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 |
|---|---|---|---|---|
4,332,748 | <p>This is slightly different question. First I need to mention that I am neither a mathematician nor a researcher. As an ordinary student the separation " with and without Axiom of choice " trouble me a lot. But I am really interested to know further.</p>
<ol>
<li><p>Every ideal in a ring contained in some ... | Lee Mosher | 26,501 | <p>Given the broad interest in models of set theory where AC fails, it seems more appropriate to clearly state the hypotheses that you are assuming for the model of set theory in which you wish to work, <em>especially</em> for a theorem which is false in certain models where AC fails.</p>
<p>By comparison, if you had a... |
194,803 | <p>I ran into some problems while doing an exercise. The problem goes as follows:</p>
<p>Suppose we have two random independent variables $X$ and $Y$. Both are distributed normally with parameters $(0, \sigma^2)$. $\mathbb{P}(dx)=\frac{1}{\sigma\sqrt{2\pi}} \exp^{-\frac{x}{2\sigma^2}}dx$. For $\gamma \in \mathbb{R}$, ... | Dilip Sarwate | 15,941 | <p>It is straightforward to compute the joint density of $(U,V)$ from that of $(X,Y)$. Jacobians and the like are involved in the standard undergraduate
treatment of this topic (which is often not understood very well by said
undergraduates). In this instance, the Jacobian approach is easier
since the transformation i... |
2,202,518 | <p>I'm trying to do exercise 9 of section 15.5 in the book Abstract Algebra written by Dummit and Foote, whose aim is to establish a homeomorphic map between the elements in the fiber over $(p)$ of the Zariski continuous map from $\mathrm{Spec}(\mathbb{Z} [x])$ to $\mathrm{Spec}(\mathbb{Z})$ and the elements in $\mathr... | Alex Provost | 59,556 | <p>In $\mathbb{Z}_7$, we have $-2 = 5$ and $-4 = 3$. Therefore, if we let $A = \left(\begin{matrix}
0&2 \\
2 & 2
\end{matrix}\right)$, the equality $A^2 = 4I + 2A$ with $\mathbb{Z}_7$ coefficients is equivalent to $$0 = A^2 - 2A - 4I = A^2 + 5A + 3I = \phi(X^2 + 5X + 3).$$</p>
|
1,980,012 | <p>I am facing issue to evaluate this particular type of definite integral,</p>
<p>$$I=\int_{0}^{\infty}\exp(-\alpha x^2)\cos(\beta x)dx$$</p>
<p>Please suggest a way to this.</p>
<p>Thanks</p>
| reuns | 276,986 | <p>The least amount of complex analysis is as follow :</p>
<ul>
<li>For $z > 0$, let $$f(z) = \int_{-\infty}^\infty e^{-x^2} e^{2zx}dx$$
Complete the square $x^2-2zx= (x-z)^2 - z^2$ so that
$$f(z) = e^{-z^2} \int_{-\infty}^\infty e^{-(x-z)^2}dx \underset{y = x-z}= e^{-z^2} \int_{-\infty}^\infty e^{-y^2}dy = Ce^{-... |
2,770,607 | <p>$\Bbb C$ is algebraically closed, does that mean there is no extension over it? Why can't we extend $\Bbb C$ like $\Bbb C(j,k)=\mathcal Q$? Or can we?</p>
| José Carlos Santos | 446,262 | <p>From the fact that $\mathbb C$ is algebraically closed you can deduce that it has no field extension of finite degree. Of course, it has field extensions of infinite degree.</p>
<p>Since the quaternions aren't a field, they don't count here.</p>
|
1,005,660 | <p>I have stuck with ODE problems. </p>
<p>\begin{align}
\frac{f''}{f'}=-2\frac{g'}{g}
\end{align}
Here i know the solution is $ g^2 f' =const$. but i don't know how to obtain those relation.
It seems simple. </p>
<p>Is my approach is correct?</p>
<p>\begin{align}
\int \frac{f''}{f'}dx= -2\int \frac{g'}{g} dx =cons... | Simon S | 21,495 | <p>$$\ell = \left\{ t\begin{pmatrix} 0 \\ 1 \end{pmatrix} : t \in \mathbb{R} \right\}$$</p>
<p>is a straight line running on the $y$-axis.</p>
|
51,895 | <p>I generated a spline function called <code>f</code> using the <code>BSplineFunction</code> with domain <code>{t, 0, 1}</code>.</p>
<pre><code>f = BSplineFunction[{{0., 5.81152}, {-0.909122, 5.73997}, {-1.86805, 5.5031},
{-1.79586, 5.52709}, {-2.40811, 5.28912}, {-3.4109, 4.70527}, {-4.68131, 3.501},
{-4.931... | Apple | 10,193 | <pre><code>Clear["Global`*"]
pts = {{1, 1}, {2, 3}, {3, -1}, {4, 1}, {5, 0}};
f1 = Interpolation[pts, InterpolationOrder -> 1];
f2 = BSplineFunction[pts];
Show[Graphics[{Red, Point[pts], PointSize[0.02],
Point[f2 /@ {0.2, 0.4}], Green, Line[pts]}, Axes -> True],
ParametricPlot[f2[t], {t, 0, 1}]]... |
3,557,252 | <p>By " same base, same exponent " rule I mean : </p>
<p><span class="math-container">$$b^x = b^n \iff x=n$$</span></p>
<p>Can the equation:
<span class="math-container">$$ 2^x+4^x = 6$$</span>
be solved by saying
<span class="math-container">$$ 2^x+4^x = 6 = 2+4= 2^1 + 4^1$$</span>
and consequently, that
<span c... | Michael Rozenberg | 190,319 | <p>No.</p>
<p>It's better:
<span class="math-container">$$(2^x)^2+2^x-6=0$$</span> or
<span class="math-container">$$(2^x-2)(2^x+3)=0$$</span> or <span class="math-container">$$2^x=2,$$</span> which gives <span class="math-container">$$x=1.$$</span>
Also, you can say that <span class="math-container">$2^{2x}+2^x$</spa... |
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,... | Dan Piponi | 1,233 | <p><a href="http://www.cs.tau.ac.il/~nachum/nachumd/Homepage.html">Nachum Dershowitz</a> has lots of <a href="http://www.cs.tau.ac.il/~nachumd/pub.html">work</a> on termination proofs and many of these make use of ordinal arithmetic. In particular, if you have a computer program that moves from one state to another, an... |
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,... | Gerald Edgar | 454 | <p>Perhaps fitting in here is Lindenstrauss's proof of Lyapunov's convexity theorem.</p>
<p><a href="http://atlas-conferences.com/c/a/e/y/06.htm" rel="noreferrer">http://atlas-conferences.com/c/a/e/y/06.htm</a> </p>
<p>"the slickest proof to end all proofs" (Halmos)</p>
<p>although the result is in $\mathbb{R}^n$, f... |
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,... | Gerald Edgar | 454 | <p>Maybe "analytic number theory" belongs here as an answer. Certain number theory results are most simply proved using complex analysis.</p>
|
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,... | Joel Adler | 26,085 | <p>What about the puzzle named escape on Joel David Hamkins' homepage?</p>
<p><a href="http://jdh.hamkins.org/escape/" rel="nofollow noreferrer">http://jdh.hamkins.org/escape/</a></p>
<p>That not all the squares occupied by the three stones in the initial configuration can be vacuated is shown with infinite series.</... |
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... | doraemonpaul | 30,938 | <p>Similar to <a href="https://math.stackexchange.com/questions/616426">Find $f(x)$ where $ f(x)+f\left(\frac{1-x}x\right)=x$</a>:</p>
<p><span class="math-container">$f(x)+f\left(\dfrac{1-x}{x}\right)=1-x$</span></p>
<p><span class="math-container">$f(x)+f\left(\dfrac{1}{x}-1\right)=1-x$</span></p>
<p><span class="... |
3,248,134 | <p>My question is, given the general solution
<span class="math-container">$$Y=C_1x+c_2\dfrac{1}{x}$$</span>
find the differential equation</p>
<p>My attempt:</p>
<p>I have derivated the equation and then look for the constants:</p>
<p><span class="math-container">$$Y'=C_1+c_2\dfrac{-1}{x^2}$$</span>
<span class="... | AsdrubalBeltran | 62,547 | <p>Note that if <span class="math-container">$y_h=c_1x+c_2/x$</span> implies that <span class="math-container">$y_1=x$</span> and <span class="math-container">$y_2=1/x$</span> are solutions of:
<span class="math-container">$$y''+P(x)y'+Q(x)y=0$$</span>
Then
<span class="math-container">\begin{cases} (y_1)'' + P(x)(y_1)... |
3,248,134 | <p>My question is, given the general solution
<span class="math-container">$$Y=C_1x+c_2\dfrac{1}{x}$$</span>
find the differential equation</p>
<p>My attempt:</p>
<p>I have derivated the equation and then look for the constants:</p>
<p><span class="math-container">$$Y'=C_1+c_2\dfrac{-1}{x^2}$$</span>
<span class="... | nmasanta | 623,924 | <p>Given that <span class="math-container">$Y=C_1x+C_2\dfrac{1}{x}\implies xY=C_1 x^2+C_2$</span></p>
<p>Differentiating with respect to <span class="math-container">$x$</span>,</p>
<p><span class="math-container">$Y+xY'=2xC_1$</span> . . . . .<span class="math-container">$(1)$</span></p>
<p>Again differentiating ... |
239,004 | <p>From some geometric computations I get a lot of 3 dimensional points. From theory I know two facts: The points define a convex polyhedron, and this polyhedron has non-regular 3-, 4- and 5-gons as faces.</p>
<p>Here is a simple example:</p>
<pre><code>data = {
{0.21737620787507367`, 0.28204133994445385`, -0.638196... | Michael E2 | 4,999 | <p>Maybe this?:</p>
<pre><code>chull = ConvexHullMesh[data];
polys = Cases[Show[chull], _Polygon, Infinity];
coords = First@
Cases[Show@chull, GraphicsComplex[c_, ___] :> c, Infinity];
Graphics3D[
