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 |
|---|---|---|---|---|
20,101 | <p>I've seen <a href="https://math.stackexchange.com/questions/1210976/how-to-factoring-a-high-degree-polynomial">this</a> question in the last minutes, and I noticed that the OP has a lot of issues understanding basic (really basic) mathematics. </p>
<p>It's not just that the question is really bad written, or that t... | flabby99 | 227,286 | <p>I would also say that part of the problem for new users (myself included) is that it can be hard to know exactly how much detail you should put in your question. Or perhaps I should say, it is hard to know a template for your question.</p>
<p>Your first question is very daunting, from the syntax of writing it, to t... |
20,101 | <p>I've seen <a href="https://math.stackexchange.com/questions/1210976/how-to-factoring-a-high-degree-polynomial">this</a> question in the last minutes, and I noticed that the OP has a lot of issues understanding basic (really basic) mathematics. </p>
<p>It's not just that the question is really bad written, or that t... | Piquito | 219,998 | <p>I agree with your feeling about the problem you mentioned, however I think it's also very useful for beginners to formulate their questions. It could be that this site does a separation of levels but this seems difficult to manage.</p>
|
65,059 | <p>Surely yes, and in more generality, but can it be proved?</p>
<p>It seems that most, if not all, statements about quadratic forms representing primes fall back on algebraic number theory (i.e. splitting of primes in $\mathbb{Q}(\sqrt{7})$) for their proofs, and so are incompatible with the condition that $0 < y ... | anonymous | 15,137 | <p>It is true, with the same proof as Iwaniec-Kowalski. Real or complex, the generators of principal primes are equidistributed modulo units. There just happen to be no units in the complex case. </p>
|
3,449,865 | <p>In an informal sense, what does a slice of the Mapping Cylinder look like? There seem to be two feasible choices:
i) Just <span class="math-container">$X$</span>, the slices are exactly the space <span class="math-container">$X \times \{t\}$</span> for some <span class="math-container">$t \in [0,1]$</span>.</p>
<p>... | Justin Barhite | 440,524 | <p>The issue I see with the second interpretation is that <span class="math-container">$f_0 = \mathrm{id}$</span> is a map <span class="math-container">$X \to X$</span>, whereas <span class="math-container">$f_1 = f$</span> is a map <span class="math-container">$X \to Y$</span>, so I'm not sure how to think of a homoto... |
2,708,684 | <p>The unit ball of $\ell^1$ has extreme points, but the unit ball of $L^1$ does not have extreme points. Also, $\ell^1$ can be isometrically embedded into $L^1$. Isn't this a contradiction, since isometric isomorphisms preserve extreme points?</p>
| Yuri Negometyanov | 297,350 | <p>\begin{cases}
\dfrac{\cos\alpha}{\cos\beta}+\dfrac{\sin\alpha}{\sin\beta}+1=0\\[4pt]
\sin2\alpha + \sin2\beta = 2\sin(\alpha+\beta)\cos(\alpha - \beta),
\end{cases}
\begin{cases}
2\sin(\alpha+\beta) = - \sin2\beta\\
2\sin(\alpha+\beta)(1+\cos(\alpha - \beta)) = \sin2\alpha.
\end{cases}
\begin{align}
\dfrac{\cos^3\be... |
3,321,367 | <p>According to Euler's Formula, <span class="math-container">$e^{ix} = \cos(x) + i\sin(x).$</span>
I'm computing the product <span class="math-container">$e^{ix} \cdot e^{iy}.$</span></p>
<p>What is the real part (that is, the term without a factor of <span class="math-container">$i$</span>)?</p>
<p>Why is it <span... | Berci | 41,488 | <p>We have
<span class="math-container">$$\frac{2^{x+2}-(4q+3^bn)}{3^{b+1}}=\frac{4\cdot 3^bn - 3^bn}{3^{b+1}}=\frac{3\cdot 3^b\cdot n}{3^{b+1}}=n$$</span></p>
|
3,321,367 | <p>According to Euler's Formula, <span class="math-container">$e^{ix} = \cos(x) + i\sin(x).$</span>
I'm computing the product <span class="math-container">$e^{ix} \cdot e^{iy}.$</span></p>
<p>What is the real part (that is, the term without a factor of <span class="math-container">$i$</span>)?</p>
<p>Why is it <span... | Connor Harris | 102,456 | <p>It's not especially deep. Given <span class="math-container">$$n = \frac{2^x - q}{3^b}$$</span> it follows that <span class="math-container">$$ \frac{4n}{3} = \frac{2^{x+2} - 4q}{3^{b+1}}$$</span> and <span class="math-container">$$\frac{n}{3} = \frac{3^b n}{3^{b+1}}.$$</span></p>
|
2,504,613 | <p>I have problem with solving the following equation:</p>
<blockquote>
<p>$$ty'=3y+t^5y^\frac{1}{3}$$</p>
</blockquote>
<p>I know it's easy without the $y^\frac{1}{3}$ term, but I'm confused now.</p>
<p>Any help would be appreciated.</p>
| Nosrati | 108,128 | <p>Let $y=u^3$ then $y'=3u'u^2$ and
$$3tu'u^2=3u^3+t^5u$$
$$tu'=u+\dfrac{t^5}{3u}$$
$$tu'-u=\dfrac{t^5}{3u}$$
$$\dfrac{tu'-u}{t^2}=\dfrac{t^3}{3u}$$
$$\left(\dfrac{u}{t}\right)'=\dfrac{t^2}{3}\dfrac{t}{u}$$
$$\left(\dfrac{u}{t}\right)\left(\dfrac{u}{t}\right)'=\dfrac{t^2}{3}$$
$$\int2\left(\dfrac{u}{t}\right)d\left(\d... |
995,551 | <p>How would one calculate the log of a number where the base isn't an integer (in particular, an irrational number)? For example:</p>
<p>$$0.5^x = 8 \textrm{ (where } x = -3\textrm{)}$$</p>
<p>$$\log_{0.5}8 = -3$$</p>
<p>How would you solve this, and how would this work for an irrational base (like $\sqrt{2}$)?</p>... | brick | 187,522 | <p>This is the base changing formula :
$$\log_{a}(x) = \frac{\log_{b}(x)}{\log_{b}(a)}$$</p>
|
995,551 | <p>How would one calculate the log of a number where the base isn't an integer (in particular, an irrational number)? For example:</p>
<p>$$0.5^x = 8 \textrm{ (where } x = -3\textrm{)}$$</p>
<p>$$\log_{0.5}8 = -3$$</p>
<p>How would you solve this, and how would this work for an irrational base (like $\sqrt{2}$)?</p>... | egreg | 62,967 | <p>Let's rewrite this in a different way:
$$
0.5^x=8
$$
Take the logarithm with respect to <em>any</em> base $a$ ($a>0$, $a\ne1$):
$$
\log_a(0.5^x)=\log_a8
$$
which becomes
$$
x\log_a 0.5=\log_a 8
$$
or
$$
x=\frac{\log_a 8}{\log_a 0.5}
$$
You would stop here weren't from the fact that $8=2^3$ and $0.5=2^{-1}$, so
$$... |
995,551 | <p>How would one calculate the log of a number where the base isn't an integer (in particular, an irrational number)? For example:</p>
<p>$$0.5^x = 8 \textrm{ (where } x = -3\textrm{)}$$</p>
<p>$$\log_{0.5}8 = -3$$</p>
<p>How would you solve this, and how would this work for an irrational base (like $\sqrt{2}$)?</p>... | hyst329 | 188,226 | <p>You can use natural logarithms, because for all $a$ and $b$ : $\log_a(b)=\frac{\ln(b)}{\ln(a)}$.</p>
|
1,219,514 | <p>If we assume that $\sum a_n$ converges conditionally then How can we comment that $\sum a_{2n} $ does not converges, While it does when $\sum a_n$ converges absolutely ?</p>
| Rolf Hoyer | 228,612 | <p>It may or may not converge. Two examples are $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots$ in which $\Sigma a_{2n}$ diverges, or $1 + 0 - \frac{1}{2} + 0 + \frac{1}{3} + 0 - \frac{1}{4}$, in which $\Sigma a_{2n}$ converges.</p>
<p>In the absolute convergence case, it's straightforward to see that $\Sigma a... |
437,775 | <p>I'm a graduate student studying now for the first time class field theory.<br />
It seems that how to teach class field theory is a problem over which many have already written on MathOverflow.<br />
For example here <a href="https://mathoverflow.net/questions/6932/learning-class-field-theory-local-or-global-first">... | Will Sawin | 18,060 | <p>Galois cohomology, which is a special case of group cohomology, is used pretty heavily in many areas of algebraic number theory: especially in Galois deformations and the Langlands program, but also in Iwasawa theory and other areas.</p>
<p>It's often necessary to do calculations with Galois cohomology in graduate-l... |
437,775 | <p>I'm a graduate student studying now for the first time class field theory.<br />
It seems that how to teach class field theory is a problem over which many have already written on MathOverflow.<br />
