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 |
|---|---|---|---|---|
3,841 | <p>Taking tori in symmetric products and "miraculously" proving that the Floer homology is independent of choices always seemed, well, miraculous. Some time ago Max Lipyanski explained to me the origins of this construction from gauge theory on surfaces, a la Atiyah-Floer conjecture, which I have then forgotten. What ... | Spinorbundle | 675 | <p>I'm far away from being an expert, but I think the Heegaard Floer homology was invented by Peter Ozsváth and Zoltán Szabó, so I would recommend the following link to you: <a href="http://www.math.princeton.edu/~szabo/clay.pdf" rel="nofollow">click me</a></p>
<p>If this Introduction is not enough, you should perhaps... |
274,672 | <p>I need a function which initially falls slowly, then quickly and then slowly again. </p>
<p>Shape should be like tan but I want to be able to control the gradient</p>
<p>Properties needed:</p>
<p>$x = 0, y=0$</p>
<p>As $x$ increases $y$ decreases</p>
<p>As $x \rightarrow \infty$, $y \rightarrow -1$</p>
<p>(No ... | Community | -1 | <p><a href="http://www.wolframalpha.com/input/?i=y+%3D+e%5E%28-x%29-1+from+0+to+5" rel="nofollow">$$y = e^{-ax}-1$$</a>
<a href="http://www.wolframalpha.com/input/?i=y+%3D%281%2Be%5E%28-2%2a5%29%29%2F%281%2Be%5E%282%2a%28x-5%29%29%29-1+from+0+to+10" rel="nofollow">$$y = \dfrac{1+e^{-ab}}{1+e^{a(x-b)}}-1$$</a>
should do... |
366,103 | <p>As a part of my current homework assignment, I am to derive the first variation of energy identity. Working out the problem with my friends, we came to exactly the same argument as presented in <a href="http://www-personal.umich.edu/%7Ewangzuoq/635W12/Notes/Lec%2024.pdf" rel="noreferrer">these notes</a> (I have cut ... | Ted Shifrin | 71,348 | <p>Remember that $\nabla_X h=dh(X)$ is just the directional derivative. $\dfrac{\partial f}{\partial s}$ is the vector field giving the $s$ variation. </p>
|
2,612,538 | <p>I want to find the Eigenvalues and Eigenvectors of the following matrix $$A=\begin{pmatrix}0&1\\-3&0\end{pmatrix}.$$</p>
<p>I'm able to calculate the Eigenvalues, which are simply the zero points of A's characteristic polynom $\lambda^2 + 3$:<br>
$$\lambda_{1,2} = \pm i\sqrt{3}\\$$ </p>
<p>But how do I cal... | Hagen von Eitzen | 39,174 | <p>Note that
$$\begin{pmatrix}0&1\\-3&0\end{pmatrix}\begin{pmatrix}1\\y\end{pmatrix}=\begin{pmatrix}y\\-3\end{pmatrix} $$
so you want $y=\lambda$ (and you also want $-3=\lambda y$).</p>
|
2,167,541 | <p>Is this expression true?$$y(x) =\lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h} = \lim_{h \rightarrow 0}\frac{f(x)-f(x-h)}{h}$$</p>
<p>I'm taking the limit of h goes to 0 for y(x) and I got the latter equation. Can I just write y(x) = df(x)/dx?</p>
| farruhota | 425,072 | <p>To see the details:
<span class="math-container">$$2\sin x\cos x=2\left(\frac x{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\frac{x^9}{9!}-\cdots \right)\left(\frac1{0!}-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-\cdots \right)=\\
2\left(\frac1{0!1!}\right)x-2\left(\frac{1}{0!3!}+\frac{1}{1!2!}... |
3,154,947 | <p>Hello MathStackExchange Community, </p>
<p>Confused about how to change the notation-formula for permutations when you have more complex processes. i.e. multiple independent action choices of different types of objects and orientation limitations. </p>
<ul>
<li><em>The generic formula is : <span class="math-contai... | Ross Millikan | 1,827 | <p>Have you plotted it? I don't find a loop. I find an arc below the <span class="math-container">$x$</span> axis from <span class="math-container">$t=0$</span> to <span class="math-container">$t=4$</span>. This is from <span class="math-container">$x=33$</span> to <span class="math-container">$x=49$</span>. Mayb... |
2,927,779 | <p>I am trying to understand the idea of a function having compact support. I was looking at this <a href="https://math.stackexchange.com/questions/1147407/definition-of-compact-support">post</a> among other places.</p>
<p>If <span class="math-container">$f$</span> has compact support on <span class="math-container">$... | CopyPasteIt | 432,081 | <p>Once a unit of measure is chosen, multiplication of <span class="math-container">$x \gt 0$</span> and <span class="math-container">$y \gt 0$</span> <em>is</em> the area of a rectangle with one side of length <span class="math-container">$x$</span> and the other side of length <span class="math-container">$y$</span>.... |
2,307,399 | <blockquote>
<p>How to solve $$\sum_{i=1}^n \lfloor e\cdot i \rfloor $$
For a given $n$. </p>
</blockquote>
<p>For example, if $n=3$, then the answer is $15$, and it's doable by hand. But for larger $n$ (Such as $10^{1000}$) it gets complicated . Is there a way to calculate this summation?</p>
| achille hui | 59,379 | <p>Following are three possible ideas, the first two are not that satisfactory.<br>
The third one is a modification of the second ideas which might work.<br>
I hope they can inspire others to create something that is useful.</p>
<p><strong>As a Fourier series</strong></p>
<p>First, we can rewrite $\lfloor x \rfloor$ ... |
3,789,584 | <p>I tried to find the limit <span class="math-container">$$\mathop {\lim }\limits_{n \to \infty } \ln (n) \cdot \int_0^1 \ln ( n^{-t} + 1 ) \, \mathrm dt$$</span>, but I couldn't find a conclusive argument.</p>
<hr />
<p>By assuming that the sequence <span class="math-container">$\displaystyle x_n = \ln (n) \cdot \int... | Axion004 | 258,202 | <p>Substitute <span class="math-container">$x=t\ln(n)$</span>, then the integral becomes</p>
<p><span class="math-container">$$\lim_{n\to\infty}\int_0^{\ln(n)}\ln\left(1+e^{-x}\right)\,dx=\int_0^{\infty}\ln\left(1+e^{-x}\right)\,dx.$$</span></p>
<p>Now</p>
<p><span class="math-container">$$\int_0^\infty \ln(1+e^{-x})dx... |
11,267 | <p>John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics. </p>
<p>Did he mean that with experience and practice, one obtains understanding? </p>
| Community | -1 | <p>In my opinion, what many people mean by the word "understand" simply isn't practical or relevant to mathematics. For example, it often carries the connotation that understanding something means reducing it to something obvious (e.g. something the <em>speaker</em> can "picture"), or that understanding is about what s... |
11,267 | <p>John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics. </p>
<p>Did he mean that with experience and practice, one obtains understanding? </p>
| jdhall | 561,165 | <p>I became aware of this quote just now and am enjoying the conversation :-)
My first impression about the quote was that we relate 'understanding' to something we experience in the physical world. But, when it comes to mathematics, it is an algebraic system beginning with some axioms and then we develop a theory arou... |
389,532 | <p>I'm working out the following problem form Ahlfors' Complex Analysis text:</p>
<p>"Let $X$ and $Y$ be compact sets in a complete metric space $(S,d)$. Prove that there exist $x \in X,y \in Y$ such that $d(x,y)$ is a minimum."</p>
