*10^-8, 0.0025}, {7.*10^-9, 0.0023}, {3.*10^-9, 0.0025},...}
</code></pre>
<p>And I wanted to remove every third pair and get</p>
<pre><code> newdata = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {8.... | Sjoerd Smit | 43,522 | <p>If you ask me, this is the most direct approach:</p>
<pre><code>Delete[data, List /@ Range[3, Length[data], 3]]
</code></pre>
|
244,433 | <p>I have a list:</p>
<pre><code>data = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {6.*10^-9, 0.0025}, {8.*10^-9, 0.0025}, {1.*10^-8, 0.0025}, {7.*10^-9, 0.0023}, {3.*10^-9, 0.0025},...}
</code></pre>
<p>And I wanted to remove every third pair and get</p>
<pre><code> newdata = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {8.... | imida k | 34,532 | <p>My two answers :-)</p>
<pre><code>data[[Select[Range[Length[data]], Mod[#, 3] != 0 &]]]
</code></pre>
<p>and</p>
<pre><code>Transpose[Select[Transpose[{data, Range[Length[data]]}], Mod[#[[2]], 3] != 0 &]][[1]]
</code></pre>
|
704,917 | <p>I need your help evaluating this integral:
<span class="math-container">$$I=\int_0^\infty F(x)\,F\left(x\,\sqrt2\right)\frac{e^{-x^2}}{x^2} \, dx,\tag1$$</span>
where <span class="math-container">$F(x)$</span> represents <a href="http://mathworld.wolfram.com/DawsonsIntegral.html" rel="nofollow noreferrer">Dawson's f... | Random Variable | 16,033 | <p>Notice that for <span class="math-container">$a>0$</span>, we have <span class="math-container">$$F(ax) = e^{-a^{2}x^{2}}\int_{0}^{ax} e^{y^{2}} \mathrm dy = e^{-a^{2}x^{2}} \int_{0}^{a} u e^{x^{2}u^{2}} \, \mathrm du . \tag{1}$$</span></p>
<p>Then using <span class="math-container">$(1)$</span>, we get</p>
<p><... |
4,459,439 | <p>Suppose <span class="math-container">$G$</span> is an abelian finite group, and the number of order-2 elements in <span class="math-container">$G$</span> is denoted by <span class="math-container">$N$</span>.</p>
<p>I have found that <span class="math-container">$N= 2^n-1$</span> for some <span class="math-container... | Mark | 470,733 | <p>By the fundamental theorem of finite abelian groups, <span class="math-container">$G$</span> can be decomposed:</p>
<p><span class="math-container">$G\cong\mathbb{Z_{2^{n_1}}}\times\mathbb{Z_{2^{n_2}}}\times...\times\mathbb{Z_{2^{n_k}}}\times H$</span></p>
<p>Where <span class="math-container">$H$</span> is a group ... |
3,526,586 | <p>1) Let <span class="math-container">$A \in \mathbb{R}^{n \times n}$</span> be a matrix with nonzero determinant. Show that there exists <span class="math-container">$c>0$</span> so that for every <span class="math-container">$v \in \mathbb{R}^{n},\|A v\| \geq c\|v\|$</span></p>
<p>My attempt:
Since <span class="... | Community | -1 | <p>It must be <span class="math-container">$1$</span>, because <span class="math-container">$(0,1,0)^t\in U$</span>.</p>
|
3,202,797 | <p>Why is solving the system of equations
<span class="math-container">$$1+x-y^2=0$$</span>
<span class="math-container">$$y-x^2=0$$</span>
the same as minimizing
<span class="math-container">$$f(x,y)=(1+x-y^2)^2 + (y-x^2)^2$$</span></p>
<p>Originally I thought it was because if you take the partial derivatives of <sp... | Rohit Pandey | 155,881 | <p>We can say <span class="math-container">$f(x,y)=g(x,y)^2+h(x,y)^2$</span>. It is clear that being the sum of two square terms, <span class="math-container">$f(x,y)\geq 0$</span>. So, the minimum value of <span class="math-container">$f(x,y)$</span> (which is <span class="math-container">$0$</span>) comes about when ... |