For example here <a href="https://mathoverflow.net/questions/6932/learning-class-field-theory-local-or-global-first">... | Timothy Chow | 3,106 | <p>This is not a full answer, but is a bit too long for a comment. I recommend the introduction to the second part of Lang's <em>Algebraic Number Theory</em>, where he discusses several different approaches to class field theory, and says that ‘no
one piece of insight which has been evolved since the beginning of the
s... |
3,652,730 | <blockquote>
<p>Without using L'Hôpital's rule, find:
<span class="math-container">$$\lim_{x\to 0}\dfrac{\cos(\frac{\pi}{2}\cos x)}{\sin(\sin x)}$$</span>
I know that the answer is <span class="math-container">$0$</span>.</p>
</blockquote>
<p>My attempt:</p>
<p>I tried by using the half-angle formula, <span cl... | Claude Leibovici | 82,404 | <p>If you want more than the limit itself, compose Taylor series
<span class="math-container">$$\cos(x)=1-\frac{x^2}{2}+\frac{x^4}{24}+O\left(x^6\right)$$</span>
<span class="math-container">$$\cos(\frac{\pi}{2}\cos (x))=\frac{\pi x^2}{4}-\frac{\pi x^4}{48}+O\left(x^6\right)$$</span>
<span class="math-container">$$\s... |
258,392 | <p>Given a fiber bundle $(E,B,p,F)$ with path connected base $B$ and fiber $F$, both closed smooth manifolds of finite dimensions. The second page $E_2^{p,q}$ of the Leray-Serre spectral sequence over $\mathbb{Z}_2$ is give by $H^p(B;\mathcal{H}^q(F;\mathbb{Z}_2))$, where $\mathcal{H}^q(F;\mathbb{Z}_2)$ is the local sy... | Will Sawin | 18,060 | <p>Let me first explain why your intuition is wrong, then let me provide a counterexample. You claim that "It is hard to imagine that $E_2^{*,*}$ has generators that cannot be expressed as a polynomial in generators of $H^*(B;\mathbb Z_2)$ and $H^∗(F;\mathbb Z_2)$". But it is not so hard to imagine, as the generators o... |
1,517,488 | <p>Let $M$ be an Invertible Hermitian matrix and let $x,y\in\Bbb R$ such that $x^2\lt 4y$,Then Prove That $M^2+xM+yI$ and $M^2-xM+yI$ are non-singular.</p>
<p>My Attempt:</p>
<p>$$(M^2+xM+yI)(M^2-xM+yI)=(M^2+yI)^2-(xM)^2$$</p>
<p>Now I Don't Know How to proceed further, I know that all the eigen values of Hermitian ... | A.Γ. | 253,273 | <p>Complete the square wrt $M$ in both matrices
$$
M^2\pm xM+yI=\underbrace{\Bigl(M\pm\frac{1}{2}xI\Bigr)^2}_{\text{pos.semidef.}}+\underbrace{\frac{4y-x^2}{4}}_{\text{pos.def.}}I>0.
$$
The matrices are positive definite, hence, invertible.</p>
|
271,078 | <p>Please guys i need help with this limit:</p>
<p>$$\lim_{n \to \infty} \left(\frac {1}{\sqrt{n^2+1}}+ \frac{1}{\sqrt{n^2+2}}+\dots +\frac{1}{\sqrt{n^2+2n}}\right)$$</p>
<p>I don't know what to do?</p>
| DonAntonio | 31,254 | <p>$$\frac{2n}{\sqrt{n^2+2n}}=\frac{2}{\sqrt{1+\frac{2}{n}}}\xrightarrow [n\to\infty]{}2$$</p>
<p>Can you do now something similar with the RHS in Marvis's answer and then use the squeeze theorem?</p>
|
1,557,531 | <p>Please help.</p>
<p>Question: Addmath (Quadratic Equations)</p>
<p>Given $\alpha$ and $\beta$ are the roots of the quadratic equation
$2x^2 - 6x + 5 = 0$,
form an quadratic equation with the roots $\alpha + 1$ and $\beta + 1$.</p>
| cheesyfluff | 287,288 | <p>To add $1$ to each root, simply shift the function right $1$ unit by replacing $x$ with $x-1$. Then you get
\begin{gather}
2(x-1)^2-6(x-1)+5=0\\
2(x^2-2x+1)-6(x-1)+5=0\\
2x^2-10x+13=0
\end{gather}</p>
|
889,690 | <p>Is there a formula (with mathematical reasoning) for separating a complex-valued function $f(z)=f(x+iy)$ into the form $
f(z)=u(x,y) + iv(x,y)$?</p>
<p>Thank You,</p>
<p>C.A</p>
| copper.hat | 27,978 | <p>$u = {1 \over 2} (f + \bar{f})$,
$v = {1 \over 2i} (f - \bar{f})$.</p>
<p>It is straightforward to add the above and see that $u+iv = f$.</p>
|
1,700,493 | <p>The following is an exercise from <em>Linear Analysis</em> by Bollobas.</p>
<p>Let $f:X\to X$, with $X$ a compact metric space. Suppose that for every $\epsilon>0$, there is a $\delta=\delta(\epsilon)$ such that if $d(x,f(x))<\delta$ then $f(B(x,\epsilon))\subset B(x,\epsilon)$. Let $x_0\in X$ and define $x_n... | Mankind | 207,432 | <p>You are asking if the function
$$f\colon B_{\epsilon}(0)\rightarrow\Bbb{R}^n$$
given by
$$f(x) = \tan\left(\frac{||x||\pi}{2\epsilon}\right)\frac{x}{||x||}$$
is well-defined. It is a matter of checking that everything you do is an allowed operation. Remember that for $\tan(y)$ to make sense, you need $y$ to be diffe... |
2,271,558 | <p>I want to prove the following statement</p>
<p>All subgroups of $Q_8 \times E_{2^n}$ are normal </p>
<p>Here $E_{p^n} = \mathbb{Z}_p \times \mathbb{Z}_p \times \cdots \times \mathbb{Z}_p$ (n times)</p>
<hr>
<p>From some comments below, i made up some informal justification. </p>
<p>My strategy are following. </... | Xichao W. Gaiser | 734,890 | <p>Let <span class="math-container">$G\leq Q_8\times E_{2^n}$</span>. Let <span class="math-container">$(x,y)\in G$</span>. Then <span class="math-container">$(x,y)^{-1}=(x^{-1},y)\in G$</span> because <span class="math-container">$|y|=2$</span>.</p>
<p>Now let <span class="math-container">$(a,b)\in Q_8\times E_{2^n}$<... |
45,398 | <p>I'm trying to learn about the Kushner-Stratonovich-Pardoux equations in filtering theory.</p>
<p>I'm familiar with Itô calculus at the level of Øksendal's book (but struggle with much of Karatzas and Shreve, for example).</p>
<p>My PDE theory is pretty weak. I know about the Fokker-Planck equations, and that's abo... | Jason Swanson | 11,867 | <p>You might have a look at Martin Hairer's "An Introduction to Stochastic PDEs", which is available on arXiv at <a href="http://arxiv.org/abs/0907.4178">http://arxiv.org/abs/0907.4178</a>. At the very least, having a look at the topics he discusses might help you to plan out a road map of things you want to learn more... |
45,398 | <p>I'm trying to learn about the Kushner-Stratonovich-Pardoux equations in filtering theory.</p>
<p>I'm familiar with Itô calculus at the level of Øksendal's book (but struggle with much of Karatzas and Shreve, for example).</p>
<p>My PDE theory is pretty weak. I know about the Fokker-Planck equations, and that's abo... | Shuhao Cao | 7,200 | <p>Rama Cont compiled a rather comprehensive resource page about SPDEs, but some of the links are broken, still this gives you an overview of what books/papers you wanna read in order to know certain aspects of SPDEs.</p>
<p><a href="http://www.cmap.polytechnique.fr/~rama/spde/articles.htm" rel="noreferrer">http://www... |
344,269 | <p>Let <span class="math-container">$M$</span> be an <span class="math-container">$n$</span>-dimensional smooth manifold and <span class="math-container">$\Theta$</span> some tensor field on <span class="math-container">$M$</span>, so a smooth section of <span class="math-container">$TM^{\otimes r} \otimes T^*M^{\otime... | Robert Bryant | 13,972 | <p>The general procedure for deciding the 'local' version of this question, at least in the real-analytic connected case, was certainly known to Élie Cartan and, probably known to Lie in some form. The basic idea is this: The condition that a vector field <span class="math-container">$X$</span> preserve <span class="... |
344,269 | <p>Let <span class="math-container">$M$</span> be an <span class="math-container">$n$</span>-dimensional smooth manifold and <span class="math-container">$\Theta$</span> some tensor field on <span class="math-container">$M$</span>, so a smooth section of <span class="math-container">$TM^{\otimes r} \otimes T^*M^{\otime... | Ben McKay | 13,268 | <p>If an analytic geometric structure is rigid in the sense of Gromov (see Gromov and d'Ambra), its symmetry vector fields form a finite dimensional Lie algebra. This is not very helpful, since we don't know too many different ways to prove rigidity, but there are lots of examples in:</p>
<p>D'Ambra, G.(F-IHES); Gromo... |
599,282 | <p>I am given a function $f(x)$.</p>
<p>I determined that $f(x)'' = 0$ precisely when $x$ is $4$ or $-3$.</p>
<p>I am asked to find the interval for which the function is concave down.</p>
