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 |
|---|---|---|---|---|
109,213 | <p>In classical complex analysis it is easy to prove that a meromorphic function has at most one analytic continuation (on an open connected subset of $\mathbb C$, say).</p>
<p>The problem of non-uniqueness of analytic continuation is one of the reasons why it is not possible (if one wants a good theory) to translate ... | Ramsey | 12,107 | <p>Let me just complement xbnv's answer with a mild generalization. If $X$ is an <em>irreducible</em> rigid space (and let's suppose we're dealing with reduced spaces from the outset), then its normalization $\tilde{X}$ is connected and normal and is equipped with a finite surjective map $\tilde{X}\to X$. Using xbnv'... |
3,910,345 | <p>Recently a lecturer used this notation, which I assume is a sort of twisted form of Leibniz notation:</p>
<p><span class="math-container">$$y\,\mathrm{d}x - x\,\mathrm{d}y \equiv -x^2\,\mathrm{d}\left(\frac{y}{x}\right)$$</span></p>
<p>The logic here was that this could be used as:</p>
<p><span class="math-container... | Henry Lee | 541,220 | <p>I believe it is being used as shorthand for:
<span class="math-container">$$d\left(\frac yx\right)=dx\frac{d\left(\frac yx\right)}{dx}=dx\left(\frac{dy}{dx}x-\frac{y}{x^2}\right)=xdy-\frac{y}{x^2}dx$$</span></p>
|
508,790 | <p>I always see this word $\mathcal{F}$-measurable, but really don't understand the meaning. I am not able to visualize the meaning of it.</p>
<p>Need some guidance on this.</p>
<p>Don't really understand $\sigma(Y)$-measurable as well. What is the difference?</p>
| Davide Giraudo | 9,849 | <p>If $f\colon (X_1,\mathcal F_1)\to (X_2,\mathcal F_2)$, $f$ is $(\mathcal F_1,\mathcal F_2)$-measurable if for all $F_2\in\mathcal F_2$, $f^{-1}(F_2)\in\mathcal F_1$. </p>
<p>In some contexts we consider the case where $X_2$ is the real line and $\mathcal F_2$ the Borel $\sigma$-algebra. Then for short, we say that ... |
3,995,986 | <p>Need help integrating:
<span class="math-container">$$\int _0^{\infty }\:\:\frac{6}{\theta}xe^{-\frac{2x}{\theta }}\left(1-e^{-\frac{x}{\theta }}\right)dx$$</span></p>
<p>I think I should multiply the <span class="math-container">$$xe^{-\frac{2x}{\theta }}$$</span> out and then use integration by parts but it is not... | Community | -1 | <p>Hint.</p>
<p>Abstractly, all you need is to find integrals of the form:
<span class="math-container">$$
\int xe^{mx}dx
$$</span>
which by a change of variables, you only need to find
<span class="math-container">$$
\int xe^{mx}dx
=\frac{1}{m^2}\int ue^udu=
\frac{1}{m^2}\int xe^{x}dx=\frac{1}{m^2}(xe^x-e^x)+C
$$</spa... |
133,418 | <p>Let $\langle R,0,1,+,\cdot,<\rangle$ be the standard model for R, and let S be a countable model of R (satisfying all true first-order statements in R). Is it true that the set 1,1+1,1+1+1,… is bounded in S? My intuition says "no", but I am yet to find a counter example. I read something about rational functions, ... | zyx | 14,120 | <p>This is not what "non-standard model of ..." means. </p>
<p>You are asking whether non-Archimedean real-closed ordered fields exist. An example is to add some algebraically independent transcendental elements to the real algebraic numbers, or the Puiseux series field over the real algebraic numbers. </p>
<p>Arch... |
2,622,583 | <blockquote>
<p>Prove that if $f:\mathbb R \to \mathbb R$ is a measurable function and $f(x)=f(x+1)$ almost everywhere, then there exists a measurable function $g:\mathbb R \to \mathbb R$ with $f=g$ almost everywhere and $g(x)=g(x+1)$ for every $x \in \mathbb R$</p>
</blockquote>
<p>I'm trying to prove this by const... | John Dawkins | 189,130 | <p>Suppose, to start, that $f$ is bounded. Define $g_n(x):=n\int_{(x,x+11/n]} f(t)\,dt$. It is easy to check that $g_n(x)=g_n(x+1)$ for all $x$. Define $g(x):=\limsup_ng_n(x)$, $x\in\Bbb R$, and notice that $g(x+1)=g(x)$ for all $x$. If $x$ is a <em>Lebesgue point</em> of $f$, then $\lim_ng_n(x) =f(x)$, and in particul... |
3,695,439 | <p>So I know that we can find <span class="math-container">$dy/dx$</span> of a curve in polar coordinates by leveraging the fact that <span class="math-container">$x=rcos\theta$</span> and <span class="math-container">$y=rsin\theta$</span>, and since <span class="math-container">$r$</span> is a function of <span class=... | Christian Blatter | 1,303 | <p>If a curve <span class="math-container">$\gamma$</span> is given in <em>polar form</em>
<span class="math-container">$$\gamma:\quad r=r(\theta)\qquad(\theta_0\leq\theta\leq \theta_1)$$</span>
this is an abbreviation for the parametric representation
<span class="math-container">$$\gamma:\quad\theta\mapsto{\bf r}(\th... |
29,255 | <p>sorry! am not clear with these questions</p>
<ol>
<li><p>why an empty set is open as well as closed?</p></li>
<li><p>why the set of all real numbers is open as well as closed?</p></li>
</ol>
| Adrián Barquero | 900 | <p>Well the definition of a <a href="http://en.wikipedia.org/wiki/Topological_space">topological space</a> $X$ specifies that both $X$ and the empty set must be open sets (if the topology is defined in terms of closed sets rather than open sets, it will stipulate that they are closed). But then it is just by definitio... |
3,350,251 | <p>The integral of velocity plots position and not change in position. But the definition of the integral is the area under the velocity curve and the area under the velocity curve is change in position. So why doesn't the integral of velocity plot change in position?</p>
| Ethan Bolker | 72,858 | <p>The definite integral of the velocity from time <span class="math-container">$a$</span> to time <span class="math-container">$b$</span> is the change in position over that time interval.</p>
<p>You can only specify the position of an object relative to some point. "The position" makes sense only in a coordinate sys... |
717,664 | <p>I need a step by step answer on how to do this. What I've been doing is converting the top to $2e^{i(\pi/4)}$ and the bottom to $\sqrt2e^{i(-\pi/4)}$. I know the answer is $2e^{i(\pi/2)}$ and the angle makes sense but obviously I'm doing something wrong with the coefficients. I suspect maybe only the real part goes ... | ZHN | 131,755 | <p>The secret is to multiply and divide by the conjugate of the denominator:</p>
<p>$$\frac{2i+2}{1-i} =\frac{2i+2}{1-i}\frac{1+i}{1+i}=\frac{2(1+i)(1+i)}{1-i^2}=\frac{(1+i^2)}{1+1}=1+2i+i^2=1+2-1=2i.$$</p>
|
1,109,853 | <blockquote>
<p><em>The below proof is incorrect. See the answers for more information.</em></p>
</blockquote>
<p>This question is in the context of exploring how to explain the process of developing a proof.</p>
<p>When reading a proof on the irrationality of $ \sqrt{3} $, I came across the following statement, wh... | Bernard | 202,857 | <p>Congruences allow for a very simple proof of the assertion: ‘ If $a^2$ is divisible by $3$, the $a$ is divisible by $3$.</p>
<p>It suffices to draw up the list of squares modulo $3$:</p>
<ul>
<li>if $a\equiv 0\mod 3$, then $a^2\equiv 0^2=0 $;</li>
<li>if $a\equiv \pm 1$, then $a^2\equiv 1 \mod 3$.
