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 |
|---|---|---|---|---|
1,775,965 | <p>I'm learning about measure theory, specifically the Lebesgue integral of nonnegative functions, and need help with the following problem. </p>
<blockquote>
<p>Let $f:\mathbb{R}\to[0,\infty)$ be measurable and $f\in L^1$. Show that $F(x)=\int_{-\infty}^x f$ is continuous.</p>
</blockquote>
<p>I know is isn't muc... | Aloizio Macedo | 59,234 | <p>Let $x \in \mathbb{R}$.
It suffices to prove that $F(x_n) \to F(x)$ for every $x_n \to x$.
Therefore, let $x_n \to x$.</p>
<p>We have $F(x_n)=\int_{-\infty}^{\infty}f \cdot \chi_{[-\infty,x_n]} $. </p>
<p>It is easy to see that $f \cdot \chi_{[-\infty,x_n]} \to f \cdot \chi_{[-\infty,x]}$ (except possibly at $x$)... |
1,154,763 | <p>I'm given this equation:</p>
<p>$$
u(x,y) =
\begin{cases}
\dfrac{(x^3 - 3xy^2)}{(x^2 + y^2)}\quad& \text{if}\quad (x,y)\neq(0,0)\\
0\quad& \text{if} \quad (x,y)=(0,0).
\end{cases}
$$</p>
<p>It seems like L'hopitals rule has been used but I'm confused because</p>
<ol>
<li>there is no limit here it's just... | TravisJ | 212,738 | <p>Probably, it is a piecewise definition. $u(x,y)$ makes sense as long as both $x$ and $y$ are not $0$. It is probably just to mean that: if $(x,y)\neq (0,0)$ then $u(x,y)$ is as defined (with the formula), otherwise $u(x,y)=0$. You certainly cannot plug in $(0,0)$ to the formula. It may be that the point is chose... |
2,258,557 | <p>Why the equation of an arbitrary straight line in complex plane is $zz_o + \bar z \bar z_0 = D$ where D $\in R$</p>
<p>I understand that a vertical straight line can be defined by the equation $z+\bar z= D$ because suppose $z =x+yi$ then $\bar z = x-yi$ Thus, $z+\bar z = x+yi+x-yi=2x$ which is an arbitrary vertic... | Mark Viola | 218,419 | <p><strong>HINT:</strong></p>
<p>$$z+\bar z=2\text{Re}(z)\implies zz_0+\bar z\bar z_0=2\text{Re}(zz_0)$$</p>
|
2,258,557 | <p>Why the equation of an arbitrary straight line in complex plane is $zz_o + \bar z \bar z_0 = D$ where D $\in R$</p>
<p>I understand that a vertical straight line can be defined by the equation $z+\bar z= D$ because suppose $z =x+yi$ then $\bar z = x-yi$ Thus, $z+\bar z = x+yi+x-yi=2x$ which is an arbitrary vertic... | dxiv | 291,201 | <p>Hint: given any two points $z_1, z_2 \in \mathbb{C}\,$, then $z$ is collinear with $z_1, z_2$ iff there exists $\lambda \in \mathbb{R}$ such that $z-z_1 = \lambda(z-z_2)$. Eliminate $\lambda$ between the following, then define $z_0, D$ appropriately:</p>
<p>$$
\begin{cases}
\begin{align}
z-z_1 &= \lambda(z-z_2)... |
21,372 | <blockquote>
<p>Let $ y = \min \{ (x + 6), (4 – x) \}$, then find $y$.</p>
</blockquote>
<p>How to solve this problem?</p>
| Isaac | 72 | <p>Try graphing $y=x+6$ and $y=4-x$ together on one graph, then highlight or otherwise mark the parts of those graphs that make up $y=\min{x+6,4-x}$. The resulting shape should be a familiar type of basic function, perhaps translated and/or reflected.</p>
|
1,989,259 | <p><strong>Can modus tollens be statement of proof by contradiction or is it just a specific case of contradiction?</strong></p>
<p>i.e we know that in general, proof by contradiction stated as follows</p>
<p><span class="math-container">$[P' \implies (q \land q')] \implies P$</span></p>
<p>And by modus tollens, we hav... | PMar | 383,670 | <p>Actually, these are two different views of the same thing. In one case you begin with q' already proven, and prove P' -> q; then you apply the logical equivalence P' -> q <==> q' -> P, and use modus-ponens on the latter (plus q') to arrive at P. In the other case you prove P' -> (q^q'), apply the same equivalen... |
37,900 | <p>I use the following code to find out the number of consecutive prime numbers using a formula $n^2+n+i$ found out by Euler (starting from n=0):</p>
<pre><code>Nbs = {};
Do[Nbs = Union[Nbs,
Select[Range[5000], (PrimeQ[#^2 + # + i] == False &), 1]], {i, 1,
5000}];
Nbs
</code></pre>
<p>How can I also get in th... | ubpdqn | 1,997 | <p><strong>EDIT</strong></p>
<p>My previous answer related to the title of the question and was directed to "reaping" or "catching" first cases of a condition being met from a loop. </p>
<p>The Euler formula, as I now understand, was a method for generating consecutive prime numbers : $n^2+n+k$, where $k$ is prime a... |
69,508 | <p>I was just wondering, when I call the <code>CopulaDistribution</code> function in Mathematica, am I calling its cumulative function or its density function?</p>
<p>I have looked up the help and am still a little bit unsure.</p>
<p>EDIT: In particular, what does it mean when I take a RandomVariate from this CopulaD... | Romke Bontekoe | 1,178 | <p>I recently asked WRI about a similar behaviour on probability distributions. The answer was that the output IS generated, but erroneously the result IS NOT displayed.</p>
<p>Try</p>
<pre><code>dist = ProbabilityDistribution[1 - Abs[x], {x, -1, 1}]
</code></pre>
<p>with </p>
<pre><code>Mean[dist]
</code></pre>
<... |
1,533,362 | <p>I need to prove this identity:</p>
<p>$\sum_{k=0}^n \frac{1}{k+1}{2k \choose k}{2n-2k \choose n-k}={2n+1 \choose n}$</p>
<p>without using the identity:</p>
<p>$C_{n+1}=\sum_{k=0}^n C_kC_{n-k}$.</p>
<p>Can't figure out how to.</p>
| user | 293,846 | <p>Rewriting the LHS as:
<span class="math-container">$$
\sum_{k=0}^n \frac{1}{1+2k}{1+2k \choose k}{2n-2k \choose n-k}
$$</span>
one observes that this is a particular case (<span class="math-container">$a=1,b=2,c=2n$</span>) of the <a href="https://en.wikipedia.org/wiki/Rothe%E2%80%93Hagen_identity" rel="nofollow nor... |
1,141,074 | <p>I need help with this integral: $$\int\frac{\sqrt{\tan x}}{\cos^2x}dx$$ I tried substitution and other methods, but all have lead me to this expression: $$2\int\sqrt{\tan x}(1+\tan^2 x)dx$$ where I can't calculate anything... Any suggestions? Thanks!</p>
| Aaron Maroja | 143,413 | <p>Hint: Let $u = \tan x$ then $du = \sec^2 x\ \ dx = \frac{1}{\cos^2 x} dx$</p>
|
1,141,074 | <p>I need help with this integral: $$\int\frac{\sqrt{\tan x}}{\cos^2x}dx$$ I tried substitution and other methods, but all have lead me to this expression: $$2\int\sqrt{\tan x}(1+\tan^2 x)dx$$ where I can't calculate anything... Any suggestions? Thanks!</p>
| Bernard | 202,857 | <p>Set $t=\tan x$; then $\,\mathrm d\mkern1.5mu t=\dfrac1{\cos^2x}\mathrm d\mkern1.5mu x$ so the integral becomes $$\int\sqrt t\,\mathrm d\mkern1.5mu t = \frac 23 (t)^{\frac32}=\frac23\tan t\sqrt{\tan t}.$$</p>
|
7,981 | <p>I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?</p>
| PAD | 27,304 | <p>Let $H_n$ be the nth harmonic number, i.e. $ H_n = 1 + \frac12 + \frac13 + \dots + \frac1n.$ Then, the Riemann hypothesis is true if and only if</p>
<p>$$ \sum_{d | n}{d} \le H_n + \exp(H_n)\log(H_n)$$</p>
|
188,102 | <p>I have the following list: </p>
<pre><code>m={{14, "extinguisher"}, {54, "virgule"}, {55, "turnoff"}, {51,
"sofa"}, {77, "beachcomber"}, {61, "stoic"}, {6,
"isomorphism"}, {34, "leftist"}, {84, "spline"}, {42,
"heartiness"}, {35, "postnatal"}, {41, "stratified"}, {66,
"silkworm"}, {95, "conformance"}, {... | JimB | 19,758 | <p>This is an answer but might likely be interpreted as just a comment: If you want to construct a spiral bubble chart as an example of poor information transfer, then by all means go for it. If not, don't do it. For such data a simple bar chart might be your best bet.</p>
<pre><code>m = Sort[m]
BarChart[m[[All, 1... |
3,615,117 | <p>I want to find the intersection of the sphere <span class="math-container">$x^2+y^2+z^2 = 1$</span> and the plane <span class="math-container">$x+y+z=0$</span>. </p>