<p>How can I do it by knowing the values $x = 4$ and $x = -3$ and without having to plot the function?</p>
| André Nicolas | 6,312 | <p>For simplicity let us assume that the second derivative is everywhere defined and continuous. You have found that $f''(x)=0$ at $x=4$ and at $x=-3$. That is not enough to determine concavity, it only locates the points where concavity <strong>might</strong> change.</p>
<p>In most simple cases, concavity will change... |
329,792 | <p>I need some suggestions to solve this integral:</p>
<p>$$\int_{1}^{3} \frac{1}{x^3 + 8} dx$$</p>
<p>Thanks.</p>
| André Nicolas | 6,312 | <p><strong>Hint:</strong> Use partial fractions: $x^3+8=(x+2)(x^2-2x+4)$. </p>
|
24,795 | <p>I have a set of points $(x, y)$ where each one comes from either one of two linear functions:
\begin{align*}
y &= m_1 x + b_1\\
y &= m_2 x + b_2
\end{align*}
Is there a fitting method to find such functions, without knowing from which function each of the points come from?</p>
<p>PS. can somebody ad... | Shiyu | 7,156 | <p>Your question reminds me a robust model fitting method: RANSAC, which is widely used in computer vision to remove image feature outliers. Suppose the data are roughly located on two lines. If traditional fitting methods are employed, you may get a linear function that is very different to the ones you expected. But ... |
642,863 | <p>Imagine there's a quiz on the internet intended for a wide audience. It contains a (unlimited) number of questions, all of them with yes/no answers. A person gets one random question and must answer it, after that he can get another one. He can continue answering any number of questions he wants. So the only data yo... | Elie Bergman | 117,102 | <p>I suppose the problem is if you have an infinite list of questions, any participant could scroll down the list indefinitely and only answer those questions they knew the answer to. But they could also do this for arbitrarily large lengths of time thus obtaining both a very large number of questions correct, and a 10... |
3,966,951 | <p>I am reading about stationary harmonic maps and I came across the following calculation.</p>
<p>Let <span class="math-container">$\mathcal{Q}_t(x) = x+ t\zeta(x),$</span> where <span class="math-container">$\zeta\in C^{\infty}_c(B_{\rho}(x_0), \mathbb{R}^n)$</span> where <span class="math-container">$x_0\in \mathbb{... | Romulus Augustulus | 865,041 | <p>Since <span class="math-container">$DQ_t=I+tD\xi$</span>, the two formulas follow from the following two facts:</p>
<ul>
<li><p>If <span class="math-container">$Inv:GL(\mathbf{R}^n)\to GL(\mathbf{R}^n)$</span> is the inversion, then
<span class="math-container">$$D(Inv)_I[A]=-A;$$</span></p>
</li>
<li><p>The differe... |
4,470,741 | <p><span class="math-container">$$
\begin{aligned}
\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(1+x)(1+2x)(1+3x)\cdots(1+nx)} && x > 0
\end{aligned}
$$</span></p>
<p>I tried using (1) some inequalities (2) Taking the coefficients of x common to get some factorials in the denominator , couldn't reach a right conc... | Svyatoslav | 869,237 | <p>We can try to get a closed form for the sum, what will simplify the further analysis.
<span class="math-container">$$S(x)=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(1+x)(1+2x)\cdots(1+nx)}=\sum_{n=0}^{\infty}\frac{(-1)^n x^{n}}{(1/x+1)(1/x+2)\cdots(1/x+n)}$$</span>
<span class="math-container">$$=\sum_{n=0}^{\infty}\f... |
603,471 | <p>Im totally new to statistics , but what is the characteristic function for ?
I do not get that. I was reading about the bell curve and the Central Limit Theorem , but I did not get what the characteristic function is suppose to be , where it comes from or what it is used for.</p>
<p>It seems to appear in the proof ... | Brian M. Scott | 12,042 | <p>Your induction hypothesis is that</p>
<p>$$\sum_{j=0}^n\left(-\frac12\right)^j=\frac{2^{n+1}+(-1)^n}{3\cdot2^n}\;,\tag{1}$$</p>
<p>and you want to prove that</p>
<p>$$\sum_{j=0}^{n+1}\left(-\frac12\right)^j=\frac{2^{n+1}+(-1)^{n+1}}{3\cdot2^{n+1}}\;.\tag{2}$$</p>
<p>The natural thing to do is to split the sum in... |
1,329,374 | <p>I'm looking for an explicit example of a BVP for a second order ODE: </p>
<blockquote>
<p>$y''+p(x)y'+q(x)y=f(x)$ (where $\,0\leq x\leq L\,$ and $\,y(0)=\alpha\,$ $\,y(L)=\beta$).</p>
</blockquote>
<p>If you also have the exact solution, the better. The reason is for test purposes, I've just finished a Mathemati... | Community | -1 | <p><strong>Hint</strong>:</p>
<p>If you increase $\alpha$ (from $0$ to $1$), the width of the hatched area goes decreasing linearly, like $1-\alpha$.</p>
<p>Hence, the fraction of the triangle area that is hatched is proportional to the integral</p>
<p>$$\int_0^\alpha(1-\alpha)\,d\alpha=\alpha-\frac{\alpha^2}2.$$</p... |
376,575 | <p>This is in part motivated from my attempt to understand tate diagonal in III.1 of Thomas Nikolaus, Peter Scholze, <em>On topological cyclic homology</em>, arXiv:<a href="https://arxiv.org/abs/1707.01799" rel="nofollow noreferrer">1707.01799</a>. I just want to make my understanding precise.</p>
<hr />
<p><strong>Pa... | Harry Gindi | 1,353 | <p>I actually worked this out a few months ago (with a hint from Denis Nardin) and wrote this in a message to a friend of mine:</p>
<blockquote>
<p>Consider SymmMonCat as a symmetric monoidal category with the cocartesian monoidal structure. I filled in the details for my own sake:
SymmMonCat itself has a symmetric mo... |
2,290,458 | <p>I wonder if someone can help me with this problem:</p>
<blockquote>
<p>Let $(X,d)$ be a connected metric space such that all continuous functions $f:(X,d) \to \mathbb{R}$ are uniformly continuous. Show that $(X,d)$ is compact.</p>
</blockquote>
<p>A hint is to work with the counter positive and assume that $(X,d... | DanielWainfleet | 254,665 | <p>If a metric space $X$ is non-compact then $X$ has a countably infinite closed discrete subspace Y. Because if $(p_n)_{n\in \mathbb N}$ is a Cauchy sequence in $X$ with no convergent subsequence, let $Y=\{p_n:n\in \mathbb N\}.$</p>
<p>We show that if $(X,d)$ is connected and non-compact then there is a continuous ... |
1,710,799 | <p>Right triangle $ABC$ is inscribed in a circle with $AC = 6$, $BC = 8$ and $AB=10$. $AC$ and $CB$ are semi-circles. Find the sum of the areas of regions $X$ and $Y$.</p>
<p><a href="https://i.stack.imgur.com/XXENo.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XXENo.png" alt="enter image descript... | Sam | 333,969 | <p>Answer is area of triangle ABC, 24 square meters. Because total area of two smaller Half circles equals big half circle area. So x+ y= area of ABC</p>
|
1,535,057 | <p>I am confused with the inductive step of this very basic induction example in the book <a href="http://rads.stackoverflow.com/amzn/click/0072899050" rel="nofollow">Discrete Mathematics and Its Applications</a>:</p>
<p>$$1 + 2+· · ·+k = k(k + 1) / 2$$</p>
<p>When we apply $k+1$, the equation becomes:</p>
<p>$$1 + ... | Community | -1 | <p>If that can help you:</p>
<p>$$1+2+3+4+5=\frac{5\cdot6}2,$$
1+2+3+4+5=5.6/2</p>
<p>and
$$1+2+3+4+5\color{green}{+6}=\frac{5\cdot6}2\color{green}{+6}=\frac{(5+2)\cdot6}2=\frac{6\cdot7}2.$$
1+2+3+4+5+6=5.6/2+6=(5+2).6/2=6.7/2</p>
|
2,943,199 | <p>This is actually a doubt I got while solving <a href="https://www.askiitians.com/forums/Discuss-with-Askiitians-Tutors/please-prove-the-following-2cos-1-3-root-13_114688.htm" rel="nofollow noreferrer">this question</a>. The thing is I know how to convert <span class="math-container">$2\arctan(3/4)$</span> to <span c... | Sort of Damocles | 478,044 | <p>I would say: "<span class="math-container">$M$</span> has exactly one entry equal to zero". Much easier to read and understand than a sea of symbols.</p>
|