<p>My attempt:</p>
<p>Define $E:=\{d(x,y): x \in X,y \in Y \}$. We will prove that $... | Plop | 71,418 | <p>The boundedness of $E$ isn't really the problem. We only need $E$ to be bounded below, which is the case : it is the set of the values of a positive function.</p>
<p>The main part is to prove that $E$ is closed. And the fact that a bounded closed subset of $\mathbb{R}$ contains its minimum isn't really a famous the... |
2,297,542 | <blockquote>
<p>Given integers $a,b,c,d$ with $d\not\equiv0\bmod5$ and $m$ an integer for which $am^3+b m^2+cm+d\equiv0\bmod5$, prove that there exists an integer $n$ for which $dn^3+cn^2+bn+a\equiv0\bmod5$.</p>
</blockquote>
| Jean Marie | 305,862 | <p>If $x \geq 0, F(x)=1$.</p>
<p>Thus we have only to consider the case $x<0$.</p>
<p>By definition of a cdf, $F_X(x):=P(X<x)=1-(1-e^x)_+$</p>
<p>Taking the contrary event: $P(X\geq x)=(1-e^x)_+$</p>
<p>which is equivalent to</p>
<p>$$P(X\geq x)=1-e^x$$</p>
<p>(because for $x<0$, $e^{x}<1,$ thus $1-e^... |
522,179 | <p>Everyone says that the dot product is interpreted as the projection of A onto B (if you are dot producting A and B), but isn't that length just equal to |A|$\cos \left( \theta \right)$? Why does the dot product have an extra |B|?</p>
| Michael Hoppe | 93,935 | <p>If the length of $B$ is $1$ then $\langle A,B\rangle$ is the coordinate of $A$ in direction $B$, not the projection of $A$ onto $B$; that would be $\langle A,B\rangle B$.</p>
<p>There is a nice interpretation of the scalar product where $B$ has arbitrary length. Let $B=(b_1,b_2)$, then define $J(B):=(-b_2,b_1)$; y... |
2,607,449 | <p>I'm trying to get intuition about why gradient is pointing to the direction of the steepest ascent. I got confused because I found that directional derivative is explained with help of gradient and gradient is explained with help of directional derivative.</p>
<p>Please explain what are the exact steps that lead fr... | N8tron | 32,820 | <p>Let $s$ be a distance $u=\left<u_1,u_2,\ldots,u_n \right>$ a unit vector, $P(a_1,a_2,\ldots,a_n)$ a point and and $f(x_1,x_2,\ldots, x_n)$ be a real valued function differentiable in an open neighborhood $D$ of $P$ </p>
<p>Then parameterize a line distance $s$ away from $P$ in the direction of $u$</p>
<p>$... |
65,865 | <p>There are so many different optimization algorithms out there, and lots of research going on. However, I have difficulties to find good comparison between them, and all articles / books / papers seem to evaluate them in different ways.</p>
<p>Isn't there some set of standard functions to run "standardized" benchmar... | Dominique | 5,155 | <p>The CUTEr collection (<del><a href="https://magi-trac-svn.mathappl.polymtl.ca/Trac/cuter/wiki" rel="nofollow noreferrer">https://magi-trac-svn.mathappl.polymtl.ca/Trac/cuter/wiki</a></del>) bundles over 1000 optimization problems including the Hock and Schittkowski collection, the Maros and Meszaros collection, and ... |
17,885 | <p>In most education systems, Mathematics is a compulsory subject from primary school all the way to the start of university. A common reason given is that essential concepts like addition and multiplication are taught to the children. </p>
<p>But for many high school students, especially those who are keen on pursuin... | Daniel R. Collins | 5,563 | <p>I'll make some observations from a Western perspective (which may or may resonate with a Chinese student, say). Consider: Mathematics has been at the forefront of all education forever -- since long before the idea of compulsory education itself was conceived. </p>
<p>In <a href="https://en.wikipedia.org/wiki/Educa... |
3,043,699 | <p>I'm having difficulty understanding how below integral is evaluated : </p>
<p><span class="math-container">$$\int_0^1\frac{(1-y)^2}{2}+(1-y)y dy = \frac{-1(1-y)^3}{6}+\frac{y^2}{2}-\frac{y^3}{3}\bigg|^1_0$$</span></p>
<p>What are steps involved in this evaluation ?</p>
<p>For <span class="math-container">$\frac{(... | clathratus | 583,016 | <p><span class="math-container">$$
\begin{align}
I=&\int\bigg(\frac{(1-x)^2}{2}+x(1-x)\bigg)dx\\
=&\frac12\int(1-x)^2dx+\int x(1-x)dx\\
=&\frac12\int(1-x)^2dx+\int xdx-\int x^2dx\\
\end{align}
$$</span>
The first one:
<span class="math-container">$$I_1=\int(1-x)^2dx$$</span>
Substitution:
<span class="math-... |
1,313,216 | <p>Its my first time on here and my maths is poor so please be kind. I am working on a Masters dissertation focused on document clustering methods in which I would like to apply a weight based on the time interval between two documents. </p>
<p>I am looking for some help coming up with a function to express a time int... | 5xum | 112,884 | <p>Do you have a maximum and minimum date? If so, I suggest you simply divide the number by the number of days between the maximum and the minimum.</p>
|
3,798,402 | <p>For any natural number <span class="math-container">$n$</span>, how would one calculate the integral</p>
<p><span class="math-container">$$ \int_{0}^{2 \pi} |1 - ae^{i\theta}|^n \ d \theta $$</span></p>
<p>where <span class="math-container">$a$</span> is a complex number such that <span class="math-container">$|a| ... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$|1-e^{i\theta}|^{2}=(1-\cos (\theta))^{2}+\sin ^{2} (\theta)=2-2\cos (\theta)=4\sin^{2}(\frac {\theta} 2)$</span> If <span class="math-container">$n=2m$</span> and <span class="math-container">$a=1$</span> the given integral becomes <span class="math-container">$4^{2m}\int_0^{2\pi} (\si... |
72,994 | <p>I am not able to solve this question from chapter "Similar & Congruent Triangles" in my book.</p>
<p>Can some one help to calculate <strong>AC</strong>?</p>
<p><img src="https://i.stack.imgur.com/5ZTn9.jpg" alt="HERE">.</p>
| Raskolnikov | 3,567 | <p>Hint: $\triangle ABC \sim \triangle CEB \sim \triangle BDE$.</p>
|
869,114 | <p>I know, stupid question. But humor me for a sec. First off, we know that all real numbers have two numbers which are infinitely close to them, right?
That would seem to be, for any given value of x,</p>
<blockquote>
<p>x ± y</p>
</blockquote>
<p>where </p>
<blockquote>
<p>y = .000...1</p>
</blockquote>
<p>B... | hmakholm left over Monica | 14,366 | <p>Your mistake is in assuming that there is a real number $y=0.000\ldots1$, but actually there is no such thing. The real number system contains no "infinitely small" elements.</p>
<p>It is also wrong when you assert that</p>
<blockquote>
<p>we know that all real numbers have two numbers which are infinitely close... |
123,675 | <p>What are some "applications to" / "connections with" topology that one could hope to reasonably cover in a first course on topos theory (for master students)? I have an idea of what parts of the theory I would like to cover, however, I would love some more nice examples and applications. Of course, I have some ideas... | Dmitri Pavlov | 402 | <p>If one is willing to stretch the question a little,
then one can cite the theorem that states
that the bicategory of Grothendieck toposes is equivalent
to the bicategory of localic groupoids.