<p><span class="math-container">$z=-(x+y)$</span> that gives <span class="math-container">$x^2+y^2+xy= \frac 12$</span></p>
<p>How do I represent thi... | Z Ahmed | 671,540 | <p><span class="math-container">$$x^2+y^2+xy=1/2 \implies \frac{(x+y)^2+(x-y)^2}{2}+\frac{(x+y)-(x-y)^2}{4}=\frac{1}{2}$$</span>
<span class="math-container">$$\implies \frac{3}{2}(x+y)^2+\frac{(x-y)^2}{2}=1$$</span>
<span class="math-container">$$\implies \frac{(\frac{x+y}{\sqrt{2}})^2}{1/3}+\frac{(\frac{x-y}{\sqrt{2}... |
2,267,005 | <p>I have been asked to evaluate the $\int{Fdr}$ over a curve $C$ where $F = yz\mathbf{i} + 2xz\mathbf{j} + e^{xy}\mathbf{k}$ and $C$ is the curve $x^2 + y^2 = 16, z =5$ with downward orientation</p>
<p>I want to use Stokes theorem, so I am thinking of parametrizing this surface as $(x, y, z)=(4 \cos t, 4\sin t,z)$ bu... | Steve Kass | 60,500 | <p>The car can certainly go at least $450$ and no more than $600$ km on one set of tires, regardless of how the tires are swapped, so the function $r=f(S)$ that gives the range $r$ of a swapping strategy $S$ is bounded and has a maximum value over all swapping strategies. Call this optimal range $K$. ($f$ is continuous... |
148,374 | <p>I have checked all Mathematica color schemes, and I think "Hue" is the most vibrant, beautiful one. However, it has one issue: the two ends of the spectrum are red (though, different reds). I like a spectrum from, say, red to blue. Is that possible to manipulate the Hue and remove the pink and second red? </p>
<p>C... | kglr | 125 | <p>From <a href="http://reference.wolfram.com/mathematica/ref/Hue.html" rel="nofollow noreferrer">Documentation >> Hue >> Details</a></p>
<p><img src="https://i.stack.imgur.com/Zy63W.png" alt="Mathematica graphics"></p>
<p>So, we need to rescale the function values (that run from <code>-1</code> to <code>1</code>) t... |
4,127,149 | <p>I understand that the addition and subtraction of complex number is the same as vector addition and subtraction. But what is the vector equivalent of multiplication and division of complex numbers?</p>
| José Carlos Santos | 446,262 | <p>In general, there is none. In some cases, as in the case of complex numbers and of quaternions, such operations can be defined. And in the case of quaternions, you have two divisions, not just one (if <span class="math-container">$q$</span> and <span class="math-container">$r$</span> are quaternions and <span class=... |
3,752,402 | <p>I want to find out the existence of the solutions in diophantine equations of the style:</p>
<p><span class="math-container">$$-259y ^2+ 2400yx + 1817y + 2122x =
$$</span>
<span class="math-container">$$1057364602723981500371957207036553770637547302056514367123547565680640946707606178926389130616$$</span></p>
<p>Th... | dan_fulea | 550,003 | <p>The general form of the "type" of the equation is (4) in</p>
<p><a href="https://mathworld.wolfram.com/PellEquation.html" rel="nofollow noreferrer">https://mathworld.wolfram.com/PellEquation.html</a></p>
<p>And there are algorithms to reduce the equation. The idea is to homogenize, thus getting an equation... |
853,659 | <p>Evaluate the integral:</p>
<p>$$\int \frac{x^6}{x^4-1} \, \mathrm{d}x$$</p>
<p>After a lot of help I have reached this point:</p>
<p>$x^2 = Ax^3 - Ax + Bx^2 - B + Cx^3 + Cx^2 + Cx + C + Dx^3 - Dx^2 + Dx - D$</p>
<p>But now I don't really know how to solve for $A, B, C$, and $D$. Please help!</p>
| Tunk-Fey | 123,277 | <p>Rewrite the integrand as
\begin{align}
\frac{x^6}{x^4-1}&\stackrel{\color{red}{[1]}}=\color{darkgreen}{x^2}+\color{blue}{\frac{x^2}{x^4-1}}\\
&=\color{darkgreen}{x^2}+\color{blue}{\frac{x^2}{(x^2-1)(x^2+1)}}\\
&\stackrel{\color{red}{[2]}}=\color{darkgreen}{x^2}+\color{blue}{\frac{1}{2}\left(\color{black}... |
259,083 | <p>There was a question asked: <a href="https://math.stackexchange.com/q/136204/8348">An open subset $U\subseteq R^n$ is the countable union of increasing compact sets.</a> There Davide gave an <a href="https://math.stackexchange.com/a/136209/">answer</a>. Can anyone tell me how the
equality holds, and the motivation ... | user642796 | 8,348 | <p>Let's first fix a $k$ and consider Davide's set (which I will slightly rewrite) $$X_k := U \cap A_k \cap B_k$$ where $$\begin{gather}A_k := \{x : \lVert x\rVert\leq k\} \\ B_k := \{x : d(x,U^c)\geq k^{-1}\} = \{ x : B ( x; k^{-1} ) \subseteq U\}.\end{gather}$$ One thing to note is that as $B_k \subseteq U$, we actu... |
168,819 | <p>I was looking for a free PDF from which I can review MV calculus.</p>
<p>Specifically:</p>
<ol>
<li>MV Limits, Continuity, Differentiation.</li>
<li>Differentiation of vector and scalar fields</li>
<li>Surface/Multiple Integrals</li>
</ol>
<p>A succinct book would be great, (coherent) course notes and presentatio... | Alex Nelson | 31,693 | <p>Michael Corral's <a href="http://www.mecmath.net/" rel="nofollow"><em>Vector Calculus</em></a> is a good free reference too.</p>
|
157,876 | <p>Can anyone tell me how to find all normal subgroups of the symmetric group $S_4$?</p>
<p>In particular are $H=\{e,(1 2)(3 4)\}$ and $K=\{e,(1 2)(3 4), (1 3)(2 4),(1 4)(2 3)\}$ normal subgroups?</p>
| Douglas S. Stones | 139 | <p>As suggested by Babak Sorouh, the answer can be found easily using <a href="http://www.gap-system.org/" rel="nofollow">GAP</a> using the <a href="http://www.gap-system.org/Packages/sonata.html" rel="nofollow">SONATA</a> library. Here's the code:</p>
<pre><code>G:=SymmetricGroup(4);
S:=Filtered(Subgroups(G),H->I... |
23,911 | <p>I am teaching a course on Riemann Surfaces next term, and would <strong>like a list of facts illustrating the difference between the theory of real (differentiable) manifolds and the theory non-singular varieties</strong> (over, say, $\mathbb{C}$). I am looking for examples that would be meaningful to 2nd year US g... | Charles Staats | 5,094 | <p>Any two compact surfaces (without boundary) of the same genus are diffeomorphic. However, if S is a surface of genus g > 0, there are uncountably many non-isomorphic complex (or, equivalently, algebraic) structures on S.</p>
|
23,911 | <p>I am teaching a course on Riemann Surfaces next term, and would <strong>like a list of facts illustrating the difference between the theory of real (differentiable) manifolds and the theory non-singular varieties</strong> (over, say, $\mathbb{C}$). I am looking for examples that would be meaningful to 2nd year US g... | Kevin H. Lin | 83 | <p>A proper variety doesn't have (non-constant) global sections. A real manifold, compact or not, has lots of global sections.</p>
<p>There are lots of maps between real manifolds. Maps between varieties are much more restricted (e.g. by Riemann-Hurwitz in the case of curves). </p>
|
683,513 | <p>There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in real analysis and the student reception of it. My impression has been that the mathematical community often holds an upbeat opinion on the success of student recepti... | Marcel Besixdouze | 29,892 | <p>The opinions are not in conflict. Something can be simple, obvious, intuitive, etc. and a person can still fail to grok that it is simple, obvious, intuitive, etc. The notion of <em>building intuition</em> is an oxymoron according to a common understanding of <em>intuition</em>, but is in fact central to the underst... |
3,543,150 | <p>My question : two indefinite integrals of a function being given , how to express one indefinite integral in terms of the other? </p>
<p><a href="https://i.stack.imgur.com/VkMzJ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VkMzJ.png" alt="enter image description here"></a></p>