3,325,749 | <p>Suppose that <span class="math-container">$A=\{y_1,...,y_r\}$</span> is a subset of a vector space <span class="math-container">$V$</span> and that every vector <span class="math-container">$x \in V$</span> can be expressed uniquely as a linear combination of the vectors of <span class="math-container">$A$</span>. S... | Pietro Paparella | 414,530 | <p>It suffices to show that <span class="math-container">$A$</span> is linearly independent. To this end, let <span class="math-container">$\mathbb{F}$</span> be the field over which <span class="math-container">$V$</span> is a vector space. If <span class="math-container">$\mathsf{T}: \mathbb{F}^r \longrightarrow V$</... |
19,876 | <p>Very important in integrating things like $\int \cos^{2}(\theta) d\theta$ but it is hard for me to remember them. So how do you deduce this type of formulae? If I can remember right, there was some $e^{\theta i}=\cos(\theta)+i \sin(\theta)$ trick where you took $e^{2 i \theta}$ and $e^{-2 i \theta}$. While I am draf... | Hans Lundmark | 1,242 | <p>It might be useful to remember that $\cos^2 x$ oscillates twice as fast as $\cos x$. This is something that people who work with alternating current know very well; the effect (which is proportional to the <em>square</em> of the current) has twice the frequency. For example, a light bulb flickers at 100 Hz if the AC... |
3,506 | <p>I am wondering what is the correct function in Mathematica to plot the true impulse function, better known as the <code>DiracDelta[]</code> function. When using this inside of a function or just the function itself when plotting, it renders output = zero. Quick example:</p>
<pre><code>Plot[DiracDelta[x], {x,-1,1}]
... | Sjoerd C. de Vries | 57 | <p>As others already have written, the Dirac delta is not a real function and it can't be plotted. Other programs that claim to plot it <strong>just fake it</strong>.</p>
<p>Having said that, you can roll a diracDelta of your own, that more or less mimics the Dirac Delta's behavior but is still continuous. Advantage w... |
6,695 | <p>The standard approach for showing <span class="math-container">$\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$</span> is to multiply by <span class="math-container">$\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$</span> and then do a substitution with <span class="math-cont... | Idris Addou | 192,045 | <p>Dual way to compute $$\int \csc x\,dx.$$</p>
<p>First, we need a trigonometry identity
\begin{eqnarray*}
\sin^2x &=&(1-\cos x)(1+\cos x) \\
\frac{1-\cos x}{\sin x} &=&\frac{\sin x}{1+\cos x} \\
\csc x &=&\cot x+\frac{\sin x}{1+\cos x}
\end{eqnarray*}
Next, it suffices to integrate each side
... |
6,695 | <p>The standard approach for showing <span class="math-container">$\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$</span> is to multiply by <span class="math-container">$\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$</span> and then do a substitution with <span class="math-cont... | mdcq | 454,513 | <p>Here is a slightly different approach to calculate</p>
<p>$$ \int \frac{1}{\cos(x)}\,dx $$</p>
<p>Define $u := \tan(\frac{x}{2}) $ so it follows $dx = \frac{2}{1+u^2}\,du$. It follows that $\cos(x) = \frac{1-u^2}{1+u^2}$ under this substitution. Now we can write the integral as:</p>
<p>$$ \int \frac{1}{\cos(x)}\,... |
2,269,144 | <p>Background: this is Arfken et al mathematical methods 12.5.4 and the answer is 1.</p>
<p>Using the infinite sin product we need the alternating terms in red to cancel when $\pi$ is plugged into z but I don't know how to do that:</p>
<p>$$\frac{\sin(z)}{z(1-z^2/\pi^2)}=\prod_{\color{red}{n=2}}^{\infty}(1-\frac{z^2}... | Robert Soupe | 149,436 | <p>Yes. Let's say $a, b, \alpha$ are all nonzero real numbers, and $\alpha$ is negative. Then, if $a = b$, then $a^\alpha = b^\alpha$ as well.</p>
<p>Remember also that $$a^\alpha = \frac{1}{a^{|\alpha|}}.$$</p>
<p>For example, let's say we're trying to solve $$x^2 + 3 = 7.$$ I suppose the obvious thing to do here wo... |
246,592 | <p>I am having trouble understanding when a function might have a removable singularity over a pole. </p>
<p>For example:
$$f(z)=\frac{\sin^2 z}{z}$$</p>
<p>I believe the pole is at $z=0$. However, if we take the taylor expansion of $f(z)$ apparently the pole vanishes. I do not understand how and where does the pole ... | DonAntonio | 31,254 | <p>Directly by arithmetic of limits:</p>
<p>$$\frac{\sin^2z}{z}=\frac{\sin z}{z}\cdot\sin z\xrightarrow [z\to 0]{} 1\cdot 0 = 0$$</p>
<p>which means, just as with real analysis, that the discontinuity (singularity) is removable.</p>
|
4,304,209 | <p>Hi it's a follow up of <a href="https://math.stackexchange.com/questions/4268913/show-this-inequality-sqrt-fracabb-sqrt-fracbaa-ge-2/4269271#4269271">show this inequality $\sqrt{\frac{a^b}{b}}+\sqrt{\frac{b^a}{a}}\ge 2$</a>:</p>
<h2>Problem :</h2>
<p>Let <span class="math-container">$a,x>0$</span> then (dis)prove... | Sean Eberhard | 23,805 | <p>I suspect <span class="math-container">$f$</span> is differentiable nowhere, because its derivative is "trying" to be <span class="math-container">$\sum_{n=0}^\infty \cos(2^n x)$</span>, which converges nowhere. Below I show that <span class="math-container">$f$</span> is differentiable <em>almost</em> now... |
301,889 | <p>For example, in a homework assignment I need to prove that a metric space $M$ is isometric to itself. If I say that $M$ maps to $M$ is that the same as saying that theres a function that takes elements from $M$ and I get a result in $M$? Or what is the "literal" meaning behind saying "map/maps/mapping".</p>
| Brian M. Scott | 12,042 | <p>It is not literally true that</p>
<p>$$\prod_{i\in I}{X_i^2}=\left(\prod_{i\in I}{X_i}\right)^2\;:$$</p>
<p>elements of the lefthand side are functions with domain $I$ such that $f(i)\in X_i\times X_i$ for each $i\in I$, while elements of the righthand side are ordered pairs of functions with domain $I$ such that ... |
1,221,487 | <p>Problem: Let V be the subspace of all 2x2 matrices over R, and W the subspace spanned by:</p>
<p>\begin{bmatrix}
1 & -5 \\
-4 & 2 \\
\end{bmatrix}
\begin{bmatrix}
1 & 1 \\
-1 & 5 \\
\end{bmatrix}
\begin{bmatrix}
2 & -4 \\
-5 & 7 \\
\end{bmatrix}
\begin{bmatrix}
1 & -7 \\
-5 & 1 \\
\e... | san | 229,191 | <p>One can describe the process of taking the digital root
by saying that at each step you take away a multiple of 9
until you reach a number less than 10.</p>
<p>In fact the operation
$$
10^k a_k+...+a_0\mapsto a_k+\dots +a_0
$$
can be broken into $k$ operations
$$
-(10^j-1)a_j=-9(10^{j-1}+10^{j-2}+\dots+ 10+1)a_j,
... |
1,232,690 | <p>As we all knew that Aryabhata (<a href="http://en.wikipedia.org/wiki/Aryabhata#Place_value_system_and_zero" rel="nofollow">http://en.wikipedia.org/wiki/Aryabhata#Place_value_system_and_zero</a>) invented zero ($0$) in our number system. I have few questions about it.</p>
<ol>
<li>How did the numeric system work bef... | user140337 | 140,337 | <p>I would start with roman numerals. This is something "average" person probably knows somewhat, so he has some ground to stand. I would explain that roman numeral system is additive: fifteen in roman numerals is XV i.e. X+V.
Then it should be explained that our common decimal system is in some sense additive as well,... |
630,697 | <p>I know that </p>
<p><1,2,3,...,10>$\cdot$<1 0,9,8,...1>=220</p>
<p><1,2,3,...,100>$\cdot$<100,99,98,...,1>=171700</p>
<p><1,2,3,...,1000>$\cdot$<1000,999,998,...,1>=167167000</p>
<p><1,2,3,...,10000>$\cdot$<10000,9999,9998,...,1>=166716670000
`</p>
<p>And 2,17,167,1667
is a part of the ... | Matthew Conroy | 2,937 | <p>We have $$<1,2,3,...,n> \cdot <n,n-1,n-2,...,1> $$
$$= \sum_{i=1}^n (n+1-i)i = \sum_{i=1}^n i(n+1) - \sum_{i=1}^n i^2$$
$$= (n+1) \frac{1}{2}n(n+1) - \frac{1}{6}n(n+1)(2n+1) = \frac{1}{6}n(n+1)(n+2).$$</p>
<p>Replacing $n$ by $10^n$ yields the general term of your sequence.</p>
|
1,369,485 | <p>$\log_2(x) = \log_x(2) $ </p>
<p>Using the change of base theorem: $\dfrac{\log(x)}{\log(2)} = \dfrac{\log(2)}{\log(x)}$</p>
<p>Multiplied the denominators on both sides: $\log(x)\log(x) = \log(2)\log(2)$</p>
<p>I kind of get stuck here. I know that you can't take the square root of both sides of the equation, bu... | Ken | 169,838 | <p>You have it, actually.
