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 |
|---|---|---|---|---|
3,098,838 | <blockquote>
<p>The displacement of a particle varies according to <span class="math-container">$x=3(\cos t +\sin t)$</span>.
Then find the amplitude of the oscillation of the particle.</p>
</blockquote>
<p>Can someone kindly explain the concept of amplitude and oscillation and how to solve it?</p>
<p>Any hints ... | mathcounterexamples.net | 187,663 | <p>You must have</p>
<p><span class="math-container">$$\langle z,z \rangle = ax^2 + 2bxy + dy^2 >0$$</span> for all <span class="math-container">$z\neq 0$</span>. In particular for all <span class="math-container">$z=(x,1)^T$</span>.</p>
<p>In particular the discriminant of the trinomial</p>
<p><span class="math-... |
3,098,838 | <blockquote>
<p>The displacement of a particle varies according to <span class="math-container">$x=3(\cos t +\sin t)$</span>.
Then find the amplitude of the oscillation of the particle.</p>
</blockquote>
<p>Can someone kindly explain the concept of amplitude and oscillation and how to solve it?</p>
<p>Any hints ... | Bernard | 202,857 | <p>You don't have to. If the homogeneous quadratic polynomial <span class="math-container">$\;ax^2+2bxy+dy^2\;$</span> takes positive values for all <span class="math-container">$(x,y)\ne (0,0)$</span>, it means its (reduced) discriminant <span class="math-container">$\;\delta'=b^2-ad <0$</span>, and this discrimin... |
158,549 | <p>Denote by $\Sigma$ the collection of all $(S, \succeq)$ wher $S \subset \mathbb{R}$ is compact and $\succeq$ is an arbitrary total order on $S$.</p>
<p>Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ such that for all $(S, \succeq) \in \Sigma$ there exists a compact interval $I$ with the properties that<... | Brian M. Scott | 12,042 | <p>If I understand correctly what you’re asking, the answer is <em>no</em>. Let $\preceq$ be a linear order on $[0,1]$ having no last element; no compact subset of $\Bbb R$ can be mapped onto $[0,1]$ in such a way that $f(x)\preceq f(y)$ whenever $x\le y$, because every compact subset of $\Bbb R$ has a last element wit... |
158,549 | <p>Denote by $\Sigma$ the collection of all $(S, \succeq)$ wher $S \subset \mathbb{R}$ is compact and $\succeq$ is an arbitrary total order on $S$.</p>
<p>Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ such that for all $(S, \succeq) \in \Sigma$ there exists a compact interval $I$ with the properties that<... | William | 13,579 | <p>If you have the property that if $x < y$, then $f(x) \prec f(y)$, all strictly, then no.</p>
<p>Let $S$ be an infinite compact subset of $\mathbb{R}$. Let $\aleph$ be a cardinal in bijection with $S$. That is there is a bijection $f : \aleph \rightarrow S$. From this bijection, you can define a well-ordering on ... |
648,607 | <p>I would like to determine whether the following series is absolut convergent or not. I´m not sure how to begin generally. I would say no, because when taking the absolut value of the fraction and add all of them together the series doesnt converge...could someone give me a general road plan how to manage this.</p>
... | Pankaj Sejwal | 33,578 | <p>Basically two tests are needed to check for convergence in case of alternating series which it is :</p>
<p>a) It is decreasing monotonically.</p>
<p>b) Its limit is 0 as n approaches infinity.</p>
<p>In this, for first check that ,</p>
<p>let $f(x)= \frac { 1 }{ (2n+1) } $. So,$f'(x) = \frac { -2 }{ (2n+1)^2 } ... |
3,466,680 | <p>I'm solving a problem in ODE:</p>
<blockquote>
<p>Solve in <span class="math-container">$\left (-\dfrac{\pi}{2},\dfrac{\pi}{2} \right )$</span> the ODE <span class="math-container">$y''(t) \cos t + y (t) \cos t=1$</span></p>
</blockquote>
<p>In my lecture, we are given three theorems:</p>
<blockquote>
<p><span class... | mathsdiscussion.com | 694,428 | <p><span class="math-container">$$y''Cosx+yCosx=1$$</span>
<span class="math-container">$$y''Cosx-y'Sinx+y'Sinx+yCosx=1$$</span>
<span class="math-container">$$(y'Cosx)'+(ySinx)'=1$$</span>
<span class="math-container">$$y'Cosx+ySinx=x+c$$</span>
<span class="math-container">$$y'+ytanx=(x+c)Secx$$</span>
Now this is 1s... |
462,983 | <h2>The Question:</h2>
<p>This is a very fundamental and commonly used result in linear algebra, but I haven't been able to find a proof or prove it myself. The statement is as follows:</p>
<blockquote>
<p>let $A$ be an $n\times n$ square matrix, and suppose that $B=\operatorname{LeftInv}(A)$ is a matrix such that... | xavierm02 | 10,385 | <p>You notation $A^{-1}$ is confusing because it makes you think of it as a two-sided inverse but we only know it's a left-inverse.</p>
<p>Let's call $B$ the matrix so that $BA=I$. You want to prove $AB=I$.</p>
<p>First, you need to prove that there is a $C$ so that $AC=I$. To do that, you can use the determinant but... |
4,339,772 | <p>The problem is stated as:</p>
<blockquote>
<p>Show that <span class="math-container">$\int_{0}^{n} \left (1-\frac{x}{n} \right ) ^n \ln(x) dx = \frac{n}{n+1} \left (\ln(n) - 1 - 1/2 -...- 1/{(n+1)} \right )$</span></p>
</blockquote>
<p><strong>My attempt</strong></p>
<p>First of all, we make the substitution <span ... | Kavi Rama Murthy | 142,385 | <p>Hint: <span class="math-container">$\sum_{k=1}^{\infty} \frac{1}{k}$</span> and <span class="math-container">$ \sum_{k=1}^{\infty} \frac{1}{n+k+1} $</span> are both infinity so you cannot write these sums separately. Instead, you should write <span class="math-container">$\lim_{N \to \infty} [-\sum_{k=1}^{N} \frac{... |
2,778,422 | <p>The generalized $\lambda-\text{eigenspace}$ is defined by:
$V^f_{(\lambda)}=\bigl\lbrace v\in V\mid\exists j\,\text{ such that }\,(f-\lambda)^jv=0 \bigr\rbrace$.
Suppose that $V$ is a vector space over the field $k$ and $f,g\in \operatorname{End}_k(V)$ satisfy $f\circ g=g\circ f$. Show that $g(V^f_{(\lambda)})\subse... | Maxime Ramzi | 408,637 | <p><strong>Hint</strong>: Show by induction that $f^n\circ g = g\circ f^n$; then conclude that $g$ commutes with all elements of $k[f]$.</p>
|
2,555,861 | <p>I am reading up on <strong>Fraleigh's</strong> <em>A First Course in Abstract Algebra</em>, and he says ($H$ subgroup of $G$) $Hg=gH$ $iff$ $i_g[H]=H$ $iff$ $H$ is invariant under all inner automorphisms. I look up invariant and I find this definition:</p>
<p>"Firstly, if one has a group G acting on a mathematical... | Michael Hardy | 11,667 | <p>I'm guessing you mean this is to be done by limits of Riemann sums.</p>
<p>If $x$ goes from $0$ to $1$ by steps of $\Delta x=1/n,$ then at the $i$th step we have $x=0 + i/n,$ and so</p>
<p>$$
\int_0^1 e^x\,dx = \lim_{n\to\infty} \frac 1 n \sum_{i=1}^n e^{0 \,+\, i/n}.