This establishes a connection with general topology
and Lie groupoids, presumably making Grothendieck toposes
easier to under... |
1,210,948 | <blockquote>
<p><strong>(Convergence of Types Theorem)</strong> Suppose that $F_n(u_nx+v_n) \Rightarrow F(x)$ and $F_n(a_nx+b_n) \Rightarrow G(x)$, where $u_n>0, a_n>0$ and $F$ an $G$ are non-degenerate. Then there exist $a>0$ an $b \in\mathbb R$ such that $a_n/u_n \to a$ and $(b_n-v_n)/u_n \to b$, and $F(ax... | user227182 | 227,182 | <p>$A^{-1} I$ is just $A^{-1}$, so your equation actually reads $$A^{-1} = \frac{1}{3}(A-2I).$$</p>
|
2,810,755 | <p>I'm trying to set up a double integral on $e^{-(x+y)}$ and the range listed is: $0 < x < y < \infty.$</p>
<p>I'm interpreting this as $0 < x < y$ for $x$'s range and $x < y < \infty$ as $y$'s range. I've put those bounds on my two integrals and proceeded with the mission. However, $y$ is stil... | Bernard | 202,857 | <p>The inductive step consists in showing that
$$\frac{n(n+1)(n+2)}3+(n+1)(n+2)=\frac{(n+1)(n+2)(n+3)}3,$$
which shouldn't be too hard.</p>
|
2,559,639 | <p>In the book of <em>General Topology</em> by Munkres, on page 104, it is given that</p>
<p><a href="https://i.stack.imgur.com/Pv9MU.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Pv9MU.png" alt="enter image description here"></a></p>
<p>However, as far as I know, the lower limit topology <span c... | Luis Prado | 485,730 | <p>$[x,b)\subset (a,b)$ for every $x\in (a,b)$ and $[x,b)$ is open.</p>
|
2,036,278 | <p>I have a question about inverse images and containment. </p>
<p>Let $\;f: A \rightarrow B, D \subseteq A$, and $E \subseteq B$. Show that $\; f^{-1}(B-E) \subseteq A-f^{-1}(E)$. </p>
<p>I have started by letting $x \in f^{-1}(B-E)$ so $f(x) \in (B-E)$, but I am unsure where to go from here.</p>
| Community | -1 | <p>Consider the grouping</p>
<p>$$\frac1{2^m}+\frac1{3^m}+\\
\frac1{4^m}+\frac1{5^m}+\frac1{6^m}+\frac1{7^m}+\\
\frac1{8^m}+\frac1{9^m}+\frac1{10^m}+\frac1{11^m}+\frac1{12^m}+\frac1{13^m}+\frac1{14^m}+\frac1{15^m}+\\\cdots
$$
with doublings of the number of terms.</p>
<p>As every term in a group is bounded above by i... |
2,204,834 | <p>I have to show by induction that this function is a multiple of 8. I have tried everything but I can only show that is multiple of 4, some hints? The function is
$$5^{n+1}+2\cdot 3^n+1 \hspace{1cm}\forall n\ge 0$$, because it is a multiple of 8, you can say that$$5^{n+1}+2\cdot 3^n+1=8\cdot m \hspace{1cm}\forall m\i... | dxiv | 291,201 | <p>Hint: the difference between consecutive terms is:</p>
<p>$$
\require{cancel}
5^{n+2}+2\cdot 3^{n+1}+ \bcancel{1} - (5^{n+1}+2\cdot 3^n+\bcancel{1})
= 5^{n+1}(5-1)+2\cdot 3^n(3-1) = 4 \cdot (5^{n+1}+3^n)
$$</p>
<p>The last factor is a sum of two odd numbers.</p>
|
356,217 | <p>Find the following infinite sum : $$q\sin a+q^2\sin 2a+\ldots+q^n\sin na+\ldots$$ where $|q|<1$ .It would be good if you could find it without the help of any auxiliary sequences using only trigonometric formulas.</p>
| Community | -1 | <p>Note that for $\vert q \vert < 1$
$$\sum_{k=1}^{\infty} (qe^{ia})^k = \dfrac{qe^{ia}}{1-qe^{ia}}$$
Now look at the imaginary part of the above expression.</p>
<hr>
<p><strong>EDIT</strong></p>
<p>If complex numbers are not allowed, then proceed as follows. We have
$$S_n(q,a) = \sum_{k=1}^n q^k \sin(ka)$$
\begi... |
2,362,485 | <blockquote>
<p>Evaluate: $$\lim_\limits{x\to \pi/4} \frac {2-\csc^2 x}{1-\cot x}$$</p>
</blockquote>
<p>My Attempt:
\begin{align}\lim_\limits{x\to \pi/4} \frac {2-\csc^2 x}{1-\cot x}&=\lim_{x\to \pi/4} \frac {2-\csc^2 x}{1-\cot x}\\\\
&=\lim_{x\to \pi/4} \frac {2-\frac {1}{\sin^2 x}}{1-\frac {\cos x}{\sin x... | N. F. Taussig | 173,070 | <p>We use the trigonometric identity $1 + \cot^2x = \csc^2x$ to simplify the expression.
\begin{align*}
\lim_{x \to \frac{\pi}{4}} \frac{2 - \csc^2x}{1 - \cot x} & = \lim_{x \to \frac{\pi}{4}} \frac{2 - (1 + \cot^2x)}{1 - \cot x}\\
& = \lim_{x \to \frac{\pi}{4}} \frac{1 - \cot^2x}{1 - \cot x}\\
& = \lim_{x ... |
4,048,223 | <blockquote>
<p><strong>Problem 20.3</strong>
If <span class="math-container">$R$</span> is a nonnegative random variable, then Markov’s Theorem gives an upper
bound on <span class="math-container">$Pr[R \geq x]$</span> for any real number <span class="math-container">$x \gt Ex[R]$</span>. If b is a lower bound on R, t... | dmh | 352,034 | <p>For part a:</p>
<p><span class="math-container">$$\mathbb P (R - b \geq t) \leq \frac{\mathbb E R - b}{t} \leq \frac{\mathbb E R}{t}$$</span></p>
<p>where the second inequality follows when <span class="math-container">$b \geq 0$</span></p>
<p>For part b:</p>
<p>The RHS of the inequality given is linear in <span cla... |
279,471 | <p>Is it true that any finite dimensional division algebra over a pseudo-algebraically closed field is trivial? We know that this is true for algebraically closed field.</p>
| user05811 | 101,929 | <p>The fact that the Brauer group of a pseudo-algebraically closed field is trivial is noted in: J.Ax, The elementary theory of finite fields, Annals Math. 88 (1968), p. 269. A more detailed proof is given in: M. Fried and M. Jarden, Field Arithmetic (3rd edition), Springer, p. 209.</p>
|
1,593,899 | <p>Let $f = u + iv$ be a holomorphic function on unit (open) disk $\mathbb D^2$.</p>
<p>and Suppose $v$ is nonnegative and $v(0)=0$.</p>
<p>Can we determine all such $f$?</p>
<p>I wanted to use schwarz lemma, but I couldn't catch anything about bound condition.</p>
<p>I think it is about liouville's theorem.</p>
<... | David C. Ullrich | 248,223 | <p>If $0<r<1$ then $$0=v(0)=\frac1{2\pi}\int_0^{2\pi}v(re^{it})dt.$$Since $v\ge0$ (and $v$ is continuous) this shows that $v=0$. So $f$ is real, hence constant.</p>
|
3,562,324 | <blockquote>
<p>Find all positive integers x such that <span class="math-container">$x^4-8x+16$</span> is a full square.</p>
</blockquote>
<p>I presented it as <span class="math-container">$(x-4)^2+x^4-x^2$</span> and found that <span class="math-container">$x = +1$</span> satisfy the condition. Then I equalized it ... | LHF | 744,207 | <p>For <span class="math-container">$x \ge 3$</span>, we have:</p>
<p><span class="math-container">$$x^4-8x+16=x^4-8(x-2)<x^4$$</span></p>
<p>and</p>
<p><span class="math-container">$$x^4-8x+16-(x^2-1)^2=2(x-2)^2+7 > 0$$</span></p>
<p>Therefore <span class="math-container">$x^4>x^4-8x+16>(x^2-1)^2$</spa... |
2,626,454 | <blockquote>
<p>I used Wolfram Alpha to try and compute the integral $$\int_0^{\pi/2}(1+\cos2x)^{-0.5}\,dx$$ and I'm told that the integral does not converge. However, if I try
$$\lim_{a\to0.5^-}\int_0^{\pi/2}\frac{dx}{(1+\cos2x)^a}$$
by substituting in values of $a$ close to $0.5$ e.g $a\in\{0.49,0.499,0.4999...... | Jack D'Aurizio | 44,121 | <p>$$\int_{0}^{\pi/2}\frac{dx}{\sqrt{1+\cos(2x)}}=\int_{0}^{\pi/2}\frac{dx}{\sqrt{2}\left|\cos x\right|}=\frac{1}{\sqrt{2}}\int_{0}^{\pi/2}\frac{dx}{\sin x}\geq\frac{1}{\sqrt{2}}\int_{0}^{\pi/2}\frac{dx}{x}=+\infty.$$
For any $\alpha<\frac{1}{2}$ we have, through Euler's Beta function,
$$ \int_{0}^{\pi/2}\frac{dx}{(... |
2,045,226 | <p>We denote with $\phi$ Euler's Phi Function.</p>
<p><strong>We want to find all $n\in \mathbb{Z^+}: \ \phi (3n)=\phi (4n)=\phi (6n).$</strong></p>
<p><strong>Answer:</strong>
Let $n=2^k3^lm$ : $k,l\in \mathbb{N}, \ m\in \mathbb{Z^+},\ \text{gcd}(m,2)=\text{gcd}(m,3)=1$. Then we have:</p>
<ol>
<li>$\phi (3n)=\phi (... | Dietrich Burde | 83,966 | <p>We have $\phi(3)=\phi(4)=\phi(6)=2$, and for any $n$ coprime to $2$, $3$, $6$, hence to $6$, we have, because $\phi$ is multiplicative
$$
\phi(3n)=\phi(4n)=\phi(6n)=2\phi(n).
$$
If $n$ is divisible by $2$, then $\phi(4n)>\phi(3n)$, so the above equality does not hold. Similarly for the other cases.</p>
|
2,045,226 | <p>We denote with $\phi$ Euler's Phi Function.</p>
<p><strong>We want to find all $n\in \mathbb{Z^+}: \ \phi (3n)=\phi (4n)=\phi (6n).$</strong></p>
<p><strong>Answer:</strong>
Let $n=2^k3^lm$ : $k,l\in \mathbb{N}, \ m\in \mathbb{Z^+},\ \text{gcd}(m,2)=\text{gcd}(m,3)=1$. Then we have:</p>
<ol>
<li>$\phi (3n)=\phi (... | Alex M. | 164,025 | <p>As you begin, let $n = 2^a 3^b m$ with $\gcd (m, 6) = 1$. The given conditions can then be rewritten as $\phi (2^a 3^{b+1}) = \phi (2^{a+2} 3^b) = \phi (2^{a+1} 3^{b+1})$, i.e. $\phi (2^a) \phi(3^{b+1}) = \phi (2^{a+2}) \phi (3^b) = \phi (2^{a+1}) \phi(3^{b+1})$.</p>
<p>As boring as it may be, you need now examine ... |
593,212 | <p><img src="https://i.stack.imgur.com/4P4gT.jpg" alt=""></p>
<p>This is a part of the group of practice problem I've been working on and I'm just lost. I'm really struggling when it comes to these ring problems. Anybody who could lay out an outline for this problem so I can study the structure to help with other prob... | SimpleLikeAnEgg | 103,072 | <p>First, you should split $$y'''+y^2y''-y'=0$$ into a system of three first order ordinary differential equations by letting $$y'= u$$ and $$ u'=v $$ such that you get $$v'+y^2v-u = 0$$</p>
<p>After taking another look, I think I might now see what you are trying to do with Simpson's rule. Starting with the definit... |
3,218,794 | <p>I have <span class="math-container">$f(x)=\cos^{2n}(x)+\sin^{2n}(x),\; n\in \mathbb{N},\; n\geq 2,\;x\in \mathbb{R}$</span>
I need to find the range of the function.