\begin{align}
(\log(x))^2 &= (\log(2))^2 \\
\log(x) &= \pm \log(2)
\end{align}</p>
<p>For the "$+$" case, you've already solved it.</p>
<p>In the "$-$" case, you have $\log(x) = -\log(2) = \log(2^{-1}) = \log(\frac12)$, from which you can get $x=\frac12$.</p>
|
3,439,084 | <p>How to use mean value theorem to show that <span class="math-container">$\sqrt{x}(1+x)\log(\frac{1+x}{x})-\sqrt{x}<1$</span> when <span class="math-container">$x$</span> is positive.</p>
| Ari Royce Hidayat | 435,467 | <p>Because what to be proved is what in Proposition 5 itself which is <span class="math-container">$uN \: vN = uvN$</span>, NOT <span class="math-container">$uvN = u_1 v_1 N$</span>. The later is what a well defined condition is.</p>
<p>By letting <span class="math-container">$u = 1$</span>, <span class="math-containe... |
838,759 | <p>It seems that a reasonable $\log$ approximation for $\pi(x)$ can be given, where</p>
<p>$f(y, x) :=
\log\left(\dfrac{\log(x)}{\log\left(e(y - \lfloor y\rfloor) + x^{1/x}(1 - y + \lfloor y\rfloor)\right)}\right)$</p>
<p>is an interpolating function between $\log(x)\text{ and }\log(\log(x)).$</p>
<p>Comparing $\p... | William C. Newman | 773,090 | <p>In the newly updated Rising Sea (August 2022), this exercise is now 19.4.M. And in this version, there is a new exercise preceding it,</p>
<blockquote>
<p>19.4.K. EXERCISE. If <span class="math-container">$\mathcal{F}$</span> is a torsion sheaf on <span class="math-container">$C$</span>, show that <span class="math-... |
184,534 | <p>Does the Gamma function $\Gamma: \mathbb{C} \to \mathbb{C}$ preserve the Kummer ring $\mathbb{Z}[\exp(2\pi\imath/m)]$? And if not, then what about the Gaussian integers $\mathbb{Z}[\imath]$ or the Eisenstein integers $\mathbb{Z}[\exp(2\pi\imath/3)]$?</p>
<p>Is it possible to characterize holomorphic function which ... | Robert Israel | 13,650 | <p>I don't know about characterization, but there are lots of such functions. In fact, for any map $g$ of the lattice $L$ into itself, there are continuum-many entire functions $f$ such that $f(z) = g(z)$ for $z \in L$. This is because you can
get entire functions that take prescribed values on any subset of $\mathbb... |
284,184 | <p>I am expected to prove by induction that any polynomial function is continuous. In which "direction" would you advise to make induction? </p>
<p>e.g. Taking $x^n$ and making induction on $n$ is not sufficient. By polynomial I understand, $\sum^{m}_{n=1}a_n x^n$. How do I prove it's continuity using epsilon delta no... | Hagen von Eitzen | 39,174 | <p>You should already know the following functions are continuous:</p>
<ul>
<li>$x\mapsto const$</li>
<li>$x\mapsto x$</li>
<li>$x\mapsto f(x)+g(x)$ where $f,g$ are continuous</li>
<li>$x\mapsto f(x)\cdot g(x)$ where $f,g$ are continuous</li>
</ul>
<p>Every polynomial function can be obtained from these in finitely m... |
493,042 | <p>$−3x+5y+7z=7$<br>
$−3x-7y+kx=8$<br>
$15x+23y-19z=-40 $</p>
<p>by using echolon form I got to this</p>
<p>\begin{bmatrix}
-3 & -7 & k & 8 \\[0.3em]
0 & -12 & 5k-19 & 0 \\[0.3em]
0 & 0 & -4k + 12 & 1
\end{bmatrix}</p>
<p>but... | Julian Kuelshammer | 15,416 | <p>Maybe it helps if you rewrite your reduced echolon form as a system of linear equations:</p>
<p>$$\begin{align*}
-3x-7y+kz&=8\\
-12y+(5k-19)z&=0\\
(-4k+12)z&=1
\end{align*}$$</p>
<p>Now start solving your system from bottom to top. Be careful not to divide by zero.</p>
|
1,859,741 | <p>How do I prove that</p>
<p>$$\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}} \approx 1$$</p>
<p>without using the calculator?</p>
| Ennar | 122,131 | <p>You can do some ad hoc evaluation:</p>
<p>$4=\sqrt{20-\sqrt{20-\sqrt{16}}} \leq\sqrt{20-\sqrt{20-\sqrt{20}}}\leq \sqrt{20-\sqrt{14-\sqrt{25}}}=\sqrt{17}$</p>
<p>$\sqrt{24}\leq\sqrt{20+\sqrt{24}}= \sqrt{20+\sqrt{20+\sqrt{16}}} \leq\sqrt{20+\sqrt{20+\sqrt{20}}}\leq \sqrt{20+\sqrt{20+\sqrt{25}}}=5$</p>
<p>And thus, ... |
1,017,026 | <blockquote>
<p><strong>Cauchy-Schwarz Inequality:</strong></p>
<p>If <span class="math-container">$\textbf{u}$</span> and <span class="math-container">$\textbf{v}$</span> are vectors in a real inner product space <span class="math-container">$V$</span>, then <span class="math-container">$$|\left\langle\textbf{u},\text... | David K | 139,123 | <p>It's disappointing that dustin's excellent answer has not yet been accepted as I write this, but to apply the $\mathrm{atan2}(y,x)$ function directly to the specific problem in the question, one would write
$$
\theta = \mathrm{atan2}(v^2\pm\sqrt{v^4-g(gx^2+2yv^2)}, gx).
$$</p>
<p>Assuming that you would prefer to g... |
729,352 | <p>I am trying to prove that $a_1$, $a_2$, $a_3$ are linearly independent.</p>
<p>I am asked to use vector product and prove that if $c_{1}a_{1} + c_{2}a_{2} + c_{3}a_{3} = 0$ then $c_1 = c_2 = c_3 = 0$</p>
<p>I am completely stuck on where to go with this problem. I would think that linearly independent then the nul... | Alijah Ahmed | 124,032 | <p>You are given </p>
<p>$$c_1a_1+c_2a_2+c_3a_3=0$$</p>
<p>As $a_1,a_2,a_3$ are orthogonal, we end up with the following three equalities, where $a\cdot b$ denotes the vector product between vectors $a$ and $b$.</p>
<p>$$(a_1\cdot c_1a_1+c_2a_2+c_3a_3)=c_1(a_1\cdot a_1)=0$$
$$(a_2\cdot c_1a_1+c_2a_2+c_3a_3)=c_2(a_2\... |
100,842 | <p>I have the following list of centers of disks.</p>
<pre><code>r=0.03;
pts = {{0.10420089319018544`, -0.024872674177014872`}, \
{0.9743669105930046`, 0.9169054125547074`}, {0.028760526736240563`,
0.45959879163736717`}, {-0.0059035632830851115`,
0.2922099255180086`}, {0.41615337459441437`,
0.9928402345... | Marvin | 35,799 | <p>You can use geometry property. Let's say your equation of line is <code>x+y-1=0</code>. You can distinguish whether a point lies on same side as that of the origin by substituting the points into the equation of line. If the value is same that when origin <code>0,0</code> is substituted, then the point lies on the s... |
1,613,172 | <p>Given three (nonempty) sets $A, B$ and $C$, and knowing that $|A| \leq |B|$, how can I prove that $|C^A| \leq |C^B|$? This problem is trivial if we want to prove that the sets are equipotent, because it is very easy to create a bijective function. </p>
<p>Here, we know that there exists an <em>injective</em> functi... | Guy | 206,544 | <p>Since $\left|A\right|\leq\left|B\right|$, we have an injection $\phi:A\to B$. Fix some $c_0\in C$ and define
$$\psi:C^A\to C^B$$
by
$$\left(\psi\left(f\right)\right)\left(b\right)=\left\{\begin{matrix}\phi\left(a\right)&\exists a\in A:\phi\left(a\right)=b\\c_0&\textrm{Otherwise}\end{matrix}\right.$$
This is ... |
258,209 | <p>Let $X$ be an infinite set and let $\text{End}(X)$ be the set of all functions $f:X\to X$. For $f\in\text{End}(X)$ let $$\text{Com}(f) = \{g\in\text{End}(X): g\circ f = f \circ g\}.$$
Is there $f\in \text{End}(X)$ such that $\text{Com}(f) = \{\text{id}_X, f\}$? </p>
<p>If not, what is $\min\{|\text{Com}(f)|:f\in\te... | Joel Adler | 26,085 | <p>Let $X:=\mathbf{N}$ and $f:X\rightarrow X$ be defined by $f(n)=n+1$, and let $g:X\rightarrow X$ commute with $f$.</p>
<p>We have $g(n+1)=g(f(n))=f(g(n))=g(n)+1$ for all $n\in X$. Thus, with $g(0)=a$ we obtain $g(n)=a+n=f^a(n)$, which means that the powers of $f$ are the only elements of $\text{End}(X)$ commuting wi... |
3,378,104 | <blockquote>
<p>Let <span class="math-container">$\mathcal{F} \subseteq \mathcal{G}$</span>.</p>
<p>Show that <span class="math-container">$$E((E(X\mid \mathcal{G})-E(X\mid \mathcal{F}))^2=E(E(X\mid \mathcal{G}))^{2}-E(E(X\mid \mathcal{F}))^{2}$$</span></p>
</blockquote>
<p>My idea:</p>
<p><span class="math-container">... | ajotatxe | 132,456 | <p><em>Hint</em>:
<span class="math-container">$$\log(x^2+y^2)\le2\max\{\log|x-y|,\log|x+y|\}$$</span></p>
|
2,324,692 | <p>The Dual Group of $\mathbb{R}$ is isomorphic to $\Bbb{R}$ itself in the following way:
The map
$$\Bbb{R} \to \hat{\Bbb{R}}, \quad y \mapsto \exp(ixy) $$
is an isomorphism. Further it is stated in the literature that this map is also an homeomorphism. See for exmaple Conway, A course in functional analysis Theorem 9.... | mcaselli | 372,147 | <p>Expand $(\cos(x)+i\sin(x))^6=\cos(6x)+i\sin(6x) $ and take raeal part of both sides</p>
|
2,324,692 | <p>The Dual Group of $\mathbb{R}$ is isomorphic to $\Bbb{R}$ itself in the following way:
The map
$$\Bbb{R} \to \hat{\Bbb{R}}, \quad y \mapsto \exp(ixy) $$
is an isomorphism. Further it is stated in the literature that this map is also an homeomorphism. See for exmaple Conway, A course in functional analysis Theorem 9.... | S.C.B. | 310,930 | <p>Note that you can use the identity $\cos^2 x+\sin^2 x=1$(or alternatively, $\sin^2 x=1-\cos^2 x$). </p>
<p>Take each side of the equation to the power of $2,3$ to get $$9 \sin^2 x=9-9\cos^2 x$$
$$24\sin^4 x=24-48 \cos x^2+24 \cos^4 x$$
$$16\sin^6 x=16-48 \cos^2 x+48 \cos^4 x-16\cos^6 x$$
Tedious, but it probably wi... |
3,284,426 | <p>The possible orders of Sylow 3 subgroups is <span class="math-container">$\{1, 13\}$</span> (if I understood correctly). But how can I check the exact number?