GraphicsComplex[
coords,
{EdgeForm[Red], polys}
]]
</code></pre>
<p><a href="https://i.stack.imgur.com/mlmVQ.... |
1,171,142 | <p>I know this question has been asked before, but I think I'm very close to a new solution and wanted to know if it is a viable approach.</p>
<p>Let $C(X)$ be the ring of continuous functions $X \rightarrow \mathbb{C}$ where $X$ is compact Hausdorff, and let $\mathfrak M$ be a maximal ideal. We want to show that $\m... | D_S | 28,556 | <blockquote>
<p>Proof of the Zorn's lemma argument:</p>
</blockquote>
<p>Suppose $I_{A_i}$ is an ascending chain of ideals (set $i < j$ if and only if $I_{A_i} \subseteq I_{A_j}$). By order reversing correspondence, $A_i$ is a descending chain of closed subsets whose intersection $A$ is nonempty, as $X$ is compa... |
73,454 | <p>Can anyone help me with a proof of the following claim (see for example the book Higher algebraic geometry of Olivier Debarre, proof of Proposition 1.43, page 31):</p>
<p>Let X be a complex manifold, and let W be a complex submanifold of X, with codimension $\geq 2$. Let $\pi :Y \rightarrow X$ be a bimeromorphic mo... | roy smith | 9,449 | <p>I am kind of a rookie at this, but what if Y is a small resolution of a double point on a threefold X, with one dimensional excepTional locus. Then it seems false to expect a factorization through the blowup since the curve exceptional locus could not map onto the two dimensional exceptional locus of the blowup of ... |
414,642 | <p><strong>Background: Evaluation of</strong> <span class="math-container">$\int_0^\infty \frac{\log(1+x^2)}{(1+x^2)^2}dx$</span></p>
<p>We can prove using the Beta-Function identity that</p>
<p><span class="math-container">$$\int_0^\infty \frac{1}{(1+x^2)^\lambda}dx=\sqrt{\pi}\frac{\Gamma \left(\lambda-\frac{1}{2} \ri... | Quanto | 686,284 | <p>The generalized results for the even and odd cases of <span class="math-container">$$I_n=\int_0^\infty \frac{\log(1+x^n)}{(1+x^2)^2}dx
$$</span>
are respectively as follows
<span class="math-container">\begin{align}
I_{2m} =& -\frac{m\pi}4+\frac{m\pi}2\ln2+\pi\sum_{k=1}^{[\frac m2]}\ln \cos\frac{(m-2k+1)\pi}{4m}... |
2,714,660 | <p>I've got a rather elemental doubt about what "mapping a subspace homeomorphically onto its image" means, but I'd like to be sure what is being implied.</p>
<p>Let $f:X\to Y$ a continuous injective map between topological spaces. Let $U\subseteq X$ be an open subspace. Then we could say in some context that $f$ maps... | user | 505,767 | <p>To avoid the problem for $x=0$, note that the <a href="https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Inverse_trigonometric_functions" rel="nofollow noreferrer"><strong>following holds</strong></a></p>
<ul>
<li>for $x>0 \implies \arctan \frac 1 x=\frac{\pi}2 -\arctan x $</li>
<li>for $x<0 \imp... |
1,868,458 | <p>Assume that we have two continuous integrable functions $f,g \in L^1(\mathbb{R})$ such that, for some $x_0 \in \mathbb{R}$, we have, $$f(x_0) \leq g(x_0) \; \; \; \; (1).$$</p>
<p>Now let us define the Gaussian function with variance $\sigma>0$, $$p_\sigma:x \mapsto \frac{1}{\sqrt{2 \pi} \sigma} e^{-\frac{x^2}{2... | Thomas Andrews | 7,933 | <p>You need $f(x_0)<g(x_0)$. If $f(x_0)=g(x_0)$ this is not true.</p>
<p>It is also not true, as you note, for all $\delta$.</p>
<p>Assuming $f(x_0)<g(x_0)$, We won't need $f,g$ positive, so we can reduce it to letting $h(x)=g(x)-f(x)$ and ask</p>
<blockquote>
<p>If $h(x)$ is continuous and $L_1$ and $h(x_0)... |
540,636 | <p>I need to find the subgroup $H \leq A_4$ generated by $(123).$ I know that this subgroup will have order 1,2,3,4,6, or 12. </p>
| Community | -1 | <p><strong>Hint</strong>: $(123)^2 = (132)$, and since it's a $3$-cycle, $(123)^3$ is the identity permutation.</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... | MathGuy | 7,550 | <p>You certainly will not find a free one, but I worked for tutor.com. It is a good site to recommend to students. The tutors are verified and readily available whenever the student needs help. In my experience as a tutor and mentor, the tutors make the students do the work and do not give them the answers. I'm not sur... |
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... | Pratyush Sarkar | 64,618 | <p>I myself did not study graph theory yet but I do know that if you consider the adjacency matrix of a graph, then there are interesting things with its eigenvalues and eigenvectors.</p>
<p>You can also show how to solve a system of ordinary differential equations of the form
$$
{x_1}'(t) = a_{11}x_1(t) + a_{12}x_2(t... |
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... | bubba | 31,744 | <p>Maybe these won't be "serious" enough for you, but I think the most interesting uses of eigensystems are:</p>
<p>(1) Understanding/classifying quadric surfaces.</p>
<p>(2) Moments of inertia (principal axes)</p>
<p>(3) Principal component analysis</p>
|
3,170,504 | <p>Let <span class="math-container">$k$</span> be an algebraically closed field. Let <span class="math-container">$X$</span> be a <span class="math-container">$k$</span>-scheme that admits a <span class="math-container">$k$</span>-immersion into projective space. Is it true that any <span class="math-container">$k$</sp... | Dr. Sonnhard Graubner | 175,066 | <p>Write your equation in the form. <span class="math-container">$$\sin(2x)(3-5\cos(2x))=0$$</span> </p>
|
1,825,016 | <p>I am trying to prove that $(p\rightarrow q)\wedge (p\rightarrow r) = p\rightarrow (q \wedge r)$. </p>
<p>This is my approach:</p>
<p>$(p\rightarrow q)\wedge(p\rightarrow r) = (-p \vee q) \wedge (-p \vee r)$</p>
<p>= ${[(-p \vee q) \wedge -p] \vee [(-p \vee q) \wedge r]}$</p>
<p>= ${[(-p \wedge -p) \vee (q \wedge... | Daniel W. Farlow | 191,378 | <p>As some others have pointed out, you used distributivity correctly but in a less than desired way (by complicating the expression further as opposed to simplifying it). As Git Gud notes, you want to "factor out the $\neg p$" where you, instead, expanded things further by distributivity. Finally, you will want to use... |
1,686,834 | <p>The equations of the lines are given:</p>
<p>$l:$ $$r = i + j + k + s(i - j + 2l)$$</p>
<p>$m:$ $$r = 4i + 6j + k + t(2i + 2j + k)$$</p>
<p>We are also told that the lines intersect, and the angle between the two is $74.2°$. That's all the information we have been given.</p>
<p>This is my working:</p>
<p>$$(1,-... | BCLC | 140,308 | <p>Do you know the multinomial theorem? How about the binomial theorem?</p>
<p>Expanding $$(x_1 + ... + x_n)^3$$</p>
<p>gives us terms that look like</p>
<p>$$x_i^3, x_ix_jx_k, x_i^2x_k, x_ix_k^2$$</p>
<p>We use linearity to split the expectation of the cube of the sum into sums of expectations that look like</p>
... |
3,617,619 | <p>Let <span class="math-container">$\textbf{P}_{2}(\textbf{R})$</span> be the real vector space of polynomials of degree less than or equal to <span class="math-container">$2$</span>.</p>
<p>Let <span class="math-container">$f:\textbf{P}_{2}(\textbf{R})\rightarrow \textbf{P}_{2}(\textbf{R})$</span> be the linear map ... | IgnoranteX | 698,646 | <p>Let's consider the first equation:
<span class="math-container">$$
f(1) = 0
$$</span>
Now, we can decompose the input and the output on the polynomial basis you suggested:
<span class="math-container">$$
f(1\times(1) + 0(x+1) + 0(x^2+1))= 0\times(1) + 0(x+1) + 0(x^2+1)
$$</span>
Reading out the coefficients both o... |
248,134 | <p>Consider a tiling of a square by convex polygons, such that every line through the square intersects at most $k$ polygons. Let $n$ be maximum number of polygons such a tiling can have. What is the dependence of $n$ on $k$?</p>
<p>Tiling the square with a grid of $k/2$ by $k/2$ smaller squares gives a simple constru... | fedja | 1,131 | <p>The correct upper bound seems to be of order $k^k$ in the sense that we can create $\ge k^k$ polygons so that each line intersects at most $Ck$ of them and the number is bounded from above by $k^{Ck}$. I'll prove the second part of this claim here.</p>
<p><strong><em>Definition.</em></strong>
Let $A,B$ be two close... |
886 | <p>Okay so <a href="https://math.stackexchange.com/questions/835/optimal-strategy-for-deal-or-no-deal">this question</a> reminded me of one my brother asked me a while back about the hit day-time novelty-worn-off-now snoozathon <a href="http://en.wikipedia.org/wiki/Deal_or_no_deal" rel="noreferrer">Deal or no deal</a>.... | Community | -1 | <p>The key difference between the Monty Hall problem and the one you're describing is:
a) the host (in the case of Monty Hall) makes an unsuccessfull choice for the contestant thus altering the probabilities.