39,684 | <p>In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).</p>
<p>For example in <em>natural number theory</em> it is good to know quadratic reciprocity and in <em>linear algebra</em> it's good... | Community | -1 | <ul>
<li>In Algebraic Number theory you have the Kronecker-Weber theorem.</li>
</ul>
|
39,684 | <p>In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).</p>
<p>For example in <em>natural number theory</em> it is good to know quadratic reciprocity and in <em>linear algebra</em> it's good... | Yuval Filmus | 1,277 | <p>In Combinatorics you have for example Szemerédi's regularity lemma and all sorts of "related" results, such as the Green-Tao theorem.</p>
|
39,684 | <p>In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).</p>
<p>For example in <em>natural number theory</em> it is good to know quadratic reciprocity and in <em>linear algebra</em> it's good... | Arturo Magidin | 742 | <p>In <strong>Finite Group Theory,</strong> while the Odd Order Theorem and the Classification are major results, I would put a major landmark at the Sylow Theorems and Hall's Theorem (a generalization of the Sylow Theorems). Especially the former come up all the time, and there are many interesting corollaries that of... |
39,684 | <p>In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).</p>
<p>For example in <em>natural number theory</em> it is good to know quadratic reciprocity and in <em>linear algebra</em> it's good... | Eric Naslund | 6,075 | <p>In subject $X$ the Fundamental Theorem of $X$ is always pretty important.</p>
<p>For example: </p>
<ul>
<li>Fundamental Theorem of Finitely Generated Abelian Groups. </li>
<li>Fundamental Theorem of Calculus.</li>
<li>Fundamental Theorem of Arithmetic.</li>
<li>Fundamental Theorem of Algebra.</li>
<li>Fundamenta... |
39,684 | <p>In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).</p>
<p>For example in <em>natural number theory</em> it is good to know quadratic reciprocity and in <em>linear algebra</em> it's good... | Community | -1 | <p>In graph theory you have <a href="http://en.wikipedia.org/wiki/K%C3%B6nig%27s_theorem_%28graph_theory%29" rel="nofollow">Konig's theorem</a>.</p>
|
39,684 | <p>In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).</p>
<p>For example in <em>natural number theory</em> it is good to know quadratic reciprocity and in <em>linear algebra</em> it's good... | Community | -1 | <ul>
<li>In Field theory The Impossibility of Trisecting the Angle and Doubling the Cube </li>
</ul>
|
2,530,458 | <p>Find Range of $$ y =\frac{x}{(x-2)(x+1)} $$</p>
<p>Why is the range all real numbers ? </p>
<p>the denominator cannot be $0$ Hence isn't range suppose to be $y$ not equals to $0$ ?</p>
| StephenG - Help Ukraine | 298,172 | <p>I think this range is more properly $\bar{\mathbb{R}}$ which is the Extended Real Number line ( or the <a href="http://mathworld.wolfram.com/AffinelyExtendedRealNumbers.html" rel="nofollow noreferrer">Affinely Extended Real Numbers</a> if you prefer ) and not $\mathbb{R}$.</p>
<p>$\mathbb{R}$ does not include $\pm\... |
410,905 | <p>If A is real, symmetric, regular, positive definite matrix in $R^{n.n}$ and $x,h\in R^n$, why is it $\langle Ah,x\rangle = \langle h,A^T x\rangle =\langle Ax,h\rangle$?