| J. W. Tanner | 615,567 | <p>Fractional powers of negative numbers aren't uniquely defined. </p>
<p>There are three cube roots of <span class="math-container">$-1$</span>: <span class="math-container">$-1$</span>, <span class="math-container">$\frac12+\frac{\sqrt{3}}2i$</span>, and <span class="math-container">$\frac12-\frac{\sqrt3}2i$</span... |
3,246,244 | <p>Consider the action of <span class="math-container">$G$</span> on <span class="math-container">$X$</span>.</p>
<p>Let it be a property of <span class="math-container">$G,X$</span> that <span class="math-container">$\forall x,y,\exists g:g\cdot x=g\cdot y$</span>. This is not quite a transitive action - it describe... | Mark Kamsma | 661,457 | <p>This is only possibly if <span class="math-container">$X$</span> is a singleton (or, vacuously, if <span class="math-container">$X$</span> is empty). To see this: let <span class="math-container">$x, y \in X$</span> and suppose <span class="math-container">$g$</span> is such that <span class="math-container">$g \cdo... |
518,627 | <p>Proof that:$ \sum\limits_{n=1}^{p} \left\lfloor \frac{n(n+1)}{p} \right\rfloor= \frac{2p^2+3p+7}{6} $
<br>
where $p$ is a prime number such that $p \equiv 7 \mod{8}$
<br>
<br>I tried to separate the sum into parts but it does not seems to go anywhere. I also tried to make a substitutions for $p$ ,but, I don't thi... | mercio | 17,445 | <p>$$\sum_{i=1}^p \frac{n(n+1)}p = \frac1 p \frac {p(p+1)(p+2)}3 = \frac{p^2+3p+2}3$$, and
$$\frac{p^2+3p+2}3 - \frac{2p^2+3p+7}6 = \frac{p-1}2$$</p>
<p>So you are asking to prove that $$\sum_{n=1}^p \frac{n(n+1)}p - \lfloor\frac{n(n+1)}p\rfloor = \frac{p-1}2$$.</p>
<p>The term being summed is $\dfrac 1 p$ times the... |
1,693,045 | <p>I know if $x=e^{\frac{2\pi i}{17}}$ then $x^{17}=1$ and $\Re(x)=\cos\left(\frac{2\pi}{17}\right)$.</p>
<p>But how do I form a polynomial which has root $\cos\left(\frac{2\pi}{17}\right)$.</p>
<p>I know you can consider de Moivre's theorem and expand the LHS using binomial theorem but that will take a long time.</p... | Darío G | 27,454 | <p>The number $x=cos\left(\frac{2\pi}{17}\right)$ is a root of the polynomial $$\sum_{k=0}^{8} \binom{17}{2k+1}x^{2k+1}\cdot i^{16-2k}\cdot (1-x^2)^{8-k}=1$$.</p>
|
1,693,045 | <p>I know if $x=e^{\frac{2\pi i}{17}}$ then $x^{17}=1$ and $\Re(x)=\cos\left(\frac{2\pi}{17}\right)$.</p>
<p>But how do I form a polynomial which has root $\cos\left(\frac{2\pi}{17}\right)$.</p>
<p>I know you can consider de Moivre's theorem and expand the LHS using binomial theorem but that will take a long time.</p... | lhf | 589 | <p>Let
$c=\cos\left(\frac{2\pi}{17}\right)$ and
$s=\sin\left(\frac{2\pi}{17}\right)$.</p>
<p>Then</p>
<p>$
1=\Re(1)=\Re ((c+s\, i)^{17})=c^{17}-136 c^{15} s^2+2380 c^{13} s^4-12376 c^{11} s^6+24310 c^9 s^8-19448 c^7 s^{10}+6188 c^5 s^{12}-680 c^3 s^{14}+17 c s^{16}
$</p>
<p>Note that $s$ appears only with even power... |
3,053,975 | <p><span class="math-container">$3^6-3^3 +1$</span> factors?, 37 and 19, but how to do it using factoring, <span class="math-container">$3^3(3^3-1)+1$</span>, can't somehow put the 1 inside </p>
| Bill Dubuque | 242 | <p>It's a special case of: <em>completing</em> the square leads to a <em>difference</em> of squares, i.e.</p>
<p><span class="math-container">$$\begin{eqnarray}\overbrace{3^{\large 6}+1}^{\rm incomplete}-\,3^{\large 3}&=\ &\!\!\!\! \overbrace{(3^{\large 3}+1)^{\large 2}}^{\rm\!\!\! complete\ the\ square\!\!\!... |
3,200,354 | <p>How can I find the maximal value in the range <span class="math-container">$[-1,1]$</span> for <span class="math-container">$x$</span> and <span class="math-container">$y$</span> of the following expression:</p>
<p><span class="math-container">$$\sin(\Pi x)(y-3)/2.$$</span></p>
<p>I tried doing the derivative of... | José Carlos Santos | 446,262 | <p>Yes, they are equivalent. Asserting that <span class="math-container">$A\subset B$</span> is equivalent to asserting that <span class="math-container">$(\forall a\in A):a\in B$</span>. And asserting that <span class="math-container">$\Phi(B)\subset B$</span>, in particular, is equivalent to <span class="math-contain... |
2,196,037 | <p>Let $E$ be a universal set and $\{A_{\alpha}\}_{\alpha \in J},$ for some index set $J$ be a family of subsets of $E.$</p>
<p>Prove that:
(a)$E-\bigcup_{\alpha \in J}A_{\alpha} = \bigcap_{\alpha \in J}($R$-A_{\alpha}).$</p>
<p>I do not know what is $R$ or it is a mistake in the question, Could anyone help me ? </p... | Graham Kemp | 135,106 | <p>What ever $R$ means should have been identified earlier in your reference book, otherwise it is a mystery.</p>
<p>As to the second, just use the definitions that $$\bigcup_{\alpha\in J} X_\alpha := \{x~:~ \exists \alpha \in J~ (x\in X_\alpha)\} = \{x~:~\bigvee_{\alpha\in J}(x\in X_\alpha)\}\\ \bigcap_{\alpha\in J}... |
2,196,037 | <p>Let $E$ be a universal set and $\{A_{\alpha}\}_{\alpha \in J},$ for some index set $J$ be a family of subsets of $E.$</p>
<p>Prove that:
(a)$E-\bigcup_{\alpha \in J}A_{\alpha} = \bigcap_{\alpha \in J}($R$-A_{\alpha}).$</p>
<p>I do not know what is $R$ or it is a mistake in the question, Could anyone help me ? </p... | Nosrati | 108,128 | <p>\begin{eqnarray*}
(\bigcap_{i\in\Lambda}A_i)^c &=& \{x|x\notin \bigcap_{i\in\Lambda}A_i\} \\
&=& \{x|\exists i\in\Lambda,~~x\notin A_i\}\\
&=& \{x|\exists i\in\Lambda,~~x\in A_i^c\} \\
&=& ... |
620,045 | <p>What is the mean and variance of Squared Gaussian: $Y=X^2$ where: $X\sim\mathcal{N}(0,\sigma^2)$?</p>
<p>It is interesting to note that Gaussian R.V here is zero-mean and non-central Chi-square Distribution doesn't work.</p>
<p>Thanks.</p>
| iballa | 116,491 | <p>Note that $X^2 \sim \sigma^2 \chi^2_1$ where $\chi^2_1$ is the Chi-squared distribution with 1 degree of freedom. Since $E[\chi^2_1] = 1, \text{Var}[\chi^2_1] = 2$ we have $E[X^2] = \sigma^2, \text{Var}[X^2] = 2 \sigma^4$.</p>
|
129 | <p>Is there some criterion for whether a space has the homotopy type of a closed manifold (smooth or topological)? Poincare duality is an obvious necessary condition, but it's almost certainly not sufficient. Are there any other special homotopical properties of manifolds?</p>
| Benjamin Antieau | 100 | <p>Jacob Lurie gave a talk last week at Peter May's birthday conference on noncommutative Poincaré duality. The idea is to take an $n$-manifold $M$ and a $(n-1)$-connected space $X$. Then, he showed that the compact mapping space $\mbox{Map}_c(M,X)$ is isomorphic to a certain homotopy colimit over a certain category of... |
796,262 | <p>So, I am computing something seemingly simple involving complex gaussians and constants, but I am getting a big contradiction in my calculations. </p>
<p><strong>The setup:</strong></p>
<ul>
<li>Let $C$ be a complex constant, that is, $C = c_r + jc_i$. </li>
<li>Let $G$ be a complex gaussian variable, $G = g_r + j... | Rebecca J. Stones | 91,818 | <p>Since the spanning trees are subgraphs of $K_{3,4}$, the degree sequences are of the form $(d_1,d_2,d_3,d_4), (d_5,d_6,d_7)$ where</p>
<p>\begin{align*}
d_1+d_2+d_3+d_4 &= 6, \\
d_5+d_6+d_7 &= 6, \\
d_i & \geq 1 & \text{for all } i \in \{1,\ldots,7\}, \text{ and} \\
d_i & \leq 4 & \text{for ... |
3,557,840 | <p>Find the quadratic polynomial <span class="math-container">$p(x)$</span> for given data points <span class="math-container">$$p(x_0)=y_0, p'(x_1)=y_1', p(x_2)=y_2 \text{ with } x_0 \neq x_2.$$</span></p>
<p><strong>My approach</strong></p>
<p>I tried the problem taking <span class="math-container">$p(x)=a+bx+c x^2... | user5713492 | 316,404 | <p>The most general quadratic that goes through <span class="math-container">$(x_0,y_0)$</span> and <span class="math-container">$(x_2,y_2)$</span> is
