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 |
|---|---|---|---|---|
2,146,929 | <p>Let $f:S^n \to S^n$ be a homeomorphism. I know the result that a rigid motion in $\mathbb R^{n+1}$ is always <a href="https://math.stackexchange.com/a/866471/185631">linear</a>, but can we get more information from the assumption that $f:S^n \to S^n$ is a homeomorphism?</p>
| Peter Franek | 62,009 | <p>The question is hard to understand in its current form, but a few remarks:</p>
<ol>
<li>Not every homeomorphism of the sphere is linear.</li>
<li>Every homeomorphism of the sphere extends to a homeomorphism of $\Bbb R^{n+1}$ (see Tsemo Aristide answer)</li>
<li>If you insisted on smoothness and diffeomorphisms exte... |
101,953 | <p>I've been asked (by a person not by a homework) about how to compute the following limit:</p>
<p>$$ \lim_{x \to 10^-} \frac{[x^3] - x^3}{[x] - x}$$</p>
<p>where $[\cdot]$ is used to denote the floor function:</p>
<p>$$ [x] := \begin{cases} x && x \in \mathbb{Z} \\ 
\text{biggest integer smaller ... | Gottfried Helms | 1,714 | <p>Different from Davide's I replace $\small x=N-\delta $ where $\small \delta \to 0$. Also I would change sign in numerator and denominator to subtract the smaller from the larger values: </p>
<p>$\qquad \small { (N-\delta)^3 - [(N-\delta)^3 ] \over (N-\delta) - [N-\delta]} $ </p>
<p>which for small e... |
566,993 | <p>Suppose $f(z)=1/(1+z^2)$ and we want to find the power series in $a=1$. I think we have to write $1/(1+z^2)=1/(1+(z-1)+1)^2=1/(1+(1+(z-1)^2+2(z-1)))$, but I'm stuck here.</p>
| Boris Novikov | 62,565 | <p>Set $b=a^{-1}x$. We have $x^2=a^{-1}x^2a$, i.e. $ax^2=x^2a$ for all $x,a$. Since $x^2$ runs all group, then $G$ is Abelian.</p>
<p><strong>Correction:</strong> This proof is valid only for a finite group. Thanks to <strong>DonAntonio</strong>.</p>
<p><strong>Addendum:</strong> I am not sure that this assertion is... |
566,993 | <p>Suppose $f(z)=1/(1+z^2)$ and we want to find the power series in $a=1$. I think we have to write $1/(1+z^2)=1/(1+(z-1)+1)^2=1/(1+(1+(z-1)^2+2(z-1)))$, but I'm stuck here.</p>
| Heno | 1,042,723 | <p><span class="math-container">$\forall x,a\in G,ax,a^{-1} \in G\Rightarrow ax^2a^{-1}=x^2\Rightarrow ax^2=x^2a$</span></p>
<p><span class="math-container">$\forall x,y\in G$</span></p>
<p><span class="math-container">$xyxy=yxyx\Rightarrow x^{-1}y^{-1}xy=yxy^{-1}x^{-1}$</span></p>
<p><span class="math-container">$(xyx... |
2,871,892 | <p><a href="https://i.stack.imgur.com/XLen7.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XLen7.png" alt="Q1"></a></p>
<p>Solution is 4. </p>
<p>Original matrix is simply [v1;v2;v3;v4]. It forms an identity matrix. Hence the only alteration of the determinant comes from row 1 operation where v1 i... | mengdie1982 | 560,634 | <p>We may prove that the non-identical function $f(x)$ which is continuous over $(0,+\infty)$ and satisfies $$f(xy)=f(x)f(y)$$ is $$f(x)=x^\alpha$$ only.</p>
|
520,046 | <blockquote>
<p>Find the smallest natural number that leaves residues $5,4,3,$ and $2$ when divided respectively by the numbers $6,5,4,$ and $3$.</p>
</blockquote>
<p>I tried
$$x\equiv5\pmod6\\x\equiv4\pmod5\\x\equiv3\pmod4\\x\equiv2\pmod3$$What $x$ value?</p>
| Shobhit | 79,894 | <p>Given</p>
<p>$x=6a+5=6(a+1)-1$</p>
<p>$x=5b+4=5(b+1)-1$</p>
<p>$x=4c+3=4(c+1)-1$</p>
<p>$x=3d+2=3(d+1)-1$</p>
<p>therefore x will be of the form $(\text{L.C.M(3,4,5,6)}k-1)$ or,</p>
<p>$x=60k-1$ for some $k$.</p>
<p>Can you guess that $k$?</p>
<blockquote>
<p>ANSWER:$k=1$, or $x=59$</p>
</blockquote>
|
2,594,669 | <p>Given the Pythagoras Theorem: <strong>a² + b² = c²</strong></p>
<p>Is there a way to get the value of <strong>b</strong> when we only have a value for <strong>a</strong> and the angle <strong>α</strong>?</p>
<p>To be frank, I have no clue about that, what I want isn't the angle of <strong>β</strong> but the length... | Enrico M. | 266,764 | <p>$$a = b\tan\theta$$</p>
<p>Where $\theta$ is the angle opposite to $a$</p>
<p>From this:</p>
<p>$$b = \frac{a}{\tan\theta}$$</p>
|
3,471,292 | <p>I need to find the value of the series <span class="math-container">$\sum_{n=0}^{\infty}\frac{(n+1)x^n}{n!}$</span>.I've computed its radius of convergence which comes out to be zero.</p>
<p>I'm not getting how to make adjustments in the general terms of the series to get the desired result...</p>
| Ben | 93,447 | <p>Is it not so that you can split the sum into two and simplify:</p>
<p><span class="math-container">$$\sum_{n=0}^\infty \frac{(n+1)x^n}{n!} = \sum_{n=0}^\infty \frac{nx^n}{n!} + \sum_{n=0}^\infty \frac{x^n}{n!} = \sum_{n=1}^\infty \frac{x^{n}}{(n-1)!} + \sum_{n=0}^\infty \frac{x^n}{n!}= \ldots$$</span></p>
<p>Consi... |
1,987,387 | <p>I don't remember any method to compute the closed from for the following series.
$$ \sum_{k=0}^{\infty}\binom{3k}{k} x^k .$$</p>
<p>I tried by putting $\binom{3k}{k}$ in Mathematica for different $k$ and asking for the generating function it deliver a complicated formula which is the following.