And how am I supposed to show that <span class="math-container">$S_3 = \Bbb Z_9$</span> or <span class="math-container">$S_3 = \Bbb Z_3\times \Bbb Z_3$</span>... | the_fox | 11,450 | <p>You seem a bit confused. You are asking about the possible orders of the Sylow <span class="math-container">$3$</span>-subgroups of <span class="math-container">$G$</span>. Do you know how a Sylow subgroup is defined? If no, then in a group of order <span class="math-container">$117$</span> a Sylow <span class="mat... |
2,911,049 | <blockquote>
<p><strong>Question:</strong>
Can we find the gradient and Hessian of $x x^T$ w.r.t. $x$, where $x \in \mathbb{R}^{n \times 1}$ ?</p>
</blockquote>
<p>EDIT:
If we can, may I know how to compute that? Thank you.</p>
| greg | 357,854 | <p>It's a cinch to calculate these things in index notation.
$$\eqalign{
F_{ij} &= x_ix_j \cr
G_{ijk} = \frac{\partial F_{ij}}{\partial x_k}
&= x_i\delta_{jk} + x_j\delta_{ik} \cr
H_{ijkl} = \frac{\partial^2F_{ij}}{\partial x_k\partial x_l}
&= \delta_{il}\delta_{jk} + \delta_{ik}\delta_{jl} \cr
}$$</p>
|
464,146 | <p>Suppose that a wooden cube, whose edge is $3$ inch, is painted red, then cut into $27$ pieces of $1$ inch edge. Find total surface area of unpainted?</p>
<p>First of all, I have tried to draw the cube using MS Paint, below is given picture:</p>
<p><img src="https://i.stack.imgur.com/1eDpC.png" alt="enter image des... | TheVal | 85,277 | <p>This answer requires some spatial examples, so I think it will be better if I generalize the case of a simple cube, which can be applied to all possible situations.</p>
<p>As you said, the total area covered by the paint in a cube with lenght $a$ is equal to:
$$
S_{\mathrm{ext}}=6\cdot a^2
$$
But then the division ... |
464,146 | <p>Suppose that a wooden cube, whose edge is $3$ inch, is painted red, then cut into $27$ pieces of $1$ inch edge. Find total surface area of unpainted?</p>
<p>First of all, I have tried to draw the cube using MS Paint, below is given picture:</p>
<p><img src="https://i.stack.imgur.com/1eDpC.png" alt="enter image des... | Blue | 409 | <p>Consider cutting-up a rectangular solid ("shoebox"). We'll say that the top and bottom faces (each of area $R$) are painted red, the front and back faces (each of area $G$) are painted green, and the left and right faces (each of area $B$) are painted green.</p>
<p>As we slice the box, it helps to think of the piec... |
4,788 | <p>Despite its name, its often claimed that the fundamental theorem of algebra (which shows that the Complex numbers are algebraically closed - this is not to be confused with the claim that a polynomial of degree n has <strong>at most</strong> n roots) is not considered fundamental by algebraists as it's not needed fo... | Andrea Mori | 688 | <p>I can immediately think of four important areas for which the FTA is actually quite fundamental. Maybe other people may come up with more contributions.</p>
<ol>
<li><p>Algebraic Geometry (as already touched in Agusti Roig's answer). In particular, we should couple the FTA with Lefschetz's Principle which basically... |
4,122,361 | <p>I am trying to show that <span class="math-container">$$\frac{1}{a^2+x^2} \ast \frac{1}{a^2+x^2}=\int_{-\infty}^\infty\frac{1}{(a^2+t^2)(a^2+(x-t)^2) }dt = \frac{2\pi}{a(4a^2+x^2)}$$</span> I wrote out the integral for the convolution and it becomes a big exercise in partial fractions and integration of rational fu... | Quanto | 686,284 | <p>Symmetrize the integrand with <span class="math-container">$y= t-\frac x2$</span></p>
<p><span class="math-container">\begin{align}
& \int_{-\infty}^\infty\frac{1}{(a^2+t^2)(a^2+(x-t)^2) }dt\\
= & \int_{-\infty}^\infty\frac{1}{(a^2+(y+\frac x2)^2)(a^2+(y-\frac x2)^2)}dy\\
=&\
\frac2{x(4a^2+x^2)}\int_{-\... |
4,122,361 | <p>I am trying to show that <span class="math-container">$$\frac{1}{a^2+x^2} \ast \frac{1}{a^2+x^2}=\int_{-\infty}^\infty\frac{1}{(a^2+t^2)(a^2+(x-t)^2) }dt = \frac{2\pi}{a(4a^2+x^2)}$$</span> I wrote out the integral for the convolution and it becomes a big exercise in partial fractions and integration of rational fu... | Oliver Díaz | 121,671 | <p>Here using Fourier transform may simplify things considerably:</p>
<p>Let <span class="math-container">$\phi(x)=\frac{1}{1+x^2}$</span>, and <span class="math-container">$\phi_a(x)=\frac{1}{a}\phi(a^{-1}x)=\frac{a}{a^2+x^2}$</span>.</p>
<p>Recall that <span class="math-container">$\pi e^{-2\pi|t|}=\int e^{-2\pi itx}... |
4,122,361 | <p>I am trying to show that <span class="math-container">$$\frac{1}{a^2+x^2} \ast \frac{1}{a^2+x^2}=\int_{-\infty}^\infty\frac{1}{(a^2+t^2)(a^2+(x-t)^2) }dt = \frac{2\pi}{a(4a^2+x^2)}$$</span> I wrote out the integral for the convolution and it becomes a big exercise in partial fractions and integration of rational fu... | Claude Leibovici | 82,404 | <p>Less elegant than @Quanto's answer, let us write
<span class="math-container">$$a^2+(x-t)^2=(t-r)(t-r)$$</span> with <span class="math-container">$r=(x+ia)$</span> and <span class="math-container">$r=(x-ia)$</span> and use partial fraction decomposition
<span class="math-container">$$\frac{1}{(a^2+t^2)(a^2+(x-t)^2) ... |
119,756 | <p>I hope this question is not completely trivial:</p>
<p>Suppose $V$ is an irreducible projective variety and $U\subset V$ is a Zariski open subset isomorphic to an affine variety. Is it true that $V\setminus U$ is a Cartier divisor in $V$? If not, what conditions should we impose on $V$? (I guess if $V$ is smooth, ... | Jason Starr | 13,265 | <p>As you say, if $V$ is smooth, then everything is fine. However, if $V$ is singular, the complement may fail to be the support of any Cartier divisor. For instance, take $V$ to be the projective cone over a smooth plane cubic $X$, and take $V\setminus U$ to be the line over any point $x$ of $X$ such that for every ... |
104,210 | <p>I have this code to plot contours:</p>
<pre><code>ContourPlot[(Cos[θ] Cos[ϕ])^(1/4), {θ, -π/2, π/2}, {ϕ, -π/2, π/2}, AxesLabel -> Automatic]
</code></pre>
<p>How would I map those contours on a unit sphere (if it is even possible) where <code>θ</code> and <code>ϕ</code> are the spherical angles for the sphere (... | eldo | 14,254 | <p>I don't know how to make this with <code>RegionFunction</code>s but you could show the vectors along the <code>Line[{{0, 1}, {1, 1}}]</code> like this:</p>
<pre><code>VectorPlot[{Sin[x], Cos[y]}, {x, 0, 1}, {y, 0.5, 1.5},
AspectRatio -> 1/5,
FrameTicks -> {True, {0.95, 1, 1.05}, False, False},
GridLines -&... |
1,540,069 | <p>If $\lim \limits _{x \to x_0} (f(x) + g(x))$ exists, can I write: $\lim \limits _{x \to x_0} (f(x) + g(x)) = \lim \limits _{x \to x_0} f(x) + \lim \limits _{x \to x_0} g(x)$ ? <br>I mean, to write this do I have to know that the other limits exist? Because they tell me that $\lim \limits _{x \to x_0} f(x)$ exists an... | Colm Bhandal | 252,983 | <p>There are actually two answers to your question, because exponentiation is the first <a href="https://en.wikipedia.org/wiki/Hyperoperation" rel="nofollow">hyperoperation</a> that is not commutative. Addition and multiplication are commutative, so they can only be <a href="https://en.wikipedia.org/wiki/Currying" rel=... |
2,729,217 | <p>For the following question, I am wondering if someone can give me some assistance:</p>