Hence the only c... |
386,649 | <p>If you were working in a number system where there was a one-to-one and onto mapping from each natural to a symbol in the system, what would it mean to have a representation in the system that involved more than one digit?</p>
<p>For example, if we let $a_0$ represent $0$, and $a_n$ represent the number $n$ for any... | MJD | 25,554 | <p>In base 10, we represent a number $n$ as a sequence of digits $n_0, n_1, \ldots$ such that $$n = \sum_{i=0}^\infty n_i 10^i\qquad\text{where } 0\le n_i<10$$</p>
<p>and we require that the sequence of $n_i$ must be eventually zero.</p>
<p>By changing the representation a little bit, we get the so-called <a href=... |
199,148 | <p><a href="https://i.stack.imgur.com/9BuHp.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9BuHp.png" alt="enter image description here"></a> </p>
<pre><code>pbdomains = <|
"Overall " -> Around[2.6, 0.04],
"PB" -> Around[4.25, 0.06]
|>;
BarChart[pbdomains, ChartStyle ... | Chris Degnen | 363 | <p>You can use <code>PlotRangePadding</code>.</p>
<pre><code>pbdomains = <|
"Overall " -> Around[2.6, 0.04],
"PB" -> Around[4.25, 0.06]
|>;
BarChart[pbdomains, ChartStyle -> "BrightBands",
LabelStyle -> {FontFamily -> "Times New Roman", 28, Bold,
GrayLevel[0]}, Frame -> Tr... |
2,840,333 | <p>I know that the easy way to evaluate the mean and variance of the Binomial distribution is by considering it as a sum of Bernoulli distributions.</p>
<p>However, I was wondering just for fun if there is a way to evaluate them directly. I got the mean easily: it only involves some fiddling around with the binomial c... | David | 119,775 | <p>The differentiation method is good, but you can if you want extend your method* of fiddling around with the binomial coefficient:
$$\eqalign{
\sum_{k=0}^{n}k(k-1) \binom{n}{k} r^k
&=\sum_{k=0}^n k(k-1)\frac{n!}{k!\,(n-k)!}r^k\cr
&=\sum_{k=2}^n n(n-1)\frac{(n-2)!}{(k-2)!\,(n-k)!}r^k\cr
&=n(n-1)\su... |
1,571,099 | <blockquote>
<p>Consider the rectangle formed by the points $(2,7),(2,6),(4,7)$ and
$(4,6)$. Is it still a rectangle after transformation by $\underline
A$= $ \left( \begin{matrix} 3&1 \\ 2&\frac {1}{2} \\ \end{matrix}
\right) $ ?By what factor has its area changed ?</p>
</blockquote>
<p>I've defined th... | Rob Arthan | 23,171 | <p>$[a]u_1$ means that after action $a$ you will be in state $u_1$.
$[a](p \land q)$ means that after $a$ you will get into some state that satisfies both the properties $p$ and $q$. So if $u_1$ and $u_2$ are distinct states, $u_1 \land u_2$ will be false so $[a](u_1 \land u_2)$ will be false regardless of how you defi... |
466,757 | <p>Suppose we have the following</p>
<p>$$ \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_{ij}$$</p>
<p>where all the $a_{ij}$ are non-negative.</p>
<p>We know that we can interchange the order of summations here. My interpretation of why this is true is that both this iterated sums are rearrangements of the same series an... | Alp Uzman | 169,085 | <p>The double sum you are asking about can be considered to be the sum of all terms of the infinite array of numbers $a_{ij}$:</p>
<p>$$\begin{pmatrix}
a_{11} & a_{12} & \cdots & a_{1j} & \cdots\\
a_{21} & a_{22} & \cdots & a_{2j} & \cdots\\
\vdots& \vdots & &\vdots\\
a_{i1}... |
2,441,894 | <p>The matrix $$\pmatrix{100\sqrt{2}&x&0\\-x&0&-x\\0&x&100\sqrt{2}},\quad x>0$$ have two equal eigenvalues. How can I find $x$?
What I tried is this. If $\lambda_1$ is doubly degenerate and $\lambda_2$ the third eigenvalue, then the characteristic equation is $(\lambda-\lambda_1)^2(\lambda-\... | Nick | 27,349 | <p>See the documentation for the <a href="https://www.mathworks.com/help/matlab/ref/reshape.html" rel="nofollow noreferrer">reshape</a> function. Here is how you can use it:</p>
<p>First, make a $2$-by-$n$ matrix which has both rows equal to $A$ with:</p>
<p><code>[A;A]</code></p>
<p>Then use the <code>reshape</code... |
2,172,399 | <p>Equation of the segment : $2x + 4y-3 = 0$
Equation of the hyperbola : $7x^2 - 4y^2 =14$</p>
<p>How do you find the equation of the two linear functions that are both perpendicular to the segment and tangent to the hyperbola?</p>
<p>Thanks</p>
| Francis Cugler | 405,427 | <p>Draw the graphs of both original equations for the segment and the hyperbola. Then visually draw the lines that satisfies being perpendicular to the segment and tangent to the hyperbola. Then from these visualizations you can use vector notation of the two lines you drew and from there once you have vectors you can ... |
3,436,515 | <p>Please help!</p>
<p>How to show that <span class="math-container">$ \lim _{n→∞} \frac{x_{(n+1)}}{x_n} =\frac{1+\sqrt 5}{2}$</span> for a dynamical system
<span class="math-container">$$x_{(n+1)}=x_n + y_n\\
y_{(n+1)}=x_n$$</span></p>
<p>Thank you!</p>
| Dinno Koluh | 519,191 | <p>We should firstly shift the second equation by <span class="math-container">$1$</span>:
<span class="math-container">$$y_{n+1}=x_n \rightarrow y_n = x_{n-1}$$</span>
Let us now substitute this into the first equation:
<span class="math-container">$$x_{n+1}=x_n + y_n = x_n + x_{n-1}$$</span>
Let us again shift the... |
442,759 | <p>I was reading a book on groups, it points out about the uniqueness of the neutral element and the inverse element. I got curious, are there algebraic structures with more than one neutral element and/or more than one inverse element?</p>
| Ronnie Brown | 28,586 | <p>They do exist but they are algebraic structures with <strong>partial</strong> operations, i.e. the multiplication $a*b$ is not defined for all $a,b$. Typical examples are "journeys": you can compose a journey from $x$ to $y$ with a journey from $z$ to $w$ if and only if $y=z$. Standard mathematical examples are cat... |
1,178,265 | <p>I'm supposed to be able to determine <strong><em>without calculations</em></strong> the determinant, inverse matrix, and n-th power matrix of the rotation matrix :</p>
<p>$\begin{pmatrix}
cos\theta & sin\theta \\
-sin\theta & cos\theta
\end{pmatrix} $</p>
<p>Can someone explain to me how I can do that ... | Community | -1 | <p><strong>HINTS</strong></p>
<p>The determinant tells you by how much a linear transformation transforms areas (for $2\times 2$)/ volumes (for $3\times 3$)/ etc. So by what factor does this transformation change the area of say a square in the plane? That'll be your determinant.</p>
<p>The inverse matrix is the ma... |
3,346,775 | <p>Do there exist non-zero expectation, dependent, uncorrelated random variables <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>? The examples that I have found have at least one of the variables have zero expectation.</p>
| Kavi Rama Murthy | 142,385 | <p>Let <span class="math-container">$X$</span> have standard normal distribution. Then <span class="math-container">$1+X$</span> and <span class="math-container">$1+X^{2}$</span> satisfy your requirements. </p>
|
144,818 | <p>Let $x_1,x_2,\ldots,x_n$ be $n$ real numbers that satisfy $x_1<x_2<\cdots<x_n$.