I took <span class="math-container">$n=2$</span> and I got <span class="math-container">$f(x)=1-\frac{1}{2}\sin^{2}(2x)$</span> and the range of this i... | Community | -1 | <p>With <span class="math-container">$t=\cos^2(x)$</span> such that <span class="math-container">$0\le t\le 1$</span>, you bracket</p>
<p><span class="math-container">$$t^n+(1-t)^n.$$</span></p>
<p>The stationary points are the roots of</p>
<p><span class="math-container">$$t^{n-1}-(1-t)^{n-1}=0$$</span> or <span cl... |
607,353 | <p>So I've recently been looking at the <a href="http://www.photographyblogger.net/stereographic-projection-photography-100-little-planets/" rel="nofollow">Tiny Planet</a> images.</p>
<p>I've been googling a few things to try and find out how images are converted from normal to a tiny planet. Some phone apps, as well ... | Aetherus | 409,252 | <p>DISCLAIMER: I'm not sure if this is the original Photoshop algorithm but this looks pretty good to me.</p>
<p>If you have a spherical panoramic photo like this (original photo taken by Javierdebe, on <a href="https://www.flickr.com/photos/javierdebe/4697199677/in/photolist-MZs5n3-PdYu7h-dB2A4G-8zFrnm-7Ve3ai-Jpop93-... |
1,820,853 | <p>How to derive the Gregory series for the inverse tangent function?
Why is Gregory series applicable only to the set $ [-\pi/4,\pi/4] $ ?</p>
| Paramanand Singh | 72,031 | <p>You need to know that $$\arctan x = \int_{0}^{x}\frac{dt}{1 + t^{2}}\tag{1}$$ In fact this equation can be used as a starting point to develop the theory of circular functions. But let's not worry about that here.</p>
<p>We know (via the formula for sum of a geometric progression) that $$1 - t^{2} + t^{4} - t^{6} +... |
3,681,254 | <p><span class="math-container">$$\int_{0}^{\frac{\pi}{2}}{e^{-x}} \cos3xdx$$</span>
using the intergral by parts formula
<span class="math-container">$$\int{f(x).g(x)'}\mathrm{d}x = (f(x).g(x)) - \int{f'(x)g(x)dx}$$</span>
so it should be
<span class="math-container">$${{{e^{-x}}.\frac{1}{3}\cos(3x)}} - \int_{0}^{\fra... | hamam_Abdallah | 369,188 | <p><span class="math-container">$$\cos(3x)=Re(e^{3ix})$$</span></p>
<p>let us compute</p>
<p><span class="math-container">$$\int_{0}^{\frac{\pi}{2}}e^{x(-1+3i)}dx=$$</span></p>
<p><span class="math-container">$$\Bigl[\frac{1}{-1+3i}e^{(-1+3i)x}\Bigr]_0^{\frac{\pi}{2}}$$</span></p>
|
2,528,313 | <p>I need the remainder of the following division.
The only thing I managed to get so far is:</p>
<p>\begin{align}
& \frac{x^{2017}}{x^2-3x+2}=\frac{x^{2017}+x^{2016}-x^{2016}+\dots+x-x-1}{(x-2)(x-1)} + \frac{1}{(x-2)(x-1)} \\[10pt]
= {} & \frac{(x-1)\left(x^{2016}+x^{2015}+\dots+x+1\right)}{(x-2)(x-1)}+\frac{... | Bernard | 202,857 | <p>Euclidean division helps: let $P(x)$ be any polynomial.</p>
<ul>
<li>Division by $x-1$ yields a polynomial $Q(x)$ such that
$$P(x)=(x-1)Q(x)+P(1).\tag{1}$$</li>
<li>Division of $Q(x)$ by $x-2$ yield a polynomial $S(x)$ such that
$$Q(x)=(x-2)S(x)+Q(2).$$</li>
</ul>
<p>We deduce that $$P(x)=(x-1)(x-2)S(x)+Q(2)(x-1)+... |
2,528,313 | <p>I need the remainder of the following division.
The only thing I managed to get so far is:</p>
<p>\begin{align}
& \frac{x^{2017}}{x^2-3x+2}=\frac{x^{2017}+x^{2016}-x^{2016}+\dots+x-x-1}{(x-2)(x-1)} + \frac{1}{(x-2)(x-1)} \\[10pt]
= {} & \frac{(x-1)\left(x^{2016}+x^{2015}+\dots+x+1\right)}{(x-2)(x-1)}+\frac{... | Gerry Myerson | 8,269 | <p>$x^{2017}=(x^2-3x+2)Q(x)+ax+b$. Now evaluate at $x=1$, then at $x=2$, geting two equations in two unknowns. </p>
|
1,544,789 | <blockquote>
<p>Calculate $\int_0^{\infty} \frac{x^{2N+1}}{a+x^{-b}} e^{-c x^2} dx$ where $a,c > 0$ and $b>1$.</p>
</blockquote>
<p>The best I could do about this integral is to find an upperbound for it:</p>
<p>$\int_0^{\infty} \frac{x^{2N+1}}{a+x^{-b}} e^{-c x^2} dx \leq \int_0^{\infty} \frac{x^{2N+1}}{x^{-... | TacoTesseract | 251,603 | <p>What your integral finds is the volume under the surface $z = x$ above your quadrilateral. If you wanted to find the area of your domain (the quadrilateral) use $z = 1$ as your integrand.</p>
|
1,990,865 | <blockquote>
<p>Given set $\{3 - x \mid x < 0\}$, find if it is bounded.</p>
</blockquote>
<p>So, from intuition it should be bounded from below, but how to show it?</p>
| carmichael561 | 314,708 | <p>If $x<0$ then $3-x>3$, so the set in question is the open interval $(3,\infty)$. Hence it is bounded below (by $3$ for instance) but not above.</p>
|
650,291 | <p>I'm not so good at combinatorics, but I want to know if my answer for this question is right. Originally this question is written in spanish and it says:</p>
<blockquote>
<p>Se dispone de una colección de 30 pelotas divididas en 5 tamaños distintos y 6 colores diferentes de tal manera que en cada tamaño hay los s... | heropup | 118,193 | <p>There are $\binom{5}{2} = 10$ ways to choose two distinct sizes. For each of those ways, there are $\binom{6}{2} = 15$ ways to choose two distinct colors for the first pair, and another $\binom{6}{2}$ ways to choose two distinct colors for the second pair. So your answer is $2250$.</p>
|
4,200,164 | <blockquote>
<p>How do you show that for <span class="math-container">$x \in (0,2 \pi)$</span> the following series converges?