Is there some rule or theorem for this?</p>
| pritam | 33,736 | <p>Note that inner product can be written as: $\langle x,y\rangle=y^Tx$. So $\langle Ah, x\rangle=x^T Ah$ and $\langle h, A^T x\rangle=(A^Tx)^Th=(x^TA)h=\langle Ah,x\rangle.$ Also $A^T=A$ as $A$ is symmetric and this gives the last equality.</p>
|
2,799,123 | <p>Prove the following equation by counting the non-empty subsets of $\{1,2,\ldots,n\}$ in $2$ different ways:</p>
<p>$1+2+2^2+2^3\ldots+2^{n-1}=2^n-1$.</p>
<p>Let $A=\{1,2\ldots,n\}$. I know from theory that it has $2^n-1$ non-empty subsets, which is the right-hand side of the equation but, how do count the left one... | user | 505,767 | <p>For the LHS we need to sum</p>
<ul>
<li>$n$ subset with $1$ elements</li>
<li>$\binom{n}{2}$ subset with 2 elements</li>
<li>$\binom{n}{3}$ subset with 3 elements</li>
<li>etc.</li>
</ul>
<p>that is by binomial theorem $\sum_{k=0}^{n} \binom{n}{k}a^kb^{n-k}=(a+b)^n$</p>
<p>$$\sum_{k=1}^{n} \binom{n}{k} =\sum_{k=0... |
145,303 | <p>Another question about the convergence notes by Dr. Pete Clark:</p>
<p><a href="http://alpha.math.uga.edu/%7Epete/convergence.pdf" rel="nofollow noreferrer">http://alpha.math.uga.edu/~pete/convergence.pdf</a></p>
<p>(I'm almost at the filters chapter! Getting very excited now!)</p>
<p>On page 15, Proposition 4.6 st... | Brian M. Scott | 12,042 | <p>Carl’s approach is almost certainly the easiest way to patch the gap in the notes. A nice variant is to let $D=\{x_n:n\in\Bbb N\}$ be a countable dense subset of $X$, define a map $$h:X\to\Bbb R^{\Bbb N}:x\mapsto\langle d(x,x_n):n\in\Bbb N\rangle\;,$$ and prove that $h$ is s homeomorphism of $X$ onto a subspace of $... |
1,156,874 | <p>How to show that $\mathbb{Z}[i]/I$ is a finite field whenever $I$ is a prime ideal? Is it possible to find the cardinality of $\mathbb{Z}[i]/I$ as well?</p>
<p>I know how to show that it is an integral domain, because that follows very quickly.</p>
| Arthur | 99,272 | <p>Let $\alpha \in I$ be an element, with norm $N(\alpha)$. Any element $x \in \mathbb{Z}[i]$ can be written as $q \alpha + r$ with $N(r) < N(\alpha)$. So every element of $\mathbb{Z}[i]/I$ (viewed as an equivalence class) contains an element of norm smaller than $N(\alpha)$, and there are only finitely many such el... |
1,123,777 | <p><strong><span class="math-container">$U$</span> here represents the upper Riemann Integral.</strong></p>
<p><img src="https://i.stack.imgur.com/GbNm2.jpg" alt="enter image description here" /></p>
<p><img src="https://i.stack.imgur.com/KtRI4.jpg" alt="enter image description here" /></p>
<p><img src="https://i.stack... | heropup | 118,193 | <p>For a given positive integer $n$, let $p_i(n) = (a_1, a_2, \ldots, a_k)$ be a given partition of $n$ that satisfies the criteria. For each such partition $p_i(n)$, how many ways are there to generate a unique partition $p_i(n+1)$? Is there a bijection? </p>
|
749,926 | <p>I have a group of 10 players and I want to form two groups with them.Each group must have atleast one member.In how many ways can I do it?</p>
| André Nicolas | 6,312 | <p>We solve first a different problem. We want to divide our people into two teams, one to wear blue uniforms, the other to wear red. Our set has $2^{10}$ subsets. Throw away the empty set and the full set. That leaves $2^{10}-2$ ways to choose the team that will wear blue uniforms.</p>
<p>However, there are no colour... |
4,112,958 | <p>This is a Number Theory problem about the extended Euclidean Algorithm I found:</p>
<p>Use the extended Euclidean Algorithm to find all numbers smaller than <span class="math-container">$2040$</span> so that <span class="math-container">$51 | 71n-24$</span>.</p>
<p>As the eEA always involves two variables so that <s... | J. W. Tanner | 615,567 | <p>You are looking for numbers such that <span class="math-container">$71n\equiv24\bmod51$</span>.</p>
<p>The extended Euclidean algorithm gives the Bezout relation <span class="math-container">$23\times71-32\times51=1$</span>,</p>
<p>so <span class="math-container">$23\times71\equiv1\bmod51$</span>. Therefore, you ar... |
