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 |
|---|---|---|---|---|
3,200,940 | <p><a href="https://i.stack.imgur.com/7k9P8.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7k9P8.jpg" alt="enter image description here"></a></p>
<p>Velleman's logic in sentence 3 under figure 4 is confusing me. He is using lines two and four of the truth table to infer what Q of line 1 should be. ... | Bram28 | 256,001 | <p>Velleman's logic is correct. Here is what he is saying. Suppose we change the first line of the truth-table for <span class="math-container">$P \to Q$</span>, so it becomes:</p>
<p><span class="math-container">\begin{array}{cc|c}
P&Q&P \to Q\\
\hline
F&F&F\\
F&T&T\\
T&F&F\\
T&T&a... |
1,446,168 | <p>Let $x$ and $y$ be two random variables. </p>
<p>Suppose $m$ is a random variable that is independent of $x$ and has the following distribution:</p>
<p>$$\text{Pr}(m = 1|x) = 0.5,$$ $$\text{Pr}(m = -1|x) = 0.5.$$</p>
<p>Let $y$ be given by: $$y= \left\{ \begin{array}{lcc}
0 & \text{if } x\geq0... | hunter | 108,129 | <p>In $\mathbb{R}$, let $X = \{0\} \bigcup \{1, \frac{1}{2}, \frac{1}{3}, \ldots \}$. Then $X$ is compact and has infinitely many isolated points.</p>
|
3,052,746 | <p>I want to solve <span class="math-container">$2x = \sqrt{x+3}$</span>, which I have tried as below:</p>
<p><span class="math-container">$$\begin{equation}
4x^2 - x -3 = 0 \\
x^2 - \frac14 x - \frac34 = 0 \\
x^2 - \frac14x = \frac34 \\
\left(x - \frac12 \right)^2 = 1 \\
x = \frac32 , -\frac12
\end{equation}$$</span>... | Hugh Entwistle | 361,701 | <p>From </p>
<p><span class="math-container">$$x^2-\frac{1}{4}x=\frac{3}{4}$$</span> to <span class="math-container">$$\left(x-\frac{1}{2} \right)^2=1$$</span> you have not completed the square correctly. </p>
<p>It should instead be </p>
<p><span class="math-container">$$\left(x-\frac{1}{8} \right)^2-\frac{1}{64}=\... |
3,052,746 | <p>I want to solve <span class="math-container">$2x = \sqrt{x+3}$</span>, which I have tried as below:</p>
<p><span class="math-container">$$\begin{equation}
4x^2 - x -3 = 0 \\
x^2 - \frac14 x - \frac34 = 0 \\
x^2 - \frac14x = \frac34 \\
\left(x - \frac12 \right)^2 = 1 \\
x = \frac32 , -\frac12
\end{equation}$$</span>... | Deepak | 151,732 | <p>Two mistakes:</p>
<p>1) mistake in completing the square. Remember, divide the coefficient of the <span class="math-container">$x$</span> term by two, not multiply. You should've got: <span class="math-container">$(x-\frac 18)^2 = \frac 34 + \frac{1}{64}$</span>.</p>
<p>2) when you square, you run the risk of intr... |
3,830,636 | <p>This is something I'm doing for a video game so may see some nonsense in the examples I provide.</p>
<p>Here's the problem:
I want to get a specific amount minerals, to get this minerals I need to refine ore. There are various kinds of ore, and each of them provide different amounts of minerals per volume. So I want... | RobPratt | 683,666 | <p>Yes, this is linear programming.
<span class="math-container">\begin{align}
&\text{minimize} & 35p+120k \\
&\text{subject to}
&15p+79k &\ge 1000 \\
&&29p+144k &\ge 500 \\
&&p &\ge 0\\
&&k &\ge 0
\end{align}</span>
The unique optimal solution is <span class="mat... |
3,956,828 | <p>I can find the nth integral of <span class="math-container">$\ln(z)$</span> as follows:
<span class="math-container">\begin{aligned}
\left(\frac d{dz}\right)^{-n}\ln(z)&=\frac1{\Gamma(n)}\int\limits_0^z(z-t)^{n-1}\ln(t)dt\\
&=\frac1{n!}\left[\int\limits_0^z\frac1t(z-t)^ndt-z^n\ln(0)\right]\\
&=\frac1{n!}... | Spador Yedi | 664,849 | <p>I realized that showing how I came up with my conjecture could probably provide the basis of a proof. I'm not sure if I should post this as a separate answer or as part of the question, but since it kind of answers the question I'm posting as an answer. I will not, however, accept this.</p>
<p>I already knew the ans... |
4,422,512 | <p>I do have a matrix of following form</p>
<p><span class="math-container">$$M:=\left(\begin{array}{c|ccc}
A & & * &\\
\hline 0 & & &\\
0 & & B &\\
0 & & &\\
\end{array}\right)$$</span></p>
<p>Here the <span class="math-container">$0$</span>'s represent matrices of ... | zwim | 399,263 | <p>Note that even if you ignore that in case of <span class="math-container">$B$</span> invertible <span class="math-container">$$\det(M)=\det(A-*B^{-1}0)\det(B)=\det(A)\det(B)$$</span> it is possible to solve the system easily by blocks:</p>
<p><span class="math-container">$MX=0\iff\left(\begin{array}{c|ccc}
A & &... |
252,870 | <p>Given a polynomial, lets say for example <span class="math-container">$f(x,y) = (1+x+y)^2 = 1+2x+x^2+2y+2xy+y^2$</span>, I'd like to be able to order the terms of the polynomial by total degree, either in increasing or decreasing order (and if alphabetical order can be taken into account within terms of the same tot... | Daniel Huber | 46,318 | <p>I am not sure if I understand your description correctly. You have a list with names in "hypatia". Then the same names appear in a list of lists named "names". And you want determine the indices of names from "hypatia" in the list of lists "names". The command "Position&q... |
542,808 | <p>I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + irrational number is irrational, then it stood to reason that was also proving that the number in question was irrati... | Ross Millikan | 1,827 | <p>Yes, it is true that $x+1$ being irrational implies $x$ is irrational. Given that $x+1$ is irrational, assume $x=\frac ab$ with $a,b$ integers. Then $x+1=\frac {a+b}b$ would be rational as well.</p>
|
1,416,275 | <p>I've just began the study of linear functionals and the dual base. And this book I'm reading says the dual space $V^{*}$ may be identified with the space of row vectors. This notion seems very important, but I'm having trouble understanding it. Here is the text:</p>
<blockquote>
<p>Let $\sigma$ be an element of t... | Berci | 41,488 | <p>Let us fix a basis $\def\b{{\bf b}} \b_1,\dots,\b_n$ for $V$. For coordinates of vectors of $V$ we use <em>column</em> vectors, i.e. $\pmatrix{x_1\\ \vdots\\ x_n}$ represents the vector $x_1\b_1+\dots+x_n\b_n$.</p>
<p>Any element of the dual space (i.e. a linear $f:V\to K$) is uniquely determined by its action on t... |
272,144 | <p>Consider a multi-value function <span class="math-container">$f(z)=\sqrt{(z-a)(z+\bar a)}, \Im{a}>0,\Re{a}>0$</span>. To make the function be single-valued, one needs to make a cut. Suppose <span class="math-container">$a=e^{i\theta}$</span>, my choice of the branch cut is <span class="math-container">$e^{it},... | I.M. | 26,815 | <p>Perhaps you can use <code>ComplexPlot</code> (since v12)</p>
<pre><code>ClearAll[f] ;
f[w_][z_] := Sqrt[(z - w)(z + Conjugate[w])] ;
Manipulate[
ComplexPlot[
f[Exp[I*theta]][z],
{z,-5 - 5I, 5 + 5I},
PlotPoints -> 100,
MaxRecursion -> 2,
ColorFunction -> "C... |
80,056 | <p>I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this. </p>
<p>Obviously, slides make it possible to produce and show clear graphs/pictu... | Ryan Reich | 6,545 | <p>I just decided this quarter to use slides for my calculus class, a large-lecture course of the sort I'd never done before; I figured it would be easier to see the "board" if it were on the big screen. Here is the progression of my mistakes and corrections:</p>
<ul>
<li><p>My first lectures had too many words. Sli... |
80,056 | <p>I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this. </p>
<p>Obviously, slides make it possible to produce and show clear graphs/pictu... | JP McCarthy | 35,482 | <p>Finally, I question in MO I feel qualified to answer!</p>