<span class="math-container">$$p(x)=y_0+\frac{(y_2-y_0)}{(x_2-x_0)}(x-x_0)+C(x-x_0)(x_2-x)$$</span>
Then we require
<span class="math-container">$$p^{\prime}(x_1)=y_1^{\... |
218,915 | <blockquote>
<p>Prove that for any integer $n$, $\gcd (3n^2+5n+7, n^2+1)=1$ or $41$.</p>
</blockquote>
<p>The following answer is convoluted because I've intentionally created excess solutions. However, I can't figure out how to eliminate them! Anyone?</p>
<p>Let $$d=\gcd (3n^2+5n+7, n^2+1).$$
Then $$d|[(3n^2+5n+7)... | Community | -1 | <p>From your last step, we get that $d = 1,3,41,123$.</p>
<p>Recall that $$n^2 \equiv 0,1 \pmod{3} \text{ (Why?)}$$ Hence, $3$ (or) $123$ does not divide $n^2+1$.</p>
<p><strong>EDIT</strong></p>
<p>Note that any $n$ is either $0 \pmod{3}$ or $\pm1 \pmod{3}$.</p>
<p>Hence, $n^2 \equiv 0,1 \pmod{3}$. (Recall th... |
218,915 | <blockquote>
<p>Prove that for any integer $n$, $\gcd (3n^2+5n+7, n^2+1)=1$ or $41$.</p>
</blockquote>
<p>The following answer is convoluted because I've intentionally created excess solutions. However, I can't figure out how to eliminate them! Anyone?</p>
<p>Let $$d=\gcd (3n^2+5n+7, n^2+1).$$
Then $$d|[(3n^2+5n+7)... | P.. | 39,722 | <p>Or, you can write
$$(-5n+4)(3n^2+5n+7)+(15n+13)(n^2+1)=41 \ .$$</p>
|
218,915 | <blockquote>
<p>Prove that for any integer $n$, $\gcd (3n^2+5n+7, n^2+1)=1$ or $41$.</p>
</blockquote>
<p>The following answer is convoluted because I've intentionally created excess solutions. However, I can't figure out how to eliminate them! Anyone?</p>
<p>Let $$d=\gcd (3n^2+5n+7, n^2+1).$$
Then $$d|[(3n^2+5n+7)... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\, $ Let $\rm\:d = gcd$, so $\rm\:d\:|\ i^2\!+1,\, 7+5\,i+3\,i^2.\:$ Then, like taking norms of Gaussian integers, $$\rm\:mod\ d\!:\,\ i^2\equiv -1\ \Rightarrow\ 0\equiv 7+5\,i+3\,i^2\equiv 4+5\,i\ \Rightarrow\ 0\equiv (4+5\,i)(4-5\,i)\equiv 4^2\!+5^2\equiv 41$$</p>
|
2,042,428 | <p>If I'm correct, hidden induction is when we use something along the lines of "etc..." in a proof by induction. Are there any examples of when this would be appropriate (or when it's not appropriate but used anyway)?</p>
| Bill Dubuque | 242 | <p>It's ubiquitous in inductive proofs by <em>telescopy</em>, e.g. multiplicative telescopic cancellation </p>
<p>$\qquad\qquad\, \displaystyle (x-1)(x+1)(x^{\large 2}\!+1)(x^{\large 4}\!+1)\qquad\! \cdots\qquad (x^{\large 2^{\rm N}}\!+\,1)$</p>
<p>$\qquad\ \ \ = \ \displaystyle \frac{\color{#0a0}{x-1}}{\color{#90f... |
1,309,728 | <p>I know what a 3x10 looks like, but I cannot seem to find a distinguishable pattern to extend it to a 3x14.</p>
<p>The 3x10 pattern I'm using looks like the one at the top right of figure 6 of <a href="http://faculty.olin.edu/~sadams/DM/ktpaper.pdf" rel="nofollow">this paper</a>.</p>
<p>Any help would be greatly ap... | rschwieb | 29,335 | <p>Just keep going with using $X-a$ and $Y-b$ to translate stuff.</p>
<p>$(X-a)(Y-b)+b(X-a)+a(Y-b)=XY-aY-bX+1+bX-1+aY-1=XY-1$</p>
|
638,875 | <p>Let $P$ be a $p$-group and let $A$ be maximal among abelian normal subgroups of $P$. Show that $A=C_P(A)$.</p>
<p>This is the second part of a problem in which I successfully proved the following:
Let $P$ be a finite $p$-group and let $U<V$ be normal subgroups of $P$. Show that there exists $W \triangleleft P$... | zcn | 115,654 | <p>Since $A$ is abelian, $A \subseteq C_P(A)$. If $A \neq C_P(A)$, then $C_P(A)/A$ would be a nontrivial normal subgroup of the $p$-group $P/A$, hence would intersect the center of $P/A$ nontrivially. Picking an nonidentity element $\overline{a} \in Z(P/A) \cap (C_P(A)/A)$, and lifting back to $P$, gives an element $a ... |
10,615 | <p>The tag <a href="https://math.stackexchange.com/questions/tagged/summation" class="post-tag" title="show questions tagged 'summation'" rel="tag">summation</a> was created less than a year ago, see <a href="https://math.meta.stackexchange.com/questions/6324/summation-tag-for-finite-and-formal-summations">&quo... | doraemonpaul | 30,938 | <p>In fact the tag wikis of <a href="https://math.stackexchange.com/questions/tagged/summation" class="post-tag" title="show questions tagged 'summation'" rel="tag">summation</a> and <a href="https://math.stackexchange.com/questions/tagged/sequences-and-series" class="post-tag" title="show questions tagged '... |
1,917,313 | <p>I am to find a combinatorial argument for the following identity:</p>
<p>$$\sum_k \binom {2r} {2k-1}\binom{k-1}{s-1} = 2^{2r-2s+1}\binom{2r-s}{s-1}$$</p>
<p>For the right hand side, I was think that would just be number of ways to choose at least $s-1$ elements out of a $[2r-s]$ set. However, for the left hand sid... | Marko Riedel | 44,883 | <p>Suppose we seek to verify that
$$\sum_{k=1}^r {2r\choose 2k-1} {k-1\choose s-1}
= 2^{2r-2s+1} {2r-s\choose s-1}$$</p>
<p>where presumably $s\ge 1$. The lower limit is set to $k=1$ as the
first binomial coefficient is zero when $k=0.$</p>
<p>Introduce
$${2r\choose 2k-1}
= \frac{1}{2\pi i}
\int_{|z|=\epsilon}
... |
78,478 | <blockquote>
<p>Prove that $\frac{1}{n} \sum_{k=2}^n \frac{1}{\log k}$ converges to $0.$</p>
</blockquote>
<p>Okay, seriously, it's like this question is mocking me. I know it converges to $0$. I can feel it in my blood. I even proved it was Cauchy, but then realized that didn't tell me what the limit <em>was</em... | N. S. | 9,176 | <p>Stolz Cezaro:</p>
<p>$$\lim \frac{1}{n} \sum_{k=2}^n \frac{1}{log k} = \lim \frac{1}{ \log (n+1)}$$</p>
<p><strong>Edit</strong>
Here is a direct proof:</p>
<p>$$0 \leq \frac{\sum_{k=2}^n \frac{1}{log k}}{n} = \frac{\sum_{k=2}^{\sqrt{n}} \frac{1}{log k}}{n}
+ \lim \frac{\sum_{k=\sqrt{n}}^n \frac{1}{log k}}{n} ... |
352,849 | <p>I have to show that $\lim \limits_{n\rightarrow\infty}\frac{n!}{(2n)!}=0$ </p>
<hr>
<p>I am not sure if correct but i did it like this :
$(2n)!=(2n)\cdot(2n-1)\cdot(2n-2)\cdot ...\cdot(2n-(n-1))\cdot (n!)$ so I have $$\displaystyle \frac{1}{(2n)\cdot(2n-1)\cdot(2n-2)\cdot ...\cdot(2n-(n-1))}$$ and $$\lim \limits_{... | Zev Chonoles | 264 | <p>It's correct, but I imagine you're expected to show a bit more work to <em>justify</em> your assertion that
$$\lim \limits_{n\rightarrow \infty}\frac{1}{(2n)\cdot(2n-1)\cdot(2n-2)\cdot ...\cdot(2n-(n-1))}=0$$
An easy way to do this is to bound this sequence of fractions with another, simpler one whose limit you know... |
2,596,213 | <p>I'm having huge troubles with problems like this. I know the following:</p>
<p>$$\frac{\sin{x}}{x}=1-\frac{x^2}{3!}+\frac{x^4}{5!}-\frac{x^6}{7!}+O(x^7)$$</p>
<p>and </p>
<p>$$\ln{(1+t)}=t-\frac{t^2}{2}+\frac{t^3}{3}+O(t^4)$$</p>
<p>So</p>
<p>$$\ln{\left(1+\left(-\frac{x^2}{3!}+\frac{x^4}{5!}-\frac{x^6}{7!}+O(x... | Jack D'Aurizio | 44,121 | <p>$$\frac{\sin x}{x}=\prod_{n\geq 1}\left(1-\frac{x^2}{n^2\pi^2}\right) \tag{1}$$
implies
$$ \log\frac{\sin x}{x} = -\sum_{n\geq 1}\sum_{m\geq 1}\frac{x^{2m}}{m n^{2m}\pi^{2m}}=-\sum_{m\geq 1}\frac{\zeta(2m)\,x^{2m}}{m\pi^{2m}} \tag{2}$$
and by recalling $\zeta(2)=\frac{\pi^2}{6},\zeta(4)=\frac{\pi^4}{90},\zeta(6)=\fr... |
109,298 | <p>I'm an applied model theorist, and open image theorems are important in the mathematical structures I study (they limit the number of types of elements being realised, and therefore keep things model theoretically nice e.g. stable). </p>