644,935 | <p>I'm having trouble integrating $3^x$ using the $px + q$ rule. Can some please walk me through this?</p>
<p>Thanks</p>
| Martín-Blas Pérez Pinilla | 98,199 | <p>And the $px+q$ rule is...?</p>
<p>You <strong>can</strong> use $3^x = e^{x\log 3}$ and the obvious change of variable.</p>
|
12,878 | <p>I wish the comments didn't have a lower bound for characters.Many times all I want to say is "yes". Can someone explain me what is the purpose of this l.b.?</p>
| user127096 | 127,096 | <p>This is the site humbly suggesting you to use more characters. I'd like to also encourage you to do this. And also to consider the following observations: </p>
<ol>
<li>Users who habitually type things like "n-mfld", "height fcn", "cohomol", "sing. coho", " Lebesgue meas."... are making the site harder to use for... |
3,219,428 | <p>Sorry for the strange title, as I don't really know the proper terminology.</p>
<p>I need a formula that returns 1 if the supplied value is anything from 10 to 99, returns 10 if the value is anything from 100 to 999, returns 100 if the value is anything from 1000 to 9999, and so on.</p>
<p>I will be translating th... | Sharky Kesa | 398,185 | <p>I think your function is
<span class="math-container">$$f(x) = 10^{\left \lfloor \log_{10}(x) \right \rfloor - 1}$$</span>
where <span class="math-container">$\lfloor r \rfloor$</span> denotes the largest integer less than or equal to <span class="math-container">$r$</span>.</p>
|
398,857 | <p>Please help me solve this and please tell me how to do it..</p>
<p>$12345234 \times 23123345 \pmod {31} = $?</p>
<p>edit: please show me how to do it on a calculator not a computer thanks:)</p>
| lab bhattacharjee | 33,337 | <p>As $10^1\equiv 10\pmod{31},$</p>
<p>$10^2=100\equiv7,$</p>
<p>$10^3\equiv10\cdot7\equiv8,$</p>
<p>$10^4\equiv49\equiv18,$</p>
<p>$10^5=10^2\cdot10^3\equiv 7\cdot8\equiv25,$</p>
<p>$10^6=(10^3)^2\equiv8^2\equiv2,$</p>
<p>$10^7=10^4\cdot10^3\equiv18\cdot8\equiv20,$</p>
<p>$$12345234=4+3\cdot10+2\cdot10^2+5\cdot... |
3,043,598 | <p>I have seen a procedure to calculate <span class="math-container">$A^{100}B$</span> like products without actually multiplying where <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are matrices. But the procedure will work only if <span class="math-container">$A$</span> is diagona... | Aaron | 9,863 | <p>When a matrix is not diagonalizable, you can instead use Jordan Normal Form (JNF). Instead of picking a basis of eigenvectors, you use "approximate eigenvectors." While eigenvectors are things in the kernel of <span class="math-container">$A-\lambda I$</span>, approximate eigenvectors are things in the kernel of <s... |
2,315,647 | <p>Compute the gravitational attraction on a unit mass at the origin due to the mass (of constant density) occupying the volume inside the sphere $r = 2a$ and above the plane $z=a$. Use spherical coordinates.</p>
<p>So I know the function should be
$$(G/r^2) dM$$
What are the limits of integration? What should the in... | Community | -1 | <p>While Rafa gave a good answer, I just wanted to expand a bit on the derivation here. That way you're less likely to get lost. Let me know if I need to explicate further on some part.</p>
<hr>
<p>A couple of physics formulas necessary for this:</p>
<ul>
<li>The gravitational force on a mass $m$ due to a mass $M$... |
1,057,675 | <p>I was asked to prove that $\lim\limits_{x\to\infty}\frac{x^n}{a^x}=0$ when $n$ is some natural number and $a>1$. However, taking second and third derivatives according to L'Hôpital's rule didn't bring any fresh insights nor did it clarify anything. How can this be proven? </p>
| Asaf Karagila | 622 | <p>When you "close" the bracket, it implicitly means that you're working in the space $\Bbb R\cup\{\pm\infty\}$ (and we omit the $+$ from the positive infinity sign), and in that space $[0,\infty)$ is not closed, since $\infty$ is indeed a limit point of that set.</p>
<p>[It might be the case that you are working in t... |