$$ \frac{2\cos[\f... | DonAntonio | 31,254 | <p>That's a power series about $\;x_0=0\;$ and whose sequence of coefficients is</p>
<p>$$a_k=\binom{3k}kx^k=\frac{(3k)!}{k!(2k)!}\implies\;\left|\frac{a_{k+1}}{a_k}\right|=\frac{(3k+3)!}{(k+1)!(2k+2)!}\cdot\frac{k!(2k)!}{(3k)!}|x|=$$</p>
<p>$$=\frac{(3k+1)(3k+2)(3k+3)}{(k+1)(2k+1)(2k+2)}|x|\xrightarrow[k\to\infty]{}... |
667,371 | <p>I try to solve this equation:
$$\sqrt{x+2}+\sqrt{x-3}=\sqrt{3x+4}$$</p>
<p>So what i did was:</p>
<p>$$x+2+2*\sqrt{x+2}*\sqrt{x-3}+x-3=3x+4$$</p>
<p>$$2*\sqrt{x+2}*\sqrt{x-3}=x+5$$</p>
<p>$$4*{(x+2)}*(x-3)=x^2+25+10x$$</p>
<p>$$4x^2-4x-24=x^2+25+10x$$</p>
<p>$$3x^2-14x-49$$</p>
<p>But this seems to be wrong! ... | Clive Newstead | 19,542 | <p>What makes you think you've done anything wrong? You can factorise
$$3x^2-14x-49 = (3x+7)(x-7)$$
to obtain a solution to your equation.</p>
<p>Beware: one of the solutions from this quadratic is not a solution because it doesn't satisfy the original equation... this often happens when you solve equations by squarin... |
2,355,579 | <blockquote>
<p><strong>Problem:</strong> James has a pile of n stones for some positive integer n ≥ 2. At each step, he
chooses one pile of stones and splits it into two smaller piles and writes the product
of the new pile sizes on the board. He repeats this process until every pile is exactly
one stone.</p>
... | Bram28 | 256,001 | <p>First, a couple of comments on how you think about, and write/present this proof.</p>
<p>When doing induction, it is always a good idea to get very clear on exactly what the claim is that you are trying to prove, and thus what the property is that you want to show all natural numbers have. So in this case, that w... |
1,800,519 | <blockquote>
<p>Let $\omega$ be an $n$-form and $\mu$ be an $m$-form where both are acting on a manifold $M$. Is the Lie derivative $L_{X}(\omega \wedge \mu)$ where $X$ is a smooth vector field acting on $M$ an exact form? </p>
</blockquote>
<p>I think it is but I've been unable to prove it, so any help would be gre... | Travis Willse | 155,629 | <p><strong>Hint</strong> Use <a href="https://en.wikipedia.org/wiki/Lie_derivative#The_Lie_derivative_of_differential_forms" rel="nofollow">Cartan's Magic Formula</a>, which says that the Lie derivative $\mathcal L_X$ of a differential form $\alpha$ satisfies
$$\mathcal L_X \alpha = \iota_X d \alpha + d (\iota_X \alpha... |
2,640,763 | <p>Let $\{x_i\}_{i=1}^{n}$ and $\{y_i\}_{i=1}^{n}$ two positive sequences, the first one is monotonic, the second one is strictly increasing .</p>
<p>I noticed that in many cases if $\{x_i\}_{i=1}^{n}$ is increasing $$\frac{\frac{1}{n}\sum_{i=1}^{n}{x_iy_i}}{\left(\frac{1}{n}\sum_{i=1}^{n}{x_i}\right)\left(\frac{1}... | Mathematician 42 | 155,917 | <p>Let $T:V\rightarrow V$ be a linear map and $\alpha=\left\{v_1, \dots, v_k, u_{k+1}, \dots u_n\right\}$ a basis of $V$ such that $\left\{v_1, \dots, v_k\right\}$ is a basis of $\ker(T)$. </p>
<p>You can easily prove that $T(u_{k+1}), \dots , T(u_n)$ are linearly independent vectors. (Mimick the technique used to pro... |
1,369,409 | <p>I have a bit of an advanced combination problem that has left me stumped for a few days. Essentially my question is if you have n sets of items, and you can select a different number of items from each set, how do you compute the combinations without first creating new sets.</p>
<p>An example in pictures:
I have th... | coldnumber | 251,386 | <p>An easy way to visualize all the combinations is to use a tree diagram. It will, however, get rather unwieldy, since as the other answers showed you'll end up with 54 branches. </p>
<p>First, list the three combinations of 2 elements of $A$, say $(a_1,a_2),(a_1,a_3)$, and $(a_2,a_3)$. From each these spring out 3 b... |
4,467,036 | <p>I have to calculate the following integral using contour integration: <span class="math-container">$$\int_0^1 \frac{dx}{(x+2)\sqrt[3]{x^2(x-1)}}$$</span></p>
<p>I've tried to solve this using the residue theorem, but I don't know how to calculate the residue of the function <span class="math-container">$$f(z) = \fra... | DinosaurEgg | 535,606 | <p>I will solve the more general integral</p>
<p><span class="math-container">$$I(z):=\int_0^1\frac{dx}{x^{2/3}(1-x)^{1/3}(x+z)}$$</span></p>
<p>which is found to be given by</p>
<p><span class="math-container">$$I(z)=\frac{2\pi}{\sqrt{3}}z^{-2/3}(1+z)^{-1/3}$$</span></p>
<p>Note that no matter your choice of branch of... |
4,467,036 | <p>I have to calculate the following integral using contour integration: <span class="math-container">$$\int_0^1 \frac{dx}{(x+2)\sqrt[3]{x^2(x-1)}}$$</span></p>
<p>I've tried to solve this using the residue theorem, but I don't know how to calculate the residue of the function <span class="math-container">$$f(z) = \fra... | Sangchul Lee | 9,340 | <p>I will instead compute</p>
<p><span class="math-container">$$ I = \int_{0}^{1} \frac{\mathrm{d}x}{(x+2)\sqrt[3]{x^2\bbox[color:red;padding:3px;border:1px dotted red;]{(1-x)}}}. $$</span></p>
<p>You will have no problem converting this to your case, depending on which branch of <span class="math-container">$\sqrt[3]{... |
1,421,740 | <p>Let $90^a=2$ and $90^b=5$, Evaluate </p>
<h1>$45^\frac {1-a-b}{2-2a}$ </h1>
<p>I know that the answer is 3 when I used logarithm, but I need to show to a student how to evaluate this without involving logarithm. Also, no calculators.</p>
| GAVD | 255,061 | <p>Let me try. </p>
<p>$$10 = 90^{a+b} \Rightarrow 3^2 = 90^{1-a-b} \Rightarrow 3 = 90^{\frac{1-a-b}{2}}.$$</p>
<p>Then, $$45 = 90^{(1-a-b)+b} = 90^{1-a}.$$</p>
<p>So, $$45^{\frac{1}{1-a}} = 90 \Rightarrow 45^{\frac{1-a-b}{2(1-a)}} = 90^{\frac{1-a-b}{2}} = 3.$$</p>
|
1,100,812 | <p>Here is the statement:
($\bf{Tonelli}$) If $f\in L^+(X,Y)$, then $\displaystyle g:x\mapsto\int_Yf_xd\nu$ is $\mathcal{M}$-measurable,\
$\displaystyle h:y\mapsto \int_Xf^yd\mu$ is $\mathcal{N}$-measurable (so $g\in L^+(X)$ and $h\in L^+(Y)$). And $$\displaystyle \int_{X\times Y}fd\mu \times\nu=\int_Xgd\mu=\int_Yhd\... | Loreno Heer | 92,018 | <p>I think that follows from $\int_{X×Y}|f|<\infty$.</p>
|
1,102,324 | <p>I could use some pointers solving this problem:</p>
<blockquote>
<p>Given a certain s.v. $X$ with cdf $F_x(x)$ and pdf $f_X(x)$. Let s.v. $Y$ be the lower censored of $X$ at $x=b$. Meaning:</p>
<p>$$Y = \begin{cases}0 & \text{if }X<b\\
X & \text{if } X \geq b\end{cases}$$</p>
<p>Find cdf $F_Y... | mickep | 97,236 | <p>Hint: $y=-ax^2+1$ is zero when $x=\pm1/\sqrt{a}$, so it might be so that you want to calculate the integral
$$
\int_{-1/\sqrt{a}}^{1/\sqrt{a}}1-ax^2\,dx.
$$</p>
|
381,566 | <p>I know practically nothing about fractional calculus so I apologize in advance if the following is a silly question. I already tried on math.stackexchange.</p>
<p>I just wanted to ask if there is a notion of fractional derivative that is linear and satisfy the following property <span class="math-container">$D^u((f)... | Iosif Pinelis | 36,721 | <p>It appears you actually want <span class="math-container">$D^u(f^n)=\alpha f^{n-1} D^u f$</span>, where <span class="math-container">$\alpha$</span> is a scalar.</p>
<p>There is no reason for this to be true, and this is indeed false in general. E.g., for <span class="math-container">$n=2$</span> and the <a href="ht... |
251,818 | <p>In other words if a graph is $3$-regular does it need to have $4$ vertices? I ask because I have been asked to prove that if $n$ is an odd number and $G$ is an $n$-regular graph then $G$ must have an even number of vertices.</p>
| yo' | 43,247 | <p>I'm not sure how much detailed answer you want. So this is a hint, and the proof itself is hidden: Consider simple graph (no parallel edges, no loops on a vector) on $n$ vertices and think how many edges from a vertex can exist. As well, what if $n=0$?</p>
<blockquote class="spoiler">
<p> Well, if you consider th... |
1,781,117 | <h1>The question</h1>
<p>Prove that:
$$\prod_{n=2}^∞ \left( 1 - \frac{1}{n^4} \right) = \frac{e^π - e^{-π}}{8π}$$</p>
<hr>
<h2>What I've tried</h2>
<p>Knowing that:
$$\sin(πz) = πz \prod_{n=1}^∞ \left( 1 - \frac{z^2}{n^2} \right)$$
evaluating at $z=i$ gives
$$ \frac{e^π - e^{-π}}{2i} = \sin(πi) = πi \prod_{n=1}^∞ \... | C.S. | 95,894 | <p>Note that $$\prod_{n=2}^{\infty} \left(1-\frac{1}{n^{2}}\right) \to \frac{1}{2}$$</p>
<p>This is because $$A_{n} =\prod_{k=2}^{n}\left(1-\frac{1}{n^2}\right) = \prod_{k=2}^{n} \frac{(k-1)(k+1)}{k^2} = \frac{n+1}{2n} \to \frac{1}{2}$$</p>
<p>We have used $\displaystyle \left(1-\frac{1}{n^4}\right) = \left(1+\frac{1... |