Define \begin{equation*}
A=%
\begin{bmatrix}
0 & x_{2}-x_{1} & \cdots & x_{n-1}-x_{1} & x_{n}-x_{1} \\
x_{2}-x_{1} & 0 & \cdots & x_{n-1}-x_{2} & x_{n}-x_{2} \\
\vdots & \vdots & \ddots... | Robert Israel | 8,508 | <p>Clearly the determinant is $0$ if $x_i = x_{i+1}$ (because two adjacent rows are identical) or $x_1 = x_n$ (last row is $-$ first row). So the determinant must be a polynomial divisible by $(x_1 - x_2)(x_2 - x_3) \ldots (x_{n-1} - x_n)(x_n - x_1)$. But the determinant has degree $n$, so it is a constant times this... |
255,252 | <p>Let $\mathfrak{S}_n$ be the permutation group on an $n$-element set. For each fixed $k\in\mathbb{N}$, consider the two sets
$$A_n(k)=\{\sigma\in\mathfrak{S}_n\vert\,\, \text{$\exists i,\,\, 1\leq i\leq n\,$ such that $\,\sigma(i)-i=k$}\}$$
and
$$B_n(k)=\{\sigma\in\mathfrak{S}_n\vert\,\, \text{$\exists i,\,\, 1\leq ... | Martin Rubey | 3,032 | <p>You can use sage and www.findstat.org to find a candidate for a bijection as follows. First define the statistics you are interested in:</p>
<pre><code>def A_num(s, k):
return len([1 for i,e in enumerate(s,1) if e-i==k])
def B_num(s, k):
return len([1 for e,f in zip(s, s[1:]) if f-e==k])
</code></pre>
<p... |
66,951 | <p>I am asked to find all rows in a matrix in reduced row echelon form which contain nothing but pivots (pivot is $1$, all other entries are $0$).</p>
<p>For example, in this matrix:</p>
<p>$$
\begin{bmatrix}
1 & 1 & 1 & 1 \\
0 & 1 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\sim
\begin... | bill s | 1,783 | <p>I'm sure there are lots of things one might try -- here is an attempt using some of the built in options that are available in ListStreamPlot:</p>
<pre><code>Show[Graphics[{Black, CountryData["USA", "Polygon"]}],
ListStreamPlot[
Table[{{x, y},
Through[{Cos, Sin}[
WeatherData[{y, x}, "WindDirection"]]... |
4,288,188 | <p>I am trying to obtain a formulae for a summation problem under section (d) given in a solutions manual for "Data Structures and Algorithm Analysis in C - Mark Allen Weiss", here's the screen shot</p>
<p><a href="https://i.stack.imgur.com/fMie6.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur... | DinosaurEgg | 535,606 | <p>I agree with the answer by @ThomasAndrews in terms of the fact that this recursion is difficult to be solved directly, although I have a feeling it can be solved using finite difference calculus, since there is at least one solution to it as I demonstrate below.</p>
<p>Define the function</p>
<p><span class="math-co... |
1,158,489 | <p>Is it the case that, as $N\to\infty$, $$\binom{2N}{N+j}_q\to (-1)^j,$$ where convergence of the $q$-binomial coefficient is seen as a convergence of formal power series in the variable $q$? </p>
| Johann Cigler | 25,649 | <p>Let ${\left( {a;q} \right)_n}=\prod\limits_{j = 0}^{n - 1} {(1-{q^j}a} ).$</p>
<p>Then $ {2n\brack {n+j}}=\frac{(q;q)_{2n}}{(q;q)_{n+j}(q;q)_{n-j}}$ converges to $\frac{1}{(q;q)_\infty}=\sum\limits_{n \ge 0} {p(n){q^n}}=1+q+2q^2+3q^3+5q^4+ \dots,$ where $p(n)$ is the number of partitions of $n$.</p>
|
1,529,324 | <p>I've read that if $\Phi$ is a Poisson point process (on $\mathbb{R}^d$, say), then conditional on there being $k$ points in some $A \subseteq \mathbb{R}^d$, the positions $X_1,\ldots,X_k$ of these points are uniformly distributed in $A$.</p>
<p>I'm having trouble making sense of what this means. "Conditional on $\P... | Michael Hardy | 11,667 | <p>For every measurable set $A\subseteq\mathbb R^d$ of finite measure, and every measurable set $B\subseteq A$, let $p$ be the conditional probability that the number of sites in $B$ is $\ell$, given that the number of of sites in $A$ is $k$.</p>
<p>Suppose $X_1,\ldots,X_k\sim\text{i.i.d. Uniform}(A)$. Let $q$ be the... |
3,624,662 | <p><a href="https://i.stack.imgur.com/kwAMn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/kwAMn.png" alt=" c"></a></p>
<p>In my mind, I can think of below example which seems to work.</p>
<p>If <span class="math-container">$(X,T) = \mathbb{R}$</span>, and <span class="math-container">$A = (0,\inf... | José Carlos Santos | 446,262 | <p>The set <span class="math-container">$(0,\infty)$</span> is not closed. But <span class="math-container">$[0,\infty)$</span> will work. For instance, the sequence <span class="math-container">$1,2,3,\ldots$</span> has no convergent subsequence. Or you can say that <span class="math-container">$\{[0,n)\mid n\in\mathb... |
1,132,003 | <blockquote>
<p><strong>Problem</strong> Find the value of $$\frac{1}{\sqrt 1 + \sqrt 3} + \frac 1 {\sqrt 3 + \sqrt 5} + \dots + \frac 1 {\sqrt {1087} + \sqrt{1089}}$$</p>
</blockquote>
<p>I cant figure out how to solve this problem. I cant use summation.</p>
| jameselmore | 86,570 | <p>To complement the other answer: </p>
<p><strong>Hint:</strong></p>
<p>$$\frac12(\sqrt{n+4} - \sqrt{n+2}) + \frac12(\sqrt{n+2} - \sqrt{n}) = \frac12(\sqrt{n+4} - \sqrt{n}) + \frac12(\sqrt{n+2} - \sqrt{n+2})$$
$$ = \frac12(\sqrt{n+4} - \sqrt{n})$$</p>
<p>A rearrangement of terms shows that they collapse</p>
<p>ED... |
1,356,545 | <p>Given a fair 6-sided die, how can we simulate a biased coin with P(H)= 1/$\pi$ and P(T) = 1 - 1/$\pi$ ?</p>
| lulu | 252,071 | <p>Well, here's an (approximate) way to do it: write P as a "hex-decimal" in base$_6$. Then toss your die repeatedly forming a hex-decimal. (if you throw a "6" mark it as "0"). A Win comes if the hex-decimal produced by the die is less than P.</p>
<p>Of course, it is theoretically possible that this takes an arbitr... |
365,631 | <p>Suppose we want to prove that among some collection of things, at least one
of them has some desirable property. Sometimes the easiest strategy is to
equip the collection of all things with a measure, then show that the set
of things with the desired property has positive measure. Examples of this strategy
appear in... | Terry Tao | 766 | <p>The <a href="https://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem" rel="noreferrer">Chevalley-Warning theorem</a> asserts that if a system of polynomial equations in <span class="math-container">$r$</span> variables over a finite field of characteristic <span class="math-container">$p$</span> has total de... |
3,068,031 | <blockquote>
<p>Let <span class="math-container">$G$</span> be a group and <span class="math-container">$H$</span> be a subgroup of <span class="math-container">$G$</span>. Let also <span class="math-container">$a,~b\in G$</span> such that <span class="math-container">$ab\in H$</span>.</p>
<p>True or false? <span cla... | Aphelli | 556,825 | <p>Let <span class="math-container">$u \in G$</span>, <span class="math-container">$v \in H$</span>.</p>
<p>Take <span class="math-container">$a=u$</span>, <span class="math-container">$b=u^{-1}v$</span>. Then <span class="math-container">$ab \in H$</span>. </p>
<p>Moreover, <span class="math-container">$a^2b^2=uvu^{... |
3,110,508 | <p>I read that implication like a=>b can be proof using the following steps :
1) suppose a true.
2) Then deduce b from a.
3) Then you can conclude that a=>b is true.</p>
<p>Actually my real problem is to understand why step 1 and 2 are sufficient to prove that a=>b is true. I mean, how can you prove the truth table of... | cansomeonehelpmeout | 413,677 | <p>Maybe not easier, but given <span class="math-container">$$\frac{1-z^n}{1-z}=\sum_{i=0}^{n-1}z^i\\1-z^n=(1-z)\sum_{i=0}^{n-1}z^i\\$$</span> let <span class="math-container">$z=\frac{y}{x}$</span>, then <span class="math-container">$$1-\left(\frac{y}{x}\right)^n=\left(1-\frac{y}{x}\right)\sum_{i=0}^{n-1}y^ix^{-i}$$</... |
2,933,375 | <p>I have a set of vectors, <span class="math-container">$M_1$</span> which is defined as the following:
<span class="math-container">$$M_1:=[\begin{pmatrix}1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix}0 \\ 1 \\ 1 \end{pmatrix}]$$</span>
I have to show that <span class="math-container">$M_1$</span> isn't a generating set ... | Dan Christensen | 3,515 | <p>There are no universally accepted standards in this case. You really must get used to slight variations in notation from one author to the next.</p>
<p>To add to the confusion, I often write <span class="math-container">$\forall x: [x \in N \implies P(x)]$</span> and <span class="math-container">$\exists x: [x \in ... |
3,572,842 | <p><strong>Context:</strong> 1st year BSc Mathematics, Vectors and Mechanics module, constant circular motion.</p>
<p>This may be trivial, but can someone tell me what's wrong with the following reasoning?</p>
<p><span class="math-container">$$\underline{e_r}=\underline{i}\cos\theta+\underline{j}\sin\theta=(1,\theta)... | Will Jagy | 10,400 | <p>I have been running some programs. It seems that the break even point, where the possible values of your <span class="math-container">$a+b$</span> are half prime and half composite for
<span class="math-container">$$ a+b < 1736495 \; , \; $$</span>
a number between one million and two million. I'm impressed. ... |
393,712 | <p>I studied elementary probability theory. For that, density functions were enough. What is a practical necessity to develop measure theory? What is a problem that cannot be solved using elementary density functions?</p>
| John Douma | 69,810 | <p>Measure theory goes beyond probability theory. It generalizes our notion of length, area and volume.</p>
|
255,374 | <p>Does there exist any noncomputable set $A$ and probabilistic Turing machine $M$ such that $\forall n\in A$ $M(n)$ halts and outputs $1$ with probability at least $2/3$, and $\forall n\in\mathbb{N}\setminus A$ $M(n)$ halts and outputs $0$ with probability at least $2/3$? What if you only require that $M(n)$ is correc... | none | 101,583 | <p>Construct $A$ as follows. Roll a 6-sided die infinitely many times, giving output $r_1,r_2\dots$.</p>
<p>Now for odd $k$, say $k\in A$ iff $r_k=6$. For even $k$, say $k\in A$ iff $r_k<6$. So $k\in A$ with probability 1/6 if $k$ is odd, and $5/6$ if $k$ is even. $A$ is obviously incomputable.</p>
<p>Turing m... |
1,253,687 | <p>I don't know how to solve this one and the question is:</p>
<p>Find the values of a at which $y = x^3 + ax^2 + 3x + 1$.</p>
<p>My solution is:</p>
<p>$y'= 3x^2 + 2ax + 3$</p>
<p>I know that if $y' \ge 0$, $y$ should be always increasing. I don't know how to make it true. Please help and explain and thank you in ... | alkabary | 96,332 | <p>Well, you made the first right step.</p>
<p>$$y^{'} = 3x^2 + 2ax + 3$$</p>
<p>Now You need to know which values for $a$ satisfy $3x^2 + 2ax + 3 \geq 0$</p>
<p>Now $$ 2ax \geq -3 -3x^2$$</p>
<p>so</p>
<p>$$ax \geq \frac{-3x^2}{2} - \frac{3}{2}$$</p>
<p>Now you should do some effort to figure out what are the po... |
2,037,030 | <p>I am studying Distribution theory. But I am curious about that why we coin compact support. In what situation is it useful? Can any one give an intuitive way to explain this concept?</p>
| reuns | 276,986 | <p>Why do we require that $\varphi(x) = 0$ for $x$ large enough ? Because it allows us to <strong>integrate by parts without fear</strong> :
$$\int_{-\infty}^\infty T(x) \varphi'(x)dx = \lim_{x \to \infty} T(x)\varphi(x)-T(-x)\varphi(-x)-\int_{-\infty}^\infty T'(x) \varphi(x)dx$$
Here $\varphi \in C^\infty_c$ and $T$... |
898,755 | <p>The function $G_m(x)$ is what I encountered during my search for approximates of Riemann $\zeta$ function:</p>
<p>$$f_n(x)=n^2 x\left(2\pi n^2 x-3 \right)\exp\left(-\pi n^2 x\right)\text{, }x\ge1;n=1,2,3,\cdots,\tag{1}$$
$$F_m(x)=\sum_{n=1}^{m}f_n(x)\text{, }\tag{2}$$</p>
<p>$$G_m(x)=F_m(x)+F_m(1/x)\text{, }\t... | Daccache | 79,416 | <p>While this is just a partial answer, I hope this serves at least as a step in the right direction for proving what you need to. </p>
<p>First, to work with something more concrete, I substituted the expressions for $f_n(x)$ into $F_m(x)$ and then that into $G_m(x)$ in order to get an explicit set of functions:<br>... |
898,755 | <p>The function $G_m(x)$ is what I encountered during my search for approximates of Riemann $\zeta$ function:</p>
<p>$$f_n(x)=n^2 x\left(2\pi n^2 x-3 \right)\exp\left(-\pi n^2 x\right)\text{, }x\ge1;n=1,2,3,\cdots,\tag{1}$$
$$F_m(x)=\sum_{n=1}^{m}f_n(x)\text{, }\tag{2}$$</p>
<p>$$G_m(x)=F_m(x)+F_m(1/x)\text{, }\t... | mike | 75,218 | <p>@GeorgeDaccache Nice results!</p>
<p>I just need the space and easy of use in this answer area to convey some thoughts that I have.</p>
<p>First let me define two integers $n_0(m)$ and $n_1(m)$ as</p>
<p>$$ n_0(m) =\lfloor{\sqrt{3/(2\pi)}\sqrt{m + 1}}\rfloor$$
$$ n_1(m) =\lceil{\sqrt{3/(2\pi)}\sqrt{m + 1}}\text{ ... |
463,139 | <p>I have this:</p>
<p>Case 1)</p>
<p><img src="https://i.stack.imgur.com/xEEFQ.png" alt="enter image description here"></p>
<p>If <em>f</em> is a pair function $f(-x)=f(x)$ then $\int_{-a}^a f(x)dx=2\int_0^af(x)dx$</p>
<p>Case 2)</p>
<p><img src="https://i.stack.imgur.com/mnHbB.png" alt="enter image description h... | Sujaan Kunalan | 77,862 | <p>Algebraically, </p>
<p>A function is odd if $f(-x)=-f(x)$ and a function is even when $f(-x)=f(x)$.</p>
<p>For example, if $f(x)=x^3$, then $f(-x)=(-x)^3=-x^3$, which is $-f(x)$. This is our original function multiplied by $-1$. That means we can say this is an <em>odd</em> function.</p>
<p>Example for an even fu... |
463,139 | <p>I have this:</p>
<p>Case 1)</p>
<p><img src="https://i.stack.imgur.com/xEEFQ.png" alt="enter image description here"></p>
<p>If <em>f</em> is a pair function $f(-x)=f(x)$ then $\int_{-a}^a f(x)dx=2\int_0^af(x)dx$</p>
<p>Case 2)</p>
<p><img src="https://i.stack.imgur.com/mnHbB.png" alt="enter image description h... | dls | 1,761 | <blockquote>
<p>How can I realize when and where this symmetry exist in a function?</p>
</blockquote>
<p>You can recognize symmetric functions by knowing basic examples and understanding how these behave under common combinations.</p>
<ol>
<li><p>The most basic examples of even functions $f(x)=f(-x)$ are the monomi... |
227,797 | <p>I have this function and I want to see where it is zero.
<span class="math-container">$$\frac{1}{16} \left(\sinh (\pi x) \left(64 \left(x^2-4\right) \cosh \left(\frac{2 \pi x}{3}\right) \cos (y)+\left(x^2+4\right)^2+256 x \sinh \left(\frac{2 \pi x}{3}\right) \sin (y)\right)+\left(x^2-12\right)^2 \sinh \left(\frac... | Michael E2 | 4,999 | <p>The following shows that <code>f[x, y]</code> is negative (not zero) along the vertical line <code>x == 3.4657284...</code> through the saddle point between the two branches, and therefore the line separates the two branches where <code>f[x, y] == 0</code> in the OP's graph:</p>
<pre><code>yAssum = 1.046786`32 < ... |
779,095 | <p>Let
$$f(x,y)=\left\{ \begin{matrix} \frac{x^2y}{x^4+y^2} & (x,y)\neq(0,0) \\0 & (x,y)=(0,0)\end{matrix}\right.$$</p>
<p>It is easy to prove that the $f$ is not continuous at $(0,0)$ (doing the limit along the curve $y=x^2$).</p>
<p>I want to know whether it is possible to define the partial derivatives of... | bof | 111,012 | <p>In the usual terminology, a <a href="http://en.wikipedia.org/wiki/Partially_ordered_set" rel="nofollow"><em>partial order</em></a> is required be <em>antisymmetric</em>: if $x\le y$ and $y\le x$, then $x=y$. Thus there are only three partial orders on a two-element set; your fourth example in not antisymmetric.</p>
... |
2,642,144 | <p>How would I prove or disprove the following statement?