b) the host can't provide that information when there is only a binomial distribution (2 choices remaining). If... |
1,634,265 | <p>I have some trouble in how to evaluate this integral:
$$
\int_{0}^{\pi}\theta\ln\left(\tan\left(\theta \over 2\right)\right)
\,\mathrm{d}\theta
$$
I think it maybe has another form
$$
\int_{0}^{\pi}\theta\ln\left(\tan\left(\theta \over 2\right)\right)
\,\mathrm{d}\theta
=
\sum_{n=1}^{\infty}{1 \over n^{2}}
\left[\ps... | FDP | 186,817 | <p><span class="math-container">\begin{align}
J&=\int_0^\pi \theta\ln\left(\tan\left(\frac{\theta}{2}\right)\right)d\theta\\
&\overset{u=\tan\left(\frac{\theta}{2}\right)}=4\underbrace{\int_0^\infty \frac{\arctan u\ln u}{1+u^2}du}_{=K}\\
K&\overset{\text{IBP}}=\underbrace{\left[\arctan u\left(\int_0^u \frac... |
1,634,265 | <p>I have some trouble in how to evaluate this integral:
$$
\int_{0}^{\pi}\theta\ln\left(\tan\left(\theta \over 2\right)\right)
\,\mathrm{d}\theta
$$
I think it maybe has another form
$$
\int_{0}^{\pi}\theta\ln\left(\tan\left(\theta \over 2\right)\right)
\,\mathrm{d}\theta
=
\sum_{n=1}^{\infty}{1 \over n^{2}}
\left[\ps... | robjohn | 13,854 | <p>I ran across this question while researching <a href="https://math.stackexchange.com/q/4609446">a duplicate question</a>.</p>
<p>I had written the following answer, which seems to be a different approach to the answers I've seen so far.</p>
<hr />
<p>As shown in <a href="https://math.stackexchange.com/a/1071367">thi... |
989,043 | <p>I'm trying to prove that, for a finite set $A$, if the map $f: A\rightarrow A$ is a surjection, then it's an injection. I've looked at this post: <a href="https://math.stackexchange.com/questions/63072/surjectivity-implies-injectivity">Surjectivity implies injectivity</a> but the arguments there seem very round-abo... | lhf | 589 | <p>Let $f: A\to A$ be any function. For each $a\in A$, let $N(a)$ be the number of elements in $A$ that are mapped to $a$. Then $\sum_a N(a) = n = |A|$ because every element of $A$ is mapped to some element of $A$. (This is the statement that the fibers of $f$ partition $A$.)</p>
<p>If $f$ is surjective, then $N(a)\ge... |
989,043 | <p>I'm trying to prove that, for a finite set $A$, if the map $f: A\rightarrow A$ is a surjection, then it's an injection. I've looked at this post: <a href="https://math.stackexchange.com/questions/63072/surjectivity-implies-injectivity">Surjectivity implies injectivity</a> but the arguments there seem very round-abo... | Lei Zhao | 147,999 | <p>I'm gonna provide some elegant direct proofs (without contradiction) that injectivity implies surjectivity and that surjectivity implies injectivity.</p>
<p>First, we introduce some notation as shown below.</p>
<p>$$f^k(x) := \begin{cases}
x & \text{if } k = 0 \\[3pt]
f(f^{k-1}(x)) & \text{if } k > 0 ... |
58,241 | <p>I am trying to export a plot generated by <code>MatrixPlot</code> to PDF but am having problems with the way the exported PDF is rendered.</p>
<pre><code> mPlot = MatrixPlot[length,
Epilog -> {Cyan,
MapIndexed[If[#1 != 0, Text[#1, Reverse[#2 - 1/2]]] &,
Reverse[number], {2}]}, Mesh -> True,... | Jens | 245 | <p>I see this problem too, and I found it is fixed in Skim and Preview by using this function:</p>
<pre><code>rasterTrick[plot_] :=
Show[plot,
Prolog -> {Opacity[0], Texture[{{{0, 0, 0, 0}}}],
VertexTextureCoordinates -> {{0, 0}, {1, 0}, {1, 1}},
Polygon[{{0, 0}, {.1, 0}, {.1, .1}}]}]
Export["reg... |
89,458 | <blockquote>
<p>the conformal structure does not see the conical singularities of a polyhedral surface.</p>
</blockquote>
<p>This is a quote from the Preface of <em><a href="http://www.springer.com/physics/book/978-3-642-24439-1">Quantum Triangulations</a></em> (eds.: Carfora, Marzuoli).
The sentiment is attributed ... | Jim Belk | 6,514 | <p>Consider the following pair of surfaces:</p>
<ol>
<li><p>$P\;$ is the plane with the origin removed.</p></li>
<li><p>$C$ is the cone $z = \sqrt{x^2+y^2}$ in $\mathbb{R}^3$, with the origin removed.</p></li>
</ol>
<p>There are several possible structures we could put on these surfaces. For example, both of the sur... |
2,293,731 | <blockquote>
<p>Given $x$ and $y$ are positive real numbers less than 9, what is the probability that the sum of them is greater than 3</p>
</blockquote>
<p>I thought that these are integers and did $\frac{_6C_2}{_8C_2}$ but then I realized that they are all the positive real numbers between $0$ and $9$, not just wh... | Dragonite | 395,130 | <p>This sort of problem is best done by drawing a plane, put a line at $x=9$ and $y=9$. Then we want to know when $x + y \geq 3$ or rather when $y \geq -x + 3$, graph this line and shade the region appropriately and find its area. Then, take this area and divide by $81$, which is the area of the square that we are cons... |
186,453 | <p>I want to get <code>x^2 + y^2</code> by evaluating <code>Abs[a + b I]^2</code>. Unfortunately, Mathematica just reproduces the same <code>Abs[a + b I]^2</code>. </p>
| Carl Woll | 45,431 | <p>Use <a href="http://reference.wolfram.com/language/ref/ComplexExpand" rel="noreferrer"><code>ComplexExpand</code></a>:</p>
<pre><code>ComplexExpand[Abs[(x + y I)]^2]
</code></pre>
<blockquote>
<p>x^2 + y^2</p>
</blockquote>
|
1,387,180 | <p>Given a selfadjoint (maybe unbounded) operator $T$ on a Hilbert space $H$, I want to calculate the adjoint of $\lambda I - T$ for a $\lambda \in \Bbb C$.</p>
<hr>
<p>I am tempted to argue as follows:</p>
<p>According to Lemma 4.26 of <a href="https://people.math.ethz.ch/~kowalski/spectral-theory.pdf" rel="nofollo... | Pedro | 23,350 | <p>Matrices are Morita equivalent to their underlying ring, and Morita equivalent rings have isomorphic Hochschild cohomology. A field as an algebra over itself has trivial cohomology for nonzero degrees.</p>
|
51,247 | <p>I have a matrix of 0's and 1's forming a number of disjoint paths:
<hr />
<img src="https://i.stack.imgur.com/pUlT7.jpg" alt="RandomPaths">
<hr />
I would like to find the lengths of the paths, and from that "spectrum,"
the longest length (in the above example: 27, starting at {1,14}).