$$
And
\begin{align}
\frac 1 n \sum_{i=1}^n e^... |
1,423,252 | <p>The proposition is:</p>
<blockquote>
<p>If $\lim S_n = L$ and for every $n$, $S_n$ is in the interval $[a,b]$, then $L$ is also in $[a,b]$.</p>
</blockquote>
<p>I have proved this effectively, but now the question is to provide a counterexample to the stronger assumption, for the interval $(a,b)$. </p>
<p>Basi... | Ludolila | 60,678 | <p><strong>Hint:</strong></p>
<p>Think about the sequence ${1\over n}$. Can you figure out the interval for the counter example?</p>
|
2,041,839 | <p>$$\int_{1}^{x}\frac{dt}{\sqrt{t^3-1}}$$ does this have a closed form involving jacobi elliptic functions of parameter $k$?</p>
<p><strong>N.B</strong> I tried with the change of variables $t=1+k\frac{1-u}{1+u}$. But this leads no where. <a href="http://mathworld.wolfram.com/JacobiEllipticFunctions.html" rel="nofoll... | Matias Heikkilä | 66,856 | <p>The notation $\mathrm{d}x$ should not be attempted to be taken too literally. <a href="https://en.wikipedia.org/wiki/Non-standard_analysis" rel="nofollow noreferrer">While it can be made precise</a> it's usually considered a relic from time when things were not as precisely defined as they are these days. You will r... |
2,688,608 | <p>Assume matrix </p>
<p>$$A=
\begin{bmatrix}
-1&0&0&0&0\\
-1&1&-2&0&1\\
-1&0&-1&0&1\\
0&1&-1&1&0\\
0&0&0&0&-1
\end{bmatrix}
$$</p>
<p>Its Jordan Canonical Form is
$$J=
\begin{bmatrix}
-1&1&0&0&0\\
0&-1&0&0&... | user | 505,767 | <p>Note that the condition $AP=PJ$ is equivalent to</p>
<ul>
<li>$Ap_1=-p_1 \to p_1$</li>
<li>$Ap_2=p_1-p_2\to p_2$</li>
<li>$Ap_3=-p_3\to p_3$</li>
<li>$Ap_4=p_4 \to p_4$</li>
<li>$Ap_5=p_4+p_5 \to p_5$</li>
</ul>
<p>Since the set up is equivalent, from you results seems that there is something wrong in the calculat... |
2,357,272 | <p>Find out the sum of the following infinite series
$$\frac{3}{2^2(1)(2)} + \frac{4}{2^3(2)(3)} +\dots+\frac{r+2}{2^{r+1}(r)(r+1)}+\cdots
$$
up to $r\to\infty$.</p>
<p>MY TRY:- I tried to split $r+2$ as $[(r+1) +{(r+1)-r}]$ so that I can cancel one term from each terms in the numerator. Then I got an expression whic... | B. Goddard | 362,009 | <p>If you do the partial fraction expansion, the summand becomes</p>
<p>$$\frac{1}{2^{r+1}}\left( \frac{2}{r} - \frac{1}{r+1}\right) = \frac{1}{2^r r} - \frac{1}{2^{r+1}(r+1)}, $$</p>
<p>So the sequence is telescoping. All terms cancel except the first and so sum equals $\frac12$.</p>
|
547,971 | <p>I have to show that for $f,g$ analytic on some domain and $a$ a double zero of $g$, we have:</p>
<p>$$\operatorname{Res} \left(\frac{f(z)}{g(z)}, z=a\right) = \frac{6f'(a)g''(a)-2f(a)g'''(a)}{3[g''(a)]^2}.$$</p>
<p>The problem is that direct calculation using the formula (for pole of order $2$):</p>
<p>$$\operato... | Ron Gordon | 53,268 | <p>Because $a$ is a double zero of $g(z)$, write</p>
<p>$$g(z) = (z-a)^2 p(z)$$</p>
<p>where $p(a) \ne 0$ and is analytic, etc. etc.</p>
<p>Then</p>
<p>$$\operatorname*{Res}_{z=a} \frac{f(z)}{g(z)} = \left [\frac{d}{dz} \frac{f(z)}{p(z)} \right ]_{z=a}$$</p>
<p>Now,</p>
<p>$$\frac{d}{dz} \frac{f(z)}{p(z)} = \frac... |
638,244 | <p>In any (simple) type theory there are <strong>base types</strong> (i.e. the type of <em>individuals</em> and the type of <em>propositions</em>) and <strong>type builders</strong> (i.e. $\rightarrow$, which takes two types $t,t'$ and yields the type of <em>functions</em> $t \rightarrow t'$). </p>
<p>For each type in... | Basil | 36,042 | <p>I don't know about "official", but from what I understand you speak of a special kind of <em>type contexts</em>, where the "holes"---that is, the type placeholders in the context---are to be filled exclusively by <em>base</em> types (I'm borrowing the term <em>context</em> from some old experience of mine in algebra... |
2,275,604 | <p>We can break up the circle into an infinite amount of rings with perimeter $2\pi r$. For a given circle $r$, the outside ring has perimeter of $2\pi r$ and the smallest one has of course perimeter $0$. We can add up all the area of the infinite rings using the arithmetic series concept; we can get the average of the... | Google X | 441,387 | <p>This would go solidly into the category of "plausibility argument" not proof. The fundamental idea could be formalized into a proof, but this goes for many(if not most) plausibility arguments. To do it properly would require lots of work to remove ambiguity from statements like "we can add up all of the infinite rin... |
2,275,604 | <p>We can break up the circle into an infinite amount of rings with perimeter $2\pi r$. For a given circle $r$, the outside ring has perimeter of $2\pi r$ and the smallest one has of course perimeter $0$. We can add up all the area of the infinite rings using the arithmetic series concept; we can get the average of the... | Community | -1 | <p>For a slightly more detailed argument (not rigorous yet, but closer), consider a large integer radius $r$ and accumulate the contributions of $r$ rings $1$ unit large.</p>
<p>$$A\approx\sum_{s=0}^{r-1}2\pi\left(s+\frac12\right)=2\pi\frac{(r-1)r+r}2=\pi r^2$$</p>
<p>by the triangular numbers formula. Then by simila... |
3,230,957 | <p>Q5. Calculate the eigenvalues and eigenvectors of the following matrix</p>
<p><span class="math-container">$$\left(\begin{matrix}
3 & \sqrt{2} \\
\sqrt{2} & 2
\end{matrix}\right)$$</span></p>
<p>It is <span class="math-container">$2 \times 2$</span> matrix and having square-root value.</p>
| zbrads2 | 655,480 | <p>The characteristic polynomial is <span class="math-container">$$(3-\lambda)(2-\lambda)-2.$$</span> Can you see how to get that?</p>
<p>Now set this polynomial equal to zero and solve for <span class="math-container">$\lambda$</span> to get the eigenvalues. If we let <span class="math-container">$A$</span> be your m... |
3,230,957 | <p>Q5. Calculate the eigenvalues and eigenvectors of the following matrix</p>
<p><span class="math-container">$$\left(\begin{matrix}
3 & \sqrt{2} \\
\sqrt{2} & 2
\end{matrix}\right)$$</span></p>
<p>It is <span class="math-container">$2 \times 2$</span> matrix and having square-root value.</p>
| Community | -1 | <p>We have for the eigenvalues:</p>
<p><span class="math-container">$$ \text{det} (\begin{bmatrix}
3 & \sqrt{2} \\
\sqrt{2} & 2 \\
\end{bmatrix} - \lambda \begin{bmatrix}
1 & 0 \\
0 & 1 \\
\end{bmatrix})$$</span></p>
<p>So then we get <span class="math-container">$\lambda^2-5\lambda+4=0$</span> so the... |
3,230,957 | <p>Q5. Calculate the eigenvalues and eigenvectors of the following matrix</p>
<p><span class="math-container">$$\left(\begin{matrix}
3 & \sqrt{2} \\
\sqrt{2} & 2
\end{matrix}\right)$$</span></p>
<p>It is <span class="math-container">$2 \times 2$</span> matrix and having square-root value.</p>
| IamKnull | 610,697 | <p>Find eigenvalues from the characteristic polynomial:</p>
<p><span class="math-container">$\left|\begin{matrix}
3-\lambda & \sqrt{2} \\
\sqrt{2} & 2-\lambda
\end{matrix}\right| =\lambda^2-5*\lambda+4=(\lambda-1)*(\lambda-4)$</span></p>
<h2><span class="math-container">$\lambda_1=1\;\;
\lambda_2=4$</span></... |
212,240 | <p>I'm a beginner of the area of free boundary problem. Let me first give some background: </p>
<p>$\Omega \subset \mathbb{R}^n$ is an open connected set, and locally $\partial \Omega$ is a Lipschitz graph.
Consider the convex set $$K:＝\{v \in L^1_{loc}(\Omega): \nabla v \in L^2(\Omega) \,, v=u^0 \mbox{on $\partial \... | jfbonder | 43,444 | <p>Since $\partial\{u>0\}$ has locally finite $H^{n-1}$ measure $|\partial\{u>0\}\cap B_r|=0$ and hence $|\partial\{u>0\}|=0$.</p>
|
4,037,697 | <p><span class="math-container">$a_n $</span> is a sequence defined this way:
<a href="https://i.stack.imgur.com/nHdiD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nHdiD.png" alt="enter image description here" /></a></p>
<p>and we define: <a href="https://i.stack.imgur.com/b01hX.png" rel="nofollow... | Chrystomath | 84,081 | <p>Let <span class="math-container">$A=RU$</span> be the polar decomposition of <span class="math-container">$A$</span>, where <span class="math-container">$R$</span> is positive semi-definite and <span class="math-container">$U$</span> is unitary.</p>
<p>Then <span class="math-container">$R^2=A^TA=\lambda^2I$</span>; ... |
1,768,100 | <p>I have started studying field theory and i have a question.somewhere i saw that a finite field with $p^m $ elements has a subfield of order $p^m $ where $m$ is a divisor of $n $.My question that if it is a field then how can it have a proper subfield.because since it is field it doesnt have any proper ideal.how can... | Martín-Blas Pérez Pinilla | 98,199 | <p>Parametrize the path:
$$z(t) = x(t) + iy(t) = -i(1-t) + (1+i)t,\qquad t\in[0,1]$$.