$ \forall a \in \mathbb{Z} \forall b \in \mathbb{N}$ , if $a < b$ then $a^2 < b^2$</p>
| Rgkpdx | 112,537 | <p>Not true. Take $a=-2$ and $b=1$. Then $a<b$ but $a^2=4> 1=b^2$.</p>
<p>Generally, it is good practice to start with easy examples to get an idea of why something might or not be true. </p>
|
559,194 | <p>$\mathscr{F}\{\delta(t)\}=1$, so this means inverse fourier transform of 1 is dirac delta function so I tried to prove it by solving the integral but I got something which doesn't converge.</p>
| L. Xu | 77,573 | <p>In the following $\langle f, \cdot \rangle$ denotes the linear functional on Schwartz space induced by $f$ and $f^\lor$ stands for the inverse Fourier transform of $f$. By definition, for any Schwartz function $\varphi$
\begin{align*}
\langle 1^\lor, \varphi \rangle=\langle 1, \varphi^\lor \rangle=&\int_\mathbb{... |
3,430,812 | <p>Consider the set of integers, <span class="math-container">$\Bbb{Z}$</span>. Now consider the sequence of sets which we get as we divide each of the integers by <span class="math-container">$2, 3, 4, \ldots$</span>.</p>
<p>Obviously, as we increase the divisor, the elements of the resulting sets will get closer and... | Brian Moehring | 694,754 | <p>The typical way to define limits of sets is via
<span class="math-container">$$\liminf_{n\to\infty} A_n = \bigcup_{n\geq 1} \bigcap_{k \geq n} A_k \\ \limsup_{n\to\infty} A_n = \bigcap_{n\geq 1} \bigcup_{k\geq n} A_k$$</span></p>
<p>Using these and <span class="math-container">$A_n = f_n(\mathbb{Z})$</span> where <... |
3,927,488 | <p>Given a random variable <span class="math-container">$X$</span> with finite expectation, I know that <span class="math-container">$$X_n\to X, a.s.\text{and} |X_n| \leq X\implies \mathbb{E}|X-X_n|\to 0 \text{ by DCT.}$$</span></p>
<p>I am wondering if it is possible to approximate <span class="math-container">$X$</s... | Claude Leibovici | 82,404 | <p>Starting from @J. W. Tanner's answer, the easy way to solve
<span class="math-container">$$\dfrac{dC}{dt}=k(A_0-C)(B_0-C)$$</span> is to write it as
<span class="math-container">$$\dfrac{dt}{dC}=\frac 1{k(A_0-C)(B_0-C)}=\frac{1}{k (A_0-B_0)}\left( \frac 1{C-A_0}- \frac 1{C-B_0}\right)$$</span> which gives (since <sp... |
954,419 | <p>I am teaching myself mathematics, my objective being a thorough understanding of game theory and probability. In particular, I want to be able to go through A Course in Game Theory by Osborne and Probability Theory by Jaynes.</p>
<p>I understand I want to cover a lot of ground so I'm not expecting to learn it in le... | Rgkpdx | 112,537 | <p>If you are more or less new to mathematics, your priority might be to train proof skills (by doing lost of easy proofs) and gain comfort with basic properties of sets and functions for example (Introduction to Metric and Topological Spaces by Sutherland gives a good quick overview within an advanced framework, I thi... |
1,395,619 | <p>One of my friend asked this doubt.Even in lower class we use both as synonyms,he says that these two concepts have difference.Empty set $\{ \}$ is a set which does not contain any elements,while null set ,$\emptyset$ says about a set which does not contain any elements.</p>
<p>I could not make out that...is his arg... | A Bc | 647,801 | <p>I would call "null sets" in the Measure Theory sense "sets of measure 0" just to avoid any confusion. In many parts of mathematics, "null set" and "empty set" are synonyms.</p>
|
792,924 | <p>If a quantity can be either a scalar or a vector, how would one call that property? I could think of scalarity but I don't think such a term exists.</p>
| Fermat | 83,272 | <p>No. Since $f$ is continuous, There exist a sequence $P_n$ of polynomials such that converges to $f$ uniformly on $[0,1]$. Therefore
$$\int_{0}^{1}x^{n}f\left(x\right)dx=\lim_{n\to \infty}\int_{0}^{1}x^{n}P_{n}\left(x\right)dx$$
Now let $$P_{n}(x)=a_nx^n+a_{n-1}x^{n-1}+....+a_0$$
convert left integral to a sum and sh... |
140,754 | <p>Pleas tell me that what a "Kink" is and what this sentence means: </p>
<blockquote>
<p>Distance functions have a kink at the interface where $d = 0$ is a local minimum.</p>
</blockquote>
| robjohn | 13,854 | <p>A "kink" in a curve would be a point where the curve is continuous, yet the first derivative (gradient) is not continuous. The curvature would be infinite at a kink because the direction changes a finite amount in an infinitesimal distance.</p>
|
140,754 | <p>Pleas tell me that what a "Kink" is and what this sentence means: </p>
<blockquote>
<p>Distance functions have a kink at the interface where $d = 0$ is a local minimum.</p>
</blockquote>
| Mikasa | 8,581 | <p>As above you can find it via the web as <a href="http://en.wikipedia.org/wiki/Cusp_%28singularity%29" rel="noreferrer">Cusp (singularity)</a>. See the following graphs:</p>
<p><img src="https://i.stack.imgur.com/R08Ey.jpg" alt="enter image description here"></p>
<p><img src="https://i.stack.imgur.com/Oy4dR.jpg" al... |
1,335,483 | <p>Given a relation $R \subseteq A \times A$ with $n$ tuples, I am trying to prove that its transitive closure $R^+$ has at the most $n^2$ elements.</p>
<p>My initial idea was to use the following definition of the transitive closure to identify an argument why the statement to be proven must be true:</p>
<p>$$R^+ = ... | D Left Adjoint to U | 26,327 | <p>Suppose that it's true for $n = 1...K-1$, then add a tuple to your $n = K-1$ tuples $R$. $(R \cup \{(a,b)\})^+ = R^+ \cup \{(a,x) : (b,x) \in R^+\} \cup \{(x,b): (x,a) \in R^+\} \cup \{(a,b)\} = R'^+$. So since by inductive assumption $|R^+| \leq (K-1)^2 = K^2 - 2K +1.$, we have that $|R'^+|$ is no greater than $... |
1,812,956 | <blockquote>
<p>Find the equation of the normal to the curve with equation $4x^2+xy^2-3y^3=56$ at the point $(-5,2)$.</p>
</blockquote>
<p>I know that the normal to a curve is $$-\frac{1}{f'(x)}$$
And when I differentiate the curve implicitly I get $$-\frac{8x-y^2}{6y^2}$$</p>
<p>Substituting that into the equation... | DooplissForce | 281,590 | <p>I believe your implicit differentiation is wrong. Given $4x^2+xy^2-3y^3=56$, we can implicitly differentiate to find $\frac{dy}{dx}$:</p>
<p>$$
4x^2+xy^2-3y^3=56 \\
8x+\color{blue}{\left(y^2+2xy\frac{dy}{dx}\right)} - 9y^2\frac{dy}{dx}=0 \\
$$</p>
<p>(The blue part is arrived at through the product rule.) After si... |
1,182,953 | <p>Does anyone know the provenance of or the answer to
the following integral</p>
<p>$$\int_0^\infty\ \frac{\ln|\cos(x)|}{x^2} dx $$</p>
<p>Thanks.</p>
| Mark Fischler | 150,362 | <p>This integral is equal to
$$
\frac{1}{2} \int_0^\infty \frac{\ln (\cos^2 x)}{x^2} dx = \frac{1}{2}(-\pi) = -\frac{\pi}2$$</p>
<p>The easiest place to remember seeing this is Gradshteyn and Ryzhik, where it appears as definite integral 4.322.6. The source quoted there is Fikhtebgik'ts, G. M. (<a href="http://en.wi... |
3,387,138 | <p>First Definition.
A modular form of level n and dimension -k is an analytic function <span class="math-container">$F$</span> of <span class="math-container">$\omega_1 $</span> and <span class="math-container">$\omega_2$</span> satisfying the following properties :</p>
<ol>
<li><span class="math-container">$F(\om... | reuns | 276,986 | <p>For <span class="math-container">$n=1$</span>, <span class="math-container">$F$</span> is a function of lattices.</p>
<p>For <span class="math-container">$a,b,c,d\in\Bbb{Z},ad-bc=1$</span>
<span class="math-container">$$f(z)=F(\Bbb{Z}+z \Bbb{Z}) = F((az+b)\Bbb{Z}+(cz+d) \Bbb{Z})$$</span> <span class="math-container... |
152,880 | <p>I know that for every $n\in\mathbb{N}$, $n\ge 1$, there exists $p(x)\in\mathbb{F}_p[x]$ s.t. $\deg p(x)=n$ and $p(x)$ is irreducible over $\mathbb{F}_p$.</p>
<blockquote>
<p>I am interested in counting how many such $p(x)$ there exist (that is, given $n\in\mathbb{N}$, $n\ge 1$, how many irreducible polynomials of... | Eugene | 31,288 | <p>With regards to your question, <a href="http://arxiv.org/pdf/1001.0409v6.pdf">this paper</a> has a formula for counting the number of monic irreducibles over a finite field.</p>
|
928,772 | <p>Let <span class="math-container">$G$</span> be a group. If <span class="math-container">$(ab)^n=a^nb^n$</span> <span class="math-container">$\forall a,b \in G$</span> and <span class="math-container">$(|G|, n(n-1))=1$</span> then prove that <span class="math-container">$G$</span> is abelian.</p>
<hr />
<p>What I hav... | James | 751 | <p>We can assume that $n>2$. Since $(ab)^n = a^nb^n$, for <em>all</em> $a,b\in G$, we can write $(ab)^{n+1}$ in two different ways:
$$(ab)^{n+1} = a(ba)^nb = ab^na^nb,$$
and
$$(ab)^{n+1} = ab(ab)^n = aba^nb^n.$$
Hence,
$$ab^na^nb = aba^nb^n.$$
Cancel $ab$ on the left and $b$ on the right to obtain
$$b^{n-1}a^n = a^... |
928,772 | <p>Let <span class="math-container">$G$</span> be a group. If <span class="math-container">$(ab)^n=a^nb^n$</span> <span class="math-container">$\forall a,b \in G$</span> and <span class="math-container">$(|G|, n(n-1))=1$</span> then prove that <span class="math-container">$G$</span> is abelian.</p>
<hr />
<p>What I hav... | Ri-Li | 152,715 | <p>I have got another answer though it can also be easily viewed from James answer too.</p>
<p>We can assume that $n>2$. Since $(ab)^n = a^nb^n$, for <em>all</em> $a,b\in G$.