(The shortest length po... | E.O. | 999 | <p>Not nearly as clean as @rm -rf's solution, but here is another image processing solution. Once again <code>m</code> is taken to be the matrix above.</p>
<pre><code>mc=MorphologicalComponents@m;
Max@Tally[Join @@ mc][[2 ;;, 2]]
(*27*)
</code></pre>
<hr>
<p>If one wishes to vizualize the exact path:</p>
<pre><cod... |
3,151,564 | <p>I have been asked to calculate a general formula for <span class="math-container">$1 + b + 2b^2 + 3b^3 + ... + Nb^N$</span></p>
<p>I think that a part of this is a geometric sequence, and I have rewritten this as </p>
<p><span class="math-container">$f(x) = 1 + \sum_{i=1}^n i\cdot x^i$</span></p>
<p>(I couldn't f... | J.G. | 56,861 | <p>The extra factor of <span class="math-container">$i$</span> or <span class="math-container">$k$</span> or whatever you call it means we no longer have a geometric series, but we can get one by differentiating with respect to <span class="math-container">$x$</span>. You'll want to check powers of <span class="math-co... |
2,987,055 | <blockquote>
<p>Let <span class="math-container">$\ V $</span> be a linear space over <span class="math-container">$\ F $</span>, <span class="math-container">$\ \dim V = n $</span> and <span class="math-container">$\ T_1,T_2 : V \rightarrow F $</span> are linear transofrmations. <span class="math-container">$\ T_1 ,... | Oldboy | 401,277 | <p>Equation of ellipse in a general form is:</p>
<p><span class="math-container">$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$</span></p>
<p>with an additional condition that:</p>
<p><span class="math-container">$$4AC - B^2 > 0$$</span></p>
<p>Such ellipse has axes rotated with respect to <span class="math-container">... |
316,046 | <p>What are some examples of (compact, say) manifolds <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> that are stably equivalent, i.e. <span class="math-container">$\Sigma^{\infty}X_+\simeq\Sigma^{\infty}Y_+$</span>, but are not homotopy equivalent?</p>
| Andy Putman | 317 | <p>The easiest examples are given by complements of open neighborhoods of distinct knots in <span class="math-container">$S^n$</span>.</p>
<p>More generally, Spanier-Whitehead duality says that if <span class="math-container">$X$</span> is a compact simplicial complex and if <span class="math-container">$f,g:X \righta... |
263,397 | <p>I'm having a trouble with this integral expression:</p>
<p>$$\int_0^{2\pi} \frac{d\theta}{A+B \cos\theta}$$</p>
<p>I've done this substitution: $t= \tan(\theta/2)$</p>
<p>and get: $\displaystyle \cos\theta= \frac{1-t^2}{1+t^2}$ and $\displaystyle d\theta=\frac{2}{1+t^2}dt$ where $\displaystyle \cos^2\theta/2=\fra... | Community | -1 | <p>Ideally, you want to do your substitution on a domain where it is defined, continuous, and invertible.</p>
<p>That's not quite true here, as <a href="http://www.wolframalpha.com/input/?i=tan%20t/2%20for%20t%20from%200%20to%202%20pi" rel="nofollow">this plot</a> shows.</p>
<p>But we can carefully break the domain u... |
48,118 | <p>I'm reading an article by Ricardo Mañé, "The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces" (<a href="https://doi.org/10.1007/BF02585431" rel="nofollow noreferrer">https://doi.org/10.1007/BF02585431</a>). I'm having a technical problem. Sorry for my ignorance, but I would like a referenc... | R W | 8,588 | <p>The $k$-dimensional Grassmannian over a vector space $V$ naturally embeds into the projectivization of the $k$-th exterior power of $V$. If you have a Euclidean structure on $V$, then it extends to its exterior products as well. Then a metric on the projectivization is provided, say, by the angle between the corresp... |
2,793,040 | <p>The definition of the compact-open topology differs slightly depending on whether one is working in the context of compactly generated topological spaces or arbitrary topological spaces.</p>
<p>If $X$ and $Y$ are topological spaces, then the compact-open topology on the set of continuous functions $C(X,Y)$ has, as ... | Henno Brandsma | 4,280 | <p><a href="https://www.researchgate.net/publication/30814746_The_Construction_of_Finer_Compact_Topologies" rel="nofollow noreferrer">This extended abtract by Künzi and van der Zypen</a> seems of interest. It mentions in passing (remark 3, page 3) a reference </p>
<blockquote>
<p>Stone, A.H.: Compact and compact Hau... |
3,430,235 | <p>If you have an arbitrary line, say a bezier curve, that has a width, is there a relationship between the length of the curve at the inner and outer edges? Given the width of the curve at the centerline, can you determine the length at ±width/2 distance from the centerline?</p>
| user289143 | 289,143 | <p>Since <span class="math-container">$f$</span> is injective, for every <span class="math-container">$y \in B$</span>, <span class="math-container">$f^{-1}(y)$</span> consists of at most one point. Fix some <span class="math-container">$a_0 \in A$</span> and define <span class="math-container">$g:B \rightarrow A$</spa... |
3,430,235 | <p>If you have an arbitrary line, say a bezier curve, that has a width, is there a relationship between the length of the curve at the inner and outer edges? Given the width of the curve at the centerline, can you determine the length at ±width/2 distance from the centerline?</p>
| fleablood | 280,126 | <p>It's best to create directly. And if you understand what is being asked pretty easy.</p>
<p>To make a function <span class="math-container">$g: B \to A$</span> we must for every <span class="math-container">$y \in B$</span> assign a value <span class="math-container">$g(y) \in A$</span>. Now if <span class="math-... |
1,195,337 | <p>If i suppose that $I$ is an interval but it is not connected,i.e there existe two open sets on $I$: $U,V$ such that $I=U\cup V$ et $U\cap V=\emptyset$ </p>
<p>Let $x\in U$ and $y\in V$ (for example $x<y$) and let $B_x=\{z\in U, [x,z]\subset U\}$ </p>
<p>Then $B_x$ is bounded by y .</p>
<p>My question is why $... | egreg | 62,967 | <p>You're wrong in your assumptions. Saying that $I$ is not connected means there exist two open sets $U$ and $V$ such that </p>
<ul>
<li>$I\subseteq U\cup V$</li>
<li>$(U\cap I)\cap(V\cap I)=\emptyset$</li>
<li>$U\cap I\ne\emptyset$ and $V\cap I\ne\emptyset$</li>
</ul>
<p>Or you have to say that $U$ and $V$ are open... |
434,585 | <p>Given a matrix <span class="math-container">$L\in \mathbb{R}^{3 \times 3}$</span>, I'm looking for a method to find the closest (in a least squares sense) product of a non-uniform scaling matrix and a rotation matrix:</p>
<p><span class="math-container">$$
\min_{s\in\mathbb{R}^3,\ R\in SO(3) } \left\| L - \text{diag... | Alec Jacobson | 23,064 | <p>I'm still not sure whether a closed form solution exists. But I recognized that most implementations of SVD will be iterative (the simplest being power iterations), so the block-coordinate descent in the original question can be understood as</p>
<pre><code>while not converged
update s ← diag(LRᵀ)
while not ... |
2,828,972 | <p>Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ such that $$(x,y,z) \mapsto (y+z,x-z,-x+y+2z) $$
Find its matrix, then show that this function is a projection parallel to a vector subspace $G$ onto a vector subspace $F$. Find the basis $B_F$ for $F$ and the basis $B_G$ for $G$ and then find the matrix of $f$ in the b... | MAOC | 343,511 | <p>$P$ is the matrix of the identity linear transformation </p>
<p>$$I: \mathbb{R}^3 \to \mathbb{R}^3$$</p>