<p>Evaluate the integral </p>
<p>$$\int_{0}^{1} \int_{0}^{x} \sqrt{{x}^{2}+{y}^{2}} \,dy\,dx$$</p>
<p>using the transformation $\ x=u, y=uv\ $</p>
<p>I plugged into the integrand $u$ for $x$ and $uv$ for $y$ with the jacobian ... | Delta-u | 550,182 | <p>To see how the domain changes with the change of variable we can notice that $u=x$ and $v=\frac{y}{x}$.</p>
<p>So $(u,v)$ is in the square $[0,1] \times [0,1]$.</p>
<p>More precisely with $\phi: (u,v) \mapsto (u,uv)$ you have to check that $\phi([0,1] \times [0,1])=\{0 \leq y \leq x \leq 1\}$.</p>
|
2,729,217 | <p>For the following question, I am wondering if someone can give me some assistance:</p>
<p>Evaluate the integral </p>
<p>$$\int_{0}^{1} \int_{0}^{x} \sqrt{{x}^{2}+{y}^{2}} \,dy\,dx$$</p>
<p>using the transformation $\ x=u, y=uv\ $</p>
<p>I plugged into the integrand $u$ for $x$ and $uv$ for $y$ with the jacobian ... | Jack D'Aurizio | 44,121 | <p>$$\int_{0}^{1}\int_{0}^{x}\sqrt{x^2+y^2}\,dy\,dx = \int_{0}^{1}\int_{0}^{1}x\sqrt{x^2+x^2 z^2}\,dz\,dx = \int_{0}^{1}x^2\,dx\int_{0}^{1}\sqrt{1+z^2}\,dz$$
by the change of variable $y\mapsto xz$ and Fubini's theorem. The RHS equals
$$ \frac{1}{3}\left\{\left[z\sqrt{1+z^2}\right]_{0}^{1}+\int_{0}^{1}\frac{z^2\,dz}{\s... |
358,075 | <p>Suppose $ \lim_m \sum_n f(n,m) = c $ and $ 0 \leq c< \infty $. Is it true that $ \lim_m \sum_n f(n,m)^k =0 $ if k >1?</p>
<p>Thank you</p>
| Lord_Farin | 43,351 | <p><strong>Hint:</strong> It is necessary and sufficient to show that around every $p = (x,y) \in S$, there is <em>some</em> ball $U_p$ around $p$ such that $U_p \subseteq S$.</p>
<p>What could the maximal radius of $U_p$ be?</p>
|
358,075 | <p>Suppose $ \lim_m \sum_n f(n,m) = c $ and $ 0 \leq c< \infty $. Is it true that $ \lim_m \sum_n f(n,m)^k =0 $ if k >1?</p>
<p>Thank you</p>
| copper.hat | 27,978 | <p>Suppose $(x,y) \in S $. Let $r = \min(x,y)$. Then $B((x,y),r) \subset S$.</p>
<p>To see this, suppose $(x',y') \in B((x,y),r)$. Then $(x'-x)^2+(y-y')^2 < r^2$. This implies $\max \ (|x'-x|,|y'-y|) < r$.</p>
<p>This gives $x' > x-r$, and since $r \le x$, we have $x'>0$. Similarly, $y'>0$. Hence $(x... |
167,846 | <p>The question is the following:</p>
<p>Let $K\subset S^{3}$ be a knot, consider the double branched cover $Y$ of $S^{3}$ over $K$. We know $Y$ has a unique spin structure $\mathfrak{s}$, now the question is: </p>
<p>When do we have $\widehat{HF}(Y,\mathfrak{s})\cong \mathbb{Z}$? If yes, what do we know about the d-... | Marco Golla | 13,119 | <p>John Baldwin said something about both questions in <a href="http://arxiv.org/abs/0804.3624" rel="nofollow">this</a> paper. He considers closures of 3-braids and classifies which of these have an $L$-space branched double cover (Theorem 4.1). He then computes the correction term for the spin structure (Theorem 5.1);... |
4,236,968 | <p>I want to define a function implicitly by <span class="math-container">$$x^y = y^x$$</span> with <span class="math-container">$$x \ne y$$</span>
such that <span class="math-container">$f: X \rightarrow Y$</span> is constrained by <span class="math-container">$X = \{ x \in \mathbb{R} | x > 1 \}$</span> and <span c... | Jin Kim | 963,060 | <p>Taking logs on both sides yields
<span class="math-container">$$
\frac{x}{\log x} = \frac{y}{\log y}
$$</span>
provided that neither of them is 1. In fact, the following function can be analyzed:
<span class="math-container">$$
f(t) = \frac{t}{\log t},\quad t \in (1,\infty)
$$</span>
This function has a convex shape... |
4,236,968 | <p>I want to define a function implicitly by <span class="math-container">$$x^y = y^x$$</span> with <span class="math-container">$$x \ne y$$</span>
such that <span class="math-container">$f: X \rightarrow Y$</span> is constrained by <span class="math-container">$X = \{ x \in \mathbb{R} | x > 1 \}$</span> and <span c... | M. Wind | 30,735 | <p>Yes, the function exists. A parametrization can be obtained by assuming that for some pair <span class="math-container">$(x,y)$</span> on the curve the relation <span class="math-container">$y = px$</span> holds. Substitution of this relation in the original expression yields:</p>
<p><span class="math-container">$$x... |
3,102,210 | <p>I'm trying to solve the following problem:</p>
<blockquote>
<p>Let <span class="math-container">$v_0$</span> be a vertex in a graph <span class="math-container">$G$</span>, and <span class="math-container">$D_0 := \{v_0\}$</span>.</p>
<ol>
<li><p>For <span class="math-container">$n = 1, 2 \dots$</span> ind... | Thomas Lesgourgues | 601,841 | <p>You are mixing closed and open neighbourhood. I think when they are stating <span class="math-container">$D_n=N(D_0\cup\ldots\cup D_{n-1})$</span>, they are implying the <strong>open neighbourhood</strong></p>
<p>The open neighbourhood <span class="math-container">$N(S)$</span> of some set of vertices <span class="... |
1,776,931 | <p>I need to find the sum of an alternating series correct to 4 decimal places. The series I am working with is: $$\sum_{n=1}^\infty \frac{(-1)^n}{4^nn!}$$ So far I have started by setting up the inequality: $$\frac{1}{4^nn!}<.0001$$Eventually I arrived at $$n=6$$ giving the correct approximation, which is approxima... | Empy2 | 81,790 | <p>Your final answer is just the sixth term in the series. You need the sum of the first six terms.<br>
On the other hand, look again at your equation $\frac1{4^nn!}<0.0001$, which is equivalent to $4^nn!>10000$. Find the first $n$ for which this is true. You can ignore that $n$ and all the later ones, and fin... |
13,582 | <p>Question: Let $a, b \in \mathbb{R}$ with $a < b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$.</p>
<p>I suppose this has to do with the basic definition of continuity. The definition I am using is that $f$ is continuous at $a$ if $... | Isaac | 72 | <p>Consider $g(x)=f(x)-x$. $f(a)\ge a$ so $g(a)=f(a)-a\ge 0$. $f(b)\le b$ so $g(b)=f(b)-b\le 0$. By the Intermediate Value Theorem, since $g$ is continuous and $0\in[g(b),g(a)]$ there exists $c\in[a,b]$ such that $g(c)=f(c)-c=0$, so $f(c)=c$ for some $c\in[a,b]$.</p>
|
13,582 | <p>Question: Let $a, b \in \mathbb{R}$ with $a < b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$.</p>
<p>I suppose this has to do with the basic definition of continuity. The definition I am using is that $f$ is continuous at $a$ if $... | JT_NL | 1,120 | <p>For a different approach then the ones above, let us take $a = 0$ and $b = 1$. So assume $f:[0,1] \to [0,1]$ has no fixed point. Then $[0,1] = \{x \in [0,1] : f(x) < x\} \cup \{x \in [0,1] : f(x) > x \}$. Now argue that this is not possible.</p>
|
13,582 | <p>Question: Let $a, b \in \mathbb{R}$ with $a < b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$.</p>
<p>I suppose this has to do with the basic definition of continuity. The definition I am using is that $f$ is continuous at $a$ if $... | goblin GONE | 42,339 | <p>You could also nuke the mosquito: in particular, this is a special case of <a href="https://en.wikipedia.org/wiki/Kakutani_fixed-point_theorem" rel="nofollow noreferrer">Kakutani's fixed point theorem.</a></p>
<p>This works because:</p>
<ul>
<li>By the <a href="https://en.wikipedia.org/wiki/Closed_graph_theorem" r... |
1,004,534 | <p>I'd like to have a translation (in English) of a paper of Klaus Doerk published