<p>I am a PhD student in Ireland doing an amount of lecturing. As a first remark, I am lucky in the sense that undergraduate maths was never especially easy for me and therefore I empathise with the average student. My second remark is that I hope for a care... |
424,675 | <p>Just one simple question:</p>
<p>Let $\tau =(56789)(3456)(234)(12)$.</p>
<p>How many elements does the conjugacy class of $\tau$ contain? How do you solve this exersie?</p>
<p>First step is to write it in disjunct cyclces I guess. What's next? :)</p>
| Han de Bruijn | 96,057 | <p>Isn't it a pity that one has to <B>choose</B> between these methods, while they all have their advantages and disadvantages? Wouldn't it be better trying to take the best of the worlds and mix ingredients together? Perhaps a decent research effort of the kind will result in just <B>one</B> numerical method for solvi... |
424,675 | <p>Just one simple question:</p>
<p>Let $\tau =(56789)(3456)(234)(12)$.</p>
<p>How many elements does the conjugacy class of $\tau$ contain? How do you solve this exersie?</p>
<p>First step is to write it in disjunct cyclces I guess. What's next? :)</p>
| Han de Bruijn | 96,057 | <p>It shall be argued in this post that the whole idea of one Numerical Method being superior to
another is merely a prejudice that rests upon insufficient in-depth analysis of the real thing.</p>
<p>The argumentation will proceed at hand of a two-dimensional example.<BR>The reader is invited
not to skip through but t... |
424,675 | <p>Just one simple question:</p>
<p>Let $\tau =(56789)(3456)(234)(12)$.</p>
<p>How many elements does the conjugacy class of $\tau$ contain? How do you solve this exersie?</p>
<p>First step is to write it in disjunct cyclces I guess. What's next? :)</p>
| Han de Bruijn | 96,057 | <p><H2>Labrujère's Problem</H2></p>
<p>In Februari 1976, Dr. Th.E. Labrujère, at the National Aerospace Laboratory
NLR, the Netherlands, wrote a memorandum which is titled, when
translated in English: <I>The "Least Squares - Finite Element" Method</I> [L.S.FEM]
<I>applied to the 2-D Incompressible Flow a... |
211,778 | <p>In</p>
<pre><code>Block[{ ds = Dataset[{
<|"task" -> "task4", "parents" -> "parent1",
"start" -> "2019-05-14 17:10", "end" -> "2019-05-14 17:15",
"utility" -> "0.9"|>}
],
hrAssociation =
KeySort[Merge[
Rule @@@
Flatten[GroupBy[
ds[All, <|"... | Rolf Mertig | 29 | <p>This works:</p>
<pre><code>Module[{ds,hrAssociatio}
,
ds = Dataset[{
<|"task" -> "task4", "parents" -> "parent1",
"start" -> "2019-05-14 17:10", "end" -> "2019-05-14 17:15",
"utility" -> "0.9"|>}
]
; hrAssociation =
KeySort[Merge[
... |
211,778 | <p>In</p>
<pre><code>Block[{ ds = Dataset[{
<|"task" -> "task4", "parents" -> "parent1",
"start" -> "2019-05-14 17:10", "end" -> "2019-05-14 17:15",
"utility" -> "0.9"|>}
],
hrAssociation =
KeySort[Merge[
Rule @@@
Flatten[GroupBy[
ds[All, <|"... | b3m2a1 | 38,205 | <p>To add to Rolf's answer, if you <em>need</em> this in the body of the declaration you can use <code>:=</code></p>
<pre><code>Block[{ds =
Dataset[{<|"task" -> "task4", "parents" -> "parent1",
"start" -> "2019-05-14 17:10", "end" -> "2019-05-14 17:15",
"utility" -> "0.9"|>}
... |
189,293 | <p><img src="https://i.stack.imgur.com/Ngfb2.jpg" alt="Vesica Pisces"></p>
<p>I have the radius and center $(x,y)$ on both circles, but how do I get the $(x,y)$ of the red circle, or in other words how do I get the $(x,y)$ position of where the circles intersect at the top or bottom?</p>
| Jakube | 38,034 | <p>Here is a nice example:
<a href="http://www.analyzemath.com/CircleEq/circle_intersection.html" rel="nofollow">http://www.analyzemath.com/CircleEq/circle_intersection.html</a></p>
<p>Just set up the two circle equations: $(X-M)^2=r^2$ and follow the instructions.</p>
|
466,576 | <p>Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the German word <em>Zahl(en)</em> 'number(s)'. What are some more examples?</p>
| MJD | 25,554 | <p>Łukasiewicz notation for logic represents $\land \lor \leftrightarrow$ with the letters $K A E$ respectively, so that for example $r\lor(p\land q)$ is $ArKpq$. $K A E$ are the initials of the Polish words <em>koniunkcja</em>, <em>alternatywa</em>, <em>ekwiwalencja</em>.</p>
<p>I don't know why Łukasiewicz used $C$... |
2,432,817 | <p>Let $X$ and $Y$ be topological spaces and let $A \subseteq X$ be a subspace of $X$. Suppose $A$ is homeomorphic to some subspace $B \subseteq Y$ of $Y$. Let $f$ explicitly denote this homeomorphism.</p>
<p>If $f : A \to B$ is a homeomorphism, does $f$ extend to a homeomorphism between $\text{Cl}_X(A)$ and $\text{Cl... | DanielWainfleet | 254,665 | <p>Suppose $X$ has a sub-space $A$ that is homeomorphic to $X$ where $A$ is not a closed subset of $X$ and $Cl_X(A)\ne A$ and where $A$ is not homeomorphic to $Cl_X(A)$.Let $X=Y=B$ and let $f:A\to B$ be any homeomorphism. Since $f$ is a bijection from $A$ onto $Y$ and $Cl_X(A)\ne A,$ therefore $f$ cannot be extended to... |
2,236,846 | <p>For $x\in A[a,b]$</p>
<p>$\sup_{x\in A}|f(x)|\ge\int_{a}^{b}f(x)dx$</p>
<p>I'm just wondering if this is an analysis result or if the result is slightly different to this?</p>
<p>Sorry I just realised it was a greater than sign not an equals!</p>
| Trevor Gunn | 437,127 | <p>If $f(x) \le g(x)$ for all $x \in [a,b]$ (and both are integrable) then $$\int_a^b f(x) \,dx \le \int_a^b g(x) \,dx.$$ If you then take $g$ to be the constant function $g(x) = \sup_{t \in [a,b]} |f(t)|$ what this says is that
$$ \int_a^b f(x) \,dx \le (b - a) \sup_{t \in [a,b]} |f(t)|. $$</p>
<p>Note that you do ne... |
4,498,203 | <p>We know that each row (and each column) of composition table of a finite group, is a rearrangement (permutation) of the elements of the group.</p>
<p>How about the other way round? If we have a composition table where each row and each column is a permutation of the elements of a set, does this composition table nec... | Arturo Magidin | 742 | <p>What you describe is a <a href="https://en.wikipedia.org/wiki/Quasigroup" rel="noreferrer">quasigroup</a>.</p>
<p>A quasigroup is an ordered pair <span class="math-container">$(A,\cdot)$</span>, where <span class="math-container">$A$</span> is a set, and <span class="math-container">$\cdot$</span> is a binary operat... |
90,673 | <p>let $M$ be a closed ortientable irreducible 3-mfd, let $T$ be a non-separating torus in $M$, we cut $M$ along $T$ and glue two solid tori along the two boundary tori, we get a new closed 3-mfd $M_1$ (we need $M_1$ also is irreducible). Now If there exists nonseperable torus $T_1$ in $M_1$, we go on the above process... | Kevin Walker | 284 | <p>Start with $S^2\times \{1\}$ inside $S^2\times S^1$. Increase the genus of this non-separating 2-sphere to obtain a non-separating torus $T\subset S^2\times S^1$. Cutting along $T$ yields $S^2\times I$ with a 1-handle attached to each boundary component. It is possible to glue solid tori to the two boundary compo... |
2,906,865 | <p>I am trying to find general formula of the sequence $(x_n)$ defined by
$$x_1=1, \quad x_{n+1}=\dfrac{7x_n + 5}{x_n + 3}, \quad \forall n>1.$$
I tried
put $y_n = x_n + 3$, then $y_1=4$ and
$$\quad y_{n+1}=\dfrac{7(y_n-3) + 5}{y_n }=7 - \dfrac{16}{y_n}, \quad \forall n>1.$$
From here, I can't solve it. How can ... | Yuta | 590,171 | <p>Here is a method that I read from a book. Yet I did not think deeply why it works in general.</p>
<hr>
<p>If there exists real numbers $\alpha$, $\beta$ and $r$ such that</p>
<p>$$\frac{a_{n+1}-\beta}{a_{n+1}-\alpha}=r\cdot\frac{a_n-\beta}{a_n-\alpha}$$</p>
<p>for all $n\in\mathbb{N}$, then the sequence $\{b_n\}... |