<p>In other words, suposse the canonical base $\{ e_1, e_2, e_3\}$ and $\{ \epsilon_1, \epsilon_2, \epsilon_3\}$ the new base. Then, $P(e_1)= \epsilon_1$, $P(e_2) = \epsilon_2$ and $P(e_3) = \epsilon_3$. </p>... |
2,828,972 | <p>Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ such that $$(x,y,z) \mapsto (y+z,x-z,-x+y+2z) $$
Find its matrix, then show that this function is a projection parallel to a vector subspace $G$ onto a vector subspace $F$. Find the basis $B_F$ for $F$ and the basis $B_G$ for $G$ and then find the matrix of $f$ in the b... | amd | 265,466 | <p>The last part of the problem is actually the easiest: with a little bit of thought about the properties of a projection, you can write down the matrix with almost no work. The restriction of any projection to its image is the identity map and it of course vanishes on its kernel. Now recall that the columns of a tran... |
3,282,383 | <p>I am trying to solve exercises from Lang's Algebra, and I am stuck on a problem about Herbrand quotients. </p>
<p>Let <span class="math-container">$G$</span> be a finite cyclic group of order <span class="math-container">$n$</span> generated by an element <span class="math-container">$\sigma$</span>. Assume that <s... | Daniel Kawai | 466,883 | <p>I will use the result in <a href="https://math.stackexchange.com/questions/66253/proving-ab-afbfa-fb-f/4038289#4038289">this page</a>. Let <span class="math-container">$\pi:A\rightarrow A/B$</span> be the canonical surjection.</p>
<p>It is sufficient to prove:</p>
<p><span class="math-container">$$[A_f:A^g][B_g:B^f]... |
4,570,552 | <p>This question was asked in my quiz of Ring theory and I was not able to prove it.</p>
<blockquote>
<p>Question: Show that if <span class="math-container">$I$</span> is not prime, then there exists ideal <span class="math-container">$I_1$</span> and <span class="math-container">$I_2$</span> such that <span class="mat... | Mathmo123 | 154,802 | <p>By definition, since <span class="math-container">$I$</span> is not prime, there exist elements <span class="math-container">$x_1, x_2\in R\setminus I$</span> such that <span class="math-container">$x_1x_2 \in I$</span>.</p>
<p>As a first guess, we could set <span class="math-container">$I_i = \langle x_i\rangle$</s... |
248,385 | <p>For example, dividing by 2, means we cut something in two.</p>
<p>But dividing by 0.5, can only be explained with multiplying something by 2.</p>
<p>So, is there a "physical" explanation of dividing by 0.5? Is it "I divide by an entity that internally multiplies' or something as so bizarre?</p>
| Austin Mohr | 11,245 | <p>Along the same lines as Pedro Tamaroff's answer:</p>
<p>If you have ten ounces of vodka and each cocktail requires one ounce, then you can make ten cocktails.</p>
<p>If you have ten ounces of vodka and each cocktail requires only half an ounce, then you can make twenty cocktails. In other words, <em>halving</em> t... |
405,368 | <p>I'm doing some work in linear algebra and these came up and I realized I don't know how to solve them as they have quadratics in them. I'm sure I've done this before but if someone could give me a crash course on how to find the values of the variables, it would be most appreciated. For the following questions, let ... | Nate Eldredge | 822 | <p>Let $X$ be an uncountable set, equipped with the $\sigma$-algebra $\mathcal{F}$ consisting of all countable sets and their complements, or any other $\sigma$-algebra that contains all singletons but is not $2^X$. Let $\mu$ be counting measure, which is semifinite. Note that any function in $L^1(\mu)$ is supported o... |
1,261,572 | <p>If $\alpha$ and $\beta$ are the zeros of the polynomial $p(s)=3s^2-6s+4$, find the value of $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}+2(\frac{1}{\alpha}+\frac{1}{\beta})+3\alpha\beta$</p>
<p>By calculating, I am finding the answer as $9$, but, the book has the answer as $8$. I did a cross check and my answer seems... | colormegone | 71,645 | <p>This is a quadratic polynomial with real coefficients and a negative discriminant [ $ \ (-6)^2 \ - \ 4 \ \cdot \ 3 \ \cdot \ 4 \ < \ 0 \ $ ] so its roots are the complex conjugates $ \ z \ $ and $ \ \overline{z} \ $ . Using Viete's formulas, much as <strong>André Nicolas</strong> has done, on $ \ 3 \ ( \ x^2... |
1,088 | <p>OK, so <a href="http://chat.stackexchange.com">chat</a> is now available... but; it has been suggested that for Mathematics we should have TeX support.</p>
<p>The current TeX processing has some non-trivial client impact. Before I even attempt trying to hack this in, is this something that the community would want ... | Zhen Lin | 5,191 | <p><strong>Revision 3.</strong> This one actually has a working loop, and uses jQuery instead of directly injecting a script element.</p>
<hr>
<p>Unfortunately, it seems that new chat messages no longer trigger the ajaxComplete event, so my hack has to be replaced with a different one. This one runs MathJax on a 1-se... |
3,125,081 | <p>We know that a line can be written as <span class="math-container">$ ⟨x,y,z⟩ = ⟨x_0,y_0,z_0⟩ + t⟨a,b,c⟩ $</span> where <span class="math-container">$ x_0, y_0, z_0 $</span> is a point on the line and <span class="math-container">$ ⟨a, b, c⟩$</span> is a direction vector of the line. In a similar way, a plane can be ... | user247327 | 247,327 | <p>Another way:
<span class="math-container">$<x, y, z>= <2, 1, 3>+ s<1, 0, 2>+ t<0, -1, 3>= <2+ s, 1- t, 3+ 3s+ 3t>$</span>. That is, x= 2+ s, y= 1- t, and z= 3+ 3s+ 3t. From the first two equations, s= x- 2 and t= 1- y. Putting those into the third equation z= 3+ 3(x- 2)+ 3(1- y)= 3+ ... |
2,587,395 | <blockquote>
<p>Find $\lim_{x \to 0} \frac{x^2e^{x^4}-\sin(x^2)}{1-\cos(x^3)}$</p>
</blockquote>
<p>By taylor polynomials we get: $e^{x^4}=1+x^4+\frac{x^8}{2}+\mathcal{O}(x^{12})$</p>
<p>$\sin(x^2)=x^2-\frac{x^6}{6}+\mathcal{O}(x^{10})$</p>
<p>$\cos(x^3)=1-\frac{x^6}{2}+\mathcal{O}(x^{12})$</p>
<p>so putting thes... | DonAntonio | 31,254 | <p>As $\;x\to\infty\;$ :</p>
<p>$$\frac{x^2e^{x^4}-\sin x^2}{1-\cos x^3}\ge\frac{x^2e^{x^4}-1}2\xrightarrow[x\to\infty]{}\infty$$</p>
|
4,366,845 | <p><span class="math-container">$$\|A\|_1=\inf\{c\ge 0: \forall x\in V,\|Ax\|\le c, \|x\| \le 1\}=\inf\ S_1$$</span>
<span class="math-container">$$\|A\|_2=\inf\{c\ge 0: \forall x\in V,\|Ax\|\le c, \|x\| = 1\}=\inf S_2$$</span>
<strong>I am trying to prove these two expresions are equal DIRECTLY</strong>, I know there ... | Paul Frost | 349,785 | <p>Let us do it very formally. We write <span class="math-container">$D = \{ x \in V \mid \lVert x \rVert \le 1 \}$</span> and <span class="math-container">$B = \{ x \in V \mid \lVert x \rVert = 1 \}$</span>. Your sets <span class="math-container">$S_1, S_2$</span> are then given as</p>
<p><span class="math-container">... |
315,051 | <p>Unlike the collection <span class="math-container">$L$</span> of Lebesgue measurable sets, the collection <span class="math-container">$J$</span> of Jordan measurable sets do not form a Sigma algebra. (A set is Jordan measurable if and only if its characteristic function is Riemann integrable, and the characteristi... | 喻 良 | 14,340 | <p>Here is an answer to your question.</p>
<blockquote>
<p><strong>Assume <span class="math-container">$V=L$</span>, there is a null <span class="math-container">$\Pi^1_1$</span>-set which is not in <span class="math-container">$\sigma(J)$</span>.</strong></p>
<p><strong>Proof</strong>: Let <span class="math-co... |