$$\int_{L}\left( \overline {z}+1\right)dz = \int_0^1(\overline{z(t)}+1)z'(t)\,dt =\cdots$$</p>
|
416,940 | <p><span class="math-container">$\DeclareMathOperator\Spec{Spec}\newcommand{\perf}{\mathrm{perf}}\DeclareMathOperator\SHC{SHC}$</span>I have just finished reading the paper "The spectrum of prime ideals in tensor triangulated categories" in which Balmer proposes his notion of spectrum which nowadays is consid... | Drew Heard | 16,785 | <p>To help with (3), let me point out that in 'nice' situations (e.g., in the derived category of a noetherian commutative ring), the Balmer spectrum classifies <em>all</em> localizing tensor ideals of the category, in terms of arbitrary subsets of the spectrum (so the topology plays no role). This uses a theory of sup... |
1,742,768 | <p>How to examine convergence of $\sum_{n=1}^{\infty}(\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2})$ for $a, b, c> 0$ using Taylor's theorem?</p>
| Martin Argerami | 22,857 | <p>You have, using Taylor's polynomial,
$$
a^{1/n}=e^{\frac1n\,\log a}=1+\frac1n\log a+\frac{e^{c_n}}{2n^2}\,\log^2 a,
$$
where $0<c_n<\frac1n\,\log a$. So
\begin{align}
\sum_{n=1}^{\infty}(\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2})
&=\sum_{n=1}^\infty\frac1n\,\left(\log a-\frac12\log b-\frac12\,\log c... |
1,760,687 | <p>Can anyone explain me why this equality is true?</p>
<p>$x^k(1-x)^{-k} = \sum_{n = k}^{\infty}{{n-1}\choose{k-1}}x^n$</p>
<p>I really don't see how any manipulation could give me this result. </p>
<p>Thanks!</p>
| marty cohen | 13,079 | <p>Yes,
with initial term 1,
difference 3,
and length 2.</p>
<p>Note that
<em>any</em> sequence
of length 2
is a linear progression,
and that
any sequence of length $n$
is a progression of order
$n-1$
(i.e.,
has its $n-1$ difference
constant).</p>
|
4,492,566 | <blockquote>
<p>To which degree must I rotate a parabola for it to be no longer the graph of a function?</p>
</blockquote>
<p>I have no problem with narrowing the question down by only concerning the standard parabola: <span class="math-container">$$f(x)=x^2.$$</span></p>
<p>I am looking for a specific angle measure. O... | 5xum | 112,884 | <p>Rotating the parabola even by the smallest angle will cause it to no longer be well defined.</p>
<p>Intuitively, you can prove this for yourself by considering the fact that the derivative of a parabola is unbounded. This means that the parabola becomes arbitrarily "steep" for large (or small) values of <s... |
4,492,566 | <blockquote>
<p>To which degree must I rotate a parabola for it to be no longer the graph of a function?</p>
</blockquote>
<p>I have no problem with narrowing the question down by only concerning the standard parabola: <span class="math-container">$$f(x)=x^2.$$</span></p>
<p>I am looking for a specific angle measure. O... | Carsten S | 90,962 | <p>A part of the question is: how much do we have to rotate <span class="math-container">$y = x^2$</span> around the origin such that it hits the <span class="math-container">$y$</span>-axis in a second point, in addition to the origin? To simplify this, instead of rotating the parabola (by <span class="math-container"... |
4,492,566 | <blockquote>
<p>To which degree must I rotate a parabola for it to be no longer the graph of a function?</p>
</blockquote>
<p>I have no problem with narrowing the question down by only concerning the standard parabola: <span class="math-container">$$f(x)=x^2.$$</span></p>
<p>I am looking for a specific angle measure. O... | whuber | 1,489 | <p>Without any loss of generality, assume the center of rotation is the origin, so that the first coordinate of the image of any point <span class="math-container">$(x,y)$</span> under a rotation by <span class="math-container">$\theta$</span> equals <span class="math-container">$x\cos\theta - y\sin\theta.$</span></p>... |
48,864 | <p>I can't resist asking this companion question to the <a href="https://mathoverflow.net/questions/48771/proofs-that-require-fundamentally-new-ways-of-thinking"> one of Gowers</a>. There, Tim Dokchitser suggested the idea of Grothendieck topologies as a fundamentally new insight. But Gowers' original motivation is to ... | Daniel Moskovich | 2,051 | <p>A <a href="http://en.wikipedia.org/wiki/Simplicial_set">simplicial set</a> is surely an idea which would be more natural to a computer. Breaking a shape up into simplices is still something a human would do, because simplices are contractible geometric objects whose gluings one can explicitly describe. But to pass f... |
2,014,366 | <p>first of all it's not an exam sheet or some kinda stuff. I'm just preparing myself about quantifiers.</p>
<p>i couldn't find similar task to this one so had to ask here. </p>
<hr>
<p>Let '$x \mathrel{\heartsuit} y$' stand for 'x loves y'. Rewrite the sentence 'Someone loves everyone' using quantifiers in two dif... | Jan Schultke | 379,035 | <p>I can only think of one way:
$$\exists x \forall y:x \heartsuit y$$ </p>
|
340,264 | <p>Given that</p>
<p>$L\{J_0(t)\}=1/(s^2+1)$</p>
<p>where $J_0(t)=\sum\limits^{∞}_{n=0}(−1)n(n!)2(t2)2n$,</p>
<p>find the Laplace transform of $tJ_0(t)$. </p>
<p>$L\{tJ_0(t)\}=$_<strong><em>_</em>__<em>_</em>__<em>_</em>___<em></strong>---</em>___?</p>
| Andreas Blass | 48,510 | <p>You want the $p$-adic infinite sum $x=x_0+x_1p+x_2p^2+\dots$ to satisfy $x^2=2$. That will require each of the finite partial sums $S_n=x_0+x_1p+x_2p^2+\dots+x_np^n$ to satisfy ${S_n}^2\equiv 2\pmod{p^{n+1}}$, because the omitted terms of the infinite series will contribute only terms divisible by $p^{n+1}$ to $x^2... |
1,230,159 | <p>Where can I find a complete proof to the fact that the integral closure of $\mathbb{Z}$ in $\mathbb{Q}(i)$ is $\mathbb{Z}[i]$ (the Gaussian integers are the integral closure of $\mathbb{Z}$ in the Gaussian rationals)? For such a seemingly standard fact, I can not seem to find a complete proof of this anywhere. Yes, ... | hunter | 108,129 | <p>Here is an elementary proof from scratch. If $\alpha = a + bi$ is an algebraic integer, then $\overline{\alpha}$ will be as well, since any polynomial over $\mathbb{Q}$ having $\alpha$ as a root also has $\overline{\alpha}$ as a root.</p>
<p>Therefore its trace $\alpha + \overline{\alpha} = 2a$ and its norm $\alpha... |
2,418,448 | <p>Let $\mathbb{N}^{\mathbb{N}}$ be the set of all sequences of positive integers. For $a=(n_1,n_2,\cdots),b=(m_1,m_2,\cdots)$ define $d(a,b)=\frac{1}{\min(i:n_i\neq m_i)}, a\neq b$, $d(a,a)=0$</p>
<hr>
<p>It can be easily shown that $d(a,b)\leq \max\{d(a,c),d(b,c)\}$, I want to prove that if two balls in this metric... | Neal | 20,569 | <p>Let $x\in \Bbb{N}^\Bbb{N}$. The ball $B_x(r)$ is equal to all the sequences $z$ such that the first $\lfloor \frac{1}{r}\rfloor$ elements of $z$ coincide with those of $x$.</p>
<p>Let us suppose $B_x(r) \cap B_y(s) \neq \emptyset$, with $r < s$. Then there exists some $z$ such that the first $\lfloor \frac{1}{r}... |
1,764,106 | <p>In my book I have the following definition for subgroups of a group $G$ generated by $A$, a subset of G:</p>
<p>$$\langle A\rangle=\{x_1^{\epsilon_1}x_2^{\epsilon_2}...x_n^{\epsilon_n}\mid x_i\in A,~\epsilon_i\in\mathbb Z,~ x_i \neq x_{i+1} ,~ n=1,2,3...\}$$</p>
<p>I have no trouble understanding this. But then we... | Ken Duna | 318,831 | <p>You've got to remember that you are assuming that $G$ is abelian from the start. Therefore the elements of $A$ commute with each other.</p>
|
293,341 | <p>My apologies if this question is more appropriate for mathisfun.com, but I can only get so far reading about combinatrics and set theory before the interlocking logic becomes totally blurred. If this is a totally fundamental concept, feel free just to name it so I can read and understand the math myself.</p>
<p>So ... | Peter Smith | 35,151 | <p>If it's the very idea of (fairly) rigorous proof that is bugging you, then can I warmly suggest looking at the excellent</p>
<blockquote>
<p>Daniel J. Velleman, <em>How to Prove it: A Structured Approach</em> (CUP, 1994 and much reprinted, and now into a second edition).</p>
</blockquote>
<p>From the blurb: "Man... |
2,414,011 | <p>In my recent works in PDEs, I'm interested in finding a family of cut-off functions satisfying following properties:</p>
<p>For each $\varepsilon >0$, find a function ${\psi _\varepsilon } \in {C^\infty }\left( \mathbb{R} \right)$ which is a non-decreasing function on $\mathbb{R}$ such that:</p>
<ol>
<li>${\psi... | Joey Zou | 260,918 | <p>Let $\psi\in C^{\infty}(\mathbb{R})$ be any non-decreasing smooth function satisfying $\psi(x) = 0$ for $x\le 1$ and $\psi(x) = 1$ for $x\ge 2$, and set $\psi_{\epsilon}(x) = \psi(x/\epsilon)$, so that $\psi_{\epsilon}(x) = 0$ for $x\le\epsilon$ and $\psi_{\epsilon}(x) = 1$ for $x\ge 2\epsilon$. Then $\psi_{\epsilon... |
69,948 | <p>Has anyone ever created a "pairing function" (possibly non-injective)
with the property to be nondecreasing wrt to product of arguments, integers n>=2, m>=2. (We can also assume that n and m are bounded by an integer K, if useful) :</p>
<p>n m > n' m' => p(n,m) > p(n',m') </p>
<p>If yes what does it look lik... | ma11hew28 | 48,031 | <p>How about these unordered pairing functions?</p>
<p>For positive integers as arguments and where argument order doesn't matter:</p>
<ol>
<li><p>Here's an <a href="http://www.mattdipasquale.com/blog/2014/03/09/unique-unordered-pairing-function/" rel="nofollow">unordered pairing function</a>:</p>
<p>$<x, y> =... |
1,611,730 | <p>I am a linguist, not a mathematician, so I apologize if there's something wrong with my terminology and/or notation.</p>
<p>I have two structures that I want to merge (partially or completely). To generate a list of all possible combinations, I compute the Cartesian product of the two sets of objects, which gives m... | bof | 111,012 | <p>If $P$ is an $n$-element set ($\bar{\bar P}=n$) then the number of $k$-element subsets of $P$ (that's unordered subsets, no repetitions) is given by the <a href="https://en.wikipedia.org/wiki/Binomial_coefficient" rel="nofollow">binomial coefficient</a>
$$\binom nk=\frac{n!}{k!(n-k)!}=\frac{n(n-1)(n-2)\cdots(n-k+1)}... |
2,094,596 | <p>I'm questioning myselfas to why indeterminate forms arise, and why limits that apparently give us indeterminate forms can be resolved with some arithmetic tricks. Why $$\begin{equation*}
\lim_{x \rightarrow +\infty}
\frac{x+1}{x-1}=\frac{+\infty}{+\infty}
\end{equation*} $$</p>
<p>and if I do a simple operation,</... | Olivier Oloa | 118,798 | <p>An <em>inderterminate form</em> just means that we have to take a closer look to understand what happens.