<span class="math-container">$$\sum_{n=1}^\infty \frac{\cos(xn)}{n}$$</span></p>
</blockquote>
<p>Of course, this series doesn't converge absolutely. For <span class="math-container">$x= \pi$</... | sirous | 346,566 | <p>Comment:</p>
<p><span class="math-container">$$y(x)=\cos x +\frac {\cos 2x}2+\frac {\cos 3x}3 +\cdots +\frac{ \cos nx}{n}.$$</span></p>
<p>Take derivative of all term and sum up you get:</p>
<p><span class="math-container">$$y'(x)=-(\sin x+ \sin 2x +\sin 3x +\cdots+ \sin nx)=-\frac{\sin\left(\frac{n+1}2x\right) \sin... |
13,230 | <p>Introduction:</p>
<p>Let A be a subset of the naturals such that <span class="math-container">$\sum_{n\in A}\frac{1}{n}=\infty$</span>. The <a href="https://en.wikipedia.org/wiki/Erd%C5%91s_conjecture_on_arithmetic_progressions" rel="nofollow noreferrer">Erdos Conjecture</a> states that A must have arithmetic progr... | Alex Rice | 3,632 | <p>For those interested, here is the most elementary proof that sets of the natural numbers of positive upper density necessarily have divergent reciprocal sums, r.e. Pete's discussion above.</p>
<p>Suppose <span class="math-container">$A \subset \mathbf{N}$</span> and <span class="math-container">$\displaystyle\limsu... |
2,579,074 | <p>When I try to factor the quadratic form, I end up with </p>
<p>$$6x^2+4y^2+2z^2+4xz-4yz = 2((x+z)^2+2x^2+(y-z)^2-z^2)$$ </p>
<p>which does not ensure that $f(x,y,z) \geq 0$ for all $x, y, z \geq 0$ since the $z$ term is negative. How should these kinds of problems be tackled?</p>
| aid78 | 511,281 | <p>$$6x^2+4y^2+2z^2+4xz−4yz=(4x^2+z^2+4xz)+(4y^2+z^2−4yz)+2x^2=(2x+z)^2+(2y-z)^2+2x^2\ge 0$$</p>
|
322,598 | <p><a href="https://en.wikipedia.org/wiki/Partially_ordered_set" rel="noreferrer">Partially ordered sets</a> (<em>posets</em>) are important objects in combinatorics (with <a href="https://gilkalai.wordpress.com/2019/02/05/extremal-combinatorics-v-posets/" rel="noreferrer">basic connections to extremal combinatorics</a... | Sam Hopkins | 25,028 | <p>Here's a problem that I believe has little chance of being resolved, and it's also not so clear to me what the motivation behind the problem is, but it involves some very pretty algebraic combinatorics.</p>
<p>Stanley (in his PhD thesis) defined a poset <span class="math-container">$P$</span> to be <em>Gaussian</em... |
2,565,932 | <p>Completion of rational number at primes is clear. But when it comes to quadratic field is little tricky.
Can any one provide a material or illustrate, how does a quadratic field, when completed at a prime ideal becomes a p-adic field.</p>
| nguyen quang do | 300,700 | <p>To your specific question about the completions of quadratic fields, because the degree is just 2,the general theory easily gives complete and neat answers. Let $=\mathbf Q (\sqrt d)$, where $d$ is a square free integer. The decomposition of a rational prime $p$ in the ring of integers $R$ of $K$ is completetely kno... |
2,509,583 | <p>How is intermediate value theorem valid for $\sin x$ in $[0,\pi]$?</p>
<p>It has max value $1$ in the interval $[0,\pi]$ which doesn't lie between values given by $\sin0$ and $\sin\pi$.</p>
| David K | 139,123 | <p>In general, any time you have a theorem, "If ____, then ____," it really means, "If ____, then ____ and maybe some other stuff not mentioned here also happens."
Because when you're working in just about any branch of mathematics,
no matter how thoroughly you describe the implications of any mathematical fact there i... |
2,662,007 | <p>I'd like to show that the group given by presentation $G=\langle a,b \ | \ a^2b^2 \rangle$ is not isomorphic to any of $\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}$, or $\mathbb{Z} \times \mathbb{Z}$.</p>
<p>For $G \ncong \mathbb{Z}$, I note that $G$ is not free because it has a relator in its presentation.</p>
<p>For $G... | Pece | 73,610 | <p>You can also invoke the following theorem : </p>
<blockquote>
<p>Two categories have equivalent categories of presheaves if and only if they have equivalent Cauchy completion.</p>
</blockquote>
<p>Then remark that a group (seen as a category with one object) is Cauchy complete : its only idempotent is the neutra... |
2,662,007 | <p>I'd like to show that the group given by presentation $G=\langle a,b \ | \ a^2b^2 \rangle$ is not isomorphic to any of $\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}$, or $\mathbb{Z} \times \mathbb{Z}$.</p>
<p>For $G \ncong \mathbb{Z}$, I note that $G$ is not free because it has a relator in its presentation.</p>
<p>For $G... | Julian Quast | 298,866 | <p>I think you already know this, but for the sake of completeness let me show how to prove the claim using your categorical approach.</p>
<p>For any category <span class="math-container">$\mathcal C$</span> equivalent to the category of left <span class="math-container">$G$</span>-sets for some group <span class="mat... |
247,622 | <p>Find the dimension of the vector subspace in $\mathbb{R}^{4}$ spanned by the columns </p>
<p>$$
\begin{pmatrix}
1 & 1 & 2 & 4 \\
t & 3 & 2 & 8 \\
0 & 0 & 2 & 2 \\
0 & 0 & 0 & 0
\end{pmatrix}
$$</p>
<p>(discussion over real $t$).</p>
<p>Thanks :) </p>
| Community | -1 | <p>If $f(x) \neq x$, then since $X$ is Hausdorff, there exist open sets $V$ and $W$ such that $x \in V$, $f(x) \in W$ and $V \cap W = \emptyset$. Moreover, since $f$ is continuous, there exists a neighborhood $V'$ of $x$ such that $f(V') \subset W$. Let $V_x = V \cap V'$. This means that $f(y) \neq y$ for all $y \in V_... |
2,258,281 | <p>I have the following relation symbolized as $\sim$, defined on $\mathbb{Z}\times\mathbb{Z}$: </p>
<p>$$x\sim y \iff |x-y|=1$$</p>
<p>I would like to know how the transitive-reflexive closure looks like in this case, because this relation isn't transitive or reflexive, is it just an empty set?</p>
<p>In that case,... | shardulc | 140,607 | <p>In mathematical logic, we have the concept of a 'contrapositive': the contrapositive of the statement $a \implies b$ is $\neg b \implies \neg a$, and the two are equivalent. To intuitively understand why, replace $a$ with 'this is a tree' and $b$ with 'this contains wood'. To rigorously prove this can be a good exer... |
2,258,281 | <p>I have the following relation symbolized as $\sim$, defined on $\mathbb{Z}\times\mathbb{Z}$: </p>
<p>$$x\sim y \iff |x-y|=1$$</p>
<p>I would like to know how the transitive-reflexive closure looks like in this case, because this relation isn't transitive or reflexive, is it just an empty set?</p>
<p>In that case,... | StuartMN | 439,545 | <p>You can not get help unless you say what rules of inference you are permitting .there are many deductive systems .</p>
|
3,691,747 | <p>Consider a random sample of 11 letters chosen from the alphabet with replacement. What is the probability that the letters can be arranged to spell ‘mississippi’.</p>
<p>For this question, the number of ways to choose 11 letters would be (26)^(11).
To spell 'mississippi',we need:
1M, 4I. 4S and 2P.</p>
<p>Hence, t... | zipirovich | 127,842 | <p>Your denominator is correct, but your numerator of "<span class="math-container">$11!$</span>" isn't.</p>
<p>This — the <code>MISSISSIPPI</code> thing — is (imho) the most classis example of permutations with repetition. You have several objects, say <span class="math-container">$n$</span> of them, and ... |
2,957,440 | <p>Knowing <span class="math-container">$f(1) = 2, f(4)=7,f'(1) = 5,f'(4)=3$</span></p>
<p>Find
<span class="math-container">$\int_1^4xf''(x)dx$</span></p>
<p>It is obvious that if there was no <span class="math-container">$x$</span> the answer would be <span class="math-container">$f'(4) - f'(1)$</span> from the fun... | Sean | 604,734 | <p>We can just use integration by parts. Just integrate double prime and differentiate the x part, and you will have one more integral with one prime. Repeat the step and you should get the answer.</p>
|
1,350,733 | <p>The following (multiple choice) problem is from a test review.</p>
<blockquote>
<p>For the given matrix $A$, find a basis for the corresponding eigenspace for the given eigenvalue.</p>
<p>$$A = \begin{bmatrix}1 & 6 & 6 \\ 6 & 1 & -6 \\ -6 & 6 & 13\end{bmatrix};\quad \lambda = 7.$$</p>... | Brian Fitzpatrick | 56,960 | <p>As @GTonyJacobs points out in the comments, you are correct but the key is also correct. Vector spaces can have many different bases!</p>
<p>Note that you have proven that $A-7I$ has rank one so the rank-nullity theorem implies that the nullspace of $A-7I$ has dimension two. This allows us to immediately eliminate ... |
1,378,769 | <p>The equation is given by</p>
<p>$$ \sum_{n=1}^N \min(\gamma, \beta a_n)=N$$
where $\beta$ is the variable with $\beta\in[0,\sqrt\gamma/\min(a_n\mid a_n>0)]$, $ \gamma $ is a constant with $1\le\gamma\le N$, and $a_n$ is an nonnegative constant.</p>
<p>I know that the function $f(\beta)=\sum_{n=1}^N \min(\gamm... | Hagen von Eitzen | 39,174 | <p>There seems to be one step missing:</p>
<ul>
<li>ordered pairs (the elements of the cartesian products) can be reduced to sets</li>
</ul>
<p>Following Kuratowski we can <a href="https://en.wikipedia.org/wiki/Ordered_pair" rel="noreferrer"><em>define</em></a>
$$ (x,y):=\{\{x\},\{x,y\}\}$$
which may look somewhat ar... |
483,602 | <p>$$
\large\sqrt{8+\sqrt{5}-\sqrt{6-2\sqrt{5}}}
$$</p>
<p>I am asked to write the following as a ratio of two integers (i.e.,) show that this value is rational
(Do not use calculator)</p>
<p>What I tried doing was putting everything to the exponent of $.5$ and then was unsure how to proceed </p>
| Pedro | 23,350 | <p>Since you have already been given an answer, let me share this nice algorithm to denest radicals due to Bill Dubuque. </p>
<p>Write $\alpha={a+b\sqrt c}$ and denote $\alpha'={a-b\sqrt c}$. We denote the product $\alpha\cdot \alpha'=a^2-cb^2$ by the symbol $|\alpha|$, and we call this the <em>norm</em> of $\alpha$. ... |
916,705 | <p>I came across an example in Chapter-2 of Dummit and Foote(page-47) which says :$D_6$ is not a subgroup of $D_8$ ,the former is not even a subset of latter.I can't understand why is it not the subset of $D_8$?