2,269 | <p>Manin <a href="http://www.ihes.fr/document?id=1704&id_attribute=48" rel="noreferrer" title="Manin talk">stressed</a> that every projective scheme should have a quantum-cohomology structure. I'd like to know more about that. And since the varieties considered in texts about monodromy resp. vanishing cycles which ... | Kevin H. Lin | 83 | <p>I'm not sure what you mean by "every projective scheme should have a quantum cohomology structure". In the talk abstract that you link to, it does not say "projective scheme" but "smooth projective variety". I don't know whether the theory generalizes to non-smooth things or to things that are not varieties.</p>
<p... |
209,619 | <p>Prove that $$n!e-2 < \sum_{k=1}^{n}(^{n}\textrm{P}_{k}) \leq n!e-1$$ where $^{n}\textrm{P}_k = n(n-1)\cdots(n-k+1)$ is the number of permutations of $k$ distinct objects from $n$ distinct objects and $e$ is the exponential constant (Euler's number).</p>
| Alexander Shamov | 37,460 | <p>Since ${}^n P_k / n! = 1/(n-k)!$, we have to prove the following:</p>
<p>$\displaystyle e - \frac{1}{n!} < \sum_{k=0}^n \frac{1}{k!} < e$</p>
<p>which follows obviously from the expansion $e = \sum_{k=0}^\infty \frac{1}{k!}$ and the estimate</p>
<p>$\displaystyle \sum_{k \ge n+1} \frac{1}{k!} \le \sum_{\ell... |
209,619 | <p>Prove that $$n!e-2 < \sum_{k=1}^{n}(^{n}\textrm{P}_{k}) \leq n!e-1$$ where $^{n}\textrm{P}_k = n(n-1)\cdots(n-k+1)$ is the number of permutations of $k$ distinct objects from $n$ distinct objects and $e$ is the exponential constant (Euler's number).</p>
| Alex | 38,873 | <p>Denote the expression in question by $S$
$$
\sum_{k=0}^{n-1}\frac{1}{k!}=e-\sum_{k=n}^{\infty}\frac{1}{k!}=e-\frac{1}{n!}\bigg(1+\frac{1}{n+1}+\frac{1}{(n+1)(n+2)} + \cdots \bigg)\\
\leq e-\frac{1}{n!} \bigg(1+\frac{1}{n+1} + \frac{1}{(n+1)^2} +\cdots \bigg)\\
=e-\frac{1}{n!} \sum_{k=0}^{\infty} \bigg(\frac{1}{n+1}... |
3,623,803 | <p>I was reading a proof of the following proposition and had one small doubt about the proof:</p>
<blockquote>
<p><strong>Proposition:</strong> Let <span class="math-container">$H$</span> be a subgroup of <span class="math-container">$G,$</span> and <span class="math-container">$\mathcal{A}$</span> a <span class="m... | Community | -1 | <p>I am going to address your title question, and let you apply it to the specific example. </p>
<p>You should be familiar with the <em>first isomorphism theorem</em>. Well I think it actually answers your question.</p>
<p>Any time you have a quotient of one group <span class="math-container">$G$</span> by a normal ... |
3,605,957 | <p>I saw similar questions with this one, but I did not find geometrical solution. Therefore, I want to share my solution and also seek other simpler solutions.</p>
<h1>Problem:</h1>
<p>I need to prove that the locus of <span class="math-container">$\frac{|z-a|}{|z-b|}=c, c<1$</span> is a circle.</p>
<h1>My solution... | Arthur | 15,500 | <p>Here is the proof I learned back in the day (in geometrical notation, not complex arithmetic, because I'm more used to that in this context):</p>
<p>Let <span class="math-container">$r$</span> be a point on the line <span class="math-container">$ab$</span> to the left of <span class="math-container">$a$</span> such... |
3,605,957 | <p>I saw similar questions with this one, but I did not find geometrical solution. Therefore, I want to share my solution and also seek other simpler solutions.</p>
<h1>Problem:</h1>
<p>I need to prove that the locus of <span class="math-container">$\frac{|z-a|}{|z-b|}=c, c<1$</span> is a circle.</p>
<h1>My solution... | Greg Markowsky | 387,394 | <p>An algebraic solution: square everything, write <span class="math-container">$z=x+yi$</span>, use <span class="math-container">$|w|^2 = Re(w)^2 + Im(w)^2$</span>, rearrange everything, and voila! The equation for a circle should pop out.</p>
<p>Greg </p>
|
1,115,006 | <p>I already tried to putting the square root like this: $\sqrt{\frac{x}{5 + x}}$ but I dont know what to do next.</p>
<p>$$\int \frac{\sqrt{x}}{\sqrt{5+x}}dx$$</p>
| user84413 | 84,413 | <p>Another method would be to let $u=\sqrt{x+5}$, so $x=u^2-5 \text{ and } dx=2udu$ to get</p>
<p>$\;\;\;\displaystyle\int\frac{\sqrt{u^2-5}}{u}\cdot 2u\;du=2\int{\sqrt{u^2-5}}\;du$ . </p>
<p>Now let $u=\sqrt{5}\sec\theta, du=\sqrt{5}\sec\theta\tan\theta d\theta$ to get $10\int\tan^{2}\theta\sec\theta\;d\theta$, and... |
2,595,186 | <p>S=$1.3.2^2+2.4.3^2+3.5.4^2+........$ upto n terms</p>
<p>Find S?</p>
<p>This was asked under the heading using method of difference and ans given was
(n)(n+1)(n+2)(n+3)(2n+3)/10</p>
| Empy2 | 81,790 | <p>The $0^{th}$, $-1^{st}$ and $-2^{nd}$ terms are all zero, so think of the sum as zero after $-3,-2,-1$ and $0$ terms. So the sum, which is a polynomial, is a multiple of $n(n+1)(n+2)(n+3)$</p>
|
1,585,960 | <p>How to prove the following?
$$ \int_0^\infty \log x \operatorname{sech}x\,dx = \frac{\pi}{2} \log\left( \frac{4\pi^3}{\Gamma(1/4)^4} \right)
$$
I obtained the right side with CAS. It seems like this function has many poles on the imaginary axis so a simple contour integral cannot be used. I also tried the following
... | Jack D'Aurizio | 44,121 | <p>The residue theorem gives:
$$ \text{sech}(z) = \sum_{n\geq 0}\frac{4\pi(-1)^n (2n+1)}{4z^2+(2n+1)^2}\tag{1}$$
hence:</p>
<p>$$ \int_{0}^{+\infty}x^s \text{sech}(x)\,dx = \pi^{s+1}2^{-s}\sec\left(\frac{\pi s}{2}\right)\sum_{n\geq 0}(-1)^n (2n+1)^s\tag{2} $$
and that can be simplified to:
$$\begin{eqnarray*}\int_{0}^... |
2,675,882 | <p>A lattice $(L, \le)$ is a <a href="https://en.wikipedia.org/wiki/Modular_lattice#Introduction" rel="nofollow noreferrer">modular lattice @ wiki</a> if the following modular law holds:
$$\forall x \in L: a \le b \implies a \lor (x \land b) = (a \lor x) \land b.$$</p>
<p>The <a href="https://en.wikipedia.org/wiki/Mod... | Chris Culter | 87,023 | <p>There are a few reasonable developments, which all have equivalent results, so it doesn't really matter which one you personally use.</p>
<hr>
<p><strong>Explicit definition, cautious version</strong></p>
<ul>
<li><em>Definition</em> $\mathbf T$: Let $10=S(9).$</li>
<li><em>Definition</em> $\mathbf C$: For digits... |
1,257,544 | <p>I have given the following problem to solve;</p>
<p>(i) Prove any two cycles in Sn of the same length are conjugate in $S_n$ for any $n\geq 3$.</p>
<p>(ii) Is the same true in $A_n$ for $n\geq 3$? </p>
<p>(iii) Prove any two cycles of length three are conjugate in $A_n$ if $n\ge 5$.</p>
<p>I have answered the fi... | Tim Raczkowski | 192,581 | <p>For the second question, consider $A_4$ and the cycles $[123]$ and $[124]$. It is true that $[124]=[34][123][34]^{-1}$, but there is no even permutation $\tau$ for which $[124]=\tau[123]\tau^{-1}$.</p>
|
4,037,740 | <p>I need to prove <span class="math-container">$\lim\limits_{x \to \infty} x \sin^2 x$</span> does not exist, and request your help verifying my argument is correct. I've got somewhat decent at making <span class="math-container">$\epsilon-\delta$</span> arguments, but proving a limit does not exist has been a bit ch... | Paul Frost | 349,785 | <p>This is just a variation of Martin Argerami's answer. He constructed a sequence <span class="math-container">$(x_k)$</span> such that <span class="math-container">$\lim_{k\to\infty}x_k\sin^2x_k=\infty$</span>. Let us show that for each <span class="math-container">$a \in [0,\infty)$</span> there exists a sequence <s... |
2,347,923 | <p>I have read a few proofs and found them all really confusing, here is one for example.