Continuing with your example, we have
$$
"\lim_{x \rightarrow +\infty}
\frac{x+1}{2x-1}=\frac{+\infty}{+\infty}"
$$ then <em>seing things in more details</em>:
$$
\lim_{x \rightarrow +\infty}
\frac{x+1}{2x-1}=\l... |
1,040,136 | <p>Just a quick question:</p>
<p>Is the size of the set of real numbers from 1 to 2 greater, or equal in size to the number of real numbers between 1 and 10?</p>
<p>I'm a Physicist so I'm not totally clued up on Mathematical jargon pertaining to set theory...</p>
| erfan soheil | 195,909 | <p>If $ a < b$ and $c < d $ then $card ([a,b]) = card ([c,d]) $</p>
<p>Proof : define $ f :[a,b]\to [c,d]$</p>
<p>$f(x)= \frac{d-c}{b-a}x- \frac{a(d-c)-c(b-a)}{b-a}$, $f$ is bijection so the proof is complete.</p>
|
1,721,565 | <p>I'm having trouble with what I have done wrong with the chain rule below. I have tried to show my working as much as possible for you to better understand my issue here.</p>
<p>So:</p>
<p>Find $dy/dx$ for $y=(x^2-x)^3$
<br> So power to the front will equal = $3(x^2-x)^2 * (2x-1)$</p>
<p>Where did the $-1$ come f... | Santiago | 326,828 | <p>Since $(x^n)' = n x^{n-1}$, we have $(x^2)' = 2x$ and $x' = 1$, therefore $(x^2 -x)' = 2x-1$ - differentiation of functions is additive.</p>
|
1,555,548 | <p>There are $8$ people and they want to sit in a bus which has $2$ single front seats and $4$ sets of $3$ seats with $1$ person that is always the designated driver. How many ways are there for the people to sit in the bus?</p>
<p>I solved it by using:</p>
<p>$6!*(\binom{9}{3}) - 4((6*5*4*3)*2(\binom{4}{2})+(6*5*4*3... | DJohnM | 58,220 | <p>Looking at another, viable(?) interpretation:</p>
<p>Take the one driver and put him/her in the driver's seat (one of the two single front seats). That leaves seven <strong>distinguishable</strong> people and $13$ <strong>distinguishable seats</strong>.</p>
<p>Add six <strong>identical</strong> empty boxes to the... |
52,364 | <p>In addition to my previous post, <a href="https://mathematica.stackexchange.com/questions/52295/problem-regarding-3d-plot-of-a-moebius-strip-from-a-set-of-2d-points">regarding plotting the surface of a Möbius strip</a>, I now realised, that some of the eigenmodes for a Möbius strip are either oscillations of a scala... | chuy | 237 | <p>Something like this?</p>
<pre><code>data2D = Import["http://pastebin.com/raw.php?i=0Liw8F1r", "NB"];
data4D = Import["http://pastebin.com/raw.php?i=zgrCRiQh", "NB"];
</code></pre>
<p>Find <code>min</code> and <code>max</code> values for the color data</p>
<pre><code>{min, max} = {Min[#], Max[#]} &@data4D[[All... |
52,364 | <p>In addition to my previous post, <a href="https://mathematica.stackexchange.com/questions/52295/problem-regarding-3d-plot-of-a-moebius-strip-from-a-set-of-2d-points">regarding plotting the surface of a Möbius strip</a>, I now realised, that some of the eigenmodes for a Möbius strip are either oscillations of a scala... | J. M.'s persistent exhaustion | 50 | <p>Nowadays, one can use <code>DelaunayMesh[]</code> directly for mesh triangulation instead of <code>ListDensityPlot[]</code>:</p>
<pre><code>data2D = Import["http://pastebin.com/raw.php?i=0Liw8F1r", "NB"];
data4D = Import["http://pastebin.com/raw.php?i=zgrCRiQh", "NB"];
pts = Drop[data4D, None, -1];
dm = DelaunayM... |
2,774,923 | <blockquote>
<p>$ABC$ is a triangle where $AE$ and $EB$ are angle bisectors, $|EC| = 5$, $|DE| = 3$, $|AB| = 9$. Find the perimeter of the triangle $ABC$.