156,479 | <p>Let $S$ be a compact oriented surface of genus at least $2$ (possibly with boundary). Let $X$ be a connected component of the space of embeddings of $S^1$ into $S$.</p>
<p>Question : what is the fundamental group of $X$? My guess is that the answer is $\mathbb{Z}$ with generator the loop of embeddings obtained by... | Will Sawin | 18,060 | <p>For $S$ the sphere, assuming smooth embeddings, any curve divides the sphere into two discs, hence is diffeomorphic to the equator. Then $X$ is a quotient space of the orientation-preserving diffeomorphism group of the sphere by the subgroup that preserves the equator. The orientation-preserving diffeomorphism group... |
2,618,804 | <p>Let $V$ be a vector space of dimension $m\geq 2$ and $ T: V\to V$ be a linear transformation such that $T^{n+1}=0$ and $T^{n}\neq 0$ for some $n\geq1$ .Then choose the correct statement(s):</p>
<p>$(1)$ $rank(T^n)\leq nullity(T^n)$</p>
<p>$(2)$ $rank(T^n)\leq nullity(T^{n+1})$</p>
<p><strong>Try:</strong></p>
<... | Community | -1 | <p>1.</p>
<p>Let $y\in Range(T)$ $\implies y=T(x)$ for some $x\in V$, $T^{n+1}(x)=T^{n}(T(x))=0 \implies y \in Ker T^n.$</p>
<ol start="2">
<li><p>Let $y \in KerT^n \implies T^n(y)=0 \implies T^{n+1}(y)=T(T^n(y))=T(0)=0 \implies y \in KerT^{n+1}$</p></li>
</ol>
|
66,370 | <p>Let $(X,\mathcal{E},\mu)$ be a measure space. Let $u,v$ be $\mu$-measurable functions. If $0 \leq u \leq v$ and $\int_X v d\mu$ exists we know that $\int_X u d\mu \leq \int_X v d\mu$.</p>
<p>I wanted to know if $0 \leq u < v$ and $\int_X v d\mu$ exists then is it true that
$\int_X u d\mu < \int_X v d\mu$? Th... | Ilya | 5,887 | <p><strong>Edit (after 11 years):</strong> Somehow nowhere (in the OP or in the previous version of the answer) I see being mentioned that strict integral inequality only holds if <span class="math-container">$v>u$</span> on some set of strictly positive measure.</p>
<hr />
<p>First of all, <span class="math-contain... |
1,613,171 | <p>On page $61$ of the book <a href="http://solmu.math.helsinki.fi/2010/algebra.pdf" rel="nofollow">Algebra</a> by Tauno Metsänkylä, Marjatta Näätänen, it states</p>
<blockquote>
<p>$\langle \emptyset \rangle =\{1\},\langle 1 \rangle =\{1\}. H\leq G \implies \langle H \rangle =H$</p>
</blockquote>
<p>where $H \leq ... | Michael Albanese | 39,599 | <p>The notation $H \leq G$ means that $H$ is a subgroup of $G$. Your proposed counterexample fails because $\emptyset$ is not a subgroup of $G$ (it doesn't contain the identity element).</p>
|
1,844,374 | <p>Why does the "$\times$" used in arithmetic change to a "$\cdot$" as we progress through education? The symbol seems to only be ambiguous because of the variable $x$; however, we wouldn't have chosen the variable $x$ unless we were already removing $\times$ as the symbol for multiplication. So why do we? I am very cu... | Fay | 530,222 | <p>The x might be used for younger children instead of the ⋅ because they might confuse it with the . in decimals especially if the equation was handwritten. The x on the other hand can't be confused with a different symbol since they are probably not doing algebra. Just a thought. (just realized other people said this... |
4,203,704 | <p>Understanding the Yoneda lemma maps.</p>
<p>I'm trying to understand the maps between the natural transformations and <span class="math-container">$F(A)$</span> in the proof of the Yoneda lemma. I've been struggling for a bit to understand the Yoneda lemma, so I'm trying to understand the mapping construction as a r... | azif00 | 680,927 | <ul>
<li>No, <span class="math-container">$\eta$</span> is a natural transformation between <span class="math-container">$h^A : C \to \textsf{Set}$</span> and <span class="math-container">$F : C \to \textsf{Set}$</span>, so for each object <span class="math-container">$X$</span> in <span class="math-container">$C$</spa... |
250,687 | <p>I'm doing a sanity check of the following equation:
<span class="math-container">$$\sum_{j=2}^\infty \frac{(-x)^j}{j!}\zeta(j) \approx x(\log x + 2 \gamma -1)$$</span></p>
<p>Naive comparison of the two shows a bad match but I suspect one of the graphs is incorrect.</p>
<ol>
<li>Why isn't there a warning?</li>
<li>H... | Michael E2 | 4,999 | <p>The sum is alternating, so you might need extra precision and <code>NSumTerms</code>:</p>
<pre><code>katsurda[x_] :=
NSum[(-x)^j/j! Zeta[j], {j, 2, Infinity}, WorkingPrecision -> 16,
NSumTerms -> Max[15, 2 x]];
katsurdaApprox[x_] := x (Log[x] + 2 EulerGamma - 1) - Zeta[0];
plot1 = DiscretePlot[katsurda[... |
3,124,285 | <p>I have written a proof, and I would appreciate verification. The problem is picked from "Set Theory and Matrices" by I. Kaplansky.</p>
<hr>
<p><em>Proof</em>. Let <span class="math-container">$a_1=f(x)$</span> and <span class="math-container">$a_2=f(y)$</span>, then</p>
<p><span class="math-container">$g(a_1)=g(a... | Sri-Amirthan Theivendran | 302,692 | <p>First be precise about what you are proving: Let <span class="math-container">$f\colon X\to Y$</span> be a surjective map and <span class="math-container">$g\colon Y\to Z$</span> be a map such that <span class="math-container">$g\circ f$</span> is injective. Then <span class="math-container">$g$</span> is injective... |
121,541 | <p>I'd like to pick <em>k</em> points from a set of points in <em>n</em>-dimensions that are approximately "maximally apart" (sum of pairwise distances is almost maxed). What is an efficient way to do this in MMA? Using the solution from C Woods, for example:</p>
<pre><code>KFN[list_, k_Integer?Positive] := Module[{kT... | C. Woods | 37,886 | <p>Here is a naive solution that can be slow for large lists of points: </p>
<pre><code>KFN[list_, k_Integer?Positive] := Module[{kTuples},
kTuples = Subsets[list, {k}];
MaximalBy[kTuples,
Total[Flatten[Outer[EuclideanDistance[#1, #2] &, #, #, 1]]] &]
]
</code></pre>
<p>(Use of <code>Subsets</code>... |
121,541 | <p>I'd like to pick <em>k</em> points from a set of points in <em>n</em>-dimensions that are approximately "maximally apart" (sum of pairwise distances is almost maxed). What is an efficient way to do this in MMA? Using the solution from C Woods, for example:</p>
<pre><code>KFN[list_, k_Integer?Positive] := Module[{kT... | Daniel Lichtblau | 51 | <p>Not necessarily best of quality but maybe could be made better with a bit of tuning.</p>
<pre><code>kDistant[pts_List, n_] := Module[
{objfun, len = Length[pts], ords, a, c1},
ords = Array[a, n];
c1 = Flatten[{Map[.5 <= # <= len + .5 &, ords],
Element[ords, Integers]}];
objfun[oo : {_Integer... |
260,516 | <p>I was inspired by <a href="https://math.stackexchange.com/questions/2062960/there-exist-infinite-many-n-in-mathbbn-such-that-s-n-s-n-frac1n2?noredirect=1#comment4336226_2062960">this</a> topic on Math.SE.<br>
Suppose that $H_n = \sum\limits_{k=1}^n \frac{1}{k}$ - $n$th harmonic number. Then</p>
<h2>Conjecture</h2>
... | Gottfried Helms | 7,710 | <p><strong><em>This is a comment at the comments of Gerhard "Still Computing Oh So Slowly" Paseman's answer, giving just indexes n for more record-holders.</em></strong> </p>
<p>Let $\small h_n$ denote the <em>n</em>'th harmonic number, $\small A_n=\{h_n\}$ its fractional part. The sequence of $A_n$ has a r... |
3,173,636 | <p>I have been trying to prove that there is no embedding from a torus to <span class="math-container">$S^2$</span> but to no avail.</p>
<p>I am completely stuck on where to start. The proof is supposed to be based on Homology theory. I know how to prove that <span class="math-container">$S^n$</span> cannot be embedde... | Camilo Arosemena-Serrato | 33,495 | <p><a href="https://math.stackexchange.com/questions/1519028/torus-cannot-be-embedded-in-mathbb-r2">Here</a> are given several reasons why the torus cannot be embedded into <span class="math-container">$\mathbb R^2$</span>; two of them use the invariance of domain theorem. </p>
<p>Now, if the torus could be embedded i... |
3,950,808 | <p><em>(note: this is very similar to <a href="https://math.stackexchange.com/questions/188252/spivaks-calculus-exercise-4-a-of-2nd-chapter">a related question</a> but as I'm trying to solve it without looking at the answer yet, I hope the gods may humor me anyways)</em></p>
<p>I'm self-learning math, and an <a href="h... | Mike Earnest | 177,399 | <p>The key to using the below identity is to look at the coefficient of <span class="math-container">$x^l$</span> on both sides. Since the polynomials in <span class="math-container">$x$</span> are equal, their coefficients must be as well.