How do we define one group as a subset of another?</p>
| Quang Hoang | 91,708 | <p>$D_{2n}$ is usually defined as the group of symmetries of the regular $2n$-gon. That is $D_6$ acts on a hexagon, while $D_8$ acts on a octagon. You can write them down explicitly and see why $D_6$ is not a subset of $D_8$.</p>
|
916,705 | <p>I came across an example in Chapter-2 of Dummit and Foote(page-47) which says :$D_6$ is not a subgroup of $D_8$ ,the former is not even a subset of latter.I can't understand why is it not the subset of $D_8$?
How do we define one group as a subset of another?</p>
| sequence | 209,695 | <p>It appears to be the case that ord $r$ in $D_6$ is 3, while ord $r$ in $D_8$ is 4, that's one reason why $D_6$ is not a subgroup of $D_6$.</p>
|
3,214,993 | <p>I understand how to find the residue for the simple pole at <span class="math-container">$z = 0$</span>, but I don't understand how to find the residues at subsequent singularities located at <span class="math-container">$z = n\pi$</span>. The solution tells me to set <span class="math-container">$w = z + n\pi$</spa... | Ian | 83,396 | <p>At <span class="math-container">$z=n\pi,n \neq 0$</span> you have a double pole, so the residue can be computed by the formula <span class="math-container">$\frac{1}{(n-1)!} \lim_{z \to n \pi} \frac{d^{n-1}}{dz^{n-1}} \left ( (z-n\pi)^n \frac{z}{\sin(z)^2} \right )$</span> with <span class="math-container">$n=2$</sp... |
3,214,993 | <p>I understand how to find the residue for the simple pole at <span class="math-container">$z = 0$</span>, but I don't understand how to find the residues at subsequent singularities located at <span class="math-container">$z = n\pi$</span>. The solution tells me to set <span class="math-container">$w = z + n\pi$</spa... | user | 293,846 | <p>You made a mistake in the last expression. The poles at <span class="math-container">$z=n\pi $</span> are of the 2nd order. Therefore, according to general <a href="https://en.m.wikipedia.org/wiki/Residue_(complex_analysis)#Limit_formula_for_higher_order_poles" rel="nofollow noreferrer">rule</a>:
<span class="math-c... |
2,925,988 | <p>Where does this series converges?</p>
<p>Using Ratio Test, I have checked that this series converges. Now I want to find the limit of this series.</p>
<p><span class="math-container">$$\sum_{n=1}^{\infty}\frac{1}{2^{n-1}+3}$$</span> </p>
<p>I would like to have some hint.</p>
| Robert Israel | 8,508 | <p>The limit can also be written as
<span class="math-container">$$ \frac{1}{4} + \frac{1}{5} + \frac{1}{2} \sum_{k=0}^\infty \frac{(-3/2)^k}{2^{k+1}-1}$$</span></p>
<p>But AFAIK there is no closed form.</p>
|
2,925,988 | <p>Where does this series converges?</p>
<p>Using Ratio Test, I have checked that this series converges. Now I want to find the limit of this series.</p>
<p><span class="math-container">$$\sum_{n=1}^{\infty}\frac{1}{2^{n-1}+3}$$</span> </p>
<p>I would like to have some hint.</p>
| Claude Leibovici | 82,404 | <p>There is no simple closed form for the summation but there is one
<span class="math-container">$$S=\sum_{n=1}^{\infty}\frac{1}{2^{n-1}+3}=\frac{2 \psi _2^{(0)}\left(-\frac{\log (3)+i \pi}{\log (2)}\right)+2 i \pi +\log
(18)}{\log (64)}$$</span> where appears the digamma function.</p>
<p>Numerically <span class="... |
2,666,390 | <p>How do I show that for $x>0$:
$$\frac{x-(x^2+1)\arctan(x)}{x^2(x^2+1)} < 0$$</p>
<p>I tried to do it somehow using the fact that $$\frac{\arctan(x)}{x} < 1$$ but still didn't figure it out...</p>
| Dr. Sonnhard Graubner | 175,066 | <p>HINT: definie $$f(x)=\arctan(x)-\frac{x}{x^2+1}$$ and Show that $$f'(x)=\frac{2 x^2}{\left(x^2+1\right)^2}>0$$ for $x>0$</p>
|
3,669,961 | <p>Let <span class="math-container">$f:[a,b] \longrightarrow \mathbb{R}$</span> be a integrable function. I know if <span class="math-container">$ f> 0 $</span> then
<span class="math-container">$$\int_{a}^{b} f(x)\; dx >0.$$</span> The converse is true? That is, if <span class="math-container">$$\int_{a}^{b} f(x... | Community | -1 | <p>Take any function that is negative. Say <span class="math-container">$-1$</span> from <span class="math-container">$0$</span> to <span class="math-container">$1$</span>. Compensate by a larger positive area, say <span class="math-container">$+1$</span> from <span class="math-container">$1$</span> to <span class="mat... |
389,912 | <p>I'm writing a paper and want to cite some references to efficiently prove that over any field <span class="math-container">$k$</span> of characteristic zero, every irreducible representation of a product of symmetric groups, say</p>
<p><span class="math-container">$$ S_{n_1} \times \cdots \times S_{n_p} $$</span></p... | Andy Putman | 317 | <p>I wrote a note giving a simple character-theory free proof that over an algebraically closed field of char 0, irreducible representations of products are tensor products of irreducible representations. See <a href="https://www3.nd.edu/%7Eandyp/notes/RepOfProducts.pdf" rel="noreferrer">here</a>. As the example afte... |
191,262 | <p>I have to fit a Cos^2 function to data I measured. The function is <span class="math-container">$a \cos^2(\frac{bx\pi}{180}+c)+d $</span> and I tried the FindFit and Linear Model Function. I have 4 datasets which I have to fit, the first one worked. The other three only yielded not usable fits. I am pretty new to Ma... | MassDefect | 42,264 | <p>Fitting arbitrary curves is actually a pretty difficult task as it's hard for a computer to make a good guess about what the initial parameters should be. We can help it out by providing some initial values that seem to make sense.</p>
<pre><code>params = FindFit[
data45,
a Cos[b x + c]^2 + d,
{{a, 300}, {b... |
191,262 | <p>I have to fit a Cos^2 function to data I measured. The function is <span class="math-container">$a \cos^2(\frac{bx\pi}{180}+c)+d $</span> and I tried the FindFit and Linear Model Function. I have 4 datasets which I have to fit, the first one worked. The other three only yielded not usable fits. I am pretty new to Ma... | OkkesDulgerci | 23,291 | <p>As always you can minimize cost function using <code>NMinimize</code>.</p>
<pre><code>ClearAll["Global`*"]
data = {{0, 132}, {20, 279.5}, {40, 289}, {60, 312}, {80, 307}, {100,
173}, {120, 92}, {140, 25}, {160, 44.5}, {180, 109.5}, {200,
230.5}, {220, 305}, {240, 339}, {260, 246.5}, {280, 181.5}, {300,
... |
370,151 | <p>Let $f: \Bbb R → \Bbb R$ be a continuous function such that $f(x)=x$ has no real solution . Then is it true that $f(f(x))=x$ also has no real solution ? </p>
| Zev Chonoles | 264 | <p><strong>Hint:</strong> We know that $f(x)\neq x$ for all $x\in\mathbb{R}$. Without loss of generality, let's say that at $x_0\in\mathbb{R}$, we have that $f(x_0)<x_0$. Can it ever be the case that $f(x_1)>x_1$ for any other $x_1\in\mathbb{R}$?</p>
|
207,433 | <p>(When one says several things in a question, then several things may get answered and others neglected. Hence this posting overlaps with one of my earlier ones, but (I hope) this one will be short, simple, and narrowly focused.)</p>