<p>So I have some idea as to why these open image theorems should hold from a ... | Barinder Banwait | 5,744 | <p>Serre's open image theorem (on page IV-20 in his book "Abelian $l$-adic representations...) for non-CM elliptic curves $E/K$ is equivalent to the statement that, for almost all $l$ (how large depending on $E$ and $K$), the $l$-adic representation attached to $T_l(E)$ is surjective. Ditto for the mod-$l$ representati... |
381,059 | <p><span class="math-container">$\newcommand\R{\mathbb R}$</span>Let <span class="math-container">$f\colon\R^p\to\R$</span> be a continuous function. For <span class="math-container">$u=(u_1,\dots,u_p)$</span> and <span class="math-container">$v=(v_1,\dots,v_p)$</span> in <span class="math-container">$\R^p$</span>, let... | Zach Teitler | 88,133 | <p>The <span class="math-container">$p=2$</span> dimensional case is an exercise in Rogawski's calculus textbook. It is exercise 47 on page 885, section 15.1 (Integration in Several Variables) in the 2008 Early Transcendentals edition.</p>
|
381,059 | <p><span class="math-container">$\newcommand\R{\mathbb R}$</span>Let <span class="math-container">$f\colon\R^p\to\R$</span> be a continuous function. For <span class="math-container">$u=(u_1,\dots,u_p)$</span> and <span class="math-container">$v=(v_1,\dots,v_p)$</span> in <span class="math-container">$\R^p$</span>, let... | Abdelmalek Abdesselam | 7,410 | <p>For an elementary fact like this, which may have been reinvented a thousand times, it is hard to find the first paper where this appeared. However, let me give some missing context. There is a whole industry in <strong>constructive quantum field theory</strong> and <strong>statistical mechanics</strong> about relate... |
1,285,941 | <p>I have a question where i couldn't find any clue.
The question is</p>
<p>$$\frac{1}{1\cdot 2}+\frac{1\cdot3}{1\cdot2\cdot3\cdot4}+\frac{1\cdot3\cdot5}{1\cdot2\cdot3\cdot4\cdot5\cdot6}+\cdots$$</p>
<p>I could get the general term as $t_n=\frac{1\cdot3\cdot5\cdot7\cdots(2n-1)}{1\cdot2\cdot3\cdot4\cdot5\cdot6\cdots2n... | wythagoras | 236,048 | <p>$$t_n=\frac{1\cdot3\cdot5\cdot7\dots(2n-1)}{1\cdot2\cdot3\cdot4\cdot5\cdot6\dots2n}=\frac{(2n-1)!!}{(2n-1)!!(2n)!!}=\frac{1}{(2n)!!}=\frac{1}{2^n n!}$$</p>
<p>Therefore this sum does not go to infinity. </p>
<p>Invoking $e^x = \sum^{\infty}_{n=0} \frac{x^n}{n!}$, we can compute the sum: $$\sum^{\infty}_{n=1} \frac... |
1,521,720 | <p>I'm working through a book on logic and wanted to check one of my steps in a derivation I'm working out. Given the two quantifier negation equalities:</p>
<ol>
<li><p>$\lnot\exists P(x) = \forall\lnot P(x)$</p></li>
<li><p>$\exists\lnot P(x) = \lnot\forall P(x)$</p></li>
</ol>
<p>I'm trying to derive (2) from (1).... | Graham Kemp | 135,106 | <p>The full statement is Second Order Logic</p>
<ol>
<li>$\forall P\; \big(\neg \exists x \; P(x) \;\leftrightarrow\; \forall x\;\neg P(x)\big)$</li>
</ol>
<p>Then we can use Universal Instantiation</p>
<p>1.1. $\neg \exists x\;\neg Q(x)\;\leftrightarrow\; \forall x\; \neg\neg Q(x)$</p>
<p>Double Negation Eliminati... |
4,530,792 | <p>I have the following sequence <span class="math-container">$\left \{k \sin \left(\frac{1}{k}\right) \right\}^{\infty}_{1}$</span>. I don't know how to show that this is monotonically increasing.</p>
<p>I tried taking the derivative of the corresponding function <span class="math-container">$f(x) = x \sin \left(\frac... | GReyes | 633,848 | <p>It is easier to look at the function <span class="math-container">$g(x)=\frac{\sin x}{x}$</span> and show that it is decreasing on <span class="math-container">$(0,1)$</span> (notice that <span class="math-container">$f(x)=g(1/x)$</span>). You have
<span class="math-container">$$
g'(x)=\frac{x\cos x-\sin x}{x^2}
$$<... |
2,012,947 | <p>I'm trying to prove that if f,g are continuous functions, and if E is a dense subset of X $(\text{or } Cl(E) = X)$ and if $f(x)=g(x) \forall x \in E$ then $f(x)=g(x) \forall x \in X$. </p>
<p>I understand that if f,g are continuous, then:</p>
<blockquote>
<p>$\exists \delta_1, \delta_2$ such that $\forall X \in ... | Firepi | 379,085 | <p>Rememer that $x\in Cl(E)$ if and only if there is a sequence of elements in $E$ which converges to $x$.</p>
<p>If $\{x_n\}$ is a sequence in $X$ such that $\lim_{n \to \infty}x_n=x$ for some $x\in X$ then $\lim_{n\to \infty}f(x_n)=f(x)$ and $\lim_{n\to \infty}g(x_n)=g(x)$ since $f$ and $g$ are continuous in $X$. Fo... |
3,391,225 | <p>What is the term for a (connected?) set <span class="math-container">$S$</span> of the plane <span class="math-container">$\mathbb{R}^2$</span> such that the intersection of <span class="math-container">$S$</span> with every horizontal line <span class="math-container">$\ell_{b}: y=b$</span> is either empty, or an i... | RobPratt | 683,666 | <p>Such a set is called <em>horizontally convex</em>.</p>
|
4,394,247 | <p>I know how to represent the sentence “there is exactly one person that is happy”,</p>
<p>∀y∀x((Happy(x)∧Happy(y))→(x=y))</p>
<p>Edit: ∃x∀y(y=x↔Happy(y))
(NOW, I actually know how to represent it)</p>
<p>Where x and y represent a person.</p>
<p>However, my problem is that I can’t figure out how to say “there are exac... | ryang | 21,813 | <blockquote>
<p>I know how to represent the sentence “there is exactly one person that is happy”: <span class="math-container">$$∀y∀x((\text{Happy}(x)∧\text{Happy}(y))→(x=y))$$</span></p>
</blockquote>
<p>Correction: <span class="math-container">$$∃x \,∀p\;\Big( p=x \leftrightarrow\text{Happy}(p) \Big).$$</span></p>
<p... |
3,794,101 | <blockquote>
<p>Show that <span class="math-container">$f: \mathbb{R^3} \to \mathbb{R}$</span> <span class="math-container">$$f(x, y, z) = xy + z^2$$</span> is continuous.</p>
</blockquote>
<p>One could just deduce that since it's a polynomial it's continuous, but how would I show this using <span class="math-container... | peek-a-boo | 568,204 | <p>Hint: Denote <span class="math-container">$\xi = (x,y,z)$</span> and <span class="math-container">$\alpha = (a,b,c)$</span>. Then,
<span class="math-container">\begin{align}
|f(\xi) - f(\alpha)| &= |(xy+z^2) - (ab+c^2)|\\
&= |(x-a)y + a(y-b) + (z-c)(z+c)|\\
& \leq |y| |x-a| + |a||y-b| + |z+c| |z-c| \\
&a... |
3,464,342 | <blockquote>
<p>For <span class="math-container">$j\in \mathbb{N}$</span> let <span class="math-container">$$M_j=\{f\in L^2([0,1]):\int_0^1 |f|^2 dx \leq j\}$$</span>
(a) Establish that <span class="math-container">$L^2([0,1])=\cup_{j\in \mathbb{N}}M_j$</span>.</p>
<p>(b) Show that each <span class="math-container">$M_... | Ian | 83,396 | <p>In b, say you have a sequence in <span class="math-container">$M_j$</span> converging in <span class="math-container">$L^1$</span>, pass to a subsequence to get an a.e. convergent subsequence, then Fatou's Lemma does what you want.</p>
<p>In c, find a family <span class="math-container">$f_{\epsilon,A}$</span> with... |
833,827 | <p>I am trying to refresh on algorithm analysis. I am looking for a refresher on summation formulas.<br>
E.g.<br>
I can derive the $$\sum_{i = 0}^{N-1}i$$ to be N(N-1)/2 but I am rusty on the and more complex e.g. something like $$\sum_{i = 0}^{N-1}{\sum_{j = i+1}^{N-1}\sum_{k=j+1}^{N-1}}$$<br>
Is there a good refreshe... | mlk | 155,406 | <p>Well, the basic time proven technique for simpler problems like this is guessing + proof by induction, something you can learn best by excercise. There certainly is way more advanced stuff even going into analytic number theory, but at least initially I would suggest you should get some problem book on induction or ... |
833,827 | <p>I am trying to refresh on algorithm analysis. I am looking for a refresher on summation formulas.<br>
E.g.<br>
I can derive the $$\sum_{i = 0}^{N-1}i$$ to be N(N-1)/2 but I am rusty on the and more complex e.g. something like $$\sum_{i = 0}^{N-1}{\sum_{j = i+1}^{N-1}\sum_{k=j+1}^{N-1}}$$<br>