3,076,253 | <p>Problem: Let <span class="math-container">$I=[0,1]$</span> be the closed unit interval. Suppose <span class="math-container">$f$</span> is a continuous mapping of <span class="math-container">$I$</span> into <span class="math-container">$I$</span>. Prove that <span class="math-container">$f(x)=x$</span> for at least... | Community | -1 | <p>The first thing you should note is that if <span class="math-container">$\tau, \sigma \in S_n$</span>, then <span class="math-container">$\tau \circ \sigma \in S_n$</span>. This means that if you have two permutations, then their product is also a permutation of the same permutation group. </p>
<p>You also know tha... |
50,994 | <p>I am trying to calculate the following integral. </p>
<pre><code>sigma1 = 10.0; sigma2 = 5.0; delta = 0.5;
t[x1_, y1_, x_, y_] := 100*HeavisideLambda[sigma1^-1*(x - x1), sigma2^-1*(y - y1)];
B2[x1_, y1_, x_, y_] := HeavisideTheta[(delta/2)^2 - (x - x1)^2, (delta/2)^2 - (y - y1)^2];
trans[x1_, y1_, x2_, y2_] :=
... | george2079 | 2,079 | <p>ah, you are right for that condition: <code>B2[x2, y2, xz, yz]</code> and <code>B2[x2, y2, xp, yp]</code> are returning 0 for every <em>sample</em> point in the domain. This is not so much an answer but I thought i may be useful to show how to use <code>EvaluationMonitor</code>and <code>Reap/Sow</code> to get at th... |
114,289 | <p>I am trying to use C++ programs through MathLink in my notebooks, but I cannot compile successfully the simple programs included in Mathematica. </p>
<p>I do not have a specific question, I am just looking for guidance.</p>
<p><code>$Version
$SystemID
"9.0 for Linux x86 (64-bit) (November 20, 2012)"
"Linux-x86-... | halirutan | 187 | <p>The easiest way to see what libraries are required is to let <em>Mathematica</em> compile it and look at the commandline. First, locate the two required files <code>addtwo.tm</code> and <code>addtwo.c</code>. This might be a bit different on your system:</p>
<pre><code>file = FileNames["addtwo.*", {$InstallationDir... |
1,111,935 | <p>For a given $n \in \Bbb N$, how do you find the minimum $m \in \Bbb N$ which satisfies the inequality below?</p>
<p>$$3^{3^{3^{3^{\unicode{x22F0}^{3}}}}} (m \text{ times}) > 9^{9^{9^{9^{\unicode{x22F0}^{9}}}}} (n \text{ times})$$</p>
<p>What I have tried to do so far is decomposing the $9$ on the right side to ... | Nathan E. | 173,608 | <p>I can tell you with certainty that <code>m+n+1</code> is the minimum value for m. Working it out is the real problem. I'm sure it's correct, though, because if you take the 3's n+1 times and subtract the 9's n times, </p>
<pre><code>3^3^3^3 (n+1 times) - 9^9^9 (n times) > 0
</code></pre>
<p>it's always positive... |
1,111,935 | <p>For a given $n \in \Bbb N$, how do you find the minimum $m \in \Bbb N$ which satisfies the inequality below?</p>
<p>$$3^{3^{3^{3^{\unicode{x22F0}^{3}}}}} (m \text{ times}) > 9^{9^{9^{9^{\unicode{x22F0}^{9}}}}} (n \text{ times})$$</p>
<p>What I have tried to do so far is decomposing the $9$ on the right side to ... | Hasit Bhatt | 261,725 | <p>$9^{9^{9^{9^{\dots}}}} n$ times $= 3^{2*3^{2*3^{2*3^2{\dots}}}} $, where the upper most $2$ is $(n+1)^{th}$ power.</p>
<p>$2*3^2 < 3^{2+1} = 3^3 $</p>
<p>$\implies 9^{9^{9^{9^{\dots}}}} n$ times $\lt 3^{3^{3^{3^{\dots}}}} n+1 $ times</p>
<p>So, $ m = n + 1 $</p>
|
2,917,299 | <p><strong>Let $h: R\to S$ be a ring homomorphism. Let $P\subset R$ be a prime ideal.</strong>
<strong>Give an example to show that in general $h(P)$ is not an ideal of $S$</strong></p>
<p>The first thing I think is to take $R=\mathbb{Z}$ and $P=(2)$ but I do not know how to take $S$ or if this works in this way, any ... | Yanko | 426,577 | <p>Consider $S=\mathbb{R}$. With the injection map $h(x)=x$, $R=\mathbb{Z}$ and any prime ideal of $\mathbb{Z}$, say $P=(2)$ as you suggested.</p>
|
2,874,763 | <p>I know for 3-D $$\nabla^2 \left(\frac1r\right)=-4\pi\, \delta(\vec{r})\,.$$
I would like to know, what is $$\text{Div}\cdot\text{Grad}\left(\frac{1}{r^2}\right)$$ in 4-Dimensions ($r^2=x_1^2+x_2^2+x_3^2+x_4^2$)?</p>
| Batominovski | 72,152 | <p>Let $\sigma_k$ denote the hypersurface area measure on the $k$-sphere. For example,
$$\text{d}\sigma_1(\varphi_1)=\text{d}\phi_1\,,\,\,\text{d}\sigma_2(\varphi_1,\varphi_2)=\sin(\varphi_2)\,\text{d}\varphi_1\,\text{d}\varphi_2\,,$$
and
$$\text{d}\sigma_3(\varphi_1,\varphi_2,\varphi_3)=\sin(\varphi_2)\,\sin^2(\varp... |
818,161 | <p>Suppose that repetitions are not allowed.</p>
<p>There are $6 \cdot 5 \cdot 4 \cdot 3 $ numbers with $4$ digits , that can be formed from the digits $1,2,3,5,7,8$.</p>
<p>How many of them contain the digits $3$ and $5$?</p>
<p>I thought that I could subtract from the total number of numbers those,that do not cont... | Community | -1 | <p>There are $6!$ combinations of six different numbers, and there are $6^6$ combinations of six rolls. The probability is then:
$$
p = \frac{6!}{6^6}
$$</p>
|
2,103,602 | <p>What is the maximum value of
$\displaystyle{{1 + 3a^{2} \over \left(a^{2} + 1\right)^{2}}}$, given that $a$ is a real number, and for what values of $a$ does it occur ?.</p>
| Simply Beautiful Art | 272,831 | <p>Let $x=a^2\ge0$. We then have to look at</p>
<p>$$\frac{1+3x}{(x+1)^2}$$</p>
<p>upon differentiating and setting equal to $0$, we get</p>
<p>$$\frac{3(x+1)^2-2(x+1)(1+3x)}{(x+1)^4}\\\implies3(x+1)^2=2(x+1)(1+3x)\\\implies3(x+1)=2(1+3x)\\\implies x=\frac13$$</p>
<p>Thus, the relative maxima (or minima) occurs at... |
2,103,602 | <p>What is the maximum value of
$\displaystyle{{1 + 3a^{2} \over \left(a^{2} + 1\right)^{2}}}$, given that $a$ is a real number, and for what values of $a$ does it occur ?.</p>
| Michael Rozenberg | 190,319 | <p>Let $a^2=x$.</p>
<p>Hence, $\frac{1+3x}{(1+x)^2}\leq\frac{9}{8}$ it's $(3x-1)^2\geq0$.</p>
<p>The equality occurs for $x=\frac{1}{3}$, which says that the answer is $\frac{9}{8}$.</p>
|
4,309,797 | <p>I have a question which asks me to compute the double integral
<span class="math-container">$$\iint_By^2-x^2\,dA$$</span> where B is the region enclosed by <span class="math-container">$$y=x,y=x+2,y=\frac{2}{x},y=\frac{2}{x}$$</span>I made a change of variables by letting <span class="math-container">$$u=xy \qquad \... | Sidvhid Hsinynjad | 866,977 | <p>Hint:</p>
<p>You may use the following identity</p>
<p><span class="math-container">$$(y+x)^2-(y-x)^2=4xy$$</span></p>
|
246,606 | <p>I have matrix:</p>
<p>$$
A = \begin{bmatrix}
1 & 2 & 3 & 4 \\
2 & 3 & 3 & 3 \\
0 & 1 & 2 & 3 \\
0 & 0 & 1 & 2
\end{bmatrix}
$$</p>
<p>And I want to calculate $\det{A}$, so I have written:</p>
<p>$$
\begin{array}{|cccc|ccc}
1 & 2 & 3 & 4 & 1 & 2 ... | Brusko651 | 50,608 | <p>The trick you are applying (Rule of Sarrus) only works for $ 3\times 3$ Matrices.</p>
|
2,195,287 | <blockquote>
<p>Knowing that $p$ is prime and $n$ is a natural number show that
$$n^{41}\equiv n\bmod 55$$
using Fermat's little theorem
$$n^p\equiv n\bmod p$$</p>
</blockquote>
<p>If the exercise was to show that
$$n^{41}\equiv n\bmod 11$$ I would just rewrite $n^{41}$ as a power of $11$ and would easily prov... | user21820 | 21,820 | <p>Actually the chinese remainder theorem is unnecessary here.</p>
<p>$n^{41} \equiv n \pmod{5}$ and hence $n^{41}-n = 5c$ for some integer $c$.</p>
<p>$n^{41} \equiv n \pmod{11}$ and hence $11 \mid n^{41}-n = 5c$.</p>
<p>$11$ is prime and does not divide $5$, so by Euclid's lemma $11 \mid c$.</p>
|
255,295 | <p>I just did one exercise stating:
Prove that the linear map $M: X \rightarrow C([0,1])$, is continuous iff for every $t\in[0,1]$, the rule $x\rightarrow (Mx)(t)$ defines a continuous linear functional on X.
the next exercise stated:
State, and prove a similar continuity criterion for linear maps $M:X\rightarrow Y$ wh... | Johan | 49,267 | <p>The map $x\mapsto \hat{\ell}(Mx)$ is a linear map $\ell \colon X \mapsto \mathbb{K}$.