<p><b>In what contexts in mathematics, at any level of sophistication, do products... | Henry | 6,460 | <p>I have no idea if this sort of thing counts $$\int_0^1 \left|\log_e x\right|^n\, dx = n!$$</p>
|
4,005,381 | <p>Why the probability of a single element event is "zero" on the continuous model?, the explanation given is based on probability additivity axioms. but how? some more explanation with the source would be helpful.</p>
<p>cool thanks!</p>
| user2661923 | 464,411 | <p>Intuition only approach (because I'm ignorant of the corresponding math).</p>
<p>You have the interval <span class="math-container">$[0,1]$</span> and are asked to choose a number at random from this interval.</p>
<p><span class="math-container">$E$</span> is the event that you choose the number <span class="math-co... |
199,889 | <p>What are applications of the theory of Berkovich analytic spaces? The analytification $X \mapsto X^{\mathrm{an}}$</p>
| Community | -1 | <p>I have found the following:</p>
<ul>
<li>Walter Gubler applied it to the Bogomolov conjecture.</li>
<li>Jérôme Poineau applied it to the inverse Galois problem: <a href="http://arxiv.org/abs/0809.2880" rel="nofollow">http://arxiv.org/abs/0809.2880</a></li>
<li>Michael Stoll uses them in <a href="http://arxiv.org/ab... |
530 | <p>Tau ($\tau = 2 \pi$) has more merits in its application, but pi is the established standard in industry and education. Is the trade-off of teach-ability of circle concepts worth the subsequent confusion due to pi's omnipresence?</p>
<p>How can both be most effectively taught in order to maximize student's understan... | Markus Klein | 114 | <p>(Disclaimer: I peronally really don't care if one uses $\tau$ or $\pi$, both are just numbers for me.)</p>
<p><strong>But I would strongly recommend to use $\pi$.</strong> Why?</p>
<p>In every technical literature, in many popular literature, the people always use $\pi$ (even worse: $\tau$ is used for different th... |
2,922,290 | <p>Lemma 2.2</p>
<p>Let $(X,d)$ be a compact metric space and suppose that $U$ is a clopen subset of $X$. </p>
<p>There exists $\delta >0 $ s.t $d(x,y) \geq \delta $ $\forall x \in U $ and $y\in X-U $</p>
<p>Proof attempt</p>
<p>Suppose that it is not true then for all $\delta >0 $ $\exists x \in U $ and $\e... | DanielWainfleet | 254,665 | <p>Let $F$ be the family of all open balls $B(x,r)$ that are disjoint from $X$ \ $U$, with $x\in U$ and $r>0.$ For every $x\in U$ there exists $B(x,r)\in F$ because $X$ \ $U$ is closed.</p>
<p>Let $C=\{B(x,r/2): x\in U\land B(x,r)\in F\}.$ Then $C$ is an open cover of $U.$ And $U$ is compact. So (assuming $U\ne \p... |
2,982,368 | <blockquote>
<p>Let <span class="math-container">$f:\mathbb R\to\mathbb R$</span> differentiable such that <span class="math-container">$f(0)=0$</span> and <span class="math-container">$f'(x)=[f(x)]^2$</span> for all <span class="math-container">$x\in\mathbb R$</span>. Prove that <span class="math-container">$f(x)=0$... | egreg | 62,967 | <p>Your idea just needs a kickstart.</p>
<p>Since <span class="math-container">$f'(x)=f(x)^2$</span> we know that <span class="math-container">$f$</span> is (weakly) increasing. Suppose <span class="math-container">$f(a)>0$</span> for some <span class="math-container">$a>0$</span> and let <span class="math-conta... |
2,869,189 | <p>Is the method of calculating determinant of $3\times 3$ matrix by diagonals,
apply also on $4\times 4$ matrix?</p>
<p>for example:</p>
<p>$$\begin{matrix}2&2&1&3|\\1&4&4&5|\\5&1&1&6|\\7&1&4&5|\end{matrix}\begin{matrix}2&2&1\\1&4&4\\5&1&1\\7&am... | user | 505,767 | <p>No the <a href="https://en.wikipedia.org/wiki/Rule_of_Sarrus" rel="nofollow noreferrer"><strong>Rule of Sarrus</strong></a> is only valid for 3-by -3 matrices, in general for n-by -n matrices we can refer to <a href="https://en.wikipedia.org/wiki/Laplace_expansion" rel="nofollow noreferrer"><strong>Laplace expansion... |
560,792 | <p>Here it is, I know the answer is 49! (I found it by trial and error!) I just need to know how I can show this in Working!</p>
<p>Thanks,
Alex</p>
| Gerry Myerson | 8,269 | <p>Dan's answer, without letters & subscripts: </p>
<p>The 7 people have numbers adding up to $7\times65$. The 8 people have numbers adding up to $8\times63$. So the eighth person must have $8\times63-7\times65$. </p>
|
14,167 | <p>I randomly pick a natural number <em>n</em>. Assuming that I would have picked each number with the same probability, what was the probability for me to pick <em>n</em> before I did it?</p>
| user3533 | 3,533 | <p>There is no uniform distribution on the natural numbers. So the question itself is self-contradictory.</p>
<p>Some explanation:</p>
<p>You say that each number is chosen with the same probability. Say probability $c$.
Let's denote the probability to choose $n$ by $p(n)$. So you say $p(n)=c$ for any natural $n$.
Fo... |
708,138 | <p>Show that the maximum of a convex function on a convex polytope occurs at one of the extreme points of the polytope. I am really stuck here and don't know where to begin.</p>
| 王思文 | 714,251 | <p>for example:
Now, note that for any fixed <span class="math-container">$t\geq0$</span>, the function <span class="math-container">$g_{t}:R^{r}\rightarrow R$</span> defined by <span class="math-container">$g_{t}(x)=-\frac{1}{2}\sum_{k=1}^{r}ln(1+2tx_{k})$</span> is convex. Hence, its maximum over the simplex <span cl... |
737,614 | <p>I'm confused as to how to finish answering this question. </p>
<p>Compute $$\sum_{i=28}^n (3i^2 - 4i + \frac{5}{7^i}) $$</p>
<p>I end up with
$$ 3[\frac{n(n+1)(2n+1)}{6}] - 4[\frac{n(n+1)}{2}] + [\frac{k^n(5(-log(7))^n}{n!}]$$</p>
<p>I'm not entirely sure if the last part of my answer is right but my main quest... | user607513 | 607,513 | <p>actually it does not exit.because according to handshaking theorem twice the edges is the degree.but five vertices of degree 3 which is equal to 3+3+3+3+3=15.it should be an even number and 15 is not an even number and also the number of odd degree vertices in an undirected graph must be an even count.</p>
|
2,165,066 | <p>I'm asked to show that if $X$ and $Y$ are independent exponential random variables with parameter <img src="https://latex.codecogs.com/gif.latex?%5Clambda&space;=1" title="\lambda =1" />, then <img src="https://latex.codecogs.com/gif.latex?X/(X+Y)" title="X/(X+Y)" /> has a Beta distribution.</p>
<p>Up to know ... | heropup | 118,193 | <p>The naive approach: we have $$f_X(x) = e^{-x}, \quad f_Y(y) = e^{-y}, \quad x, y > 0.$$ Define $Z = X/(X+Y)$, which implies $Y = X(1/Z - 1)$, and note that we must have $0 < Z < 1$. Then
$$\begin{align*} F_Z(z) &= \Pr[X/(X+Y) \le z] \\
&= \Pr[Y > X(1/z - 1)] \\
&= \int_{x=0}^\infty \Pr[Y ... |
974,207 | <p>Given the sequence <span class="math-container">${b_n}$</span>, let <span class="math-container">$lim_{n \to \infty}\ b_n = b$</span>.</p>
<p>Suppose that the sequence <span class="math-container">${a_n}$</span> and the number <span class="math-container">$a$</span> have the property for which there exists <span c... | hickslebummbumm | 168,882 | <p>Take $\epsilon$ fixed since $b_n \rightarrow b$,</p>
<p>\begin{align*}
\exists N_0 \in \mathbb{N} \quad \text{ s.t. } \forall n \geq N_0, \quad \vert b_n - b \vert \leq \frac{\epsilon}{M}.