Then we will get $$b^{n-1}a^n = a^nb^{n-1}.$$ this is true for <em>all</em> $a,b\in G$. </p>
<p>Now see Since the order of $G$ is prime t... |
4,344,571 | <p>In a previous exam assignment, there is a problem that asks for a proof that <span class="math-container">$\mathbb{Z}_{24}$</span> and <span class="math-container">$\mathbb{Z}_{4}\times\mathbb{Z}_6$</span> are <strong>not</strong> isomorphic.</p>
<p>We have <span class="math-container">$\mathbb{Z}_{24}$</span> is is... | Kandinskij | 657,309 | <p>These rings are not isomorphic because their additive groups are not isomorphic. In <span class="math-container">$(\mathbb{Z}_{24},+)$</span> there is an element whom order is <span class="math-container">$24$</span>(i.e. <span class="math-container">$[1]_{24}$</span>). In <span class="math-container">$(\mathbb{Z}_4... |
2,155,589 | <p>I'm reading a computer science book that gives several functions, in the mathematical sense. There are two that are the basis of this question.</p>
<p>These are equations used to convert a number represented in base ten to a bit representation using two's complement and back.</p>
<p>One function makes the conversi... | Mark Viola | 218,419 | <p>Let $f(z)=z^3-z_0^3$. Then, we can write $f(z)$ as </p>
<p>$$f(z)=(z-z_0)^3+3z_0(z-z_0)^2+3z_0^2(z-z_0)$$ </p>
<p>Hence from $(1)$, if $|z-z_0|<1$, then given $\epsilon>0$</p>
<p>$$\begin{align}
f(z)&=|z^3-z_0^3|\\\\
&=|(z-z_0)^3+3z_0(z-z_0)^2+3z_0^2(z-z_0)|\\\\
&\le |z-z_0|\left(1+3|z_0|+3|z_... |
904,041 | <p>$$tx'(x'+2)=x$$
First I multiplied it:
$$t(x')^2+2tx'=x$$
Then differentiated both sides:
$$(x')^2+2tx'x''+2tx''+x'=0$$
substituted $p=x'$ and rewrote it as a multiplication
$$(2p't+p)(p+1)=0$$
So either $(2p't+p)=0$ or $p+1=0$</p>
<p>The first one gives $p=\frac{C}{\sqrt{T}}$
The second one gives $p=-1$. My questi... | Kelenner | 159,886 | <p>Your differential equation is a complicated one. I give some hints on $I=]0,+\infty[$ (the study has to be done also on $]-\infty,0[$).</p>
<p>A) First note that on $I$, your equation is $\displaystyle (x^{\prime}(t)+1)^2=\frac{x(t)+t}{t}$. We see that $x_0(t)=-t$ is a solution. For any solution, we must have $x(t... |
1,122,926 | <p>Question: The product of monotone sequences is monotone, T or F?</p>
<p>Uncompleted Solution: There are four cases from considering each of two monotone sequences, increasing or decreasing.</p>
<p>CASE I: Suppose we have two monotonically decreasing sequences, say ${\{a_n}\}$ and ${\{b_n}\}$. Then, $a_{n+1}\leq a_... | paoloff | 211,137 | <p>A simple counter example to "The product of two monotone sequences is a monotone sequence" is the product of the monotone sequence $\{...,-3,-2,-1,0,1,2,3,...\}$ (which you can picture as a sequence of points in the $y$ axes of the the graph of $f(x) =x$) with itself. The product of these sequences is again a sequen... |
197,603 | <p>I'm a newcomer in topology, so I have many things chaotic in my minds, so I hope you could help me.
In order topology, an basis has structure $(a,b)$, right. This is no problem when considering a topology like R, but, what if the number of elements between a and b is finite, so we can write $$(a,b) = [a_1, b_1]$$ wh... | Neal | 20,569 | <p>In the case of a finite ordered set $X$, the order topology is discrete. In particular, this implies that for any $a,b\in X$, $[a,b]$ is open (as a union of open sets). It is closed, yes, but it is also open. (Perhaps your point of difficulty is thinking that closed sets cannot also be open - this is not true, si... |
2,038,189 | <p>(Note: I didn't learn how to solve equations the conventional way; instead I was just taught to "move numbers from side to side", inverting the sign or the operation accordingly. I am learning the conventional way though because I think it makes the process of solving equations clearer. That being said, I apologize ... | Eff | 112,061 | <p>We have that
$$5 = \frac2x.$$
Now multiply by $x$ on each side, and get
$$5x=2. $$
Next, divide by $5$ on each side, and get
$$x=\frac25. $$</p>
|
2,038,189 | <p>(Note: I didn't learn how to solve equations the conventional way; instead I was just taught to "move numbers from side to side", inverting the sign or the operation accordingly. I am learning the conventional way though because I think it makes the process of solving equations clearer. That being said, I apologize ... | kub0x | 309,863 | <p>Other way for solving it is by the reciprocal or inverse process:</p>
<p>$5 = \frac{2}{x} \Rightarrow 5^{-1} = (\frac{2}{x})^{-1}$</p>
<p>$\frac{1}{5} = \frac{x}{2} \Rightarrow \frac{2}{5} = x$</p>
|
229,966 | <p>I want to put a title to the plotlegends I am using. I get a solution <a href="https://mathematica.stackexchange.com/questions/201353/title-for-plotlegends">here</a> which says to use <code>PlotLegends -> SwatchLegend[{0, 3.3, 6.7, 10, 13, 17, 20}, LegendLabel -> "mu"]</code>. But I also want to p... | tad | 70,428 | <p>You can wrap Placed around the legend. Here's an example modified from the SwatchLegend documentation:</p>
<pre><code>Plot[{Sin[x], Cos[x]}, {x, 0, 5},
PlotLegends -> Placed[SwatchLegend[{"first", "second"},
LegendLabel -> "legend title"], {0.2, 0.3}]]
</code></pre>
<p><a hre... |
4,157,841 | <p>Q is to prove that integer just above(<span class="math-container">$\sqrt{3} + 1)^{2n}$</span> is divisible by <span class="math-container">$2^{n+1}$</span> for all n belongs to natural numbers.</p>
<p>In Q , by integer just above means that:
For an example , which is the integer just above 7.3 . It is 8. Then , Q w... | Mark Bennet | 2,906 | <p>Note that <span class="math-container">$0\lt \sqrt 3 -1 \lt 1$</span></p>
<p>Now look at the numbers <span class="math-container">$a=1+\sqrt 3, b=1-\sqrt 3$</span> with <span class="math-container">$a+b=2, ab=-2$</span> which are roots of the quadratic <span class="math-container">$x^2-2x-2=0$</span></p>
<p>Then wit... |
126,120 | <p>Python has generators which save memory, is there a technique for generating in memory examples for your training set "on the fly".</p>
<p>For example purposes, I constructed here a regressor for blur:</p>
<pre><code>randomMask[img_] :=
Module[{t, h, g, d = ImageDimensions[img]},
t = Table[{PointSize@RandomRe... | Alexey Golyshev | 23,402 | <p>You can do out-of-core classification with the new function <code>File</code> (<a href="http://reference.wolfram.com/language/ref/File.html" rel="noreferrer">link1</a>, <a href="https://www.wolfram.com/language/11/neural-networks/out-of-core-image-classification.html" rel="noreferrer">link2</a>).</p>
<p>I will simp... |
22,753 | <p>I've learned the process of orthogonal diagonalisation in an algebra course I'm taking...but I just realised I have no idea what the point of it is.</p>
<p>The definition is basically this: "A matrix <span class="math-container">$A$</span> is orthogonally diagonalisable if there exists a matrix <span class="math-co... | Arturo Magidin | 742 | <p>When you have a matrix $A$ that is diagonalisable, the matrix $U$ such that $U^{-1}AU=D$ is diagonal is the matrix whose columns are a basis of eigenvectors of $A$. Having a basis of eigenvectors makes understanding $A$ much easier, and makes computations with $A$ easy as well.</p>
<p>What the basis of eigenvectors... |
1,452,943 | <p>I'm working on problem where I want to use the continuity of $f'$ to assert that $f'(x)$ cannot be zero ("bounded away from zero"?) near $x = 0$. We know that $(f'(0))^2 >3$.</p>
<p>So, I think that what I really want to ask is this: if $f'$ is cts, must $f'$-squared also be continuous? </p>
<p>Can I use t... | davidlowryduda | 9,754 | <p>Very generally, if $f$ and $g$ are both continuous functions, then $f \circ g$ is a continuous function. (If you haven't proved this, then you should). Here, you are composing the square-function with the derivative of $f$.</p>
|
275,775 | <p>For the FrameLabel, I have:</p>
<pre><code>Style["\[NumberSign] Humans per city", FontFamily -> "Latin Modern Math"]
</code></pre>
<p>How can I make only "Humans" in the label to be italic?</p>
| Szabolcs | 12 | <p>Select "Humans" and press Ctrl-I (or Command-I on macOS) to format it in italics. This is by far the simplest way.</p>
<p>Note that Latin Modern Math is intended only for math, not text, and does not have an italics version. Your OS will likely fake the italics style by simply slanting letters. Install Lat... |
19,495 | <p>I was told that one of the most efficient tools (e.g. in terms of computations relevant to physics, but also in terms of guessing heuristically mathematical facts) that physicists use is the so called "Feynman path integral", which, as far as I understand, means "integrating" a functional (action... | Kevin H. Lin | 83 | <p>Recently I have been reading Kevin Costello's book (draft) <a href="http://www.math.northwestern.edu/%7Ecostello/renormalization" rel="nofollow noreferrer">Renormalization of Quantum Field Theories</a>, which claims to work out some foundations of perturbative quantum field theory following the "Wilsonian philo... |
3,570,688 | <p>For example, if a ball can be any of 3 colors, then the number of configurations (with repetition of colors) of 2 balls is <span class="math-container">$(3+2-1)C_{2} = 4C_{2} = 6$</span> Why?</p>
| Hanno | 316,749 | <p>An alternative solution relies on the Cauchy–Bunyakovsky–Schwarz inequality (CBS).</p>
<p>The given constraint <span class="math-container">$\,ab+bc+ca+abc=4\,$</span> can be written as
<span class="math-container">$(a+1)(b+1)(c+1) = 2+(a+1)+(b+1)+(c+1)$</span>, which in turn is equivalent to
<span class="math-cont... |
1,876,732 | <p>What is
$$\int_{S}(x+y+z)dS,$$ where $S$ is the region $0\leq x,y,z\leq 1$ and $x+y+z\leq 2$?</p>
<p>We can change the region to $0\leq x,y,z\leq 1$ and $x+y+z\geq 2$, because the total of the two integrals is just
$$\int_0^1\int_0^1\int_0^1(x+y+z)dxdydz=3\int_0^1xdxdydz=\frac{3}{2}.$$</p>
<p>Now, can we write the... | ChrisT | 353,893 | <p>You could write
\begin{equation}
\int_S x \, dx dy dz = \int_0^1 \left( \int_{y+z \leq 2-x; \, 0\leq y,z \leq 1} dy dz \right) x dx
\end{equation}</p>
<p>Now, you can interpret $y+z \leq 2-x$ with $y,z \geq 0$ as a triangle in the plane, whose area is $\frac{(2-x)^2}{2}$.