1,599,886 | <p>What is the proper way of proving : the density operator $\hat{\rho}$ of a pure state has exactly one non-zero eigenvalue and it is unity, i.e,</p>
<p>the density matrix takes the form (after diagonalizing):
\begin{equation}
\hat{\rho}=
{\begin{bmatrix}
1 & 0 & \cdots & 0 \\
0 & 0 & \cdots &... | user247327 | 247,327 | <p>Do you not know how to find the eigenvalues of a matrix by solving the "characteristic equation" of the matrix? The characteristic equation of any diagonal matrix is just the product of linear terms, each the number on the diagonal minus x so the eigenvalues [b]are[/b] the numbers on the diagonal.</p>
|
1,556,298 | <p>If we have $p\implies q$, then the only case the logical value of this implication is false is when $p$ is true, but $q$ false.</p>
<p>So suppose I have a broken soda machine - it will never give me any can of coke, no matter if I insert some coins in it or not.</p>
<p>Let $p$ be 'I insert a coin', and $q$ - 'I ge... | hmakholm left over Monica | 14,366 | <p>Your distinction between "true" and "logical value 1" is not one that formal logic generally observes. Here "1" and "true" are synonyms for the same concept.</p>
<p>The meaning of the $\Rightarrow$ connective is what its truth table says it is, neither more nor less -- the truth table <em>defines</em> the connectiv... |
1,556,298 | <p>If we have $p\implies q$, then the only case the logical value of this implication is false is when $p$ is true, but $q$ false.</p>
<p>So suppose I have a broken soda machine - it will never give me any can of coke, no matter if I insert some coins in it or not.</p>
<p>Let $p$ be 'I insert a coin', and $q$ - 'I ge... | CiaPan | 152,299 | <p>The fact some specific values satisfy the formula doesn't mean the formula is true in general. "It is noon now and it rains" is true right now and right here, but at other place or in other time this will appear false.</p>
<p>Your implication will turn out true if you check it keeps satisfied by any possible combin... |
539,448 | <p>$a,b,c,d,e>o$. Show that</p>
<p>$$ a^{b+c+d+e}+ b^{c+d+e+a}+c^{ d+e+a+b}+ d^{e+a+b+c}+e^{a+b+c+d}>1$$ </p>
| math110 | 58,742 | <p>oh,I ask my teacher(tian27546),he told me this is he inequality：<a href="http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=484816&p=2718780#p2718780" rel="nofollow">http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=484816&p=2718780#p2718780</a></p>
|
1,734,419 | <p>I have tried to show that : $2730 |$ $n^{13}-n$ using fermat little theorem but i can't succeed or at a least to write $2730$ as $n^p-n$ .</p>
<p><strong>My question here</strong> : How do I show that $2730$ divides $n^{13}-n$ for $n$ is integer ?</p>
<p>Thank you for any help </p>
| Quentchen | 325,651 | <p>First you calculate the prime factorisation of $2730$. You will find that it splits in 5 prime factors. Then use the Chinese Remainder theorem and show that $n^{13}\equiv n \pmod{p}$ for the prime factors. </p>
|
8,695 | <p>I have a parametric plot showing a path of an object in x and y (position), where each is a function of t (time), on which I would like to put a time tick, every second let's say. This would be to indicate where the object is moving fast (widely spaced ticks) or slow (closely spaced ticks). Each tick would just be... | s0rce | 65 | <p>I couln't get nice little lines so I've used filled circles instead. I hope this works.</p>
<p>I've spaced the points out but Pi/4 on the curve in this example.</p>
<pre><code>f = {Sin[#], Sin[2 #]} &
Show[
ParametricPlot[f[u], {u, 0, 2 \[Pi]}],
ListPlot[f /@ Range[0, 2 \[Pi], \[Pi]/4],
PlotStyle -> Di... |
974,656 | <p><img src="https://i.stack.imgur.com/LyqzL.jpg" alt="enter image description here"></p>
<p>One way to solve this and my book has done it is by : </p>
<p><img src="https://i.stack.imgur.com/2wYSn.jpg" alt="enter image description here"></p>
<hr>
<p>This is a well known way, but I have a different method, and it se... | André Nicolas | 6,312 | <p>Without loss of generality we may assume that $a_n\gt 0$. For if it is not, we can multiply $P(x)$ by $-1$ wiithout changing the roots. Then $a_0\lt 0$.</p>