Is there a good refreshe... | acegs | 52,463 | <p>Based on your code, we can say that the possible codes inside the innermost loop executes in a constant time $A$. Then it's overall execution time can be approximated by:
$$T_{N} = \sum_{i = 0}^{N-1}{\sum_{j = i+1}^{N-1}\sum_{k=j+1}^{N-1}}A$$</p>
<p>Using the summation properties given on Perry Iverson's answer, we... |
1,376,651 | <p>To be specific here is the system:</p>
<p>$$x-2y=0 \tag{1}$$
$$x-2(k+2)y=0 \tag{2}$$
$$x-(k+3)y=-k \tag{3}$$ </p>
<p>I have already solved it for equations $(1)$ and $(2)$... what should I do with the 3rd equation?</p>
<p>Just to make sure everything goes well here is my method:</p>
<p>$D=-2(k+2)$ and $D_x=D_y=0... | mvw | 86,776 | <p>The volume of the big box is $V_B = 7\cdot 9 \cdot 11 = 693$, the total volume of the small boxes is $V_b = 77 \cdot 3 \cdot 3 \cdot 1 = 693$.</p>
<p>This means the volume of the small boxes is sufficient and we need to use all small boxes.</p>
<p>Let us try to model this problem Tetris style: </p>
<ul>
<li>We ha... |
613,940 | <p>Given two parameters $a$ and $b$ (both positive integers), please estimate
the order of growth of the following function:</p>
<p>$$F(t)=\left\{\begin{array}{ll} 1, \, &t\le a \\ F(t-1) + b\cdot F(t-a),&t>a\end{array}\right.$$ </p>
<p>My guess is $\Theta\left(b^{t/a}\right)$. Any answer that might hel... | Slade | 33,433 | <p>We have $F(n) = \Theta (\kappa^n)$, where $\kappa$ is the unique positive root of $p(x) = x^a - x^{a-1} -b$.</p>
<p>This <em>can</em> be close to the crude guess of $b^{n/a}=(\sqrt[a]{b})^n$, but it can also be quite far. For example, if $a=1$, then this gives the correct asymptotics of $\Theta((b+1)^n)$, while th... |
19,521 | <p>I am trying to integrate a hat function for a project that I am doing and have found a method to do so but I find it sloppy. Currently I have the basis function</p>
<pre><code>\[Psi][z_] := z - Subscript[Z, i]/ \[CapitalDelta]z + 1;
</code></pre>
<p>which I am trying to integrate from $z_{i-1}$ to $z_{i+1}$. I bre... | Niki Estner | 242 | <p>Your integral would have worked too, if you had added assumptions that all variables are real:</p>
<pre><code>(* New assumptions so Integrate can do its work *)
Assuming[{z \[Element] Reals,
c \[Element] Reals, \[CapitalDelta]z \[Element] Reals,
Subscript[Z, i - 1] \[Element] Reals,
Subscript[Z, i + 1] \[E... |
1,585,408 | <p>I have the equation: (1-x<sup>2</sup>)u<sup>''</sup> -xu<sup>'</sup>+ku=0,
where ' represents differentiation with respect to x and k is a constant.</p>
<p>I am asked to show that cos(k<sup>1/2</sup>cos<sup>-1</sup>x) is a solution to this equation.</p>
<p>I assumed to show this you need to set u=cos(k<sup>1/2</su... | Jan Eerland | 226,665 | <p>HINT:</p>
<p>$$(1-x^2)u''(x)-xu'(x)+ku(x)=0\Longleftrightarrow$$</p>
<hr>
<p>Let $t=i\sqrt{k}\ln(\sqrt{x^2-1}+x)$, which gives $x=\frac{1}{2}e^{-\frac{it}{\sqrt{k}}}\left(1+e^{\frac{2it}{\sqrt{k}}}\right)$:</p>
<hr>
<p>$$\left(-\frac{1}{4}e^{-\frac{2it}{\sqrt{k}}}\left(1+e^{\frac{2it}{\sqrt{k}}}\right)^2+1\righ... |
432,964 | <p>Let $X\in \mathbb{R}^{n \times n}$.
Then, is the function</p>
<p>$$ \text{Tr}\left( (X^T X )^{-1} \right)$$ </p>
<p>convex in $X$? ($\text{Tr}$ denotes the trace operator)</p>
| MathIsArt | 450,658 | <p>The previous answer is unfortunately not complete due to a nontrivial sign mistake. In fact, even the local convexity result does not hold in this case. To see the obstruction, let <span class="math-container">$X$</span> be an invertible matrix, <span class="math-container">$V$</span> arbitrary, and compute
<span ... |
426,499 | <p>Let <span class="math-container">$X$</span> be a separable metric space which is <em>homogeneous</em>, i.e. for every two points <span class="math-container">$x,y\in X$</span> there is a homeomorphism <span class="math-container">$h$</span> of <span class="math-container">$X$</span> onto itself such that <span class... | Nate Eldredge | 4,832 | <p>The countable discrete space <span class="math-container">$\omega$</span> is a counterexample.</p>
<p>Suppose <span class="math-container">$Y$</span> is a homogeneous compactification of <span class="math-container">$\omega$</span>, with <span class="math-container">$X \subset Y$</span> being homeomorphic to <span c... |
636,467 | <p>What is it that makes something a paradox? It seems to me that paradoxes are just, in many cases, misunderstandings about the properties some object can have and so misunderstandings about definitions. Is there something I might be missing? How is this kind of thought handled in logic?</p>
| Carl Mummert | 630 | <p>"Paradox" is not a formally defined term. Many modern authors use "paradox" for all sorts of surprising or unexpected results; you can see examples by searching for "paradox" on Google News.</p>
<p>A more substantial use of the word "paradox" refers to a result that shows that a particular naive intuition is not so... |
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | Eric O. Korman | 9 | <p>The wikipedia article on the Collatz conjecture gives these three examples of conjectures that were disproved with large numbers:</p>
<p><a href="http://en.wikipedia.org/wiki/P%C3%B3lya_conjecture">Polya conjecture</a>.</p>
<p><a href="http://en.wikipedia.org/wiki/Mertens_conjecture">Mertens conjecture</a>.</p>
<... |
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | Larry Wang | 73 | <p>A famous example that is not quite as large as these others is the <a href="http://mathworld.wolfram.com/ChebyshevBias.html">prime race</a>. </p>
<p>The conjecture states, roughly: Consider the first n primes, not counting 2 or 3. Divide them into two groups: A contains all of those primes congruent to 1 modulo 3 ... |
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | Charles Stewart | 100 | <p>For an old example, Mersenne made the following <a href="http://en.wikipedia.org/wiki/Mersenne_conjectures" rel="noreferrer">conjecture</a> in 1644:</p>
<p><i>The Mersenne numbers, <span class="math-container">$M_n=2^n − 1$</span>, are prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and no others.</i></... |
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | Jon Bannon | 941 | <p>I don't know if I would consider this accessible or 'large', but the counterexample of Adyan to the famous <a href="http://en.wikipedia.org/wiki/Burnside%27s_problem">General Burnside Problem</a> in group theory requires an odd exponent greater than or equal to 665. The "shorter" counterexample (proof) due to Olshan... |
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | P.. | 39,722 | <p>It is well known that <a href="http://en.wikipedia.org/wiki/Goldbach's_conjecture" rel="noreferrer">Goldbach's conjecture</a> is one of the oldest unsolved problems in mathematics. A counterexample if it exists it will be a number greater than $4\cdot10^{18}$. </p>
<p>What is not well-known is that Goldbach mad... |
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | Joe K | 64,292 | <p>My favorite example, which I'm surprised hasn't been posted yet, is the conjecture:</p>
<blockquote>
<p>$n^{17}+9 \text{ and } (n+1)^{17}+9 \text{ are relatively prime}$</p>
</blockquote>
<p>The first counterexample is $n=8424432925592889329288197322308900672459420460792433$</p>
|
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | Yuriy S | 269,624 | <p>In this paper <a href="http://arxiv.org/abs/math/0602498" rel="nofollow noreferrer">http://arxiv.org/abs/math/0602498</a> a sequence of integers is proposed, which, when started with <span class="math-container">$1$</span> begins like this:</p>