in Journal of Algebra: <strong><a href="http://www.sciencedirect.com/science/article/pii/S0021869384711999" rel="nofollow">http://www.sciencedirect.com/science/article/pii/S0021869384711999</a></strong></p>
<p>It is 4,5 pages long with ... | Dietrich Burde | 83,966 | <p>I started as follows:</p>
<p><em>On solvable finite groups, which behave towards the Frattini group like nilpotent groups.</em></p>
<p>We denote by $\phi(G)$ the Frattini-subgroup of a group $G$. If $G$ is a finite nilpotent group, and if $U\le G$ is a subgroup, $N\le U$ a normal subgroup, it is well known that we... |
9,801 | <p>If I write:</p>
<pre><code>Histogram[{1, 1, 1, 2, 2, 2}]
</code></pre>
<p>I get a nice histogram chart</p>
<p>but if I write</p>
<pre><code>Histogram[{"A", "A", "A", "B", "B", "B"}]
</code></pre>
<p>I get an <strong>empty</strong> chart!!</p>
<p>How can I tell mathematica to generate an histogram from <strong>... | Yves Klett | 131 | <p>Perhaps like this:</p>
<pre><code>data = {"A", "A", "B", "B", "B"};
elements = DeleteDuplicates[data];
rep = MapIndexed[# -> #2[[1]] &, elements];
Histogram[data /. rep, ChartStyle -> "Pastel",
ChartLegends -> elements]
</code></pre>
<p><img src="https://i.stack.imgur.com/9Sqrq.png" alt="Mathemat... |
364,394 | <p>I was asked to find the minimum and maximum values of the functions:</p>
<blockquote>
<ol>
<li>$y=\sin^2x/(1+\cos^2x)$;</li>
<li>$y=\sin^2x-\cos^4x$.</li>
</ol>
</blockquote>
<p>What I did so far:</p>
<ol>
<li><p>$y' = 2\sin(2x)/(1+\cos^2x)^2$<br />
How do I check if they are suspicious extrema points? ... | Aloizio Macedo | 59,234 | <ol>
<li><p>By taking $S^n$ to be the union of </p></li>
</ol>
<p>$$A_1:=\{x=(x_1,...,x_{n+1}) \mid ||x||=1, ~ x_{n+1} \geq 0\},$$
$$A_2:=\{x=(x_1,...,x_{n+1}) \mid ||x||=1, ~x_{n+1} \leq 0\},$$</p>
<p>we have that $S^n$ is the union of two connected sets with one point in common (in fact, a lot of points in common)... |
2,972,235 | <p>Which function grows faster </p>
<p><span class="math-container">$()= 2^{^2+3}$</span> and <span class="math-container">$() = 2^{+1}$</span></p>
<p>by using the limit theorem I will first simplify </p>
<p>then I will just get <span class="math-container">$$\lim_{n \to \infty} \dfrac{2^{n^2+3n}}{2^{n+1}}=\lim_{n \... | user | 505,767 | <p><strong>HINT</strong></p>
<p>You conclusion is correct but that step is wrong</p>
<p><span class="math-container">$$\lim_{n \to \infty}= \dfrac{2^{n^2+3n}}{2^n+1}\color{red}{=\lim_{n \to \infty} 2^{n^2+3n-n-1}}$$</span></p>
<p>you could use that <span class="math-container">$2^n+1\le 2^{n+1}$</span> and therefore... |
2,972,235 | <p>Which function grows faster </p>
<p><span class="math-container">$()= 2^{^2+3}$</span> and <span class="math-container">$() = 2^{+1}$</span></p>
<p>by using the limit theorem I will first simplify </p>
<p>then I will just get <span class="math-container">$$\lim_{n \to \infty} \dfrac{2^{n^2+3n}}{2^{n+1}}=\lim_{n \... | Dr. Sonnhard Graubner | 175,066 | <p>It is <span class="math-container">$$\frac{2^{n^2}\cdot 2^{3n}}{2^n\left(1+\frac{1}{2^n}\right)}=\frac{2^{n^2+2n}}{1+\frac{1}{2^n}}$$</span></p>
|
4,642,102 | <blockquote>
<p><strong>Edits:</strong> Parcly Taxel first discovered that the length of cycle cannot be constrained by the 3-connected condition. However, I think
the construction proposed by kabenyuk later is wonderful. Therefore, I
choose it as the best answer. But this does not mean that Parcly
Taxel's construction... | freakish | 340,986 | <p>Denote by <span class="math-container">$[X,Y]$</span> the set of homotopy classes of maps between <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>. Then</p>
<p><span class="math-container">$$[S^2,\mathbb{T}]=[S^2,S^1\times S^1]\equiv [S^2,S^1]\times [S^2,S^1]=\{*\}$$</span></p>
<p>... |
205,871 | <p><strong>Question.</strong> Do you know a specific example which demonstrates that the tensor product of monoids (as defined below) is not associative?</p>
<hr>
<p>Let $C$ be the category of algebraic structures of a fixed type, and let us denote by $|~|$ the underlying functor $C \to \mathsf{Set}$. For $M,N \in C$... | Zhen Lin | 5,191 | <p><strong>EDIT.</strong> This doesn't work: see the comments.</p>
<p>I'm not sure what you're thinking of when you say "usual proof of the associativity", but the one I have in mind doesn't use commutativity.</p>
<p>Define a multihomomorphism of algebras to be a function of finitely many variables that is a homomorp... |
154,215 | <p>Prove that</p>
<ol>
<li>Each field of characteristic zero contains a copy of the rational number field.</li>
<li>For an $n$ by $n$ matrix $A,$ if it is not invertible, then there exists an $n$ by $n$ matrix $B$ such that $AB=0$ but $B\ne0.$</li>
</ol>
<p>For (1), I think I have to use the fact that each subfield o... | Simon Markett | 30,357 | <p>For the first question you can start right at the definition and build your way up from there:</p>
<p>a) Any field contains $1\neq 0$</p>
<p>b) Since the characteristic is zero it contains $1\neq 1+1\neq 1+1+1\neq \cdots$ hence a copy of the natural numbers.</p>
<p>c) It contains inverses wrt addition and multipl... |
3,214,662 | <p><strong>Q1</strong> Prove that every simple subgroup of <span class="math-container">$S_4$</span> is abelian.</p>
<p><strong>Q2</strong> Using the above result, show that if <span class="math-container">$G$</span> is a nonabelian simple group then every proper subgroup of <span class="math-container">$G$</span> has... | Walter Tross | 134,496 | <p>Your question is a bit misleading, since it hints to a possible solution of your problem. To solve your problem this way, you would apply, e.g., the answer by CY Aries twice. But your original problem is to determine whether point A is within the green strip (defined by points B and C) or not. To solve it, you can c... |
3,043,996 | <p><strong>Exercise :</strong></p>
<blockquote>
<p>Let <span class="math-container">$X$</span> be a normed space. Prove that for all <span class="math-container">$x \in X$</span> there exists <span class="math-container">$f \in X^*$</span>, such that <span class="math-container">$f(x) = \|x\|^2$</span> and <span cla... | supinf | 168,859 | <p><strong>Hint</strong>:</p>
<p>Fix an <span class="math-container">$x\in X$</span>. If <span class="math-container">$x=0$</span> then the answer is rather easy, so we can assume <span class="math-container">$x\neq 0$</span>.</p>
<p>Next, define a suitable functional on the linear hull of <span class="math-container... |
67,171 | <p>I am sure <a href="http://en.wikipedia.org/wiki/Modular_multiplicative_inverse">all those symbols</a> are really easy for you guys to understand, but I would appreciate it if someone could bring it down to earth for me.</p>
<p>How could I do this on a basic calculator? or with a few lines of programmer's code which... | J. M. ain't a mathematician | 498 | <p>Here's old JS code I had for computing the modular inverse of <code>a</code> with respect to the modulus <code>m</code>, based on a modification of the usual Euclidean algorithm. I must admit that I've forgotten the provenance of this algorithm, so I'd appreciate if somebody could point me to where this modification... |
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