<a href="https://i.stack.imgur.com/4nsiM.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/4nsiM.jpg" alt="enter image description here"></a></p... | Intelligenti pauca | 255,730 | <p>A simpler proof that such a triangle cannot exist, as also claimed by @g.kov.</p>
<p>The <a href="https://www.cut-the-knot.org/Curriculum/Geometry/LocusCircle.shtml" rel="nofollow noreferrer">locus of points</a> $P$ such that $PC/PD=5/3$ is a circle of radius $7.5$ passing through $E$ and with center on line $CD$. ... |
1,192,357 | <blockquote>
<p>What is the limit of $f(a,b) =\frac{a^\beta}{a^2 + b^2}$ as $(a,b) \to (0,0)$?</p>
</blockquote>
<p>Clearly the answer depends on the value of $\beta$. For $\beta > 0$, we can deduce via inequalities that $\lim_{(a,b) \to (0,0)} f(a,b) = 0$.</p>
<p>However, for $\beta < 0$, the answer is less ... | LaBird | 220,474 | <p><strong>Added: Assume $\beta$ is an integer.</strong></p>
<p>Case $1$: If $\beta > 2$, then the numerator is more dominant than the denominator, hence the limit of $f(a, b)$ when $(a, b) \rightarrow (0,0)$ is $0$.</p>
<p>Case $2$: If $\beta < 2$, then the denominator is more dominant than the numerator, and ... |
1,001,320 | <p>I was wondering how to do an inequality problem involving QM-AM-GM-HM.</p>
<p>Question: For positive $a$, $b$, $c$ such that $\frac{a}{2}+b+2c=3$, find the maximum of $\min\left\{ \frac{1}{2}ab, ac, 2bc \right\}$.</p>
<p>I was thinking maybe apply AM-GM, however, I'm not sure what to plug in. Any help would be app... | vadim123 | 73,324 | <p>Applying AM-GM
$$1=\frac{(a/2)+b+2c}{3}\ge \sqrt[3]{(a/2)b(2c)}=\sqrt[3]{abc}$$
We square both sides to get $$\sqrt[3]{a^2b^2c^2}\le 1 ~~~~~~(1)$$</p>
<p>Now suppose $\min\{ab/2,ac,2bc\}=k$. We take the geometric mean of these three:
$$\sqrt[3]{a^2b^2c^2}=\sqrt[3]{(ab/2)ac(2bc)}\ge \sqrt[3]{k^3}=k ~~~~~~~(2)$$</p... |
2,026,486 | <p>Let $h,g$ be <em>not</em> injective functions, can function $f:\mathbb{R}\rightarrow \mathbb{R}^2$ such that $f(x) = (h(x), g(x))$ be injective?</p>
<p>I know that, if I pick polynomials for $h$ and $g$, then may be not injective, if picked carefully. For example, I can check zeros of the polynomials, whether they ... | marco2013 | 79,890 | <p>Let $h(x)=x^2$, and $g(x)=(x+1)^2$. $h$ and $g$ are not injective.</p>
<p>But,if $f(x)=f(y)$, we have $x^2=y^2$ and $(x+1)^2=(y+1)^2$.</p>
<p>So, $(x+1)^2-x^2-1=(y+1)^2-y^2-1$.</p>
<p>And then $2x=2y$. So $x=y$.</p>
<p>So, $f$ is injective.</p>
|
4,521,199 | <blockquote>
<p><strong>Theorem 8.15</strong>: If <span class="math-container">$f$</span> is a continuous and <span class="math-container">$2\pi$</span>-periodic function and if <span class="math-container">$\epsilon>0$</span> is fixed, then there exists a trigonometric polynomial <span class="math-container">$P$</s... | Paul Frost | 349,785 | <p>Define
<span class="math-container">$$\phi : \mathbb R \to S^1, \phi(x) = e^{ix} = \cos x + i \sin x. $$</span>
It is well-known that this map is a continuous surjection such that <span class="math-container">$\phi(x) =\phi(y)$</span> iff <span class="math-container">$x - y = 2k \pi$</span> for some <span class="mat... |
52,841 | <p>In classical Mechanics, momentum and position can be paired together to form a symplectic manifold. If you have the simple harmonic oscillator with energy $H = (k/2)x^2 + (m/2)\dot{x}^2$. In this case, the orbits are ellipses. How is the vector field determined by the (symplectic) gradient, then? </p>
<p>Also, ... | Vít Tuček | 6,818 | <p>You can view $\mathbb{R}^{2n}$ as a quotient of the real Heisenberg group $\mathcal{H}^{2n+1}$ modulo its center. For a closed loop $\alpha$ in $\mathbb{R}^{2n}$ and a point in $\mathcal{H}^{2n+1}$ over $\alpha(0)$ there's unique lift $\tilde{\alpha}$ of $\alpha$ to $\mathcal{H}^{2n+1}$ going through this point. Th... |
2,826,850 | <p>$x,y$ and $z$ are consecutive integers, such that $\frac {1}{x}+ \frac {1}{y}+ \frac {1}{z} \gt \frac {1}{45} $, what is the biggest value of $x+y+z$ ?.</p>
<p>I assumed that $x$ was the smallest number so that I could express the other numbers as $x+1$ and $x+2$ and in the end I got to a cubic function but I didn'... | fleablood | 280,126 | <p>Just do it. If $z\le {3*45} $ than $\frac 1x+\frac 1y+\frac 1z >3\frac 1 z\ge \frac 1 {45}$ any such $z-2=x <z-1=y <z\le 135$ will do and the largest $x+y+z $ will be when $z=135$.</p>
<p>Alternatively if $x\ge 135$ then $\frac 1x+\frac 1y+\frac 1z <3\frac 1 x\le \frac 1 {45}$ so none with $ z=x+2>... |
4,577,651 | <p>From Gallian's "Contemporary Abstract Algebra", Part 2 Chapter 5</p>
<p>It looks like using Lagrange's theorem would work, since <span class="math-container">$|S_n| = n!$</span> and <span class="math-container">$\langle\alpha\rangle$</span> is a subgroup of <span class="math-container">$S_n$</span>. Howeve... | cigar | 1,070,376 | <p><strong>Hint</strong></p>
<p>The order of any cycle of size <span class="math-container">$k$</span> is <span class="math-container">$k$</span>, and thus less than or equal to <span class="math-container">$n$</span>. And thus divides <span class="math-container">$n!$</span>.</p>
<p>Then there's the <em>universal pro... |
1,204,128 | <p>Two sides of a triangle are 6 m and 8 m in length and the angle between them is increasing at a rate of $0.06$ rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is $\large\frac {\pi}{3}$ rad. </p>
| egreg | 62,967 | <p>No, it's not sufficiently rigorous: either you make an $\varepsilon$-$\delta$ proof or appeal to a known theorem.</p>
<p>For instance, you can observe that
$$
|f(x)|\le|2x|
$$
and appeal to the squeeze theorem.</p>
<hr>
<p>For an $\varepsilon$-$\delta$ proof, consider $\varepsilon>0$ and set $\delta=\varepsilo... |
1,204,128 | <p>Two sides of a triangle are 6 m and 8 m in length and the angle between them is increasing at a rate of $0.06$ rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is $\large\frac {\pi}{3}$ rad. </p>
| Jack M | 30,481 | <p>First of all, your conclusion should not be that $f(0)=0$, it should be that $\lim_{x\to 0} f(x) = 0$.</p>
<p>Your proof isn't really rigorous, although the idea is correct. What you're implicitly assuming is something like the following proposition:</p>
<blockquote>
<p>If $\lim_{x\to a} g(x) = L$ and $\lim_{x\t... |
2,262,094 | <p>In Sheldon Axler's <em>Linear Algebra Done Right</em> Page 35, he gave a proof of "Length of linearly independent list <=length of spanning list"(see reference below) by using a process of adding a vector from independent list to the spanning list and then cancels a vector form the spanning list to form a new spa... | Tengu | 58,951 | <p>The main thing to notice is the order of writing the list.</p>
<p>Your step 1 said we can remove $u_1$ is false. Note that in Linear Dependence Lemma, it said there exists a vector $v_j$ in the list so it belongs to span of all <strong>previous</strong> vectors $v_1, \ldots, v_{j-1}$. Notice all $v_1, \ldots, v_{j-... |
903,656 | <p>An urn has $2$ balls and each ball could be green, red or black. We draw a ball and it was green, then it was returned it to the urn. What is the probability that the next ball is red? </p>
<p>My attempt: I think it is just a probability of $1/4$ because we have 4 colors in total but on the other hand I think i ne... | Jack D'Aurizio | 44,121 | <p>There are just three colors, not four. Anyway, half of the times you re-draw the first ball, that is green. In the other case, you draw the other ball that may be green, red or black with equal probability, $\frac{1}{3}$. Hence the probability to draw a red ball the second time is $\frac{1}{2}\cdot\frac{1}{3}=\frac{... |
2,410,517 | <p>I feel like I'm missing something very simple here, but I'm confused at how Rudin proved Theorem 2.27 c:</p>
<p>If <span class="math-container">$X$</span> is a metric space and <span class="math-container">$E\subset X$</span>, then <span class="math-container">$\overline{E}\subset F$</span> for every closed set <spa... | Perturbative | 266,135 | <p>It's not too hard to prove that for any sets <span class="math-container">$A \subseteq B$</span> in a metric space <span class="math-container">$(X, d)$</span>, it follows that <span class="math-container">$A' \subseteq B'$</span>. With the above result at hand, we can prove Theorem <span class="math-container">$2.2... |
2,410,517 | <p>I feel like I'm missing something very simple here, but I'm confused at how Rudin proved Theorem 2.27 c:</p>
<p>If <span class="math-container">$X$</span> is a metric space and <span class="math-container">$E\subset X$</span>, then <span class="math-container">$\overline{E}\subset F$</span> for every closed set <spa... | Avik Chakravarty | 630,919 | <p>Let <span class="math-container">$x \in E'$</span>, then <span class="math-container">$\forall r > 0$</span>, <span class="math-container">$N_r(x) \cap(E-\{x\}) \ne \emptyset$</span>. Since <span class="math-container">$E \subset F$</span>, then <span class="math-container">$\forall r > 0$</span>, <span class=... |
894,159 | <p>I was assigned the following problem: find the value of $$\sum_{k=1}^{n} k \binom {n} {k}$$ by using the derivative of $(1+x)^n$, but I'm basically clueless. Can anyone give me a hint?</p>
| quid | 85,306 | <p>Recall that
$$
(1+x)^n = \sum_{k=0}^{n} x^k \binom {n} {k}
$$
and thus
$$
((1+x)^n)' = \sum_{k=1}^{n} k x^{k-1} \binom {n} {k}
$$
Now, calculate the left-hand side, and then think which value of $x$ could be a good choice.</p>
|
2,303,106 | <p>I was looking at this question posted here some time ago.
<a href="https://math.stackexchange.com/questions/1353893/how-to-prove-plancherels-formula">How to Prove Plancherel's Formula?</a></p>
<p>I get it until in the third line he practically says that $\int _{- \infty}^{+\infty} e^{i(\omega - \omega')t} dt= ... | Conversely | 394,051 | <p>I think he used that $$ 1 = \hat{\delta(w)} $$ so, $$\int _{-\infty}^{+\infty} e^{i(\omega-\omega ')t} dt $$ is the antitransform of $\delta$ values in $(\omega - \omega') $ plus $2\pi$ for definition of antitransform.</p>
|
2,303,106 | <p>I was looking at this question posted here some time ago.