<span class="math-container">$$\left(\sum_{i=0}^n \binom{n}{i} x^i\right)\left(... |
3,950,808 | <p><em>(note: this is very similar to <a href="https://math.stackexchange.com/questions/188252/spivaks-calculus-exercise-4-a-of-2nd-chapter">a related question</a> but as I'm trying to solve it without looking at the answer yet, I hope the gods may humor me anyways)</em></p>
<p>I'm self-learning math, and an <a href="h... | Ben | 754,927 | <p>The solution uses some facts about polynomials that, unfortunately, Spivak hasn't yet proven.</p>
<p>He hasn't formally introduced polynomials yet. This happens in Chapter 3 (3rd Ed.). You may wish to return to this problem after reading chapter 3.</p>
<p>A polynomial function <span class="math-container">$f$</span>... |
3,308,291 | <p>I have an array of numbers (a column in excel). I calculated the half of the set's total and now I need the minimum number of set's values that the sum of them would be greater or equal to the half of the total. </p>
<p>Example:</p>
<pre><code>The set: 5, 5, 3, 3, 2, 1, 1, 1, 1
Half of the total is: 11
The least a... | Ahmed Hossam | 430,756 | <p>The group operation ( = group law ) <span class="math-container">$\color{red}{+}$</span> here is <span class="math-container">$a\color{red}{+}b = (a+b)\bmod 5$</span> and <span class="math-container">$+$</span> is the normal addition of integers. The set <span class="math-container">$G=\{0,1,2,3,4\}$</span> together... |
3,308,291 | <p>I have an array of numbers (a column in excel). I calculated the half of the set's total and now I need the minimum number of set's values that the sum of them would be greater or equal to the half of the total. </p>
<p>Example:</p>
<pre><code>The set: 5, 5, 3, 3, 2, 1, 1, 1, 1
Half of the total is: 11
The least a... | user692616 | 692,616 | <p>The smallest subgroup containing <span class="math-container">$1$</span>, should also contains <span class="math-container">$1+1, 1+1+1, 1+1+1+1 \dots$</span> and also their inverses.</p>
|
1,783,458 | <blockquote>
<p>Prove that the equation
$$z^n + z + 1=0 \ z \in \mathbb{C}, n \in \mathbb{N} \tag1$$
has a solution $z$ with $|z|=1$ iff $n=3k +2, k \in \mathbb{N} $.</p>
</blockquote>
<hr>
<p>One implication is simple: if there is $z \in \mathbb{C}, |z|=1$ solution for (1) then $z=cos \alpha + i \cdot sin\al... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$z^n+1=-z$$</p>
<p>As $z\ne0,$ $$z^{n/2}+z^{-n/2}=-z^{1-n/2}$$</p>
<p>Let $z=r(\cos t+i\sin t)$</p>
<p>$$2r^{n/2}\cos\dfrac{nt}2=-r^{(2-n)/2}\left(\cos\dfrac{(2-n)t}2+i\sin\dfrac{(2-n)t}2\right)$$</p>
<p>We need $\sin\dfrac{(2-n)t}2=0\iff\dfrac{(2-n)t}2=m\pi$ where $m$ is any integer</p>
<p>So, ... |
1,783,458 | <blockquote>
<p>Prove that the equation
$$z^n + z + 1=0 \ z \in \mathbb{C}, n \in \mathbb{N} \tag1$$
has a solution $z$ with $|z|=1$ iff $n=3k +2, k \in \mathbb{N} $.</p>
</blockquote>
<hr>
<p>One implication is simple: if there is $z \in \mathbb{C}, |z|=1$ solution for (1) then $z=cos \alpha + i \cdot sin\al... | the_candyman | 51,370 | <p>If $z^n + z + 1=0 $ then $z^n = -(1+z)$ and $$|z|^n = |1+z|.$$</p>
<p>If $|z| = 1$, then $z = \cos a + i \sin a$, with $a \in [0, 2\pi]$. Moreover:</p>
<p>$$1^n = |1 + \cos a + i \sin a| \Rightarrow \sqrt{(1+\cos a)^2 + \sin^2 a} = 1 \Rightarrow \\
1 + \cos^2 a + 2 \cos a + \sin^2 a = 1\Rightarrow
\cos a = -\frac... |
2,294,548 | <p><strong>Problem:</strong> Solve $y'=\sqrt{xy}$ with the initial condition $y(0)=1$.</p>
<p><strong>Attempt:</strong> Using $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$, I get that the DE is separable by dividing both sides by $\sqrt{y}:$ $$y'=\sqrt{x}\cdot\sqrt{y}\Leftrightarrow\frac{y'}{\sqrt{y}}=\sqrt{x}$$</p>
<p>which can... | Rigel | 11,776 | <p>Since $y(0) = 1 > 0$, your solution will be positive near $x=0$ (where it is defined).
Hence, the inequality $x y(x) \geq 0$ can be satisfied (near $x=0$) only for $x\geq 0$.</p>
<p>At this point you can look for a solution $y\colon [0,a)\to\mathbb{R}$ such that $y(x) > 0$ for every $x\in [0,a)$.</p>
<p>If w... |
2,294,548 | <p><strong>Problem:</strong> Solve $y'=\sqrt{xy}$ with the initial condition $y(0)=1$.</p>
<p><strong>Attempt:</strong> Using $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$, I get that the DE is separable by dividing both sides by $\sqrt{y}:$ $$y'=\sqrt{x}\cdot\sqrt{y}\Leftrightarrow\frac{y'}{\sqrt{y}}=\sqrt{x}$$</p>
<p>which can... | Community | -1 | <p>As $y(0)>0$, $xy$ is negative for $x=0^-$ and $y$ can not be defined in the negatives. On another hand, you certainly have $y(x)>0$ in some neighborhood of $x=0^+$.</p>
<p>As the initial condition is given, you can use definite integrals,</p>
<p>$$\int_1^y\frac{dy}{\sqrt y}=\int_0^x\sqrt xdx,$$
giving
$$2(\s... |
386,073 | <p>For which values of a do the following vectors for a <strong><em>linearly dependent</em></strong> set in $R^3$?</p>
<p>$$V_1= \left(a,\, \frac{-1}{2}, \,\frac{-1}{2}\right),\;\; V_2= \left(\frac{-1}{2},\, a, \,\frac{-1}{2}\right),\; \;V_3= \left(\frac{-1}{2}, \,\frac{-1}{2},\, a\right)$$</p>
<p>Please would it be ... | colormegone | 71,645 | <p>To be linearly dependent in $\mathbb{R}^3$, the three vectors would have to be co-planar. One test would be that the triple product of the three vectors (in any order) would be zero (since a vector perpendicular to the mutual perpendicular of any two others would have to be in the same plane as those two). In Cart... |
445 | <p>Under what circumstances should a question be made community wiki?</p>
<p>Probably any question asking for a list of something (e.g. <a href="https://math.stackexchange.com/questions/81/list-of-interesting-math-blogs">1</a>) must be CW. What else? What about questions asking for a list of applications of something ... | Casebash | 123 | <p>Due to the nature of community wiki, this is quite a difficult question to answer. I think that we should be conservative in our enforcement of CW, so that when we do enforce it, there is a general consensus. Without consensus CW will cause unnecessary confusion and turn users off our site.</p>
<p>I think it is goo... |
17,480 | <p>I have asked a question at <a href="https://academia.stackexchange.com/">academia.stackexchange</a> with three sub-questions recently and I was told that it was not proper there. I just wonder if it is acceptable if one asks multiple (related) questions at math.stackexchange? </p>
<p>To mathematicians, if the answe... | Community | -1 | <p>Please do not ask multiple questions within one. There is a closing reason for those (<em>too broad</em>) which says that the author should reduce the scope of the question. I see it has been applied to your <a href="https://academia.stackexchange.com/q/32407/">Academia question</a>.</p>
<p>The "clever" packaging ... |
3,309,511 | <p>Prove that there exists infinitely many pairs of positive real numbers <span class="math-container">$x$</span> and <span class="math-container">$y$</span> such that <span class="math-container">$x\neq y$</span> but <span class="math-container">$ x^x=y^y$</span>.</p>
<p>For example <span class="math-container">$\tfr... | Ian | 83,396 | <p>It is easier than that, you need only examine the derivative: the derivative is <span class="math-container">$x^x \left ( \frac{d}{dx} x \log(x) \right ) = x^x \left ( \log(x) + 1 \right )$</span> which changes sign precisely at <span class="math-container">$x=e^{-1}$</span>. This point is a minimum, and of course <... |
224,970 | <p>$\newcommand{\Int}{\operatorname{Int}}\newcommand{\Bdy}{\operatorname{Bdy}}$
If $A$ and $B$ are sets in a metric space, show that:
(note that $\Int$ stands for interior of the set)</p>
<ol>
<li>$\Int (A) \cup \Int (B) \subset \Int (A \cup B)$.</li>
<li>$(\overline{ A \cup B}) = (\overline A \cup \overline B )$. (n... | Sixtina Aquafina | 416,396 | <blockquote>
<p>The basic rule is S→AABSBCC|ABSC|ASBC|1 where AA, BB and CC got to 0|1, and really only have different labels to highlight the counting. Not that we're not really keeping track of the order, but the step-wise counting ensures we have enough 0s and 1s to either side (remember that the "middle" 11 is no... |
224,970 | <p>$\newcommand{\Int}{\operatorname{Int}}\newcommand{\Bdy}{\operatorname{Bdy}}$
If $A$ and $B$ are sets in a metric space, show that:
(note that $\Int$ stands for interior of the set)</p>
<ol>
<li>$\Int (A) \cup \Int (B) \subset \Int (A \cup B)$.</li>
<li>$(\overline{ A \cup B}) = (\overline A \cup \overline B )$. (n... | Kianoosh Boroojeni | 1,130,118 | <p>Here is a simpler grammar that generates the same language:</p>
<p>S-> ABSC | ASBC | A1C</p>
<p>A -> 0 | 1</p>
<p>C -> 0 | 1</p>
<p>B -> 0 | 1 | epsilon</p>
<p>where variable A generally makes parts of the first third and variable C generally makes parts of the last third of the string.</p>
|
94,525 | <p>I am trying to solve the equation
$$z^n = 1.$$</p>
<p>Taking $\log$ on both sides I get $n\log(z) = \log(1) = 0$.</p>
<p>$\implies$ $n = 0$ or $\log(z) = 0$</p>
<p>$\implies$ $n = 0$ or $z = 1$.</p>
<p>But I clearly missed out $(-1)^{\text{even numbers}}$ which is equal to $1$.</p>
<p>How do I solve this equati... | Dustan Levenstein | 18,966 | <p>You can't take the logarithm of a negative number, unless you consider the multivalued <a href="http://en.wikipedia.org/wiki/Logarithm#Complex_logarithm" rel="nofollow">complex logarithm</a>.</p>
<p>If you are willing to expand to complex numbers in that manner, then you can take the log of both sides. $\log(1) = 2... |
1,114 | <p>Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?</p>
| Josh | 343 | <p>A groupoid is a generalization of a group. The easiest definition, IMO, is as a category in which all arrows are isomorphisms. So a group is just a groupoid with one object and arrows the elements of the group.</p>
<p>The best example is the fundamental groupoid of a topological space. Build a groupoid by taking th... |
1,114 | <p>Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?</p>
| Aleks Kissinger | 800 | <p>To follow on from what Qiaochu said, one of the interesting things about groupoids is their cardinality. Whereas the cardinality of a set is a natural number, the cardinality of a groupoid is a positive rational. This gives us a combinatorial way to inject "numbers" into an abstract system.</p>
<p>For example, a wa... |
1,114 | <p>Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?</p>
| Pete L. Clark | 1,149 | <p>I (mildly) disagree with David Brown's assertion that a set is an example of a groupoid. Given any set, you can put a groupoid structure on it, even "canonically", but not <em>uniquely</em> canonically. (By way of analogy, you wouldn't say that a set is an example of a topological space, would you?) Thus if I giv... |
1,114 | <p>Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?</p>
| Ali Taghavi | 36,688 | <p>The holonomy groupoid of a foliation is another example of a useful groupoid</p>
<p>it is described here:</p>
<p><a href="http://www.ams.org/journals/bull/2005-42-01/S0273-0979-04-01036-5/S0273-0979-04-01036-5.pdf">http://www.ams.org/journals/bull/2005-42-01/S0273-0979-04-01036-5/S0273-0979-04-01036-5.pdf</a>... |
2,409,918 | <p>I need your help in evaluating the following integral in <strong>closed form</strong>. <span class="math-container">$$\displaystyle\int\limits_{0.5}^{1}
\frac{\mathrm{Li}_{2}\left(x\right)\ln\left(2x - 1\right)}{x}\,\mathrm{d}x$$</span></p>
<p>Since the function is singular at <span class="math-container">$x = 0.5$... | user90369 | 332,823 | <p>$\displaystyle \int\limits_{0.5}^1 \frac{Li_2(x)\ln(2x-1)}{x}dx=$</p>
<p>$\displaystyle =\sum\limits_{k=1}^\infty \frac{1}{k^2 2^k}\sum\limits_{v=0}^{k-1} {\binom {k-1} v} \lim\limits_{h\to 0}\frac{1}{h}\left(\frac{(2x-1)^{v+h+1}}{v+h+1}-\frac{(2x-1)^{v+1}}{v+1}\right)|_{0.5}^1$</p>
<p>$\displaystyle =-\sum\limits... |
2,409,918 | <p>I need your help in evaluating the following integral in <strong>closed form</strong>. <span class="math-container">$$\displaystyle\int\limits_{0.5}^{1}
\frac{\mathrm{Li}_{2}\left(x\right)\ln\left(2x - 1\right)}{x}\,\mathrm{d}x$$</span></p>
<p>Since the function is singular at <span class="math-container">$x = 0.5$... | Quanto | 686,284 | <p>Note that</p>
<p><span class="math-container">\begin{align}
I_1=&\int_0^1 \frac{\ln^2(1-x)\ln(1+x)}{1+x}dx =\int_0^1 \frac{\ln^2x \ln(2-x)}{2-x}dx\\
=& \>\ln2 \int_0^1 \frac{\ln^2x }{2-x}dx +\int_0^1 \frac{\ln^2x }{2-x} \left( -\int_0^1 \frac x{2-xy}dy\right) dx\\
=&\>2\ln2 Li_3(\frac12)+\int_0^1 \... |
3,780,959 | <p>Consider a connected, unweighted, undirected graph <span class="math-container">$G$</span>. Let <span class="math-container">$m$</span> be the number of edges and <span class="math-container">$n$</span> be the number of nodes.</p>
<p>Now consider the following random process. First sample a uniformly random spanning... | Misha Lavrov | 383,078 | <p>Let <span class="math-container">$\tau(G)$</span> denote the number of spanning trees in <span class="math-container">$G$</span>, and let <span class="math-container">$G \bullet vw$</span> denote edge contraction: it is the <em>multigraph</em> in which adjacent vertices <span class="math-container">$v$</span> and <s... |
3,780,959 | <p>Consider a connected, unweighted, undirected graph <span class="math-container">$G$</span>. Let <span class="math-container">$m$</span> be the number of edges and <span class="math-container">$n$</span> be the number of nodes.</p>
<p>Now consider the following random process. First sample a uniformly random spanning... | Marcus M | 215,322 | <p>While the other answer is correct, it requires the computation of <span class="math-container">$|E| + 1$</span> many determinants. There is a faster route when <span class="math-container">$|E|$</span> is large. The first thing to note is Kirchoff's theorem which states that if <span class="math-container">$T$</sp... |
1,341,505 | <p>Let U: $\mathbb R$ -> $\mathbb R$ be a concave function, let X be a random variable with a finite expected value, and let Y be a random variable that is independent of X and has an expected value 0. Define Z=X+Y. Prove that $E[U(X)] \ge E[U(Z)]$</p>
<p>I know that $E(X)=E(Z)$, and by Jensen's inequality $U[E(X)] \g... | Amit | 378,131 | <p>Note that the following equality holds: <span class="math-container">$\mathbb{E}(Z|X = x) = \mathbb{E}(X +Y|X = x) = \mathbb{E}(x +Y|X = x) = x + \mathbb{E}(Y|X = x) =x + \mathbb{E}(Y) = x $</span>,</p>
<p>We are given that <span class="math-container">$u$</span> is concave, so by Jensen's inequality:
<span class="m... |
929,532 | <p>Okay so I want some hints (not solutions) on figuring out whether these sets are open, closed or neither.</p>
<p>$A = \{ (x,y,z) \in \mathbb{R}^3\ \ | \ \ |x^2+y^2+z^2|\lt2 \ and \ |z| \lt 1 \} \\ B = \{(x,y) \in \mathbb{R}^2 \ | \ y=2x^2\}$</p>
<p>Okay so since this question is the last part of the question where... | mm-aops | 81,587 | <p>Take $\Omega$ to be the set of all natural numbers, $F$ to be the family of all subsets of $\Omega$ and let $\mu(A) = 0$ if $A$ is a finite set and $\mu(A) = \infty$ if $A$ is infinite, I leave it to you to check that it's additive but not $\sigma$-additive.</p>
|
3,238,914 | <p>When is the <a href="https://en.wikipedia.org/wiki/Euler_line" rel="nofollow noreferrer">Euler line</a> parallel with a triangle's side?</p>
<p>I have found that a triangle with angles <span class="math-container">$45^\circ$</span> and <span class="math-container">$\arctan2$</span> is a case.</p>
<p>Is there any o... | Parcly Taxel | 357,390 | <p>From <a href="https://math.stackexchange.com/q/2912551/357390">here</a> we find a relation between the slopes of the three sides <span class="math-container">$p,q,r$</span> and that of the Euler line <span class="math-container">$m$</span>:
<span class="math-container">$$m=-\frac{3+pq+pr+qr}{p+q+r+3pqr}$$</span>
Wit... |
2,845,085 | <p>Find $f(5)$, if the graph of the quadratic function $f(x)=ax^2+bx+c$ intersects the ordinate axis at point $(0;3)$ and its vertex is at point $(2;0)$</p>
<p>So I used the vertex form, $y=(x-2)^2+3$, got the quadratic equation and then put $5$ instead of $x$ to get the answer, but it's wrong. I think I shouldn't hav... | rogerl | 27,542 | <p>Hints: Plug $x=0$ into $ax^2+bx+c=3$ to find the value of $c$. Then note that the vertex of a parabola is at $x$-coordinate $-\frac{b}{2a}$. Can you take it from there?</p>
|
1,057 | <p>Suppose a finite group has the property that for every $x, y$, it follows that </p>
<p>\begin{equation*}
(xy)^3 = x^3 y^3.