<p>Note that $P(0)\lt 0$. If we can show that $P(a)\gt 0$ for some positive $a$, it will follow by the Intermediate Value Theorem that $P(x)=0$ for some $x$ be... |
28,532 | <p><code>MapIndexed</code> is a very handy built-in function. Suppose that I have the following list, called <code>list</code>:</p>
<pre><code>list = {10, 20, 30, 40};
</code></pre>
<p>I can use <code>MapIndexed</code> to map an arbitrary function <code>f</code> across <code>list</code>:</p>
<pre><code>{f[10, {1}],... | amr | 950 | <p>Here's a form similar to Kuba's approach:</p>
<pre><code>mapAtIndexed[f_, list_, pos_] :=
ReplacePart[list, # :> f[list[[Sequence @@ #]], #] & /@ pos];
</code></pre>
<p>A pure pattern version:</p>
<pre><code>mapAtIndexed[f_, list_, pos_] :=
ReplacePart[list,
i : (Alternatives @@ pos) :> f[list[[S... |
2,337,583 | <p>I cannot understand the inductive dimension properly. I read something on Google but mostly there only are conditions or properties. Not a definition. I got to know about it from the book “ The fractal geometry of nature”. ( I am a 12 grader.)</p>
| Theo Bendit | 248,286 | <p>It's a recursive definition. We define all the spaces of dimension $-1$: we define there to be only one, the empty set with its one and only topology). We then define the spaces of dimension $0$ to be the spaces which have the following properties:</p>
<ul>
<li>It's not of dimension $-1$ (i.e. non-empty), and</li>
... |
325,964 | <p>This question may be trivial, or overly optimistic. I do not know (but I guess the latter...). I am a group theorist by trade, and the set-up I describe cropped up in something I want to prove. So this question is out of my comfort zone, but I am happy to clarify anything if needed.</p>
<p>I have a countable set <s... | Dirk | 109,932 | <p>You can't just do induction on chains without adding extra info. If you could, I could give you the chain <span class="math-container">$0 < z$</span> and thus see without much inductive work that the property holds. To have a chance at classical induction, you need a concept of successor, i.e. a function <span cl... |
325,964 | <p>This question may be trivial, or overly optimistic. I do not know (but I guess the latter...). I am a group theorist by trade, and the set-up I describe cropped up in something I want to prove. So this question is out of my comfort zone, but I am happy to clarify anything if needed.</p>
<p>I have a countable set <s... | Andrej Bauer | 1,176 | <p>You seem to be asking about <em>well-founded induction</em>. It generalizes many forms of induction, including the usual induction on numbers and transfinite induction on ordinals.</p>
<p>Consider a relation <span class="math-container">$<$</span> on a set <span class="math-container">$A$</span>. Say that <span ... |
43,172 | <p>I am trying to solve $\frac{dx}{dt} + \alpha x = 1$, $x(0) = 2$, $\alpha > 0$ where $\alpha$ is a constant. </p>
<p>[some very badly done mathematics deleted]</p>
<p>Continuing with Gerry's suggestion:</p>
<p>$\log|1-\alpha x | = -t\alpha + \log|1-2\alpha|$</p>
<p>$1-\alpha x = e^{-t\alpha}(1-2\alpha)$</p>
<... | Gerry Myerson | 8,269 | <p>joriki's approach is fine. Alternatively, the equation is "variables separable" and can be solved by rewriting as $${1\over1-\alpha x}\,dx=dt$$ and then integrating; $$\int{1\over1-\alpha x}\,dx=\int\,dt,\qquad -{1\over\alpha}\log|1-\alpha x|=t+C$$ stick in $t=0$ to get $$C=-{1\over\alpha}\log|1-2\alpha|,\qquad -{1\... |
464,426 | <p>Find the limit of $$\lim_{x\to 1}\frac{x^{1/5}-1}{x^{1/3}-1}$$</p>
<p>How should I approach it? I tried to use L'Hopital's Rule but it's just keep giving me 0/0.</p>
| Amr | 29,267 | <p><strong>Hint:</strong> $$\frac{x^{\frac{1}{5}}-1}{x^{\frac{1}{3}}-1}=\frac{(x^{\frac{1}{15}})^3-1}{(x^{\frac{1}{15}})^5-1}=\frac{((x^{\frac{1}{15}})-1)(x^{\frac{1}{15}})^2+(x^{\frac{1}{15}})+1)}{((x^{\frac{1}{15}})-1)((x^{\frac{1}{15}})^4+(x^{\frac{1}{15}})^3+(x^{\frac{1}{15}})^2+(x^{\frac{1}{15}})+1)}$$</p>
<p>Thi... |
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