<a href="https://math.stackexchange.com/questions/1353893/how-to-prove-plancherels-formula">How to Prove Plancherel's Formula?</a></p>
<p>I get it until in the third line he practically says that $\int _{- \infty}^{+\infty} e^{i(\omega - \omega')t} dt= ... | Disintegrating By Parts | 112,478 | <p>A classical way to interpret what you have is through the Fourier transform and its inverse. If $f$ is continuous at $x$ where it has left- and right-hand derivatives, and if $f$ is suitably integrable on $\mathbb{R}$, then
$$
\lim_{R\rightarrow\infty}\frac{1}{\sqrt{2\pi}}\int_{-R}^{R}\hat{f}(s)e^{isx}ds = f(x)... |
1,826,964 | <blockquote>
<p>A fair die is tossed n times (for large n). Assume tosses are independent. What is the probability that the sum of the face showing is $6n-3$?</p>
</blockquote>
<p>Is there a way to do this without random variables explicitly? This is in a basic probability theory reviewer, and random variables was n... | drhab | 75,923 | <p>If $Y_i:=6-X_i$ then you are looking for $P(Y_1+\cdots+Y_n=3)$. </p>
<p>Observe that (preassuming the die is $6$-sided) the $Y_i$ take values in $\{0,1,2,3,4,5\}$</p>
<p>The answer is:</p>
<p>$$\left[\binom{n}3+n(n-1)+n\right]\left(\frac16\right)^n$$</p>
<p>$\binom{n}3$ corresponds with $n$-tuples having on $3$ ... |
2,909,480 | <p>Please notice the following before reading: the following text is translated from Swedish and it may contain wrong wording. Also note that I am a first year student at an university - in the sense that my knowledge in mathematics is limited.</p>
<p>Translated text:</p>
<p><strong>Example 4.4</strong> Show that it ... | BCLC | 140,308 | <p>We want to show the following proposition</p>
<p>$$k^3 - k \ \text{is always divisible by 3 for positive integers} \ k \tag{*}$$.</p>
<p>The set of positive integers has a special property that if some proposition, such as Proposition (*), is</p>
<ol>
<li><p>true for the first positive integer, $n=1$ (analogy: th... |
2,325,565 | <p>Is it possible to calculate probability density function from a data set of values? I assume this should be some kind of a function fitting exercise. </p>
| Nick Peterson | 81,839 | <p>No, not quite. It's easy to come up with infinitely many density functions that COULD have lead to a given finite set of observations; it's even possible to come up with infinitely many that make the outcome you observed 'likely'. So, without making more assumptions, you're pretty much stuck.</p>
<p>That said, we ... |
2,325,565 | <p>Is it possible to calculate probability density function from a data set of values? I assume this should be some kind of a function fitting exercise. </p>
| spaceisdarkgreen | 397,125 | <p>The simplest way is to make a histogram of the data and then normalize it so it has area one. A more sophisticated way is to use a <a href="https://en.wikipedia.org/wiki/Kernel_density_estimation" rel="nofollow noreferrer">kernel density estimator</a>.</p>
|
2,626,920 | <p>I am working on a homework problem and I am given this situation:</p>
<p>Let $A$ be the event that the $r$ numbers we
obtain are all different from each other. So, for example, if $n = 3$ and $r = 2$ the sample space is
$S = \{(1, 1),(1, 2),(1, 3),(2, 1),(2, 2),(2, 3),(3, 1),(3, 2),(3, 3)\}$
and the event $A$ is
$A... | Ethan Bolker | 72,858 | <p>Hint. You'll do better by thinking rather than (brute force) programming.</p>
<p>Can you calculate the size of the sample space? That depends only on $n$. You've already done $n=3$ and found $9$ elements.</p>
<p>Now how many ways can you choose $r$ different elements in order from among $n$? You found $6$ when $n=... |
2,626,920 | <p>I am working on a homework problem and I am given this situation:</p>
<p>Let $A$ be the event that the $r$ numbers we
obtain are all different from each other. So, for example, if $n = 3$ and $r = 2$ the sample space is
$S = \{(1, 1),(1, 2),(1, 3),(2, 1),(2, 2),(2, 3),(3, 1),(3, 2),(3, 3)\}$
and the event $A$ is
$A... | Mostafa Ayaz | 518,023 | <p>It is in fact $\Large\binom{n}{k}=\dfrac{n!}{k!(n-k)!}$</p>
|
2,626,920 | <p>I am working on a homework problem and I am given this situation:</p>
<p>Let $A$ be the event that the $r$ numbers we
obtain are all different from each other. So, for example, if $n = 3$ and $r = 2$ the sample space is
$S = \{(1, 1),(1, 2),(1, 3),(2, 1),(2, 2),(2, 3),(3, 1),(3, 2),(3, 3)\}$
and the event $A$ is
$A... | Community | -1 | <p>I get:</p>
<p>The sample size is $n^r$.</p>
<p>The number of possibilities for success is: $n(n-1)(n-2)\cdots(n-(r-1))=\frac{n!}{(r-1)!}$...</p>
|
114,909 | <p>What is known about normal subgroups of $SL_2(\mathbb{C}[X])$? Can one hope for a congruence subgroup property, i.e. that every (non-central) normal subgroup contains the kernel of the reduction modulo some ideal of $\mathbb{C}[X]$?</p>
| Andrei Smolensky | 5,018 | <p>Most likely, the analog of the standard "sandwich classification" of normal subgroups of $SL_2(\mathbb{C}[X])$ involves the notion of a radix, which gives the more careful form of a congruence subgroup.</p>
<p>See the paper <a href="http://www.ams.org/journals/bull/1991-24-01/S0273-0979-1991-15968-9/S0273-0979-1991... |
25,363 | <p>In what way and with what utility is the law of excluded middle usually disposed of in intuitionistic type theory and its descendants? I am thinking here of topos theory and its ilk, namely synthetic differential geometry and the use of topoi in algebraic geometry (this is a more palatable restructuring, perhaps), w... | Andrej Bauer | 1,176 | <p>You make a couple of basic mistakes in your question. Perhaps you should correct them and ask again because I am not entirely sure what it is you are asking:</p>
<ol>
<li><p>Topos theory does <em>not</em> "freely use $P \lor \lnot P$", and neither does synthetic differential geometry. In fact, topos theorists are q... |
87,948 | <p>Let $\mu_t, t \geq 0,$ be a family of probability measures on the real line. One can assume whatever one wishes about them, although typically they will be continuous in some topology (usually at least the topology of weak convergence of measures), and they will be absolutely continuous with respect to Lebesgue meas... | Community | -1 | <p>If you are willing to drop continuity in the parameter $t$, then you could let $(X_t)$ be independent with distribution $\mu_t$. </p>
|
295,618 | <p>Problem A: Please fill each blank with a number such that all the statements are true:</p>
<p>0 appears in all these statements $____$ time(s)<br>
1 appears in all these statements $____$ time(s)<br>
2 appears in all these statements $____$ time(s)<br>
3 appears in all these statements $____$ time(s)<br>
4 appears ... | JB King | 8,950 | <p>Note this does carry the assumption of counting the occurrence of the number at the start of the statement and not strictly in the boxes as there is a bit of interpretation there:</p>
<p>The first one has at least one solution. 1732111211 would be the values where there are 7 1s in the statements occurring in the a... |
295,618 | <p>Problem A: Please fill each blank with a number such that all the statements are true:</p>
<p>0 appears in all these statements $____$ time(s)<br>
1 appears in all these statements $____$ time(s)<br>
2 appears in all these statements $____$ time(s)<br>
3 appears in all these statements $____$ time(s)<br>