By the continuity of $\hat{\ell}$ and uniformed boundedness we have $\sup_{\ell} \|\ell\| = c$.</p>
<p>\begin{equation}
\begin{split}
\|M\| =& \sup_{\|x\| = 1} \|Mx\| = \sup_{\|\hat{\ell}\| \leq 1}\sup_{\|x\| = 1} \|\hat{\ell}... |
255,295 | <p>I just did one exercise stating:
Prove that the linear map $M: X \rightarrow C([0,1])$, is continuous iff for every $t\in[0,1]$, the rule $x\rightarrow (Mx)(t)$ defines a continuous linear functional on X.
the next exercise stated:
State, and prove a similar continuity criterion for linear maps $M:X\rightarrow Y$ wh... | Martin Argerami | 22,857 | <p>Your answer seems to be circling around the right ideas, but I'm not sure if you are using them in the right way. </p>
<p>Fix $x\in X$. Consider the map $T_x:Y'\to\mathbb K$, given by $T_x(\ell)=\ell(Mx)$. Note that
$$
\|T_x\|=\sup\{|T_x(\ell)|:\ \|\ell\|=1\}=\sup\{|\ell(Mx)|:\ \|\ell\|=1\}=\|Mx\|
$$
where the last... |
292,518 | <p>Can someone help me with the following proof involving positive definite matrices:</p>
<p>Suppose $X\succ 0$ positive definite. Show that $X-v{v^T}\succ 0$ if and only if ${v^T}X^{-1}v \le 1$.</p>
<p>Thanks in advance. </p>
| copper.hat | 27,978 | <p>Here is another approach. I assume $X$ is symmetric.</p>
<p>To simplify life, all square matrices below are assumed symmetric.</p>
<p>Since $X>0$ it has a square root satisfying $X= (X^{\frac{1}{2}})^2$.</p>
<p>Note that if $B$ is invertible, then $A>0$ iff $BAB>0$. Also note that $I-u u^T>0$ iff $\... |
292,518 | <p>Can someone help me with the following proof involving positive definite matrices:</p>
<p>Suppose $X\succ 0$ positive definite. Show that $X-v{v^T}\succ 0$ if and only if ${v^T}X^{-1}v \le 1$.</p>
<p>Thanks in advance. </p>
| user1551 | 1,551 | <p>I prefer copper.hat's approach, but it doesn't hurt to view the problem from other perspectives.</p>
<p>The assertion in the problem statement is obvious if $v=0$. So, assume $v\not=0$. Since $X$ is positive definite, it defines an inner product $\langle u,w\rangle=w^TXu$ on $\mathbb{R}^n$. Therefore, every nonzero... |
3,676,911 | <p>I'm trying to understand what is the Hessian matrix of <span class="math-container">$f\colon\mathbb{R}^{n}\to\mathbb{R}$</span>
defined by <span class="math-container">$f\left(x\right)=\left\langle Ax,x\right\rangle \cdot\left\langle Bx,x\right\rangle $</span>
where <span class="math-container">$A,B$</span> are syme... | J. Heller | 309,909 | <p>We can write formulas for <span class="math-container">$f_i$</span> and <span class="math-container">$f_{ij}$</span> (individual first and second partial derivatives) of <span class="math-container">$f$</span>:
<span class="math-container">$$
f_i(x) = g_i(x)h(x) + g(x)h_i(x)
$$</span>
and
<span class="math-contain... |
408,601 | <p>I am asked to find the derivative of $\left(x^x\right)^x$. So I said let $$y=(x^x)^x \Rightarrow \ln y=x\ln x^x \Rightarrow \ln y = x^2 \ln x.$$Differentiating both sides, $$\frac{dy}{dx}=y(2x\ln x+x)=x^{x^2+1}(2\ln x+1).$$</p>
<p>Now I checked this answer with Wolfram Alpha and I get that this is only correct whe... | Fly by Night | 38,495 | <p>If $y=(x^x)^x$ then $\ln y = x\ln(x^x) = x^2\ln x$. Then apply the product rule:</p>
<p>$$ \frac{1}{y} \frac{dy}{dx} = 2x\ln x + \frac{x^2}{x} = 2x\ln x + x$$</p>
<p>Hence $y' = y(2x\ln x + x) = (x^x)^x(2x\ln x + x).$</p>
<p>This looks a little different to your expression, but note that $\ln(x^x) \equiv x\ln x$.... |
3,500,418 | <p>I am fining the pointwise limit of the function <span class="math-container">$f_n(x) = \frac{x^n}{3-x^n}$</span> for <span class="math-container">$x ∈ [0,1]$</span> and <span class="math-container">$n ∈ N$</span></p>
<p>In order to do this I first divided through by <span class="math-container">$x^n$</span>, yieldi... | Paultje | 466,494 | <p>One possible way would be to use the definition of <span class="math-container">$\cos(x)$</span> that is <span class="math-container">$$ \cos(x) = \sum_{k=0}^\infty \frac{(-1)^k x^{2k}}{(2k)!}$$</span></p>
|
39,424 | <p>I need to teach an intro course on number theory in 1 month. I was just notified. Since I have never studied it, what are good books to learn it quickly?</p>
| Zev Chonoles | 1,916 | <p>I think Jones + Jones </p>
<p><a href="http://rads.stackoverflow.com/amzn/click/3540761977" rel="nofollow">http://www.amazon.com/Elementary-Number-Theory-Gareth-Jones/dp/3540761977</a></p>
<p>would be a good all-around introduction. It has solutions to every problem in the back, which can be helpful for self-study... |
2,611,382 | <p>Solve the equation,</p>
<blockquote>
<p>$$
\sin^{-1}x+\sin^{-1}(1-x)=\cos^{-1}x
$$</p>
</blockquote>
<p><strong>My Attempt:</strong>
$$
\cos\Big[ \sin^{-1}x+\sin^{-1}(1-x) \Big]=x\\
\cos\big(\sin^{-1}x\big)\cos\big(\sin^{-1}(1-x)\big)-\sin\big(\sin^{-1}x\big)\sin\big(\sin^{-1}(1-x)\big)=x\\
\sqrt{1-x^2}.\sqrt{2x... | lab bhattacharjee | 33,337 | <p>Like Barry Cipra,</p>
<p>$$2\arcsin x=\arccos(1-x)$$</p>
<p>Now $0\le\arccos(1-x)\le\pi$ and $-\pi\le2\arcsin x\le\pi$</p>
<p>$\implies\arcsin x\ge0\iff x\ge0$</p>
<p>Now for $\arcsin x\ge0,2\arcsin x=\begin{cases}\arccos(1-2x^2)&\mbox{if }x\ge0\\
-\arccos(1-2x^2)& \mbox{if }x<0\end{cases}$</p>
<p>$... |
3,030,753 | <p>Let <span class="math-container">$f:\mathbb R \rightarrow \mathbb R$</span> be a continuous function and <span class="math-container">$x_0 \in \mathbb R$</span> such that f is differentiable on both intervals <span class="math-container">$(-\infty, x_0]$</span> and <span class="math-container">$[x_0, +\infty)$</span... | copper.hat | 27,978 | <p>Here is a hint:</p>
<p>Suppose <span class="math-container">$\phi(x) = \begin{cases} ax, & x < 0 \\
bx, & x \ge 0 \end{cases}$</span>, note that we can write
<span class="math-container">$\phi(x) = {b-a \over 2} |x| + {a+b \over 2} x$</span>.</p>
|
97,340 | <p>Fellow Puny Humans, </p>
<p>A <em>geometric net</em> is a system of points and lines that obeys three axioms:</p>
<ol>
<li>Each line is a set of points.</li>
<li>Distinct line has at most one point in common.</li>
<li>If $p$ is a point and $L$ is a line with $p \notin L$, then there is exactly one line $M$ such th... | Brian M. Scott | 12,042 | <p>Let $C_1,\dots,C_m$ be your equivalence classes. Suppose first that there is a point $p$ that is in more than $m$ lines; then there is some class $C_i$ such that $p$ is in two distinct lines in $C_i$. But that’s impossible, since the lines in $C_i$ are mutually parallel, so every point is in at most $m$ lines.</p>
... |
3,949,580 | <p>Is it possible to set this integral up without using substitution?</p>
<p><span class="math-container">$$\iint_D e^{x+y} \,\mathrm{d}x\,\mathrm{d}y\,,$$</span> where</p>
<p><span class="math-container">$$D = \left\{-1\le x+y \le 1, -1 \le -x + y \le 1\right\}$$</span></p>
<p>The answer is: <span class="math-containe... | Bram28 | 256,001 | <p>It has nothing to do with the operators being binary, but rather how those operations deals with the operands.</p>
<p>Let's assume the jar has <span class="math-container">$n$</span> balls, and let that be the left operand. Then, adding <span class="math-container">$2$</span> balls to the jar would correspond to the... |
1,779,965 | <p>Given the numbers $x = 123$ and $y = 100$ how to apply the Karatsuba algorithm to multiply these numbers ?</p>
<p>The formula is </p>
<pre><code>xy=10^n(ac)+10^n/2(ad+bc)+bd
</code></pre>
<p>As I understand $n = 3$ (number of digits) and I tried writing the numbers as </p>
<pre><code>x = 10*12+3 , y = 10*10 +0 t... | Peter Phipps | 15,984 | <p>What you have writtem is almost correct.