From this triangle, you subtract two small... |
1,876,732 | <p>What is
$$\int_{S}(x+y+z)dS,$$ where $S$ is the region $0\leq x,y,z\leq 1$ and $x+y+z\leq 2$?</p>
<p>We can change the region to $0\leq x,y,z\leq 1$ and $x+y+z\geq 2$, because the total of the two integrals is just
$$\int_0^1\int_0^1\int_0^1(x+y+z)dxdydz=3\int_0^1xdxdydz=\frac{3}{2}.$$</p>
<p>Now, can we write the... | Christian Blatter | 1,303 | <p>You already have made two good moves: (i) replacing $S$ by the pyramidal region $S':=[0,1]^3\setminus S$, and (ii) replacing the integrand by $3x$. The integral in question then comes to
$$Q:=\int_S (x+y+z)\>{\rm d}(x,y,z)={3\over2}-3\int_{S'}x\>{\rm d}(x,y,z)\ .$$
From a figure we read off that
$$\eqalign{\i... |
363,166 | <p>For valuation rings I know examples which are Noetherian. </p>
<blockquote>
<p>I know there are good standard non Noetherian Valuation Rings. Can anybody please give some examples of rings of this kind? </p>
</blockquote>
<p>I am very eager to know. Thanks.</p>
| Alex Youcis | 16,497 | <p>This was bumped to the front page for some reason, so I apologize for resurrecting this. But I think that there is an exceedingly natural example. In fact, it comes up all the time in 'nature'. Namely, consider $\mathbb{Q}_p$ with the standard valuation $v_p$. Then, there is a unique extension of this valuation to $... |
363,166 | <p>For valuation rings I know examples which are Noetherian. </p>
<blockquote>
<p>I know there are good standard non Noetherian Valuation Rings. Can anybody please give some examples of rings of this kind? </p>
</blockquote>
<p>I am very eager to know. Thanks.</p>
| Pramathanath Sastry | 444,395 | <p>Let <span class="math-container">$(K, \lvert\cdot\rvert)$</span> be a complete algebraically closed field with a non trivial absolute value. Let <span class="math-container">$R$</span> be its valuation ring and <span class="math-container">$\mathfrak{m}$</span> the maximal ideal of <span class="math-container">$R$</... |
2,771,240 | <p>Let $\mathbb F$ be a field and $\mathbb K $ be an extension field of $\mathbb F$ such that $\mathbb K$ is algebraically closed. </p>
<p>Let $\mathbb L$ be the field of all elements of $\mathbb K$ which are algebraic over $\mathbb F$. Then $\mathbb L_{|\mathbb F}$ is an algebraic extension. </p>
<p>My question is... | vadim123 | 73,324 | <p>Suppose that $n=kT+n'$, where $0\le n'<T$. Then:</p>
<p>$$\frac{1}{n}\int_0^n f(x)dx = \frac{k\int_0^T f(x)dx + \int_{kT}^{kT+n'}f(x)dx}{n}=\frac{k}{kT+n'}\int_0^T f(x)dx+\frac{1}{n}\int_{kT}^{kT+n'}f(x)dx$$</p>
<p>As $n\to\infty$, we have $k\to \infty$, while $n'$ and $\int_{kT}^{kT+n'}f(x)dx$ remain bounded.... |
2,626,506 | <p><strong>Proof: There is no other prime triple then $3,5,7$</strong></p>
<p>There are already lots of questions about this proof, but I can't find the answer to my question.</p>
<p>The complete the proof, we consider mod $3$ so $p=3k; p=3k+1; p=3k+2$ </p>
<p>But why do we look at divisibility by $3$?</p>
<p>Do we... | nonuser | 463,553 | <p>If $p=2$ then we have no solution.</p>
<p>If $p, p+2,p+4$ are primes then exactly one of them is divisible by 3, so it must be $p=3$.</p>
|
3,624,524 | <p>I want to figure out the process for showing why the function <span class="math-container">$\cos(1-\frac{1}{z})$</span> has an essential singularity at <span class="math-container">$z=0$</span> without using knowledge of the Laurent expansion. I know the process should be to rule out the possibility of removable sin... | Basel J. | 545,344 | <p>There always exists an open cover <span class="math-container">$\{U_\alpha\}$</span> of <span class="math-container">$M$</span> such that <span class="math-container">$TU_\alpha$</span> is homeomorphic to <span class="math-container">$U_\alpha \times \mathbb{R}^{\text{dim } M}$</span>. The collection of these maps ... |
2,353,190 | <p>Let $f(x)=\dfrac{1+x}{1-x}$ The nth derivative of f is equal to:</p>
<ol>
<li>$\dfrac{2n}{(1-x)^{n+1}} $</li>
<li>$\dfrac{2(n!)}{(1-x)^{2n}} $</li>
<li>$\dfrac{2(n!)}{(1-x)^{n+1}} $</li>
</ol>
<p>by Leibniz formula </p>
<p>$$ {\displaystyle \left( \dfrac{1+x}{1-x}\right)^{(n)}=\sum _{k=0}^{n}{\binom {n}{k}}\ (1+x... | lab bhattacharjee | 33,337 | <p>$$y=\dfrac{1+x}{1-x}=\dfrac{2-(1-x)}{1-x}=\dfrac2{1-x}-1$$</p>
<p>$$\implies\dfrac{dy}{dx}=\dfrac{2(-1)}{(1-x)^2}$$</p>
<p>$$\dfrac{d^2y}{dx^2}=\dfrac{2(-1)(-2)}{(1-x)^3}=\dfrac{2(-1)^22!}{(1-x)^{2+1}}$$</p>
<p>Can you follow the pattern?</p>
|
2,353,190 | <p>Let $f(x)=\dfrac{1+x}{1-x}$ The nth derivative of f is equal to:</p>
<ol>
<li>$\dfrac{2n}{(1-x)^{n+1}} $</li>
<li>$\dfrac{2(n!)}{(1-x)^{2n}} $</li>
<li>$\dfrac{2(n!)}{(1-x)^{n+1}} $</li>
</ol>
<p>by Leibniz formula </p>
<p>$$ {\displaystyle \left( \dfrac{1+x}{1-x}\right)^{(n)}=\sum _{k=0}^{n}{\binom {n}{k}}\ (1+x... | Angina Seng | 436,618 | <p>If we let $f_n$ denote the $n$-th derivative, then $f_{n+1}=f_n'$.
That is the case only for <strong>one</strong> of the three possible solutions you
have there.</p>
|
45,662 | <p>Does this undirected graph with 6 vertices and 9 undirected edges have a name?