4 appears ... | Erick Wong | 30,402 | <p>For the revised B there are exactly 3 solutions consisting of single digits: $173311121291$, $174121121291$, $191311111391$. No single-digit solutions were found to the original B, and the solution to A is unique within this class.</p>
|
2,096,711 | <p>I need help finding a closed form of this finite sum. I'm not sure how to deal with sums that include division in it.</p>
<p>$$\sum_{i=1}^n \frac{2^i}{2^n}$$</p>
<p>Here's one of the attempts I made and it turned out to be wrong:</p>
<p>$$\frac{1}{2^n}\sum_{i=1}^n {2^i} = \frac{1}{2^n} (2^{n +1} - 1) = \frac{2^{n... | fonfonx | 247,205 | <p>Your first attempt was a good idea but you made some mistakes in your computations.</p>
<p>Since the denominator does not depend on $i$ you can take it out of the sum and you get</p>
<p>$$\sum_{i=1}^n \frac{2^i}{2^n}=\frac{1}{2^n}\sum_{i=1}^n {2^i} = \frac{1}{2^n} 2 (2^{n} - 1) = \frac{2^{n} - 1} {2^{n-1}}=2-2^{1-... |
2,353,272 | <p>Suppose that we are given the function $f(x)$ in the following product form:
$$f(x) = \prod_{k = -K}^K (1-a^k x)\,,$$
Where $a$ is some real number. </p>
<p>I would like to find the expansion coefficients $c_n$, such that:
$$f(x) = \sum_{n = 0}^{2K+1} c_n x^n\,.$$</p>
<p>A closed form solution for $c_n$, or at le... | G Cab | 317,234 | <p>We have that
$$ \bbox[lightyellow] {
\eqalign{
& f(x) = \prod\limits_{k\, = \, - K}^K {\left( {1 - a^{\;k} x} \right)} = \prod\limits_{k\, = \, - K}^K {a^{\;k} \left( {a^{\; - k} - x} \right)} = \cr
& = \left( {\prod\limits_{k\, = \, - K}^K {a^{\;k} } } \right)\;\prod\limits_{k\, = \, - K}^K {\l... |
359,277 | <p>Can you find function which satisfies $f(ab)=\frac{f(a)}{f(b)}$? For example $log(x)$ satisfies condition $f(ab)=f(a)+f(b)$ and $x^2$ satisfies $f(ab)=f(a)f(b)$?</p>
| Andreas Caranti | 58,401 | <p>Assuming the function is defined on non-zero real numbers, and takes all non-zero values (but please do see below for a generalization), one has first
$$
f(1) = f(1 \cdot 1) = \frac{f(1)}{f(1)} = 1,
$$
and then for all $x$
$$
f(x) = f( 1 \cdot x) = \frac{1}{f(x)},
$$
so that $f(x) \in \{1, -1 \}$. Then
$$
f(x y) = \... |
2,524,809 | <p>I am trying to solve the following problem:</p>
<blockquote>
<p>Show that $\frac{dy}{dt}=f(y/t)$ is equal to $t\frac{dv}{dt}+v=f(v)$, (which is a separable differential equation) by using substitution of $y = t \cdot v$ or $v =\frac{y}{t}$. </p>
</blockquote>
<p>I did the following:</p>
<p>By using the chain-ru... | aleden | 468,742 | <p>To implicitly differentiate, you must apply the chain rule:</p>
<p>$$\frac{\rm d}{\rm dx}(x^2+y^2)=\frac{\rm d}{\rm dx}(16)$$
$$2x+2y\frac{\rm dy}{\rm dx}=0$$
$$\frac{\rm dy}{\rm dx}=-\frac{x}{y}$$</p>
|
2,524,809 | <p>I am trying to solve the following problem:</p>
<blockquote>
<p>Show that $\frac{dy}{dt}=f(y/t)$ is equal to $t\frac{dv}{dt}+v=f(v)$, (which is a separable differential equation) by using substitution of $y = t \cdot v$ or $v =\frac{y}{t}$. </p>
</blockquote>
<p>I did the following:</p>
<p>By using the chain-ru... | Faraad Armwood | 317,914 | <p>Use the chain rule. Here you are assuming $y$ is a function of $x$ i.e you have $y = y(x)$. It now follows that, $y^2 = (y(x))^2 = x^2 \circ y(x)$ and so by the chain rule,</p>
<p>$$ \frac{d y^2}{dx} = \frac{d}{dx}(x^2) \cdot \frac{d}{dx}(y(x)) = 2x \cdot \frac{d y}{dx}$$</p>
|
3,534,566 | <p>I want to know if my answer is equivalent to the one in the back of the book. if so what was the algebra? if not then what happened?</p>
<p><span class="math-container">$$x^2y'+ 2xy = 5y^3$$</span></p>
<p><span class="math-container">$$y' = -\frac{2y}{x} + \frac{5y^3}{x^2}$$</span></p>
<p><span class="math-contai... | nonuser | 463,553 | <p>Write:<span class="math-container">$$(x^2y)' = 5y^3$$</span> Now let <span class="math-container">$z=x^2y$</span> then we get <span class="math-container">$$z' = {5z^3\over x^6}\implies {z'\over z^3} = 5x^{-6}$$</span>
So after integrating both sides we get <span class="math-container">$$ -{z^{-2}\over 2} = -x^{-5}+... |
3,534,566 | <p>I want to know if my answer is equivalent to the one in the back of the book. if so what was the algebra? if not then what happened?</p>
<p><span class="math-container">$$x^2y'+ 2xy = 5y^3$$</span></p>
<p><span class="math-container">$$y' = -\frac{2y}{x} + \frac{5y^3}{x^2}$$</span></p>
<p><span class="math-contai... | Lutz Lehmann | 115,115 | <p>You switched one sign too many in
<span class="math-container">$$
-\frac12v'+\frac2xv=\frac5{x^2}
$$</span>
Then
<span class="math-container">$$
\left(\frac{v}{x^4}\right)'=\frac{v'}{x^4}-\frac{4v}{x^5}=-\frac{10}{x^6}
\implies
\frac{v}{x^4}=\frac2{x^5}+C
$$</span>
etc.</p>
|
63,525 | <p>I asked this question in math.stackexchange but I didn't have much luck. It might be more appropiate for this forum. Let $z_1,z_2,…,z_n$ be i.i.d random points on the unit circle ($|z_i|=1$) with uniform distribution on the unit circle. Consider the random polynomial $P(z)$ given by
$$
P(z)=\prod_{i=1}^{n}(z−z_i).
$... | Johan Wästlund | 14,302 | <p>Here is a more careful (EDIT: even more careful!) argument that gives an affirmative answer to the weaker version of the question (as stated in the edit to my previous post, I doubt that the stronger version is true).</p>
<p>The argument uses the following lemma, which ought to be known. If someone has a reference,... |
2,214,137 | <p>How many positive integer solutions does the equation $a+b+c=100$ have if we require $a<b<c$?</p>
<p>I know how to solve the problem if it was just $a+b+c=100$ but the fact it has the restriction $a<b<c$ is throwing me off.</p>
<p>How would I solve this?</p>
| Bérénice | 317,086 | <p>$a<b<c$ can be translated by $b=a+x$ and $c=b+y=a+(x+y)$, where $a,x,y>0$.
$$a + b + c = 100\iff a+a+x+a+x+y=100 \iff3a+2x+y=100$$</p>
<p>Let $a_k$ the number of solutions of the diophantine equation $3a+2x+y=k$, where $a,x,y>0$. </p>
<p>$$\sum_{d=0}^\infty a_k t^k=\sum_{k=0}^\infty(\sum_{3a+2x+y=k\\a,... |
2,214,137 | <p>How many positive integer solutions does the equation $a+b+c=100$ have if we require $a<b<c$?</p>
<p>I know how to solve the problem if it was just $a+b+c=100$ but the fact it has the restriction $a<b<c$ is throwing me off.</p>
<p>How would I solve this?</p>
| Vik78 | 304,290 | <p>First we count the number of triples of positive integers $(a, b, c)$ with $a + b + c = 100$. There are $\binom{99}{2}$ of them, which results from an elementary application of stars and bars. Just line up a hundred dots, place a bar in between the $a$th and $(a +1)$th dot, and then place another bar $b$ dots to the... |
892,758 | <p>Theorem: Let $(X,\mathscr{T})$ be a topological space. If $E$ is connected and $K$ is such that $E\subseteq K\subseteq\mathrm{cl}(E)$, then $K$ is connected.
(Cl(E) is closure of E)</p>
<p>Question: Consider the standard topology on $\mathbf{R}$. Let $\mathbf{E}$ = (2,4). Then cl($\mathbf{E}$)= [2,4]. Let $\mathbf{... | Mohammad Khosravi | 87,886 | <p>The point is that $[3,4)$ is not open set in $[2,4)$. In the definition of connectivity it is needed that there exist no two open sets such that $X = A\cup B$ and if there exist such sets $X$ is not connected.</p>
|
892,758 | <p>Theorem: Let $(X,\mathscr{T})$ be a topological space. If $E$ is connected and $K$ is such that $E\subseteq K\subseteq\mathrm{cl}(E)$, then $K$ is connected.