The Karatsuba Algorithm expands the multiplication of $123$ and $100$ to
$$\begin{aligned}
(12\times 10+3)(10\times10+0)&= 10^2\times(12\times10) + 10^1\times (12\times0 + 3 \times 10) + 3\times 10\\
&= 12000 + 300 + 0\\
&= 12300
\end{aligned}$$</p>
<p>The <a hr... |
1,338,980 | <p>Suppose you have a set of data $\{x_i\}$ and $\{y_i\}$ with $i=0,\dots,N$. In order to find two parameters $a,b$ such that the line
$$
y=ax+b,
$$
give the best linear fit, one proceed minimizing the quantity
$$
\sum_i^N[y_i-ax_i-b]^2
$$
with respect to $a,b$ obtaining well know results. </p>
<p>Imagine now to desi... | JJacquelin | 108,514 | <p>If you want to fit the function
$$y=a+b\:X^c$$
with a set of data $(X_1,y_1),(X_2,y_2),...,(X_k,y_k),...,(X_n,y_n)$, this is possible on various ways. The usual methods are iterative and require guessed initial values $a_0,b_0,c_0$ to start the process.</p>
<p>A non-usual method is described in the paper : <a href=... |
1,619,371 | <p>I was working on a problem and reduced it to evaluating</p>
<p>$$\int_{0}^{1}\sqrt{1+x^a}\,dx~~a>0$$</p>
<p>your suggestion? Thanks</p>
| Jack D'Aurizio | 44,121 | <p>If $a>0$ we have:
$$ \color{red}{I(a)}=\int_{0}^{1}\sqrt{1+x^a}\,dx = \phantom{}_2 F_1\left(-\frac{1}{2},\frac{1}{a};1+\frac{1}{a};-1\right) \tag{1}$$
by just expanding $\sqrt{1+z}$ as a Taylor series. Approximations for the RHS can be computed from:
$$ I(a) = \frac{1}{a}\int_{0}^{1} z^{\frac{1}{a}-1}\sqrt{1+z}\,... |
1,334,527 | <p>The integral in hand is
$$
I(n) = \frac{1}{\pi}\int_{-1}^{1} \frac{(1+2x)^{2n}}{\sqrt{1-x^2}}\, dx
$$
I dont know whether it has closed-form or not, but currently I only want to know its asymptotic behavior. Setting $x=\cos\theta$, then
$$
I(n) = \frac{1}{\pi}\int_{0}^{\pi/2} \Big[(1+2\cos\theta)^{2n}+(1-2\cos\theta... | Claude Leibovici | 82,404 | <p><em>This is not an answer but it is too long for a comment</em></p>
<p>For the antiderivative $$J_n = \frac{1}{\pi}\int \frac{(1+2x)^{2n}}{\sqrt{1-x^2}}\, dx$$ there is a "closed" for which involves the Appell hypergeometric function of two variables $\frac 2{1-x}$ and $\frac 3{2(1-x)}$ which not very useful.</p>... |
2,779,152 | <p>Consider a Poisson process with rate $\lambda$ in a given time interval $[0,T]$. The inter-arrival time between successive arrivals is negative exponential distributed with mean $\frac{1}{\lambda}$ such that $X_1 >0$, and $\sum_{i=1}^\text{Last} X_i < T$, where $X$ represents inter-arrival time.</p>
<p>What a... | Atmos | 516,446 | <p>HINT : </p>
<p>An idea is to use the D'Alembert criteria which states that if
$$
\frac{a_{n+1}}{a_n} \underset{n \rightarrow +\infty}{\rightarrow} \ell
$$
then</p>
<ul>
<li><p>$\ell>1$ makes the series $\sum_{n \in \mathbb{N}}^{ }a_n$ diverge.</p></li>
<li><p>$\ell<1$ makes the series $\sum_{n \in \mathbb{N}... |
19,261 | <p>Every simple graph $G$ can be represented ("drawn") by numbers in the following way:</p>
<ol>
<li><p>Assign to each vertex $v_i$ a number $n_i$ such that all $n_i$, $n_j$ are coprime whenever $i\neq j$. Let $V$ be the set of numbers thus assigned. <br/></p></li>
<li><p>Assign to each maximal clique $C_j$ a unique p... | Charles Siegel | 622 | <p>You might want to look up some things about index theorems (particularly Atiyah-Singer). They tend to relate topological and geometric data, so you can put geometric data in and topological data out.</p>
|
3,997,532 | <p>Please help me with this. I can't prove the result. Tried integral by parts or notations, nothing working</p>
<p><span class="math-container">$$\int_{-1}^{1}{\frac{x^2}{e^x+1}}dx$$</span></p>
| Quanto | 686,284 | <p>Note <span class="math-container">$\frac{1}{e^x+1}
= \frac12+\frac12\tanh^{-1}\frac x2 $</span> and the odd function <span class="math-container">$\tanh^{-1}\frac x2$</span> does not contribute to the integral over <span class="math-container">$(-1,1)$</span>. Thus</p>
<p><span class="math-container">$$\int_{-1}^{1}... |
61,933 | <p>Consider a lattice in R^3.
Is the some "canonical" way or ways to choose basis in it ?</p>
<p>I mean in R^2 we can choose a basis |h_1| < |h_2| and |(h_2, h_1)| < 1/2 |h_1|.
Considering lattices with fixed determinant and up to unitary transformations we get standard picture of the PSL(2,Z) acting on the upp... | Richard Borcherds | 51 | <p>The book by Terras "Harmonic analysis on symmetric spaces and applications" volume 2 has some stereoscopic pictures of the fundamental domains for some similar groups.</p>
|
2,725,697 | <p>A weird question that has me confused. Suppose I have a symmetric matrix $A$, which has to be computed somehow. For example, the Hessian matrix is a symmetric matrix that is computed by taking the gradient twice. A covariance matrix is also symmetric as another example. $A$ will have $n^2$ entries but really only ne... | user3658307 | 346,641 | <p>There are some cases where this is true. For instance, if $A=ab^T=a\otimes b$ is a rank 1 matrix, then you should definitely not compute $A$, or store it. Just keep $a$ and $b$ around instead and it will be much faster/cheaper to compute $Av$ (or almost anything really).</p>
<p>Another interesting case is <em>Hessi... |
1,465,627 | <p>The problem is to maximize the determinant of a $3 \times 3$ matrix with elements from $1$ to $9$.<br>
Is there a method to do this without resorting to brute force?</p>
| mathreadler | 213,607 | <p><strong>HINT:</strong></p>
<p>The product of eigenvalues are the determinant. The trace is the sum of eigenvalues. It may be reasonable to believe that a large eigenvalue sum increases the chance of a large eigenvalue product. If that is the case then 9,8,7 would be good candidates for diagonal elements. Then some s... |
966,798 | <p>How I solve the following equation for $0 \le x \le 360$:</p>
<p>$$
2\cos2x-4\sin x\cos x=\sqrt{6}
$$</p>
<p>I tried different methods. The first was to get things in the form of $R\cos(x \mp \alpha)$:</p>
<p>$$
2\cos2x-2(2\sin x\cos x)=\sqrt{6}\\
2\cos2x-2\sin2x=\sqrt{6}\\
R = \sqrt{4} = 2 \\
\alpha = \arctan \f... | John | 105,625 | <p>Because here $\epsilon$ is an arbitrary positive number, it could be 2. Hence we use minimum so that $|x|> \frac{1}{2}$ (bounded away from 0) to control the denominator.</p>
|
966,798 | <p>How I solve the following equation for $0 \le x \le 360$:</p>
<p>$$
2\cos2x-4\sin x\cos x=\sqrt{6}
$$</p>
<p>I tried different methods. The first was to get things in the form of $R\cos(x \mp \alpha)$:</p>
<p>$$
2\cos2x-2(2\sin x\cos x)=\sqrt{6}\\
2\cos2x-2\sin2x=\sqrt{6}\\
R = \sqrt{4} = 2 \\
\alpha = \arctan \f... | CopyPasteIt | 432,081 | <p>If challenged with any <span class="math-container">$\varepsilon \gt 0$</span>, by setting</p>
<p><span class="math-container">$\tag 1 \Large{\delta = 1-\frac{1}{\varepsilon+1}}$</span></p>
<p>the statement</p>
<p><span class="math-container">$\tag 2 \Large{|x - 1| \lt \delta \text{ implies } |\frac{1}{x} - 1| \lt ... |
869,268 | <p>I am asked if $\{n, n^{2}, n^{3}\}$ forms a group under multiplication modulo $m$ where $m = n + n^{2} + n^{3}.$</p>
<p>As an example we see that $\{2, 4, 8\}$ does form a group modulo $14,$ with identity $8,$ but am stuck starting the proof for the general case. Thanks in advance.</p>
| Ragnar | 91,741 | <p>We work modulo $m=n+n^2+n^3$. Since $m|n^4-n$, we have that $n^4=n$. This implies* $n^3=1$, so $n^3$ is the identity. Also, when $x,y\in\{n,n^2,n^3\}=G$, we have $xy\in G$, because $xy=n^an^b=n^{a+b}=n^c$, where $c-1=a+b-1\mod 3$. Take for example $x=n^2$ and $y=n^3$. Then, $xy=n^5=n^4\cdot n=n\cdot n=n^2$. So in th... |
1,068,631 | <p>I want to find the solutions of the equation $$\left[z- \left( 4+\frac{1}{2}i\right)\right]^k = 1 $$ </p>
<p>in terms of roots of unity.</p>
<p>When I try to solve this, I get
\begin{align*}z - 4 - \dfrac i2 &= 1\\
z-\dfrac{i}{2}&=5\\
\dfrac{2z-i}2 &= 5\\
z&= 5 + \dfrac i2\end{align*}</p>