<a href="https://i.stack.imgur.com/OcOOD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/OcOOD.png" alt="enter image description here"></a></p>
<p>That small and simple-looking proof was really hard for me to t... | egreg | 62,967 | <p>You're making it much more complicated than it should be.</p>
<p>We work under the same assumptions: $G$ is a group and $H$ is a subgroup thereof.</p>
<p><strong>Lemma.</strong> <em>For all $a\in G$, $a\in aH$.</em></p>
<p><em>Proof.</em> This follows from the fact that $e\in H$ (which is a subgroup) and so, by d... |
506,556 | <p><strong>Problem:</strong> </p>
<blockquote>
<p>Find the number of points which lies inside the triangle : $(0,0) , (0,21); (20,0) $</p>
</blockquote>
<p>Approach : </p>
<p>Let us take point $A = (0,0)$, $B = (0,21)$, and $C = (20,0)$.</p>
<p>Since the figure formed by joining these points $ABC$ is a triangle ... | Michael Hoppe | 93,935 | <p>Let $P$ a lattice polygon, $A$ it's area, $I$ the number of the interior lattice points, $B$ number the lattice points on the boundary of $P$. Then
$$A=I+B/2-1.$$
This is known as Pick's theorem. From here we conclude $211=I+B/2$. Obviously there are $42$ points on both axis and no lattice point on the line segment... |
3,837,047 | <p><img src="https://i.stack.imgur.com/2sM1y.png" alt="" /></p>
<p>I am currently taking a linear algebra class in university. For part v wouldn't the matrix D just be the matrix C with the row r appended to it?</p>
| Community | -1 | <p>The two results you quote do indeed involve the same idea and they certainly need to be rigorously proved. As an example, let's look at the intersection of a circle and a line. This will illustrate the general idea in a simple context.</p>
<p>The proof is elementary but needs a few prerequisites:</p>
<p>(1) The gene... |
1,874,363 | <p>For example, how would you respond to this question?</p>
<blockquote>
<p>The earnings of one-hundred workers in a company are normally distributed. If the mean of this data set is 24 and the standard deviation is 4, find an approximate value for the range.</p>
</blockquote>
| Community | -1 | <p>One can see by induction that:</p>
<p>$$\int_0^1 x_1 dx_1 \int_0^{x_2 } dx_2 \cdots = \int_0^1 x_1 \frac{x_{n - (n-1)}^{n-1}}{(n-1)!} dx_1 =\frac{1}{(n+1) \times (n-1)!}$$</p>
<p>Therefore, the original integral is:</p>
<p>$$n! \times \frac1{(n+1) \times (n-1)!} = \frac{n}{n+1}$$</p>
|
3,386,069 | <p>Let <span class="math-container">$$A=\{\phi \in \mathcal C ([0,1], \mathbb R) \mid \forall k \in\mathbb N^+:\int^1_0 x^k\phi(x) =0\}$$</span></p>
<p>Clearly, the function <span class="math-container">$[0,1] \to \mathbb R, \, x \mapsto 0$</span> belongs to <span class="math-container">$A$</span>. I would like to ask... | triple_sec | 87,778 | <p>Suppose <span class="math-container">$\phi\in A$</span>. Let <span class="math-container">$K>0$</span> be such that <span class="math-container">$|\phi(x)|\leq K$</span> for each <span class="math-container">$x\in[0,1]$</span> (Weierstrass). Fix <span class="math-container">$\varepsilon>0$</span> and find a po... |
369 | <p>This question has been kicking around in the back of my head for a couple of years, but the impetus to post it now came from reading the related question at <a href="https://matheducators.stackexchange.com/questions/244/when-did-the-american-school-systems-progression-of-math-classes-take-its-curre">When did the Ame... | Bob | 15,220 | <p>Dolciani eventually went on to write an explicit 'Algebra 2' book, used in the 1970's and 1980's commonly. That Algebra 2 book has an exact replica of the material in the SMSG Intermediate Mathematics. So I think that should resolve it. 'Algebra 2' is the SMSG 'Intermediate Mathematics.'</p>
|
369 | <p>This question has been kicking around in the back of my head for a couple of years, but the impetus to post it now came from reading the related question at <a href="https://matheducators.stackexchange.com/questions/244/when-did-the-american-school-systems-progression-of-math-classes-take-its-curre">When did the Ame... | nowradioguy | 15,592 | <p>The SMSG textbook for algebra II and trig was entitled "Modern Algebra and Trigonometry, Book 2" by Dolciani, Berman, and Wooton, published by Houghton-Mifflin. We used this during my junior year in high school (1965-66). I wish I'd kept the book instead of returning it to our bookstore for credit, because... |
476,070 | <p>We consider $T:\mathbb{C}\to\mathbb{C}$ defined by $T(z)=\lambda z+\mu\overline{z}$, where $\lambda ,\mu\in\mathbb{C}$. I want to prove that if $T$ is injective, then $\lambda\cdot\overline{\lambda}\neq\mu\cdot\overline{\mu}$.</p>
<p>Clearly, if $\mu =0$, then $\lambda\neq 0$ because $T$ is injective.</p>
<p>Now, ... | Michael Albanese | 39,599 | <p>Here's an alternative proof. </p>
<p>Suppose that $T$ is injective. Note that $\mu \neq \bar{\lambda}$, otherwise $T(z) = \lambda z + \overline{\lambda z}$ in which case $T(z) = 0$ for any $z \in \mathbb{C}$ such that $\Re(\lambda z) = 0$. Now note that $T(\bar{\lambda}) = |\lambda|^2 + \mu\lambda$ and $T(\mu) = \l... |
835,806 | <p>Let $\{f_n\}_{n=1}^\infty$ a sequence of second order differentiable functions on the interval [0,1].</p>
<p>If $\forall n\in \Bbb N$ $f_n(0)=f_n'(0)=0$ and for all $n\in \Bbb N$ and $x \in [0,1]$ , $|f_n''(x)|\le 1$ then show $\{f_n\}_{n=1}^\infty$ has uniformly convergent subsequence on [0,1].</p>
<p>D... | Community | -1 | <p>By Lagrange Theorem we have $$|f'_n (x) |=|f'_n (x) -f'_n (0) | =|f''_n (\xi ) ||x|\leq 1 \mbox{ for } 0\leq x\leq 1.$$
Analogously $$|f_n (x) |=|f_n (x) -f_n (0) | =|f'_n (\xi_1 ) ||x|\leq 1 \mbox{ for } 0\leq x\leq 1.$$
Hence the family $\{ f_n :n\in\mathbb{N}\}$ is uniformly bounded.
Moreover $$ |f_n (u) -f_n (v)... |
3,138,956 | <p>For the question</p>
<blockquote>
<p>The objective is to derive a formula for <span class="math-container">$\sum_{k=1}^{n} k^2$</span>. Find the integers <span class="math-container">$a$</span> and <span class="math-container">$b$</span> such that <span class="math-container">$$ k^2 = a {k \choose 2} + b {k \choo... | Dr. Sonnhard Graubner | 175,066 | <p>I got the equation<span class="math-container">$$ab=\frac{2}{k-1}$$</span> I have found <span class="math-container">$13$</span> Solutions.</p>
|
4,294,775 | <p>I'm struggling to figure out a solution to a real world problem I've encountered in work that I'm sure mathematics has already solved for me. I don't have any math education (beyond high school, apologies) and have spent a long time searching for an answer to this, with no success. I guess that this is a permutation... | TravorLZH | 748,964 | <p>In essence, we are finding the number of ways to partition <span class="math-container">$n$</span> students into <span class="math-container">$k$</span> different groups. Let <span class="math-container">$a_{n,k}$</span> denote this number. Then it is certain that <span class="math-container">$a_{n,1}=1$</span> and ... |
4,294,775 | <p>I'm struggling to figure out a solution to a real world problem I've encountered in work that I'm sure mathematics has already solved for me. I don't have any math education (beyond high school, apologies) and have spent a long time searching for an answer to this, with no success. I guess that this is a permutation... | Vaekor | 349,878 | <p>While @TravorLZH provided a perfectly reasonable explanation of how to calculate these numbers, the discussion was about the real world application to students, so I wanted to add a notable addendum.</p>
<p>This distribution of numbers is known as the Bell numbers, and while they are reasonable for small inputs, the... |
3,146,055 | <p>I am currently an undergrad student and recently changed my major to mathematics (after taking my first proof based math course). Unfortunately, now that I am taking several upper division math courses, I feel that my mathematical foundations are a bit weak and atrophied. I am considering taking some time off to try... | rehctawrats | 407,385 | <p>For any foundational topic of mathematics, there is a multitude of elementary textbooks. You can easily find them via Google, on Amazon, in your faliclity's library, etc. </p>