\end{align*} It follows that $\forall n \geq N_0$, $\vert a_n - a \vert \leq \epsilon$.</p>
<p>As $\epsilon$ was chosen arbitr... |
1,892,367 | <p>In Euclidean geometry two polygons are said to be similar if, by rotation and scaling, one can be transformed into the other, and vice versa. If we consider a general metric space, does this notion exist? I can think of how this works by scaling: you consider $d(a_i , a_{I+1} )$, where $a_i$ are the vertices of the ... | Emilio Novati | 187,568 | <p>In a metric space $(X,d)$ a similitude can be defined as a function $f:X\to X$ such that
$
d(f(x),f(y))=rd(x,y)
$
for $r>0$.
If $X$ is a vector space and the metric is derived from an inner product (so that the notion of angle can be defined) than similarities preserve the angles.</p>
|
754,644 | <p>This question is related to a solved problem in Gilbert Strang's 'Introduction to Linear Algebra'(Chapter 3,Question 3.6A, Page 190).</p>
<blockquote>
<p>Q) Find bases and dimensions for all four fundamental subspaces of A if you know that</p>
<p><span class="math-container">$A = \begin{bmatrix}1 & 0 & 0\\2 ... | fgp | 42,986 | <p>If you look closely at the right matrix, you'll see that</p>
<ol>
<li><p>It conflates the 2-dimensional subspace generated by $\{(1,0,0,0),(0,1,0,0)\}$ into the
1-dimensional subspace generated by $\{(1,0,0)\}$, because it's first two columns are linearly dependent. In particular, it maps vectors of the form $(3x,-... |
3,358,168 | <p><span class="math-container">$x_n$</span> is bounded, <span class="math-container">$\lim_{n \rightarrow \infty}(x_{n+1}-x_n)=0$</span>, <span class="math-container">$\liminf_{n \rightarrow \infty}=l$</span>, <span class="math-container">$\limsup_{n \rightarrow \infty}=L$</span>, show that every point in <span class=... | Mirko | 188,367 | <p>Suppose that <span class="math-container">$p\in[l,L]$</span>. If <span class="math-container">$p=l$</span> or <span class="math-container">$p=L$</span> we are done, so assume <span class="math-container">$l<p<L$</span>. We want to show that for every <span class="math-container">$\varepsilon>0$</span> and e... |
2,006,870 | <p>Is the following inequality true?</p>
<p>$\left( \sum \limits_{i=1}^\infty \sum \limits_{j=1}^\infty \sum \limits_{k=1}^\infty \sum \limits_{l=1}^\infty a_{ij}\,a_{ik}\,a_{jl}\,a_{kl} \right) \leq \left( \sum \limits_{i=1}^\infty \sum \limits_{j=1}^\infty a_{ij}^2 \right)^{1/2}\left( \sum \limits_{i=1}^\infty \sum ... | Jam | 161,490 | <h3>Equivalent Problem</h3>
<p>To expand on the other answers posted, if you want to find a unique number for each <span class="math-container">$(x,y)$</span> pair of integers, you're really looking for a one-to-one function <span class="math-container">$f$</span>, between <span class="math-container">$(x,y)$</span> an... |
65,083 | <p>Rotman's book <em>An Introduction to the Theory of Groups</em> (Fourth Edition) asks, on page 22, Exercise 2.8, to show that <span class="math-container">$S(n)$</span> cannot be embedded in <span class="math-container">$A(n+1)$</span>, where <span class="math-container">$S(n)$</span> = the symmetric group on <span c... | Dario | 489,363 | <p>Using concepts that Rotman have introduced before that exercise, I think we could provide the following counterexample: if <span class="math-container">$S_2$</span> could be embedded in <span class="math-container">$A_3$</span>, then in <span class="math-container">$A_3$</span> there would be a subgroup <span class=... |
1,712,481 | <p><span class="math-container">$$I=\displaystyle\int\frac{dx}{(3 + 2\sin x - \cos x)}$$</span></p>
<p>If <span class="math-container">$$\tan\left(\frac{x}{2}\right)=u$$</span></p>
<p>or <span class="math-container">$$x=2\cdot\tan^{-1}(u)$$</span></p>
<p>Then,</p>
<p><span class="math-container">$$\sin{x}=\dfrac{2u}{1+... | Quanto | 686,284 | <p>With <span class="math-container">$a=\tan^{-1}\frac1{\sqrt5}$</span> and <span class="math-container">$t=\frac12(x+a)$</span></p>
<p><span class="math-container">\begin{align}
\int\frac{dx}{3 + 2\sin x - \cos x}
& = \int \frac{dx}{3-\sqrt5\cos(x+a)}\\
&= \int \frac{2d(\tan^2t)}{(3-\sqrt5)+(3+\sqrt5)\tan^2t}=... |
1,233,700 | <p>A infinitely generated additive group G and its subgroup K, when they are isomorphic to each other? Is there any theorem on this?</p>
| Elaqqad | 204,937 | <p>There is no any general theorem to solve your problem</p>
<p>Clearly if a group is isomorphic to a proper subgroup of itself then $|G|$ must be infinite, but being infinite is not sufficient to be isomorphic to a proper subgroup of itself, there exist some groups which are infinite and isomorphic to a subgroup of t... |
2,696,320 | <p>If $\gcd(m,15)= \gcd(n,15)=1$, show that either $15|(m^4 + n^4)$ or $15|(m^4 - n^4)$. </p>
<p>I'm really stuck on this proof. This is what I know: </p>
<p>Since the $\gcd(m,15)= 1$ we can write it as $mx+15y=1$ where $x,y$ are integers. Also, since $\gcd(n,15)=1$ it can also be written as $nu + 15v = 1$ where $u,v... | lab bhattacharjee | 33,337 | <p>Hint</p>
<p>As $\phi(15)=8,a^8\equiv1\pmod{15}$ for $(15,a)=1$</p>
<p>$\implies15$ divides $a^8-b^8=?$ for $(15,a)=(15,b)=1$</p>
<p>We can prove something more:$15$ will actually divide $a^4-b^4$</p>
<p>as $\lambda(15)=4, a^4\equiv1\pmod{15}$ for $(a,15)=1$</p>
|
767,282 | <blockquote>
<p><strong>Problem</strong>: Show that $f: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ with $f(x,y)=\|x-y\|_2^2$ is differentiable and compute its differential at every point in the domain of $f$<br><strong>Note</strong>: $\| \cdot \|_2$ denotes the euclidian norm $d(x,y)_2 =\sqrt{\sum\limits_{i=1}^... | Siminore | 29,672 | <p>When solving this kind of problems, your eyes are often more important than your memory. I mean: look at the function and try to collect as much information as you can. For instance, your function is sum of squares, and this may be really helpful.
Then you can try to apply the criterion of the directional derivativ... |
3,197,301 | <p>Let <span class="math-container">$OABC$</span> be a tetrahedron such that <span class="math-container">$|OA|=|OB|=|OC|$</span>. Denote by <span class="math-container">$D$</span> and <span class="math-container">$E$</span> the midpoints of segments <span class="math-container">$AB$</span> and <span class="math-contai... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>It will be easier to write if you think of the plane as the Argand plane.</p>
<p>So let <span class="math-container">$z_0$</span> be the affix of the point <span class="math-container">$M_0(x_0,y_0)$</span>. The affix of a point in the open ball with centre <span class="math-container... |
3,864,364 | <p>Let <span class="math-container">$g \colon \mathbb{N} \rightarrow \{0,1\}$</span> such that <span class="math-container">$g(n)=0$</span>, if <span class="math-container">$n \equiv 0$</span> or <span class="math-container">$n \equiv 1 ~(\bmod~4)$</span>; or <span class="math-container">$g(n)=1$</span>, if <span clas... | J.G. | 56,861 | <p>For another proof of convergence, we can rewrite the sum with <span class="math-container">$a^{-1/2}=\pi^{-1/2}\int_0^\infty x^{-1/2}e^{-ax}dx$</span> as<span class="math-container">$$\pi^{-1/2}\sum_{m\ge0}\int_0^\infty x^{-1/2}e^{-(4m+1)x}(1-e^{-x})(1-e^{-2x})dx=\pi^{-1/2}\int_0^\infty\frac{x^{-1/2}e^{-x}(1-e^{-x})... |
146,394 | <p>Here is a nice figure</p>
<p><a href="https://i.stack.imgur.com/e6Kuf.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/e6Kuf.jpg" alt="enter image description here"></a></p>
<p>Now let's create a grid</p>
<pre><code>plot = Show[GraphicsGrid[{{im1, im1, im1},
{SpanFromLeft, im1, im1}... | Bill N | 44,484 | <p>Follow the Instructions linked on the xAct Homepage first to load the main xAct package, which contains the Invar kernel. Then, the database of Riemann relations in Invar must be downloaded and installed separately. Follow the Invar link on the xAct page, and on the Invar page there is a Readme link that provides ... |
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