\end{equation*}</p>
<p>How do you prove that it is abelian?</p>
<hr>
<p>Edit: I recall that the correct exercise needed in addition that the order of the group is not divisible by 3.</p>
| Mariano Suárez-Álvarez | 274 | <p>You don't, as the group is not necessarily abelian! The group of upper triangular 3-by-3 matrices with ones along the diagonal and coefficients in the three-element field
$\mathbb {Z}/3\mathbb{Z}$ has exponent three, so your equation holds, but it is not abelian.</p>
<p>There are lots of examples: the most famous ... |
1,057 | <p>Suppose a finite group has the property that for every $x, y$, it follows that </p>
<p>\begin{equation*}
(xy)^3 = x^3 y^3.
\end{equation*}</p>
<p>How do you prove that it is abelian?</p>
<hr>
<p>Edit: I recall that the correct exercise needed in addition that the order of the group is not divisible by 3.</p>
| Arturo Magidin | 742 | <p>Both proofs so far are for finite groups. However, the problem (with the complete assumptions) holds for not-necessarily-finite groups, provided that the group have no element of order <span class="math-container">$3$</span>.</p>
<p>Here is a proof:</p>
<p>From <span class="math-container">$(ab)^3 = a^3b^3$</span>, ... |
514,922 | <p>I need to prove the following affirmation: If $ \lim x_{2n} = a $ and $ \lim x_{2n-1} = a $, prove that $\lim x_n = a $ (in $ \mathbb{R} $ )</p>
<p>It is a simple proof but I am having problems how to write it. I'm not sure it is the right way to write, for example, that the limit of $(x_{2n})$ converges to a:</p>
... | drhab | 75,923 | <p>Hint: for $\epsilon>0$ find some $n_{1}$ such that for $n>n_{1}$ we have $|x_{2n}-a|<\epsilon$ and also some $n_{2}$ such that for $n>n_{2}$ we have $|x_{2n-1}-a|<\epsilon$. Based on that try to find some $n_{0}$ such that for $n>n_{0}$ we have $|x_{n}-a|<\epsilon$</p>
|
40,920 | <p>We have functions $f_n\in L^1$ such that $\int f_ng$ has a limit for every $g\in L^\infty$. Does there exist a function $f\in L^1$ such that the limit equals $\int fg$? I think this is not true in general (really? - why?), then can this be true if we also know that $f_n$ belong to a certain subspace of $L^1$?</p>
| t.b. | 5,363 | <p>Another way of phrasing your question: Is $L^{1}$ <em>weakly sequentially complete</em>? That is to say: does every weak Cauchy sequence in $L^1$ converge? </p>
<p>The answer is <strong>yes</strong>.</p>
<p>(Added: See Nate's answer for a definition of <em>weak Cauchy sequence</em> and why this "completeness" of $... |
108,372 | <p>Given a map $\psi: S\rightarrow S,$ for $S$ a closed surface, is there any algorithm to compute its translation distance in the curve complex? I should say that I mostly care about checking that the translation distance is/is not very small. That is, if the algorithm can pick among the possibilities: translation dis... | Autumn Kent | 1,335 | <p>In the braid group, Ko and Lee have given a polynomial time test of reducibility using the Garside structure. (See <a href="http://arxiv.org/abs/math/0610746" rel="nofollow">http://arxiv.org/abs/math/0610746</a>)</p>
|
1,928,149 | <p>I have the following general question about geodesics.
I know the following equation for a geodesic $\sigma$ on a manifold $M\subset R^n$ of dimension $m$, written in local coordinates:
$${\sigma^k}^{''} (t) + \Gamma_{i,j}^k {\sigma^{i}}'{\sigma^{j}}'=0,$$</p>
<p>for $i,j,k=1, \dots, m$.</p>
<p>Now, if I have a cu... | Futurologist | 357,211 | <p>If you have your $M$ to be an $m$-dimensional submanifold of $\mathbb{R}^n$ and the Riemannian metric on $M$ is the Euclidean metric of $\mathbb{R}^n$ restricted to $M$, then there are two things you need to check.</p>
<p>1) Make sure the curve $\gamma : (a,b) \to \mathbb{R}^n$ lies on $M$, i.e. $\gamma(t) \in M$... |
141,823 | <p>I am thinking about the simplest version of Hensel's lemma. Fix a prime $p$. Let $f(x)\in \mathbf{Z}[x]$ be a polynomial. Assume there exists $a_0\in \mathbf{F}_p$ such that $f(a_0)=0\mod p$, and $f'(a_0)\neq 0\mod p$. Then there exists a unique lift $a_n\in \mathbf{Z}/p^{n+1}\mathbf{Z}$ for every $n$. I know there ... | Community | -1 | <p>One can also show that any complete local ring $(R,\mathfrak{m})$ is Henselian using the infinitesimal lifting criterion for étale morphisms: Let $S$ be an étale $R$-algebra with a section $S \to R/\mathfrak{m}$. We want to show that there is a lift $S \to R = \varprojlim_n R/\mathfrak{m}^n$. But we can construct s... |
1,885,068 | <p>Prove $$\int_0^1 \frac{x-1}{(x+1)\log{x}} \text{d}x = \log{\frac{\pi}{2}}$$</p>
<p>Tried contouring but couldn't get anywhere with a keyhole contour.</p>
<p>Geometric Series Expansion does not look very promising either.</p>
| Marco Cantarini | 171,547 | <p>We have $$I=\int_{0}^{1}\frac{x-1}{\log\left(x\right)\left(x+1\right)}dx=\sum_{k\geq0}\left(-1\right)^{k}\int_{0}^{1}\frac{x^{k+1}-x^{k}}{\log\left(x\right)}dx
$$ $$\stackrel{x=e^{-u}}{=}\sum_{k\geq0}\left(-1\right)^{k+1}\int_{0}^{\infty}\frac{e^{-\left(k+2\right)u}-e^{-\left(k+1\right)u}}{x}du
$$ and now we can a... |
1,836,190 | <p>I've been working on a problem and got to a point where I need the closed form of </p>
<blockquote>
<p>$$\sum_{k=1}^nk\binom{m+k}{m+1}.$$</p>
</blockquote>
<p>I wasn't making any headway so I figured I would see what Wolfram Alpha could do. It gave me this: </p>
<p>$$\sum_{k=1}^nk\binom{m+k}{m+1} = \frac{n((m+2... | Matthew Conroy | 2,937 | <p>You can prove this by induction.</p>
<p>Here is the induction step:
$$
\begin{align*}
\sum_{k=1}^{n+1} k \binom{m+k}{m+1} &= \frac{n((m+2)n+1)}{(m+2)(m+3)}\binom{m+n+1}{m+1} + (n+1)\binom{m+n+1}{m+1} \\
&=\frac{(m+n+2)(m(n+1)+2n+3)}{(m+2)(m+3)} \binom{m+n+1}{m+1} \\
&=\frac{(n+1)(m(n+1)+2n+3)}{(m+2)(m+... |
3,306,089 | <p>I came across this meme today:</p>
<p><a href="https://i.stack.imgur.com/RfJoJ.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RfJoJ.jpg" alt="enter image description here"></a></p>
<p>The counterproof is very trivial, but I see no one disproves it. Some even say that the meme might be true. Wel... | ZAF | 609,023 | <p>Let <span class="math-container">$a_{n} = tr_{n}(\pi)$</span></p>
<p><span class="math-container">$a_{1} = 3.1$</span></p>
<p><span class="math-container">$a_{2} = 3.14$</span></p>
<p><span class="math-container">$a_{3} = 3.141$</span></p>
<p><span class="math-container">$a_{4} = 3.1415$</span></p>
<p><span cla... |
2,713,311 | <p>$ \lim_{x \to \infty} [\frac{x^2+1}{x+1}-ax-b]=0 \ $ then show that $ \ a=1, \ b=-1 \ $</p>
<p><strong>Answer:</strong></p>
<p>$ \lim_{x \to \infty} [\frac{x^2+1}{x+1}-ax-b]=0 \\ \Rightarrow \lim_{x \to \infty} [\frac{x^2+1-ax^2-ax-bx-b}{x+1}]=0 \\ \Rightarrow \lim_{x \to \infty} \frac{2x-2ax-a-b}{1}=0 \\ \Righta... | egreg | 62,967 | <p>Let $f(x)=\dfrac{x^2+1}{x+1}$. If
$$
\lim_{x\to\infty}(f(x)-ax-b)=0
$$
then also
$$
\lim_{x\to\infty}\frac{f(x)-ax-b}{x}=0
$$
Thus we must have
$$
\lim_{x\to\infty}\left(\frac{x^2+1}{x(x+1)}-a\right)=0
$$
and therefore $a=1$. Now