<p>Is t... | DeepSea | 101,504 | <p>Let $w = z - 4 + \dfrac{i}{2} \to w^k = 1 \to w = e^{\left(\dfrac{2\pi in}{k}\right)}, n = 0,1,\cdots ,(k-1) \to z = e^{\left(\dfrac{2\pi in}{k}\right)} + 4 - \dfrac{i}{2}$,</p>
<p>For b), $w^k = 2 = \left(\sqrt[k]{2}\right)^k \to \left(\dfrac{w}{\sqrt[k]{2}}\right)^k = 1 \to \dfrac{w}{\sqrt[k]{2}} = e^{\left(\dfra... |
2,502,963 | <p>How do you prove that $e=\sum_{n=0}^{\infty}\frac{1}{n!}$? Here I am assuming $e:=\lim_{n\to\infty}(1+\frac{1}{n})^n$. Do you have any good PDF file or booklet available online on this? I do not like how my analysis text handles this...</p>
| Adayah | 149,178 | <p>First prove by the ratio test that the series $\displaystyle \sum_{k=0}^{\infty} \frac{1}{k!}$ converges and denote the sum by $S$.</p>
<p>Then note that</p>
<p>$$\begin{align*}
\left( 1 + \frac{1}{n} \right)^n & = \sum_{k=0}^n \binom{n}{k} \cdot 1^{n-k} \cdot \left( \frac{1}{n} \right)^k = \sum_{k=0}^n \frac{... |
1,699,752 | <p>Does $ 1 + 1/2 - 1/3 + 1/4 +1/5 - 1/6 + 1/7 + 1/8 - 1/9 + ...$ converge? </p>
<p>I know that $(a_n)= 1/n$ diverges, and $(a_n)= (-1)^n (1/n)$, converges, but given this pattern of a negative number every third element, I am unsure how to determine if this converges. </p>
<p>I tried to use the comparison test, but ... | Robert Israel | 8,508 | <p>Hint: $1 + 1/2 - 1/3 > 1$, $1/4 + 1/5 - 1/6 > 1/4$, $1/7 + 1/8 - 1/9 > 1/7$, ...</p>
|
829,449 | <p>I am confused on the concept of extensionality versus intensionality. When we say 2<3 is True, we say that 2<3 can be demonstrated by a mathematical proof. So, according to mathematical logic, it is true. Yet, when we consider x(x+1) and X^2 + X, we can say that the x is the same for = 1. However, we call this... | m.g. | 154,039 | <p>I remember that we did some geometrical optics in high school. Maybe it is a little to advanced, but maybe you'd like to judge by yourself. So what I concretly remember is:</p>
<ul>
<li>Snells law</li>
<li>Lens optics, especially the "thin lense formula"</li>
</ul>
<p>Snells law uses very basic trigonometry (actua... |
64,646 | <p>In $\triangle{ABC}$, given $\angle{A}=80^\circ$, $\angle{B}=\angle{C}=50^\circ$, D is a point in $\triangle{ABC}$, which $\angle{DBC}=20^\circ,\angle{DCB}=40^\circ$. Then how to find find $\angle{DAC}$?</p>
<p>thanks.</p>
| K. Raghavendran | 32,872 | <p>I tried the geometric method but could not succeed. However the trigonometric route yields the result.
Here it is:
Extend $CD$ to meet $AB$ at $G$. Angle $B$ is $50$. Angle $BCD$ is $40$. So $CG$ is perpendicular to $AB$.
Let $AH$ be the right bisector of angle $A$ resting on $BC$ at $H$. Let the required angle $DAC... |
631,388 | <p>If $\lim_{n\rightarrow \infty }{a_n}=\alpha (\neq 0) $ and $\lim_{n\rightarrow \infty }{b_n}=\beta$, then $\lim_{n\rightarrow \infty }{a_n}^{b_n}=\alpha ^\beta $?</p>
<p>I unconsciously used this but I realized I'd never seen this theorem before. Is it true?</p>
| Community | -1 | <p>Notice that
<span class="math-container">$$a_n^{b_n}=e^{b_n\log(a_n)}$$</span>
so by the <strong>continuity of the exponential and logarithmic functions</strong> you have the result, of course with the <strong>assumption</strong> <span class="math-container">$\boldsymbol{a_n>0}$</span> and <span class="math-conta... |
2,227,047 | <p>For any $x=x_1, \dotsc, x_n$, $y=y_1, \dotsc, y_n$ in $\mathbf E^n$, define $\|x-y\|=\max_{1 \le k \le n}|x_k-y_k|$. Let $f\colon\mathbf E^n \to \mathbf E^n$ be given by $f(x)=y$, where $y_k= \sum_{i=1}^n a_{ki} x_i + b_k$ where $k =1,2, \dotsc,n$. Under what conditions is $f$ a contraction mapping?</p>
<p>Any hint... | David H | 55,051 | <hr>
<p>Suppose $\alpha\in\mathbb{C}\setminus\left(-\infty,0\right]$, and set $\left|\alpha\right|=:\rho\in\left(0,\infty\right)\land\arg{\left(\alpha\right)}=:\theta\in\left(-\pi,\pi\right)$. Given $x\in\mathbb{R}$, we have the following expression for the modulus of the complex expression $\frac{1}{1+\alpha\,x^{2}}$... |
7,108 | <p>I need help to make a diagram(square), someone can teach me how to do? </p>
<p>I know that I could look at the posts to see a model, but I am stopped for 7 days to edit questions</p>
<p>Thanks in advance.</p>
| apnorton | 23,353 | <p>I would suggest also posting a short/abbreviated (textual) answer. If I came across an answer that was simply a YouTube video, I would skip by the answer rather than upvoting. (This is not to say I wouldn't watch the video if I asked the question--rather, if I didn't ask the question, but just stumbled upon it.)</... |
724,900 | <p>Assuming $y(x)$ is differentiable. </p>
<p>Then, what is formula for differentiation ${d\over dx}f(x,y(x))$?</p>
<p>I examine some example but get no clue....</p>
| Michael Hoppe | 93,935 | <p>You're substituting $c(t)=(t,y(t))$ in $f$. Hence $(f\circ c)'(t)=\langle \nabla f(c(t),c'(t)\rangle$.</p>
|
7,025 | <p>Many commutative algebra textbooks establish that every ideal of a ring is contained in a maximal ideal by appealing to Zorn's lemma, which I dislike on grounds of non-constructivity. For Noetherian rings I'm told one can replace Zorn's lemma with countable choice, which is nice, but still not nice enough - I'd lik... | Richard Dore | 27 | <p>Unlike full choice, you probably use countable choice all over the place without even recognizing it. Every time you do something iteratively and then take some sort of limit to your construction, you're using countable choice. In many cases, if you do very careful bookkeeping, you can eliminate it on a case by case... |
634,890 | <blockquote>
<p><strong>Moderator Notice</strong>: I am unilaterally closing this question for three reasons. </p>
<ol>
<li>The discussion here has turned too chatty and not suitable for the MSE framework. </li>
<li>Given the recent pre-print of <a href="http://arxiv.org/abs/1402.0290" rel="noreferrer">T. Ta... | myw01 | 121,064 | <p>I have started to translate the paper so that English speakers can explore it. I've only had time for the abstract, introduction, and main result statement, but that already gives an important part of the picture. Any further contributions are welcome. <a href="https://github.com/myw/navier_stokes_translate">https:/... |
137,794 | <p>I'm plotting the electric field of a charged ring based a solution from Jackson's <em>Electrodynamics</em>. </p>
<p><em>Mathematica</em> handles <code>VectorPlot3D</code> and <code>SliceVectorPlot3D</code> for the field without a hitch, and <code>SliceContourPlot3D</code> of the field magnitude as well. </p>
<p>Ho... | Jason B. | 9,490 | <p>Extended comment here, since it doesn't answer the underlying question of <em>why does this happen</em>, hopefully someone else may have an answer about why you get those errors. This is a workaround I would use.</p>
<p>For something like this (where the computations seem to take forever and I don't know how long i... |
4,496,815 | <blockquote>
<p>For <span class="math-container">$n, m \in \mathbb{N}, m \leq n$</span>, let <span class="math-container">$P(n, m)$</span> denote the number of permutations of length <span class="math-container">$n$</span> for which <span class="math-container">$m$</span> is the first number whose position is left unch... | M1183 | 531,544 | <p>The function <span class="math-container">$f: \mathbb{R}_+ \rightarrow \mathbb{R}, x\mapsto f(x)=x+1/x$</span> is continuously differentiable for all <span class="math-container">$x$</span> in its domain. Its derivative is</p>
<p><span class="math-container">$$f'(x)=1-1/x^2.$$</span></p>
<p>Thus an extremum <span cl... |
70,429 | <p>For a $n$-dim smooth projective complex algebraic variety $X$, we can form the complex line bundle $\Omega^n$ of holomorphic $n$-form on $X$. Let $K_X$ be the divisor class of $\Omega^n$, then $K_X$ is called the canonical class of $X$.</p>