(Cl(E) is closure of E)</p>
<p>Question: Consider the standard topology on $\mathbf{R}$. Let $\mathbf{E}$ = (2,4). Then cl($\mathbf{E}$)= [2,4]. Let $\mathbf{... | Henno Brandsma | 4,280 | <p>A set is only disconnected when we can partition it into two disjoint non-empty <em>open</em> sets, not just two sets. And $[2,3)$ is not open in the standard topology on $\mathbb{R}$. Neither is $[3,4)$.</p>
<p>Why don't you partition $(2,4)$ in $(2,3)$ and $[3,4)$, e.g.? The same would hold, and still you believe... |
2,405,905 | <p>Let $R$ be a commutative semi-local ring (finitely many maximal ideals) such that $R/P$ is finite for every prime ideal $P$ of $R$ ; then is it true that $R$ is Artinian ring ? From the assumed condition , we get that $R$ has Krull dimension 0 ; so it is enough to ask : Is $R$ a Noetherian ring ? From the semi-loc... | Elle Najt | 54,092 | <p>As you say, the ring is zero dimensional because each $R/P$ is a finite domain, hence a field. Hence we get a map $R \to \prod R/m_i$, running over the maximal ideals. The target here is a product of finite fields, hence the image is Noetherian. The kernel is the nilradical of $R$, since the nilradical is the inters... |
398,176 | <p>I had a calculus course this semester in which I was taught that the integration of the area gives the size (volume):</p>
<p>$$V = \int\limits_a^b {A(x)dx}$$</p>
<p>But this doesn't seem to work with the square. Since the size of the area of the square is $x^2$ then $A(x) = {x^2}$, then: </p>
<p>$$V = \int\limits... | response | 76,635 | <p>It should be:</p>
<p>$$V = \int_0^a a^2 dz$$</p>
<p>where $a$ is the length of one of the sides of the square.</p>
<p>Or using your notation:</p>
<p>$$V = \int_0^x x^2 dz$$</p>
<p>where $z$ is the dimension over which you are integrating.</p>
|
398,176 | <p>I had a calculus course this semester in which I was taught that the integration of the area gives the size (volume):</p>
<p>$$V = \int\limits_a^b {A(x)dx}$$</p>
<p>But this doesn't seem to work with the square. Since the size of the area of the square is $x^2$ then $A(x) = {x^2}$, then: </p>
<p>$$V = \int\limits... | Harish Chandra Rajpoot | 210,295 | <p>In general if a solid has cross-sectional area $A$ which is constant along its normal length say $L$ then the volume of such solid is $$\color{blue}{V=\int_{0}^L Adx=A\int_{0}^L dx=A[L-0]=AL}$$
In fact, the area of the square is not varying with the distance $x$ in a cube. It is $a^2$ which is constant along entire ... |
27,965 | <p>I'm looking at <a href="https://math.stackexchange.com/questions/2669893/calculating-the-sums-of-series">this question</a>. I gave the answer that was accepted. Please bear in mind that, when I answered this question, it was a different edit. In particular, there were more parts to the question.</p>
<p>The reason I... | Ethan Bolker | 72,858 | <p>Interesting question.</p>
<p>In general, I try not to answer short questions showing no work - commenting instead with boilerplate welcoming a new user (often that's the case, though not here) and a "show your work" prompt. There are folks here who will jump in with quick correct answers. I will often comment disap... |
2,508,508 | <p>Let $x_1$ be in $R$ with $ x_1>1$, and let $x_{k+1}=2- \frac{1}{x_k}$ for all $k$ in $N$. Show that the sequence $(x_k)_k$ is monotone and bounded and find its limit.</p>
<p>I am not sure how to start this problem.</p>
| Doug M | 317,162 | <p>by induction $x_k>1$</p>
<p>$x_1 > 1$</p>
<p>If $x_k > 1$ then $\frac {1}{x_{k}} < 1$ and $x_{k+1} = 2 - \frac {1}{x_k} > 1$</p>
<p>$\{x_k\}$ is montonic</p>
<p>We will show that $x_{k+1} - x_{k} < 0$</p>
<p>$x_{k+1} - x_{k}\\
2 - \frac {1}{x_k} - x_{k}\\
\frac {2x_k -1 + x_k^2}{x_k}\\
-\frac... |
3,521,224 | <p>Let <span class="math-container">$(U_1,U_2,...) , (V_1,V_2,...)$</span> be two independent sequences of i.i.d. Uniform (0, 1) random variables.
Define the stopping time
<span class="math-container">$N = \min\left(n\geqslant 1\mid U_n \leqslant V^2_n\right)$</span>.</p>
<p>Obtain <span class="math-container">$P(N ... | user8675309 | 735,806 | <p>To solve with minimal calculation, and focusing on your comment "I know that I should use conditioning in order to get the probability". </p>
<p>It is common to try to do "first step analysis" in these sorts of problems. Letting <span class="math-container">$A$</span> be the event that <span class="math-container... |
4,316,780 | <p>Let <span class="math-container">$X$</span> be a real Banach space. Let <span class="math-container">$J \colon X \to 2^{X^*}$</span> be its (normalized) duality map,
<span class="math-container">$$ J(x) = \{ x^* \in X^* \colon \langle x^* , x \rangle =||x|| \ ||x^*||, \ || x^* ||=||x|| \} , \ x \in X.$$</span>
As... | daw | 136,544 | <p>Let <span class="math-container">$f(x) = \|x\|$</span>. Then (1) implies
<span class="math-container">$$
f'(x; y) \ge 0.
$$</span>
Now <span class="math-container">$f$</span> is a continuous convex function, <span class="math-container">$J(x)=\{j(x)\}$</span> is the subdifferential of <span class="math-container">$f... |
1,171,980 | <p>I would like to show the following</p>
<blockquote>
<p>$$-x-x^2 \le \log(1-x) \le -x, \quad x \in [0,1/2].$$</p>
</blockquote>
<p>I know that for $|x|<1$, we have $\log(1-x)=-\left(x+\frac{x^2}{2}+\cdots\right)$. The inequality on the right follows because the difference is $\frac{x^2}{2}+ \frac{x^3}{3} + \cd... | Jack D'Aurizio | 44,121 | <p>$$\frac{d}{dx}\left(\log(1-x)+x+x^2\right) = 1+2x-\frac{1}{1-x}=\frac{x(1-2x)}{1-x} $$
is a non-negative function on $\left[0,\frac{1}{2}\right]$, hence the LHS-inequality follows.</p>
|
1,171,980 | <p>I would like to show the following</p>
<blockquote>
<p>$$-x-x^2 \le \log(1-x) \le -x, \quad x \in [0,1/2].$$</p>
</blockquote>
<p>I know that for $|x|<1$, we have $\log(1-x)=-\left(x+\frac{x^2}{2}+\cdots\right)$. The inequality on the right follows because the difference is $\frac{x^2}{2}+ \frac{x^3}{3} + \cd... | marty cohen | 13,079 | <p>$\begin{array}\\
-\ln(1-x)
&=\sum_{k=1}^{\infty} \frac{x^k}{k}\\
&=x+\sum_{k=2}^{\infty} \frac{x^k}{k}\\
&=x+x^2\sum_{k=2}^{\infty} \frac{x^{k-2}}{k}\\
&=x+x^2\sum_{k=0}^{\infty} \frac{x^{k}}{k+2}\\
&\le x+x^2\sum_{k=0}^{\infty} \frac{x^{k}}{2}\\
&\le x+\frac{x^2}{2}\sum_{k=0}^{\infty} x^{k}\... |
562,707 | <p>This is a famous rudimentary problem : how to use mathematical operations (not any other temporary variable or storage) to swap two integers A and B. The most well-known way is the following:</p>
<pre><code>A = A + B
B = A - B
A = A - B
</code></pre>
<p>What are some of the alternative set of operations to achieve... | Mark S. | 26,369 | <p>I'm not sure if you're asking for all solutions or not, but one of the most famous solutions is <a href="http://en.wikipedia.org/wiki/XOR_swap_algorithm" rel="nofollow">by using binary xor three times</a>. $A=A\oplus B,B=A\oplus B,A=A\oplus B$.</p>
|
2,645,948 | <p>I was studying neighbourhood methods from Overholt's book of Analytic Number theory(P No 42). There to estimate $Q(x)=\sum_{n \leq x}\mu^2(n)$ they have used a statement that </p>
<p>$$\sum_{j^2\leq x} \mu(j)\left[\frac x {j^2}\right]=x\sum_{j\leq \sqrt x}\frac {\mu(j)} {j^2}+ O(\sqrt x).$$</p>
<p>I am not getting... | Matthew Conroy | 2,937 | <p>Since $[x]=x-\{x\}$, we have $[x]=x+O(1)$, and so
$$
\sum_{j^2\leq x} \mu(j)\left[\frac x {j^2}\right]=
\sum_{j^2\leq x} \mu(j) \left(\frac{x}{j^2} + O(1) \right)
=\sum_{j^2\leq x} \mu(j) \frac{x}{j^2} + O(\sqrt{x})
= x \sum_{j \le \sqrt{x}} \frac{\mu(j)}{j^2} +O(\sqrt{x}).
$$</p>
|
533,399 | <p>Starting with the classical propositional logic, is there a rather canonical way to prove that $$p\wedge q=q\wedge p$$ for the commutativity of the conjunction and analogously for the other properties and connectives, other than using truth tables, visualizing with Venn diagrams akin <a href="http://en.wikipedia.org... | user43208 | 43,208 | <p>I'm not sure this will satisfy you, but a categorically-minded way to characterize meets $a \wedge b$ and joins $a \vee b$ is via universal properties: </p>
<p>$$x \leq a \wedge b \;\;\; \text{iff}\;\;\; x \leq a,\; x \leq b$$ </p>
<p>$$a \vee b \leq x\;\;\; \text{iff}\;\;\; a \leq x,\; b \leq x$$</p>
<p>for any ... |
2,222,514 | <p>A straight line OL rotates around the point O with a constant angular velocity !. A point M moves along the line OL with a speed proportional to the distance OM. Find the equation of the curve described by the point M</p>
<p>As it says angular velocity is constant which i think means
$$... | Jens | 307,210 | <p>We know that $$\frac{dr}{dt}=kr\tag{1}$$</p>
<p>There is a <a href="http://tutorial.math.lamar.edu/Classes/DE/Linear.aspx" rel="nofollow noreferrer">long way</a> to determine that this means $$r=Ce^{kt}\tag{2}$$</p>
<p>or there is the shorter way of simply inserting answer $2$ into the differential equation $1$ an... |
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