$$
\frac{x^2+1}{x+1}-x=\frac{x^2+1-x^2-x}{x+1}=\frac{-x}{x+1}
$$
so
$$
\lim_{x\to\infty... |
92,967 | <p>Let <span class="math-container">$d(n)$</span> be the number of divisors function, i.e., <span class="math-container">$d(n)=\sum_{k\mid n} 1$</span> of the positive integer <span class="math-container">$n$</span>. The following estimate is well known
<span class="math-container">$$
\sum_{n\leq x} d(n)=x \log x + (2 ... | Dimitris Koukoulopoulos | 4,003 | <p>The key is to count integers with a given number of prime factors: if $\omega(n)=\sum_{p|n}1$ and $\Omega(n)=\sum_{p^a|n,\,a\ge1}1$, then $2^{\omega(n)}\le\tau(n)\le2^{\Omega(n)}$ and there are results that control the number of integers with a given value of $\omega(n)$, or of $\Omega(n)$. The simplest one of them ... |
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Gerald Edgar | 454 | <p><a href="http://www.dimensions-math.org/">Dimensions</a></p>
<p><a href="http://www.youtube.com/watch?v=JX3VmDgiFnY&fmt=18">Möbius Transformations Revealed</a></p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Any | 950 | <p>'Not Knot' is also a nice vid</p>
<p><a href="http://www.youtube.com/watch?v=AGLPbSMxSUM">http://www.youtube.com/watch?v=AGLPbSMxSUM</a></p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Nikita Kalinin | 4,298 | <p>NMU(<a href="https://en.wikipedia.org/wiki/Independent_University_of_Moscow" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Independent_University_of_Moscow</a>) and MIAN lectures 2009-2010 (in Russian)</p>
<p><a href="http://erb-files.narod.ru/" rel="nofollow noreferrer">http://erb-files.narod.ru/</a></p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | DamienC | 7,031 | <p>The <a href="http://www.ihes.fr/jsp/site/Portal.jsp">IHES</a> also has a lot of <a href="http://www.dailymotion.com/user/Ihes_science/">on-line videos</a>. In particular, I like very much the ones from the "Colloque Grothendieck". </p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | euklid345 | 4,709 | <p>The famous proof of the snake lemma in the 1980's movie <em>It's my turn</em> (can be found on utube). </p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Carl Najafi | 19,027 | <p>At the time of writing, <a href="http://www.math.rutgers.edu/~russell2/expmath/" rel="nofollow">Rutgers experimental mathematics seminar</a> has over 200 <a href="http://www.youtube.com/user/kgnang" rel="nofollow">videos</a> up on youtube. I wish more seminars would do this!</p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Strongart | 9,946 | <p>I make some maths videos at home,Here is an English video:<a href="http://video.yayun2010.sina.com.cn/v/b/49393046-1215048895.html" rel="nofollow">Visible Fibre Bundle</a> </p>
<p>maybe that can help some begginners.</p>
<p>All my maths vedios at <a href="http://blog.sina.com.cn/s/articlelist_1215048895_12_1.html"... |
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Sniper Clown | 20,215 | <p>I am quite surprised to see <a href="http://www.youtube.com/watch?v=p3DOGo_XF2o" rel="nofollow">Dan Freed's lecture of Hodge Conjecture</a> has not been mentioned. (Although it is an old thread I believe this should be in here. Before there was a QuickTime video but I am grateful to find that it has been youtubed.) ... |
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Boris Bukh | 806 | <p>As of today, the digitized tapes of CBMS Lectures on Probability Theory and Combinatorial by Michael Steele <a href="http://sms.cam.ac.uk/collection/1189351" rel="nofollow">are online</a>. I heartily recommend them — the style is informal, but educating: there are jokes, juggling lessons, speculations about the stoc... |
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Saikat Biswas | 13,628 | <p>'Selmer Ranks of Elliptic Curves in Families of Quadratic Twists' by Karl Rubin</p>
<p><a href="http://research.microsoft.com/apps/video/default.aspx?id=140581" rel="nofollow">http://research.microsoft.com/apps/video/default.aspx?id=140581</a></p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Community | -1 | <p>Any video on </p>
<p><a href="http://www.josleys.com/galleries.php?showdate=1" rel="nofollow">Jos Leys "Mathematical Imagery"</a></p>
<p>is a true masterpiece, and has a non-trivial mathematical content...</p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Alexey Ustinov | 5,712 | <p><a href="http://sms.cam.ac.uk/collection/533691" rel="nofollow">Discrete Integrable Systems</a> at Isaac Newton Institute for Mathematical Sciences</p>
|
4,099,649 | <p>I’m trying to solve two (in my opinion, tough) integrals which appear in part of my problem. I tried different ways but in the end I failed. See them below, please.</p>
<p><span class="math-container">$${\rm{integral}}\,1 = \int {{{\left( {\frac{A}{{{x^\alpha }}}\, + \sqrt {B + \frac{C}{{{x^{2\alpha }}}}\,} } \right... | Yuri Negometyanov | 297,350 | <p><span class="math-container">$\color{brown}{\textbf{Simple cases.}}$</span></p>
<p>If <span class="math-container">$\;\underline{B=0},\;$</span> then the integration looks trivial.</p>
<p>If <span class="math-container">$\;\underline{C=0},\;$</span> then, by Wolfram Alpha,</p>
<p><a href="https://i.stack.imgur.com/V... |
1,452,425 | <p>From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not <a href="https://math.stackexchange.com/a/11155/171192">$0^0$ is $1$ is a simple matter of definition</a>.</p>
<p><strong>My question is what the definition of a set is?<... | Carl Mummert | 630 | <blockquote>
<p>So is it, for example, a right definition to say that a set is anything that satisfies the ZFC list of axioms?</p>
</blockquote>
<p>That is almost correct, but not quite. A set on its own does not satisfy the ZFC axioms, any more than a vector on its own can satisfy the vector space axioms or a point... |
1,722,948 | <blockquote>
<p>$$\frac{1}{x}-1>0$$</p>
</blockquote>
<p>$$\therefore \frac{1}{x} > 1$$</p>
<p>$$\therefore 1 > x$$</p>
<p>However, as evident from the graph (as well as common sense), the right answer should be $1>x>0$. Typically, I wouldn't multiple the x on both sides as I don't know its sign, bu... | DeepSea | 101,504 | <p>Hint: $x(1-x)>0$.Can you continue?</p>
|
1,722,948 | <blockquote>
<p>$$\frac{1}{x}-1>0$$</p>
</blockquote>
<p>$$\therefore \frac{1}{x} > 1$$</p>
<p>$$\therefore 1 > x$$</p>
<p>However, as evident from the graph (as well as common sense), the right answer should be $1>x>0$. Typically, I wouldn't multiple the x on both sides as I don't know its sign, bu... | GoodDeeds | 307,825 | <p>Continuing from the step:
$$\frac1x\gt1$$
Now, to multiply the inequality by any non zero number we need to know its sign. So, taking two cases,</p>
<hr>
<p>Case 1: $x\gt0$</p>
<p>Multiplying by $x$ on both sides will not affect the sign. Thus,
$$1\gt x$$
Due to the assumption,
$$1\gt x\gt0$$</p>
<hr>
<p>Case ... |
1,893,280 | <p>How to show $\frac{c}{n} \leq \log(1+\frac{c}{n-c})$ for any positive constant $c$ such that $0 < c < n$?</p>