<p><strong>Question</strong>: Is homology class of $K_X$ in $H_{2n-2}(X)$ ... | Dmitri Panov | 943 | <p>It is well known that in dimension $3$ and higher there exist complex structures on diffeomerphic manifolds with totally different Chern classes (and Chern numbers).</p>
<p>For the case of complex manifolds you can check </p>
<p><a href="https://mathoverflow.net/questions/26586/can-one-bound-the-todd-class-of-a-3-... |
2,934,238 | <p>Let <span class="math-container">$a\in \mathbb{Q}$</span> such that <span class="math-container">$18a$</span> and <span class="math-container">$25a$</span> are integers, then we wish to prove that <span class="math-container">$a$</span> must be an integer itself. What that means is that <span class="math-container">... | José Carlos Santos | 446,262 | <p>All you know is that there are <em>some</em> <span class="math-container">$x$</span> and <span class="math-container">$y$</span> with that property, but that doesn't imply that you can take <span class="math-container">$x=y=a$</span>.</p>
<p>Note that <span class="math-container">$\gcd(18,25)=1$</span>. Therefore, ... |
201,820 | <p>Suppose we have in <code>~/time-data/time-data.org</code> the following data:</p>
<pre><code>* Parent1
:LOGBOOK:
CLOCK: [2019-07-09 Tue 00:00]--[2019-07-09 Tue 00:20] => 0:20
:END:
** Child1
:LOGBOOK:
CLOCK: [2019-07-10 Wed 00:02]--[2019-07-10 Wed 00:40] => 0:38
:END:
** Child2
:LOGBOOK:
CLOCK: [2019-07-11 ... | kglr | 125 | <pre><code>csv = "task,parents,category,start,end,effort,ishabit,tags
Parent1,,,2019-07-07 00:00,2019-07-07 00:20,,,
Child1,Parent1,,2019-07-8 00:02,2019-07-8 00:40,,,
Child2,Parent1,,2019-07-9 00:02,2019-07-9 06:40,,,
Parent2,,,2019-07-08 00:00,2019-07-08 00:20,,,
Child21,Parent2,,2019-07... |
3,858,414 | <p>I need help solving this task, if anyone had a similar problem it would help me.</p>
<p>The task is:</p>
<p>Calculate using the rule <span class="math-container">$\lim\limits_{x\to \infty}\left(1+\frac{1}{x}\right)^x=\large e $</span>:</p>
<p><span class="math-container">$\lim_{x\to0}\left(\frac{1+\mathrm{tg}\: x}{... | QED | 91,884 | <p>There are only two possibilities: either <span class="math-container">$[x]\cap[y]=\emptyset$</span> or <span class="math-container">$[x]\cap[y]\neq\emptyset$</span>. Now <span class="math-container">$[x]\cap[y]\neq\emptyset$</span> means that there exists an element <span class="math-container">$z\in[x]\cap[y]$</spa... |
10,468 | <p>I know that many graph problems can be solved very quickly on graphs of bounded degeneracy or arboricity. (It doesn't matter which one is bounded, since they're at most a factor of 2 apart.) </p>
<p>From Wikipedia's article on the clique problem I learnt that finding cliques of any constant size k takes linear time... | Carter Tazio Schonwald | 426 | <p>I think you might want to look at the related/(same?) concept of treewidth. Its a much stronger sparseness requirement than constant degree, planar, etc, and if you have something like $\mathcal O(\log n)$ tree width, many NP-Hard problems on graphs become easy (such as computing graph cutes). Unfortunately in gene... |
1,412,594 | <p>I'm a statistics teacher at a college. One day a student came with a doubt about an exercise about probability. The text goes like this:</p>
<blockquote>
<p>A person has two boxes $A$ and $B$. In the first one has $4$ white balls and $5$ black balls and in the second has $5$ white balls and $4$ black balls. This ... | joriki | 6,622 | <p>The calculation on the right-hand side is correct; just the notation is bad, because as you say the right-hand factors in both terms are conditional probabilities. A better way to write this would be</p>
<p>$$
P(\text{C})=P(\text{W})+P(\text{Bl})= P(\text{Wb1})P(\text{Wb2}\mid\text{Wb1})+P(\text{Blb1})P(\text{Blb2}... |
1,412,594 | <p>I'm a statistics teacher at a college. One day a student came with a doubt about an exercise about probability. The text goes like this:</p>
<blockquote>
<p>A person has two boxes $A$ and $B$. In the first one has $4$ white balls and $5$ black balls and in the second has $5$ white balls and $4$ black balls. This ... | heropup | 118,193 | <p>As has already been pointed out, the precise and careful choice of notation is paramount.</p>
<p>The event $C$ of choosing the same color is the disjoint union of two separate events, so it is better to choose the following notation: Let $(b_1, b_2)$ be the random outcome of the two ball draws in order, where $b_i... |
4,531,652 | <p>In my school book, I read this theorem</p>
<blockquote>
<p>Let <span class="math-container">$n>0$</span> is an odd natural number (or an odd positive integer), then the equation <span class="math-container">$$x^n=a$$</span> has exactly one real root.</p>
</blockquote>
<p>But, the book doesn't provide a proof, onl... | user | 505,767 | <p>We can assume wlog <span class="math-container">$x$</span> and <span class="math-container">$a$</span> both positive such that <span class="math-container">$x^n=a$</span> indeed</p>
<p><span class="math-container">$$x^n=a \iff (-x)^n =(-1)^nx^n =-a$$</span></p>
<p>Then assume by contradiction <span class="math-conta... |
664 | <p>Erdős's 1947 probabilistic trick provided a lower exponential bound for the Ramsey number $R(k)$. Is it possible to explicitly construct 2-colourings on exponentially sized graphs without large monochromatic subgraphs?</p>
<p>That is, can we explicitly construct (edge) 2-colourings on graphs of size $c^k$, for some... | user1347 | 1,347 | <p>As was mentioned in the previous answers, the answer is no. Or more accurately I'd say that the answer is <em>currently no</em>, but possibly yes. </p>
<p>Also, consider the related question of constructing a bipartite graph with parts of size $2^n$, which contains no $K_{k,k}$ and whose complement contains no $K_{... |
175,723 | <p>I am reading Goldstein's Classical Mechanics and I've noticed there is copious use of the $\sum$ notation. He even writes the chain rule as a sum! I am having a real hard time following his arguments where this notation is used, often with differentiation and multiple indices thrown in for good measure. How do I get... | robjohn | 13,854 | <p>This demonstrates what Robert Israel suggests.</p>
<p>Suppose that
$$
\theta^3+11\theta-4=0
$$
and
$$
\alpha=\frac{\theta^2-\theta}{2}
$$
Then
$$
\begin{align}
\alpha^0&=\frac22\\
\alpha^1&=\frac{\theta^2-\theta}{2}\\
\alpha^2&=\frac{-5\theta^2+13\theta-4}{2}\\
\alpha^3&=\frac{19\theta^2-107\theta+3... |
2,128,182 | <p>I've been looking for a definition of game in game theory. I'd like to know if there is a definition shorter than that of Neumann and Morgenstern in <em>Theory of Games and Economic Behavior</em> and not so vague like "interactive decision problem" or "situation of conflict, or any other kind of interaction". I've s... | Anna SdTC | 410,766 | <p>This definition is from Osborne and Rubinstein, "A Course in Game Theory", section 1.1:</p>
<blockquote>
<p>Game theory is a bag of analytical tools designed to help us understand
the phenomena that we observe when decision-makers interact. The basic assumptions that underlie the theory are that decision-makers... |
3,809,127 | <blockquote>
<p>Determine if the sequence <span class="math-container">$x_k \in \mathbb{R}^3$</span> is convergent when <span class="math-container">$$x_k=(2, 1, k^{-1})$$</span></p>
</blockquote>
<p>Our professor gave a hint that one should look at <span class="math-container">$||2k-a||$</span> and try to find a contr... | Batominovski | 72,152 | <p>Write <span class="math-container">$f(\pi):=\sum\limits_{i=1}^n\,\big|\pi(i)-i\big|$</span> for each <span class="math-container">$\pi\in S_n$</span>. Decompose a permutation <span class="math-container">$\pi\in S_n$</span> as a product of disjoint cycles <span class="math-container">$$\pi=\gamma_1\gamma_2\cdots\ga... |
2,211,075 | <p>I don't understand the following example from Math book.</p>