<p>Picking one over the other is a matter of taste. They differ in the order of topics, the detailedness of explanations, amount of exercise... |
4,320,591 | <p>Recently I came across a question <a href="https://gate.iitkgp.ac.in/documents/gatepapers/2016/ee_2016.pdf" rel="nofollow noreferrer">here</a> p.25 question 7. This is a question from the GATE EE 2016 exam. But I was having difficulty in analyzing the question as to whether it has a definite answer or not based on t... | zwim | 399,263 | <p>The solution by your first peer tells you that eigenvalues can only be <span class="math-container">$0$</span>, <span class="math-container">$1$</span> or <span class="math-container">$-1$</span>.</p>
<p>But this doesn't mean they are exactly that, in fact any combination <span class="math-container">$(\lambda_1,\la... |
430,126 | <p>I've asked two years ago a post on Mathematics Stack Exchange, were provided two excellent answers. I'm asking on MathOverflow in the hope that some professor can to expand/improve (if it is possible) these results answering my question. The post has the same title and identifier <a href="https://math.stackexchange.... | Joe Silverman | 11,926 | <p>You can make a lot of progress if you're willing to assume a deep conjecutre. The <span class="math-container">$N$</span>-variable generalization of the <span class="math-container">$abc$</span>-conjecture (<a href="https://en.wikipedia.org/wiki/N_conjecture" rel="noreferrer">https://en.wikipedia.org/wiki/N_conjectu... |
2,570,627 | <blockquote>
<p>$p$,$q$,$r$ are $3$ non-negative real numbers less than or equal to $1.5$ such that $p+q+r = 3$, what will be the maximum of $p^3 + q^3 + r^3 + 4pqr$ ?</p>
</blockquote>
<hr>
<p>I tried AM-GM on $p,q,r$ to get the maximum of $pqr$ as $1$, but on doing it for $p^3 + q^3 + r^3 $, I get the minimum, so... | robjohn | 13,854 | <p>To maximize
$$
p^3+q^3+r^3+4pqr\tag1
$$
while maintaining
$$
p+q+r=3\tag2
$$
Means that we need to find $p,q,r$ so that for all variations so that
$$
\delta p+\delta q+\delta r=0\tag3
$$
we also have
$$
\left(3p^2+4qr\right)\delta p+\left(3q^2+4pr\right)\delta q+\left(3r^2+4pq\right)\delta r=0\tag4
$$
Orthogonality ... |
3,063,097 | <p>On the German Wikipedia page of the Hurwitz Zeta Function I have come across the <a href="https://de.wikipedia.org/wiki/Hurwitzsche_Zeta-Funktion#Weitere" rel="nofollow noreferrer">following formula</a></p>
<blockquote>
<p><span class="math-container">$$\zeta\left(2,\frac14\right)~=~\pi^2+8G\tag1$$</span></p>
</bloc... | mrtaurho | 537,079 | <p>Inspired by <em>Zacky</em>'s comment on the <a href="https://en.wikipedia.org/wiki/Hurwitz_zeta_function#Integral_representation" rel="nofollow noreferrer">integral representation of the Hurwitz Zeta Function</a> I have found another pretty straightforward way. First we notice that</p>
<blockquote>
<p><span class... |
2,121,004 | <p>Prove that if A and B are doubly-stochastic matrices of order n, then AB is also a doubly-stochastic matrix.</p>
<p>My attempt:</p>
<p>Since the size of A is nxn and the size of B is nxn, then the size of AB is also nxn.</p>
<p>Let A = ($a_{ij}$) and B = ($b_{ij}$). Consider the sum of the entries in the $i$th ro... | Joffan | 206,402 | <p>Base case, $n=0$: $\quad 3^{0+2}+2^{0+1} = 9+2 = 11 \quad \checkmark$</p>
<p>Induction hypothesis: $\quad 3^{3n+2}+2^{4n+1} = 11k$</p>
<p>$\begin{align} 3^{3(n+1)+2}+2^{4(n+1)+1} &= 27\cdot3^{3n+2}+16\cdot 2^{4n+1} \\
&= 22\cdot3^{3n+2}+5\cdot3^{3n+2}+11\cdot 2^{4n+1}+5\cdot 2^{4n+1}\\
&= 22\cdot3^{3n+... |
1,020,122 | <p>Let $a>-1$ be a real number. The sequence $a_n$ is defined as $a_o=a$ and $a_n=\sqrt{2^{n+1}a_{n-1}+4^n} -2^n$. Find the limit of $a_n$</p>
<p>I have problems here because of $2^{n+1}, 4^n, 2^n$ I don't know what to do with it.</p>
<p>My try : </p>
<p>OK I have something but I'm not sure if it's good. First w... | Community | -1 | <p>$$a_n=\sqrt{2^{n+1}a_{n-1}+4^n} -2^n$$
is equivalent to
$$\frac{a_n}{2^n}=\sqrt{\frac{a_{n-1}}{2^{n-1}}+1} -1,$$
or
$$b_n=\sqrt{b_{n-1}+1}-1.$$</p>
<p>As $b_n+1=\sqrt{b_{n-1}+1}$, by recurrence $b_n+1=(b_0+1)^{1/2^n}=(a+1)^{1/2^n}$, and</p>
<p>$$a_n=2^n\left((a+1)^{1/2^n}-1\right)=2^n(1+\frac a{2^n}+o(\frac a{2^n}... |
2,288,535 | <p>Three fair coins are flipped. Let x be the number of heads. Let y equal to 1 if all coins land the same outcome I.e hhh or ttt, and 0 if otherwise. Calculate Cov(X,Y)</p>
| Batman | 127,428 | <p>Note $Y=1$ iff $X=3$ or $X=0$. </p>
<p>$X$ is Binomial(3,1/2), so $P(X=x) = \binom{3}{x} (1/2)^x (1-1/2)^{3-x}$ for $x=0,1,2,3$.</p>
<p>You can write down $P(X=x,Y=y)$ from the statements above. $P(X=1,Y=1)=P(X=2,Y=2)=P(X=0,Y=0)=P(X=3,Y=0)=0$. Now, $P(X=1,Y=0) = P(X=1), P(X=2,Y=0)=P(X=2). And $P(X=3,Y=1)=P(X=3), P... |
2,903,478 | <p>I really cant underatand what is exactly difference. By their it can be verified that undefined has no value at all but inderminant form we can able to evaluate its limit by various way(such as L'Hospital's rule) whether it exists. </p>
| Ross Millikan | 1,827 | <p>Indeterminate forms generally refer to combining limits. If you have two functions $f(x)$ and $g(x)$ such that $\lim_{x \to \infty} f(x)=a, \lim_{x \to \infty} g(x)=b$ you can say $\lim_{x \to \infty} f(x)+g(x)=a+b$. Other operations can be done similarly. It is also true that if $\lim_{x \to \infty} f(x)=+\infty... |
2,841,922 | <p>Let $P$ be a point on line segment $XY$. On the perpendicular line to $XY$ through $P$ lies point $Z$. Prove that for any position of $Z$ there is a unique point $Q$ on line segment $XZ$ for which $\angle ZYQ = \angle ZXY $. The perpendicular line on $XZ$ through $Q$ then always passes through the same (fixed) point... | John Glenn | 528,976 | <p>WLOG, suppose $XY$ is on the $x$-axis. Let us define points $X,Y$ as $(-a,0)$ and $(a,0)$, respectively. Then, $Z$ should be defined as point $(b,z)$. Additionally, point $Q=\left(q,-\frac{z}{a+b}\left(q+a\right)\right)$. For $a,b,z,q \in \Bbb R$.</p>
<p>We have the following equations for lines $XZ, QY$ and $ZY$</... |
2,841,922 | <p>Let $P$ be a point on line segment $XY$. On the perpendicular line to $XY$ through $P$ lies point $Z$. Prove that for any position of $Z$ there is a unique point $Q$ on line segment $XZ$ for which $\angle ZYQ = \angle ZXY $. The perpendicular line on $XZ$ through $Q$ then always passes through the same (fixed) point... | Maxim | 491,644 | <p><img src="https://i.imgur.com/zauqQFv.png" alt=""></p>
<p>$\triangle XYZ \sim \triangle YQZ$, therefore $QZ/YZ = YZ/XZ$.</p>
<p>$PQRZ$ is cyclic, therefore
$$PX \cdot RX = QX \cdot XZ = (QZ-XZ) \cdot XZ = YZ^2 - XZ^2 = \\
(PY^2 + PZ^2) - (PX^2 + PZ^2) = PY^2 - PX^2.$$
Note that the argument doesn't go through if b... |
2,841,922 | <p>Let $P$ be a point on line segment $XY$. On the perpendicular line to $XY$ through $P$ lies point $Z$. Prove that for any position of $Z$ there is a unique point $Q$ on line segment $XZ$ for which $\angle ZYQ = \angle ZXY $. The perpendicular line on $XZ$ through $Q$ then always passes through the same (fixed) point... | mengdie1982 | 560,634 | <h1>Solution</h1>
<p>It's easy to have that the circumcircle of <span class="math-container">$\triangle QXY$</span> touches <span class="math-container">$ZY$</span> at <span class="math-container">$Y$</span>,and <span class="math-container">$Z,P,Q,R$</span> are cyclic.</p>
<p>Hence, <span class="math-container">$$ZY^2=... |
24,319 | <p>This semester, I am teaching discrete math for computer science students. Today I taught solving linear recurrence equations. The way I did it was not rigors. Instead I used the method of advanced operators described <a href="https://www.rellek.net/book/s_recurrence_solving.html" rel="nofollow noreferrer">here</a>. ... | Daniel R. Collins | 5,563 | <p>My philosophy is <em>schedule über alles</em>. As an academic, time is your most constrained resource. This goes for balancing the hours in your work vs. research, teaching, and service requirements. And in this case it goes for the limited number of class sessions in your semester.</p>
<p>Do you not have a day-by-d... |
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