<p>I'm considering the Taylor expansion, but it does not work...</p>
| JimmyK4542 | 155,509 | <p><strong>Hint</strong>: For all $n-c \le x \le n$, we have $\dfrac{1}{n} \le \dfrac{1}{x}$. Hence, $$\displaystyle\int_{n-c}^{n}\dfrac{1}{n}\,dx \le \int_{n-c}^{n}\dfrac{1}{x}\,dx.$$</p>
|
2,948,118 | <p>I understand that for a function or a set to be considered a vector space, there are the 10 axioms or rules that it must be able to pass. My problem is that I am unable to discern how exactly we prove these things given that my book lists some weird general examples.</p>
<p>For instance: the set of all third- degre... | Fred | 380,717 | <p>Your proof is not correct: if <span class="math-container">$ \epsilon$</span> is "small", then <span class="math-container">$ \sigma \notin [-1,2]$</span>.</p>
<p>Your function <span class="math-container">$f$</span> is increasing ! Let <span class="math-container">$n \in \mathbb N$</span> and let <span class="math... |
58,060 | <p>I have been looking at Church's Thesis, which asserts that all intuitively computable functions are recursive. The definition of recursion does not allow for randomness, and some people have suggested exceptions to Church's Thesis based on generating random strings. For example, using randomness one can generate str... | none | 34,896 | <p>There is an article by Leonid Levin that the OP might like:</p>
<ul>
<li><a href="http://arxiv.org/pdf/cs.CC/0203029.pdf" rel="nofollow">http://arxiv.org/pdf/cs.CC/0203029.pdf</a></li>
<li>informal overview: <a href="http://www.cs.bu.edu/fac/lnd/expo/gdl.htm" rel="nofollow">http://www.cs.bu.edu/fac/lnd/expo/gdl.htm... |
2,764,221 | <p>Let $A$ be a symmetric invertible $n \times n$ matrix, and $B$ an antisymmetric $n \times n$ matrix. Under what conditions is $A+B$ an invertible matrix? In particular, if $A$ is positive definite, is $A+B$ invertible? </p>
<p>This isn't homework, I am just curious. Assume all matrices have entries in $\mathbb{R}$... | orangeskid | 168,051 | <p>If $A$ with positive determinant then in the case $n=2$, $A+B$ will be invertible. For $n\ge 3$, you can adjust the $2\times 2$ examples given in the other answers by adding a piece $(-1,1,\ldots)$ on the diagonal to make $A$ $n\times n$ with positive determinant. </p>
<p>Maybe you are thinking about the followin... |
2,358,385 | <p>I had a test and I couldn't solve this problem:<br></p>
<p>Given $f: \mathbb R^2 \rightarrow \mathbb R$.<br>For every constant $y_0$, $f(x,y_0)$ is known to be continuous.<Br>Also, $\frac{\partial f}{\partial y}(x,y)$ is defined and bounded for all $(x,y)$.
<br><br>I needed to prove that $f$ is continuous for all $... | Fred | 380,717 | <p>Let $(x_0,y_0) \in \mathbb R^2$. For $(x,y) \in \mathbb R^2$ we have</p>
<p>$|f(x,y)-f(x_0,y_0)|=$</p>
<p>$|f(x,y)-f(x,y_0)+f(x,y_0)-f(x_0,y_0)| \le |f(x,y)-f(x,y_0)|+|f(x,y_0)-f(x_0,y_0)| $</p>
<p>There is $L \ge 0$ such that $|\frac{\partial f}{\partial y}(x,y)| \le L$ for all $(x,y) \in \mathbb R^2$. By the Me... |
2,358,385 | <p>I had a test and I couldn't solve this problem:<br></p>
<p>Given $f: \mathbb R^2 \rightarrow \mathbb R$.<br>For every constant $y_0$, $f(x,y_0)$ is known to be continuous.<Br>Also, $\frac{\partial f}{\partial y}(x,y)$ is defined and bounded for all $(x,y)$.
<br><br>I needed to prove that $f$ is continuous for all $... | Mundron Schmidt | 448,151 | <p><strong>Hint:</strong><p>
For the continuity you have to consider $|f(x_0,y_0)-f(x,y)|$. The information that $x\mapsto f(x,y_0)$ is continuous says how $f$ behaves in the $x$-direction while $\partial_yf(x,y)$ is bounded says how it behaves in the $y$-direction. Therefore you have to write your term such that you c... |
2,622,092 | <p>I want to study the convergence of the improper integral $$ \int_0^{\infty} \frac{e^{-x^2}-e^{-3x^2}}{x^a}$$To do so I used the comparison test with $\frac{1}{x^a}$ separating $\int_0^{\infty}$ into $\int_0^{1} + \int_1^{\infty}$.</p>
<p>For the first part, $\int_0^{1}$, I did $$\lim_{x\to0} \frac{\frac{e^{-x^2}-e^... | user | 505,767 | <p>For the first note that for $x\to0$</p>
<p>$$e^{-x^2}=1-x^2+o(x^2) \quad \quad e^{-3x^2}=1-3x^2+o(x^2)$$</p>
<p>$$\implies \frac{e^{-x^2}-e^{-3x^2}}{x^a}= \frac{2x^2+o(x^2)}{x^a}\sim \frac{2}{x^{a-2}}$$</p>
<p>thus $\int_0^{1}$ converges by comparison with $\frac{1}{x^{a-2}}$ for $a-2<1$ that is $a<3$.</p>
... |
1,843,662 | <p>Let C be that part of the circle $z=e^{i\theta}$, where $0\le\theta\le\frac\pi2$. Evaluate $\int_{c}\frac{z}{i}dz$.</p>
<p>This is my first time posting my question here. I'm really poor in writing English. for that reason please understand my bad explanation. proceed to the main issue I have no idea on solving thi... | hmakholm left over Monica | 14,366 | <p>You can always make a <em>non-deterministic</em> automaton with a single accepting state for any regular language (even without $\varepsilon$-transitions) -- <em>unless</em> the language contains the empty string and is not closed under concatenation. Just take an automaton without this restriction and create a new ... |
1,901,244 | <p>Let $f:[0,1] \to \mathbb{R}$ $$f(x) = \begin{cases}
1 & \text{if x }=\frac {1}n, n\in \mathbb{N} \\
0 & \text{otherwise }
\end{cases}$$
I need to prove that $f$ is integrable over $[0,1]$ but I'm failing to understand how that is true. If it is indeed integrable, then we know that for all $ε$ > 0, there exis... | zhw. | 228,045 | <p>I'll assume that this is known: If $g=0$ on $[a,b]$ except for finitley many points, then $\int_a^bg=0.$</p>
<p>Let $\epsilon > 0.$ Choose $a, 0<a<\epsilon.$ Our $f$ has only finitely many points in $[a,1]$ where $f$ is nonzero. From the above, there is a partition $P$ of $[a,1]$ such that $U(f,P) < \ep... |
1,901,244 | <p>Let $f:[0,1] \to \mathbb{R}$ $$f(x) = \begin{cases}
1 & \text{if x }=\frac {1}n, n\in \mathbb{N} \\
0 & \text{otherwise }
\end{cases}$$
I need to prove that $f$ is integrable over $[0,1]$ but I'm failing to understand how that is true. If it is indeed integrable, then we know that for all $ε$ > 0, there exis... | Farewell | 278,893 | <p>$\int_{0}^{1}f(x)dx=-\int_{1}^{0}f(x)dx=-\sum_{i=1}^{\infty}\int_{\frac {1}{i}}^{\frac {1}{i+1}}f(x)dx=-\sum_{i=1}^{\infty}0=0$</p>
|
2,628,149 | <p>I am having trouble finding the general solution of the following second order ODE for $y = y(x)$ without constant coefficients: </p>
<p>$3x^2y'' = 6y$<br>
$x>0$</p>
<p>I realise that it may be possible to simply guess the form of the solution and substitute it back into the the equation but i do not wish to us... | Jack D'Aurizio | 44,121 | <p>For $x$ close to the origin we have $(1-x)^{1/4} \approx 1-\frac{x}{4}-\frac{3x^2}{32}$ and an even better approximation is $(1-x)^{1/4} \approx 1-\frac{x}{4}-\frac{9x^2}{96-56x}$. By evaluating at $x=\frac{1}{16}$ and multiplying by $2$ we get the approximation $\sqrt[4]{15}\approx \color{red}{\frac{23301}{11840}}$... |
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