<img src="https://i.stack.imgur.com/XwuUB.png" alt="enter image description here">
I know a few names that are not right. It is not a complete graph because all the vertices are not connected. It is close to K<sub>3,3</sub> the utility gr... | Fixee | 7,162 | <p>Take two opposing vertices (the leftmost and rightmost will do). Now swap them and draw the resulting picture.</p>
<p>You should get a very clear $K_{3,3}$ as a result.</p>
|
3,208,412 | <p>I have to prove the following:</p>
<p><span class="math-container">$$ \sqrt{x_1} + \sqrt{x_2} +...+\sqrt{x_n} \ge \sqrt{x_1 + x_2 + ... + x_n}$$</span></p>
<p>For every <span class="math-container">$n \ge 2$</span> and <span class="math-container">$x_1, x_2, ..., x_n \in \Bbb N$</span></p>
<p>Here's my attempt:<... | DanielWainfleet | 254,665 | <p>Without induction. For <span class="math-container">$any$</span> non-negative reals <span class="math-container">$x_1,...,x_n$</span> let <span class="math-container">$x_j=(y_j)^2$</span> for each <span class="math-container">$j,$</span> with each <span class="math-container">$y_j\ge 0.$</span> The inequality is the... |
4,136,082 | <p><span class="math-container">$$f(x) =
\begin{cases}
\cos(\frac{1}{x}) & \text{if $x\ne0$} \\
0 & \text{if $x=0$} \\
\end{cases}$$</span></p>
<p>How do I prove this function has Darboux's property? I know it has it because it has antid... | José Carlos Santos | 446,262 | <p>Take <span class="math-container">$a,b\in\Bbb R$</span> with <span class="math-container">$a<b$</span>. You want to prove that, if <span class="math-container">$y$</span> lies between <span class="math-container">$f(a)$</span> and <span class="math-container">$f(b)$</span>, then there is some <span class="math-co... |
3,413,261 | <p>I know this was answered before but I'm having one particular problem on the proof that I'm not getting.</p>
<p>My Understanding of the distribution law on the absorption law is making me nuts, by the answers of the proof it should be like this.</p>
<p>A∨(A∧B)=(A∧T)∨(A∧B)=A∧(T∨B)=A∧T=A</p>
<p>This should prove th... | Wlod AA | 490,755 | <p>An example of a Lindelöf non-second countable space, which has some additional nice properties, was constructed/discovered during the Prague 1961 Topological Conference (by wh). The point-set is the unit disc</p>
<p><span class="math-container">$$\ B(\mathbf 0\,\ 1)\ :=
\ \{p\in\mathbb R^2: |p|\le 1\} $$</span><... |
38,586 | <p>The $n$-th Mersenne number $M_n$ is defined as
$$M_n=2^n-1$$
A great deal of research focuses on Mersenne primes. What is known in the opposite direction about Mersenne numbers with only small factors (i.e. smooth numbers)? In particular, if we let $P_n$ denote the largest prime factor of $M_n$, are any results kn... | Qiaochu Yuan | 290 | <p>I can give you a slightly better upper bound. Recall that $2^n - 1 = \prod_{d | n} \Phi_d(2)$ where $\Phi_d$ is a cyclotomic polynomial. Now,</p>
<p>$$\Phi_d(2) = \prod_{(k, d) = 1} (2 - \zeta_d^k) \le 3^{\varphi(d)}$$</p>
<p>so that in particular the largest prime factor of $2^n - 1$ is at most $3^{\varphi(n)}$... |
2,467,327 | <p>How to prove that $441 \mid a^2 + b^2$ if it is known that $21 \mid a^2 + b^2$.<br>
I've tried to present $441$ as $21 \cdot 21$, but it is not sufficient.</p>
| Tengu | 58,951 | <blockquote>
<p><strong>Lemma.</strong> If a prime $p \equiv 3\pmod{4}$ and $p$ divides $a^2+b^2$ then $p$ divides $a,b$.</p>
</blockquote>
<p><em>Proof.</em> Assume the contrary that $p \nmid a,p \nmid b$.</p>
<p>By Fermat's little theorem, we have $a^{p-1} \equiv 1 \pmod{p}, b^{p-1} \equiv 1 \pmod{p}$ so $a^{p-1}... |
2,132,936 | <p>How do you simplify this problem?
$$ \frac {\mathrm d}{\mathrm dx}\left[(3x+1)^3\sqrt{x}\right] $$
$$= \frac {(3x+1)^3}{2\sqrt {x}} + 9\sqrt{x} (3x+1)^2 $$
$$\frac{(3x+1)^2(21x+1)}{2\sqrt x} $$</p>
| Deepak | 151,732 | <p>Just offering another tool here, logarithmic differentiation. A specific use of implicit differentiation.</p>
<p>Put $y= (3x+1)^3\sqrt x$</p>
<p>$$\log y = 3\log(3x+1) + \frac 12 \log x$$</p>
<p>Observe that $\frac{d}{dx} \log y = \frac{d}{dy} (\log y) \cdot \frac{dy}{dx}$ by the chain rule. Use $y'$ to represent... |
1,455,348 | <p>Recently, having realized I did not properly internalize it (shame on me!), I went back to the definition of continuity in metric spaces and I found a proposition for which I was looking for a proof.</p>
<p>Here there is the result and my "proof" (in the hope to get rid of the quotation marks).</p>
<p><em>... | air | 181,046 | <p>Ok now the proof is basically correct (when we are working in metric spaces)! Some remarks:</p>
<p>As Umberto P. also noted in his answer in the <a href="https://math.stackexchange.com/questions/1456524/proof-based-on-convergence-arguments-that-if-phi-in-mathbbrx-is-continu">related question</a> you asked, I am not ... |
652,660 | <p>Show $\lnot(p\land q) \equiv \lnot p \lor \lnot q$</p>
<p>this is my solution . Check it please </p>
<p><img src="https://i.stack.imgur.com/1y7DB.jpg" alt="enter image description here"></p>
| Newb | 98,587 | <p>Your solution is right apart from the second line in $(\Leftarrow)$: it should be </p>
<blockquote>
<p>$p$ $\color{red}{\text{ and }} q$ is false. Then $(p\land q)$ is false.$\ldots$</p>
</blockquote>
|
4,308,316 | <p>I am given the question.</p>
<p>Suppose X and Y are iid uniform random variables on the interval (-2,2). Let <span class="math-container">$Z=\frac{Y}{X}$</span>.
Does the expectation of Z exist? If it exists, calculate <span class="math-container">$\mathbb{E}[Z]$</span>. If it does not exist, explain why.</p>
<p>I h... | tommik | 791,458 | <p>Both methods are valid. The second one is faster (and suggested). If you calculate <span class="math-container">$f_Z(z)$</span> you will realize that <span class="math-container">$Z$</span> does not have expectation as the corresponding integral diverges.</p>
<hr />
<p>The calculation of Z distribution, even not nec... |
4,308,316 | <p>I am given the question.</p>
<p>Suppose X and Y are iid uniform random variables on the interval (-2,2). Let <span class="math-container">$Z=\frac{Y}{X}$</span>.
Does the expectation of Z exist? If it exists, calculate <span class="math-container">$\mathbb{E}[Z]$</span>. If it does not exist, explain why.</p>
<p>I h... | StratosFair | 857,384 | <p>Both of the statements you have written are correct. If you compute <span class="math-container">$f_z$</span>, you will find out that the integral
<span class="math-container">$$E[Z]=\int_{-\infty}^\infty zf_{z}(z)dz $$</span>
diverges.</p>
<p>Your second approach using the fact that <span class="math-container">$X$... |
2,113 | <p>With respect to the stated reason for closure, I'd like to get some clarification as to what, precisely, "too localized" encompasses (at least a definition, or explanation, that is more specific and objective than the current "definition"). It just strikes me that some questions which might appear to some as being ... | Jeff Atwood | 153 | <p>Well, I can tell you what it means in the context of programming...</p>
<ol>
<li><p>Small geographic area</p>
<blockquote>
<p>Are there any Python user group meetings in Peoria, IL?</p>
</blockquote></li>
<li><p>Specific moment in time</p>
<blockquote>
<p>When will Visual Studio 2010 be released? </p>
</block... |
75,875 | <p>I am asking in the sense of isometry groups of a manifold. SU(3) is the group of isometries of CP2, and SO(5) is the group of isometries of the 4-sphere. Now, it happens that both manifolds are related by Arnold-Kuiper-Massey theorem: $\mathbb{CP}^2/conj \approx S^4$; one is a branched covering of the other, the quo... | Joseph Wolf | 18,505 | <p>$SU(3)$ has center of order 3 and $SO(5)$ has center reduced to the identity. In fact they are not even locally isomorphic: $SU(3)$ is of Cartan type $A_2$ and $SO(5)$ is of type $B_2$.</p>
|
217,291 | <p>I am trying to recreate the following image in latex (pgfplots), but in order to do so I need to figure out the mathematical expressions for the functions</p>
<p><img src="https://i.stack.imgur.com/jYGNP.png" alt="wavepacket"></p>
<p>So far I am sure that the gray line is $\sin x$, and that
the redline is some ver... | Qiaochu Yuan | 232 | <p><a href="http://en.wikipedia.org/wiki/Persi_Diaconis" rel="nofollow">Persi Diaconis</a> showed that it takes about $7$ shuffles to shuffle a $52$-card deck. I'm not going to explain how he proved this result, but I'm going to explain some relevant ideas. </p>
<p>Given a <a href="http://en.wikipedia.org/wiki/Regular... |
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