<p>Solve for the equation <code>sin(theta) = -0.428</code> for <code>theta</code> in <code>radians</code> to 2 decimal places. where <code>0<= theta<= 2PI</code>.</p>
<p>And this is the answer:</p>
<p><code>theta=-0.44 + 2PI = 5.84rad and theta = ... | Community | -1 | <p>If the limits exist and the denominator is nonzero, then</p>
<p>$$ \lim_{x \to a} \frac{ \frac{f(x) - f(a)}{x-a} }{g(x) }
= \frac{\lim_{x \to a} \frac{f(x) - f(a)}{x-a}}{\lim_{x \to a} g(x)} $$</p>
<p>and so you could conclude</p>
<p>$$ \lim_{x \to a} \frac{ \frac{f(x) - f(a)}{x-a} }{g(x) } = \frac{f'(a)} {\lim_{... |
2,747,578 | <p>Let $S,T$ be sets with $|S|>|T|$ and $R$ some relations on $T$.<br>
Why is then $\langle S|-\rangle$ not isomorphic to $\langle T|R\rangle$ </p>
<p>This came up when I wanted to solve a different problem, <a href="https://math.stackexchange.com/q/2747383/506844">which I also asked on this site</a>. Unfortunately... | Hagen von Eitzen | 39,174 | <p>For completeness, the following includes a proof (admittedly, using the axiom of choice) of the closely related fact that free groups are isomorphic only if they are over sets of the sae cardinality. </p>
<p>Assume $\phi\colon\langle T\mid R\rangle\to \langle S\rangle $ is an isomorphism (or just an epimorphism). ... |
2,823,758 | <p>I was learning the definition of continuous as:</p>
<blockquote>
<p>$f\colon X\to Y$ is continuous if $f^{-1}(U)$ is open for every open $U\subseteq Y$</p>
</blockquote>
<p>For me this translates to the following implication:</p>
<blockquote>
<p>IF $U \subseteq Y$ is open THEN $f^{-1}(U)$ is open</p>
</blockq... | Ittay Weiss | 30,953 | <p>The definition of continuity at a point $a$ for a function $f\colon A\to B$ (say between metric spaces) is: for all $\varepsilon >0$ there exists $\delta>0$ such that if $d(x,a)<\delta$, then $d(fx,fa)<\varepsilon$. Now, notice that the $\varepsilon$ is used for a condition in the codomain and the $\delt... |
2,823,758 | <p>I was learning the definition of continuous as:</p>
<blockquote>
<p>$f\colon X\to Y$ is continuous if $f^{-1}(U)$ is open for every open $U\subseteq Y$</p>
</blockquote>
<p>For me this translates to the following implication:</p>
<blockquote>
<p>IF $U \subseteq Y$ is open THEN $f^{-1}(U)$ is open</p>
</blockq... | John Bollinger | 404,964 | <blockquote>
<p>I would have expected the definition to be the other way round</p>
</blockquote>
<p>I take you to be proposing this:</p>
<blockquote>
<p>$f\colon X\to Y$ is continuous if $f(U)$ is open for every open $U\subseteq X$</p>
</blockquote>
<p>But that does not serve. In particular, consider constant f... |
205,671 | <p>How would one go about showing the polar version of the Cauchy Riemann Equations are sufficient to get differentiability of a complex valued function which has continuous partial derivatives? </p>
<p>I haven't found any proof of this online.</p>
<p>One of my ideas was writing out $r$ and $\theta$ in terms of $x$ a... | James S. Cook | 36,530 | <p><em>I happen to have some notes on this question. What follows here is the usual approach, it's just multivariate calculus paired with the Cauchy Riemann equations. I have an idea for an easier way, I'll post it as a second answer in a bit if it works.</em></p>
<p>If we use polar coordinates to rewrite $f$ as follo... |
205,671 | <p>How would one go about showing the polar version of the Cauchy Riemann Equations are sufficient to get differentiability of a complex valued function which has continuous partial derivatives? </p>
<p>I haven't found any proof of this online.</p>
<p>One of my ideas was writing out $r$ and $\theta$ in terms of $x$ a... | user55789 | 200,806 | <p>I understand this topic is somewhat old, but I still feel like I can contribute to it with a derivation in my eyes much simpler.</p>
<p>The key is to understand the <em>goal</em>: we want to derive some analogue of the CR-equations, except in polar form. We therefore wish to relate $u_\theta$ with $v_r$ and $v_\the... |
2,214,030 | <p>$\mathbb{R}^{13}$ has two subspaces such that dim(S)=7 and dim(T)=8 <br/></p>
<p>⒜ max dim (S∩T)=?<br/>
⒝ min dim (S∩T)=?<br/>
⒞ max dim (S+T)=?<br/>
⒟ min dim (S+T)=?<br/>
⒠ dim(S∩T) + dim (S+T)=?</p>
| Benjamin Dickman | 37,122 | <p>The absolute value is greater than <strong>or equal to</strong> zero.</p>
<p>The formal definition of a limit is written precisely to work outside of the case in which $x = a$.</p>
<p>E.g. consider an example of a function $f(x) = 0$ for all $ x\neq 0$, $f(0) = 1$. Because we are not considering $f(0)$ for $a=0$, ... |
121,362 | <p>I have a set of sample time-series data below of monthly prices for two companies. </p>
<p>Q1. I want to calculate monthly and quarterly log returns.what is the most expedient way to do this? <code>TimeSeriesAggregate[]</code> only has the standard <code>Mean</code>, etc. </p>
<p>Q2. With the returns from Q1, wha... | Mr.Wizard | 121 | <p>I am late to see this question but here is a solution closely based on my answer to <a href="https://mathematica.stackexchange.com/q/7360/121">Creating a Sierpinski gasket with the missing triangles filled in</a>.</p>
<pre><code>tri[n_] :=
Table[{2 j - i, Sqrt[3] i}, {i, 0, n}, {j, i, n}] //
Partition[Riffle... |
121,362 | <p>I have a set of sample time-series data below of monthly prices for two companies. </p>
<p>Q1. I want to calculate monthly and quarterly log returns.what is the most expedient way to do this? <code>TimeSeriesAggregate[]</code> only has the standard <code>Mean</code>, etc. </p>
<p>Q2. With the returns from Q1, wha... | kglr | 125 | <p>Using the trick in <a href="https://mathematica.stackexchange.com/a/61807/125">this answer</a> to use <code>MeshFunctions</code> and <code>Dynamic</code> <code>MeshShading</code> with random colors:</p>
<pre><code>coloredTriangles = ParametricPlot[{x, y Sqrt[3] Min[x, 2 - x]}, {x, 0, 2}, {y, 0, 1},
MeshFuncti... |
26,893 | <p>Does there exists a function $f \in C^2[0,\infty]$ (that is, $f$ is $C^2$ and has finite limits at $0$ and $\infty$) with $f''(0) = 1$, such that for any $g \in L^p(0,T)$ (where $T > 0$ and $1 \leq p < \infty$ may be chosen freely) we get
$$
\int_0^T \int_0^\infty \frac{u^2-s}{s^{5/2}} \exp{\left( -\f... | Willie Wong | 1,543 | <p>I'm pretty sure the answer is no, there exists no such $f$. Here I first give a physical argument. </p>
<p>First define the function $K(u,s) = \frac{1}{\sqrt{s}} \exp (-u^2 / 2s )$. Up to some constant normalisation factors, this is the <a href="http://en.wikipedia.org/wiki/Heat_equation#Fundamental_solutions" rel=... |
936,138 | <p>I need help approaching a proof which deals with inequalities:</p>
<p>If p and r are the precision and recall of a test, then the F1 measure of the test is
defined to be
$$F(p, r) = \frac{2pr}{p+r}$$</p>
<p>Prove that, for all positive reals p, r, and t, if t ≥ r then F(p, t) ≥ F(p, r)</p>
<p>What's the first ste... | André Nicolas | 6,312 | <p>A standard way is to look at $F(p,t)-F(p,r)$, which is
$$\frac{2pt}{p+t}-\frac{2pr}{p+r}.$$
Bring to a common denominator and simplify. We get
$$\frac{2pt(p+r)-2pr(p+t)}{(p+t)(p+r)},$$
which simplifies to
$$\frac{2p^2(t-r)}{(p+t)(p+r)}.$$
This is clearly $\ge 0$. </p>
|
4,242,765 | <p>Let <span class="math-container">$X_k = 1$</span> with probability <span class="math-container">$0.5$</span> and <span class="math-container">$X_k = -1$</span> with probability 0.5, and let <span class="math-container">$X_k$</span> be independent random variables <span class="math-container">$k = (1,2,...,n)$</span>... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$$\lim_{t \to 0} \frac {\ln (\cos (\omega \sqrt t)} t$$</span> <span class="math-container">$$=\lim_{t \to 0} \frac {-\omega \sin (\omega (\sqrt t)\frac 1 2 t^{-1/2}} {{\cos (\omega (\sqrt t)}}$$</span> <span class="math-container">$$=-\frac {\omega^{2}} 2$$</span> using the fact that ... |
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