<p><span class="math-container">$$1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1,... |
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | Trevor | 493,232 | <p>I had a <a href="https://math.stackexchange.com/questions/3305125/is-it-true-that-for-any-two-integers-with-the-same-least-prime-factor-there-mus/3459621#3459621">conjecture</a> that for any two natural numbers with the same least prime factor, there must be at least one number in between them with a higher least pr... |
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | Enzo Creti | 820,969 | <p>I have a conjecture that can be maybe disproved by an extremely huge counterexample:</p>
<p>Consider sequence <a href="https://oeis.org/A301806" rel="nofollow noreferrer">https://oeis.org/A301806</a></p>
<p>I conjecture that if n divides a(n), then a(n) +1 is prime.</p>
|
256,649 | <p>I am trying to plot phase diagram and poincare map. But I cannot get the poincare map as shown in the image below</p>
<p>Phase Space</p>
<pre><code>sol = NDSolve[{v'[t] ==
0.320 x[t] - 1.65 x[t]^3 - 0.005*v[t] + 0.855 Cos[1.2*t],
x'[t] == v[t], x[0] == 0, v[0] == 0}, {x, v}, {t, 0, 1500}];
ParametricPlot[... | Chris K | 6,358 | <p>If you run your first code for the same amount of time as the Poincare section, you'll see that they do agree (and that they're not particularly interesting parameter values):</p>
<pre><code>T = 2*Pi/1.2;
sol = NDSolve[{v'[t] == 0.320 x[t] - 1.65 x[t]^3 - 0.005*v[t] + 0.855 Cos[1.2*t],
x'[t] == v[t], x[0] == 0, ... |
4,002 | <p>I'm trying to obtain a series of points on the unit sphere with a somewhat homogeneous distribution, by minimizing a function depending on distances (I took $\exp(-d)$). My points are represented by spherical angles $\theta$ and $\phi$, starting by choosing equidistributed random vectors:</p>
<pre><code>pts = Apply... | Daniel Lichtblau | 51 | <p>I played around with a few plausible variants on the energy. In all cases I compared the result using the original energy function. Some things I learned:</p>
<p>(1) Some variants will tend to give results that do quite well when gauged via the original energy.</p>
<p>(2) Others (not shown below) will do poorly be... |
3,805,989 | <p>I'm doing Exercise 4 in textbook Algebra by Saunders MacLane and Garrett Birkhoff.</p>
<p><a href="https://i.stack.imgur.com/JQww8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/JQww8.png" alt="enter image description here" /></a></p>
<blockquote>
<p>Show that, if <span class="math-container">$F$... | Community | -1 | <p>As stated in a comment by OP, it remains to show that for all automorphisms <span class="math-container">$\sigma: F[x] \to F[x]$</span> that leave every element of <span class="math-container">$F$</span> fixed, we have that <span class="math-container">$\sigma(x) = ax + b$</span> for some <span class="math-container... |
103,776 | <p>I am curious as to why Wolfram|Alpha is graphing a logarithm the way that it is. I was always taught that a graph of a basic logarithm function $\log{x}$ should look like this:</p>
<p><img src="https://i.stack.imgur.com/3SRqI.png" alt="enter image description here"></p>
<p>However, Wolfram|Alpha is graphing it lik... | N. S. | 9,176 | <p>$\ln(x)$ is formally defined as the solution to the equation $e^y=x$.</p>
<p>If $x$ is positive, this equation has an unique real solution, anyhow if $x$ is negative this doesn't have a <strong>real</strong> solution. But it has <strong>complex</strong> roots.</p>
<p>Indeed, $\ln(x)= a+ib$ is equivalent to </p>
<... |
3,753,474 | <p><strong>Question:</strong></p>
<blockquote>
<p>If <span class="math-container">$\alpha,\beta,\gamma$</span> are the roots of the equation, <span class="math-container">$x^3+x+1=0$</span>, then find the equation whose roots are: <span class="math-container">$({\alpha}-{\beta})^2,({\beta}-{\gamma})^2,({\gamma}-{\alpha... | Jean Marie | 305,862 | <p>Final constant term <span class="math-container">$1+3+27=31$</span> can be obtained at once (or checked at once) by considering that it is the opposite of the product of roots</p>
<p><span class="math-container">$$(({\alpha}-{\beta})({\beta}-{\gamma})({\gamma}-{\alpha}))^2$$</span></p>
<p>which is the classical <str... |
3,753,474 | <p><strong>Question:</strong></p>
<blockquote>
<p>If <span class="math-container">$\alpha,\beta,\gamma$</span> are the roots of the equation, <span class="math-container">$x^3+x+1=0$</span>, then find the equation whose roots are: <span class="math-container">$({\alpha}-{\beta})^2,({\beta}-{\gamma})^2,({\gamma}-{\alpha... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>Let <span class="math-container">$y=(a-b)^2=(a+b)^2-4ab=(-c)^2-\dfrac4{-c}$</span> as <span class="math-container">$abc=-1, a+b=-c$</span></p>
<p><span class="math-container">$$\iff c^3-cy+4=0\ \ \ \ (1) $$</span></p>
<p>Again we have <span class="math-container">$$c^3+c+1=0\ \ \ \ (0)$$</span></p>
<p>... |
3,349,206 | <p>This comes theorem 17.1 of commutative ring theory by Matsumura:</p>
<p><a href="https://i.stack.imgur.com/TtGXD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/TtGXD.png" alt="enter image description here"></a></p>
<blockquote>
<p>It is easy to see that if <span class="math-container">$\text{... | Angina Seng | 436,618 | <p>From the short exact sequence
<span class="math-container">$$0\to N_1/N_2\to N_0/N_2\to N_0/N_1\to0$$</span>
we get a long exact sequence
<span class="math-container">$$
\cdots\to\text{Ext}_A^i(N_0/N_1,M)
\to\text{Ext}_A^i(N_0/N_2,M)
\to\text{Ext}_A^i(N_1/N_2,M)\to\cdots$$</span>
and as the outer groups here are zer... |
3,349,206 | <p>This comes theorem 17.1 of commutative ring theory by Matsumura:</p>
<p><a href="https://i.stack.imgur.com/TtGXD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/TtGXD.png" alt="enter image description here"></a></p>
<blockquote>
<p>It is easy to see that if <span class="math-container">$\text{... | E.R | 325,912 | <p>Let we have a sequence <span class="math-container">$0=N_n\subset N_{n-1}\subset \cdots \subset N_1\subset N_0 =N $</span>. Thus we have the following exact sequences: <span class="math-container">$$0\to N_1\to N_0\to N_0/N_1\to 0$$</span>
and <span class="math-container">$$0\to N_2\to N_1\to N_1/N_2\to 0$$</span... |
755,227 | <p>Would it be possible for a ring to have elements that are their own additive inverses? What I mean is, would it be possible to have a ring $K$ of mathematical objects $A$ such that:
$$A+A=i,\;\forall A\in K$$</p>
<p>Where $i$ is the additive identity?</p>
| Omran Kouba | 140,450 | <p>Yes, consider $K=\Bbb{Z}/2\Bbb{Z}=\{0,1\}$, with addition and multiplication Modulo 2.</p>
|
755,227 | <p>Would it be possible for a ring to have elements that are their own additive inverses? What I mean is, would it be possible to have a ring $K$ of mathematical objects $A$ such that:
$$A+A=i,\;\forall A\in K$$</p>
<p>Where $i$ is the additive identity?</p>
| Robert Lewis | 67,071 | <p>Yes, it is possible; consider any ring of characteristic $2$; since $1 + 1 = 0$ in such a ring by definition, we have for all $a$ in the ring $a + a = a(1 + 1) = a0 = 0$; this implies $-a = a$; an example, as Dietrich Burde mentioned in his comment, is $\Bbb Z_2 = \Bbb Z/2\Bbb Z$. A more complex example is $\Bbb Z_... |
3,953,153 | <p>I need help to prove this:
If <span class="math-container">$\gcd(a,b)=1$</span> then <span class="math-container">$\gcd(a+b, a^2-ab+b^2)$</span> is equal to <span class="math-container">$1$</span> or <span class="math-container">$3$</span>.
I have done this:</p>
<p>Let <span class="math-container">$d$</span> be the ... | nonuser | 463,553 | <p>You started right. Let <span class="math-container">$d =\gcd(a+b, a^2-ab+b^2)$</span>, then <span class="math-container">$d$</span> divides <span class="math-container">$a+b$</span> and <span class="math-container">$a^2-ab+b^2$</span>. But from here it is pretty messy.</p>
<p>We can say <span class="math-container">... |
1,821,411 | <p>$f:[a,b]\rightarrow R$ that is integrable on [a,b]</p>
<p>So we need to prove:</p>
<p>$$\int_{-b}^{-a}f(-x)dx=\int_{a}^{b}f(x)dx$$</p>
<p>1.) So we'll use a property of definite integrals: (homogeny I think it's called?)</p>
<p>$$\int_{-b}^{-a}f(-x)dx=-1\int_{-b}^{-a}f(x)dx$$</p>
<p>2.) Great, now using the fun... | Dinesh.hmn | 308,545 | <p>Let f(x) = x. then f(-x) = -x. Substituting -a and -b in the limits of the integration will lead to be f(a) = a then f(-a) = -(-a) = a.</p>
<p>it's simply this that you are multiplying the limits and the function by -1. if both are multiplied then they would get neutralized,</p>
|
4,374,521 | <p>In the <span class="math-container">$(x,t)$</span>- plane, the characteristic of the initial value problem <span class="math-container">$$u_t+uu_x=0$$</span> with <span class="math-container">$$u(x,0)=x,0\leq x\leq 1$$</span> are</p>
<p><span class="math-container">$1$</span>. parallel straight lines .</p>
<p><span ... | EditPiAf | 418,542 | <p>The resolution of the initial value problem is discussed in <a href="https://math.stackexchange.com/q/305727/418542">this post</a>. So we end up with the set of curves
<span class="math-container">$$
u=C_1, \qquad x-ut=C_2
$$</span>
where <span class="math-container">$C_1$</span>, <span class="math-container">$C_2$<... |
3,888,146 | <p>When we give a proof that the tangent is the sine to cosine ratio of an oriented angle,</p>
<p><span class="math-container">$$\bbox[5px,border:2px solid #C0A000]{\tan \alpha=\frac{\sin\alpha}{\cos \alpha}}$$</span>
with <span class="math-container">$\cos \alpha \neq 0$</span>, we take the tangent <span class="math-c... | Tbw | 498,717 | <p>As Christian said, the center has equal distance to <span class="math-container">$A$</span> and <span class="math-container">$B$</span> and therefore lies on the perpendicular bisector of <span class="math-container">$AB$</span>. This center is therefore at the intersection of <span class="math-container">$AB$</span... |
3,755,288 | <p>I'm trying to solve this:</p>
<blockquote>
<p>Which of the following is the closest to the value of this integral?</p>
<p><span class="math-container">$$\int_{0}^{1}\sqrt {1 + \frac{1}{3x}} \ dx$$</span></p>
<p>(A) 1</p>
<p>(B) 1.2</p>
<p>(C) 1.6</p>
<p>(D) 2</p>
<p>(E) The integral doesn't converge.</p>
</blockquot... | IPPK | 719,117 | <p>Let's try to use integration by parts to <span class="math-container">$I = \int\limits_0^1 \sqrt{1 + \frac{1}{3x}}dx$</span>. First, transform integral into <span class="math-container">$\frac{2}{\sqrt3}\int\limits_0^1\frac{\sqrt{1 + 3x}}{2\sqrt{x}}$</span>. Now <span class="math-container">$u = \sqrt{3x+1}$</span> ... |
1,182,844 | <p>I have completed Velleman's book, 'How to prove it'. I have also worked through Apostol Vol.1.
I have messed about with many rigorous single variable calculus textbooks, e.g.,Apostol, Spivak, Courant, Lang, etc. I had started working through Lang's, 'Calculus of several variables' but put it up to do a book like Edw... | Moya | 192,336 | <p>I honestly don't think skipping vector calculus to be a particularly good idea. Yes, you can certainly do it and make do by jumping immediately to calculus on manifolds, but you're definitely going to have a severe gap in your understanding by having not taken the time to sit down and do computations that you might ... |
115,821 | <p>Here are some sample data:</p>
<pre><code>data = {{1}, {50., 53, 52, 52}, {100., 105, 104, 104}, {150., 157, 156,
156}, {200., 209, 208, 208}, {250., 261, 260, 260}, {300., 313, 312,
313}, {2}, {50., 53, 52, 51}, {100., 106, 105, 102}, {150., 158,
157, 153}, {200., 211, 210, 204}, {250., 265, 264, 256}, {300., 31... | demm | 30,122 | <p>Maybe if you consider the min and max of the individual tables within normTable.</p>
<pre><code>min=Map[Min,normData];max=Map[Max,normData];
Table[ListVectorPlot[data[[i]],PlotRange->All,PlotLegends ->BarLegend[{"Rainbow",{min[[i]],max[[i]]}}],VectorColorFunction ->Function[{x,y,vx,vy,n},ColorData["Rainbo... |
571,941 | <p>I know that $\sum _{ n=1 }^{ \infty }{ { (-1) }^{ n+1 }\frac { 1 }{ n } =\ln(2) }$ .</p>
<p>How about the series $\sum _{ n=1 }^{ \infty }{ { (-1) }^{ n+1 } } \frac { 1 }{ \sqrt { n } }$ </p>
<p>To what number does it converge?</p>
| Adam | 93,671 | <p>Lets consider partial sums. $A(x)$ is a sum of problems after $x$ days. Now what is the remainder of $A(x)$ divided by $229$? $A(x)$ changes as days go by, there are only $229$ possible remainders and $365$ days - so there must be two days $x$ and $y$, where $A(x)$ and $A(y)$ have the same remainder. $A(x)$ and $A(y... |
39,387 | <p>It is usual to use second, third and fourth moments of a distribution to describe certain properties. Do partial moments or moments higher than fourth describe any useful properites of a distribtution.</p>
| Community | -1 | <p>That depends on what you mean by "useful properties". For instance, all of the even moments are required to characterize sub-Gaussian random variables, which are those variables whose tail is majorized by that of a Gaussian random variable. A random variable X is sub-Gaussian if and only if there exists a non-negati... |
39,387 | <p>It is usual to use second, third and fourth moments of a distribution to describe certain properties. Do partial moments or moments higher than fourth describe any useful properites of a distribtution.</p>
| Yemon Choi | 763 | <p>Not sure if this is what you had in mind, but a bounded random variable (i.e. one which almost surely takes values in some bounded subset of $\mathbb R$) is uniquely determined by its moments.</p>
<p>Off the top of my head, one way to see this - probably not the most efficient! - is that the distribution of a rando... |
4,188,656 | <p><strong>Problem</strong>: How many strings are there of length <span class="math-container">$ n $</span> over <span class="math-container">$ \{ 1,2,3,4,5,6 \} $</span> s.t. the sum of all characters in the string divide by <span class="math-container">$ 3 $</span>.</p>
<p><strong>Attempt</strong>:
Initially I though... | Arctic Char | 629,362 | <p>It's not true even when <span class="math-container">$A$</span> is finite. Example: <span class="math-container">$f_1 = \chi_{[0,1]}$</span> and <span class="math-container">$f_2 = \chi_{[1,2]}$</span>. Then</p>
<p><span class="math-container">$$\sup \left\{ \int f_1, \int f_2\right\} = 1 < 2 = \int \sup \{ f_1, ... |
2,363,840 | <blockquote>
<p>As the heading states, find all $p(x)$ for $p(x+1)=p(x)+3x(x+1)+1$ for all real $x$.</p>
</blockquote>
<p>I have no idea how to approach this. Any solution or guide how to solve these kinds of questions would be appreciated! </p>
| Jack D'Aurizio | 44,121 | <p>For any $x\in\mathbb{N}$ we have
$$ p(x+1)-p(x) = 6 \binom{x+1}{2}+1 \tag{1} $$
hence by applying $\sum_{x=0}^{n}$ to both sides and exploiting the <a href="https://en.wikipedia.org/wiki/Hockey-stick_identity" rel="nofollow noreferrer">hockey stick identity</a>:
$$ p(n+1)-p(0) = 6\binom{n+2}{3}+(n+1) = (n+1)^3 \tag{... |
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