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 |
|---|---|---|---|---|
136,121 | <p>I would like an efficient algorithm for square root of a positive integer. Is there a reference that compares various square root algorithms?</p>
| Dan Piponi | 1,233 | <p>If you want the square root of an odd perfect square then one approach is to find the square root 2-adically. </p>
<p>Suppose you want the square root of $n$. You can use the usual Newton-Raphson algorithm for the equation $f(x)=x-n/x=0$, working modulo $2^N$, for some large enough $N$. That's the iteration $x \rig... |
2,110,079 | <p>Is it true that every infinite finitely generated group contains an element of infinite period?</p>
<p>I think it is true, but I can't prove it.</p>
| Dietrich Burde | 83,966 | <p>The <a href="http://en.wikipedia.org/wiki/Grigorchuk_group">Grigorchuk group</a> is finitely generated, and every element has finite order, but it is still infinite. </p>
|
450,142 | <p>the equation $x^3-5x+1=0$ has a root in $(0,1)$. Using a proper sequence for which $$|a(n+1)-a(n)|\le c|(a(n)-a(n-1)|$$ with $0<c<1$ , find the root with an approximation of $10^{-4}$.</p>
| Tunococ | 12,594 | <p>You can attempt this: define a sequence $x_n$ by
$$
x_{n+1} = \frac 15\left(1 + x_n^3\right).
$$
We know that if $x_1 \in (0, 1)$, then $x_n \in (0, 1)$ for all $n$. Next, we show that it satisfies the said condition:
\begin{align*}
x_{n+1} - x_n & = \frac 15(1 + x_n^3 - 5x_n)\\
& = \frac 15\left(1 + x_n^3 -... |
36,792 | <p>I need to play with a lot of powers such as <code>10^-3</code>. <code>1E-3</code> does not work for it. Is there any short form for it?</p>
| Community | -1 | <p>I'm surprised there isn't a question about this (<em>i.e.</em> entering numbers in scientific notation) already.</p>
<p>To enter $3\times10^{-3}$, you can write <code>3*^-3</code>.</p>
<p>For further reference, see <a href="http://reference.wolfram.com/language/tutorial/InputSyntax.html#7977" rel="noreferrer">Inpu... |
36,792 | <p>I need to play with a lot of powers such as <code>10^-3</code>. <code>1E-3</code> does not work for it. Is there any short form for it?</p>
| Ratch | 44,601 | <p>Sure, easy.</p>
<pre><code>100,000 = 1*^5
1/10 = 1*^-1
6/10 = 6*^-1
3200 = 32 *^ 2
</code></pre>
<p>Try it and see!</p>
|
3,928,842 | <p>Let <span class="math-container">$B$</span> and <span class="math-container">$W$</span> be two independent Brownian motions on <span class="math-container">$(\mathcal{F}_{t})$</span>. Is <span class="math-container">$\tau$</span> defined as:
<span class="math-container">$$\tau = \inf \{ t \ge 0: B_{t} \ge W_{t} + e^... | the_candyman | 51,370 | <p>If <span class="math-container">$A$</span> is invertible, then:</p>
<p><span class="math-container">$$AB = AC \Rightarrow A^{-1}AB = A^{-1}AC \Rightarrow B = C.$$</span></p>
<p>If <span class="math-container">$A$</span> is not invertible, <span class="math-container">$AB = AC \Rightarrow B =C $</span> does not work.... |
4,204,506 | <p>We can write <span class="math-container">$4x^2+8x+5$</span> in the form <span class="math-container">$a(x+b)^2+c$</span> as <span class="math-container">$4(x+1)^2+1$</span>. However, the question I am doing asks me to write it in the form <span class="math-container">$(ax+b)^2+c$</span>. How do I change it to that ... | Jose Arnaldo Bebita Dris | 28,816 | <p><strong>Hint:</strong> <span class="math-container">$$4(x + 1)^2 = \bigg(2(x + 1)\bigg)^2$$</span>
where I have used the identity <span class="math-container">$u^2 v^2 = (uv)^2$</span> as <a href="https://math.stackexchange.com/u/483139">coffeemath</a> has noted.</p>
<p>Can you finish?</p>
|
2,026,143 | <p>What is need to be proven, in a proof by contradiction that a set is closed?</p>
<p>If we have to show that $K$ is closed that mean that we need to show that $K^{C}$ is open.
Let there be $x\in K^{C}$ we need to show that $B(x,\delta)\subset K^{C}$</p>
<p>What is the contrary assumption?</p>
<p>For all $\delta>... | marco2013 | 79,890 | <p>$\exists x \notin K, \forall \delta>0, B(x, \delta) \cap K \neq \emptyset$</p>
|
1,367,966 | <p>How do you find the exact values of the following without using a calculator?</p>
<p>$$\tan(105^\circ) \qquad \tan(11\pi/12)$$</p>
| th ie | 255,773 | <p>$$\tan 105^\circ = \tan (90+15)^\circ = -\cot 15^\circ = -1/\tan 15^\circ$$</p>
<p>Now, $\tan 2x = 2\tan x / (1-\tan^2 x)$</p>
<p>Put $x=15^\circ$ in the above equation and calculate $\tan 15^\circ$ from here and put in
$$\tan 105^\circ = -1/\tan15^\circ$$</p>
<p>Do the same thing for your other question. </p>
|
1,830,799 | <p>1There are $\frac{21!}{2!3!} = 120$ total positions (disregarding order within same colour). I imagine labelling the people Y (yellow) and NY (not yellow), so I imagine I have $4$ copies of the letter Y and $5$ of NY. So I draw out $9$ slots and want to arrange so that at least $2$ Y are together.
I get $30$ arrang... | true blue anil | 22,388 | <p>To count <em>unfavorable</em> ways, arrange the reds and blues in $\dfrac{5!}{3!2!} = 10$ ways,<br>
and insert the yellows at the uparrows in $\binom64 = 15$ ways</p>
<p>$\uparrow\fbox{red or blue}\uparrow\fbox{red or blue}\uparrow\fbox{red or blue}\uparrow\fbox{red or blue}\uparrow\fbox{red or blue}\uparrow$</p>
... |
1,040,505 | <p>Apparently,
$$(1-\cot 37^\circ)(1-\cot 8^\circ)=2.00000000000000000\cdots$$<br>
Since it is a $2.0000000000\cdots$ instead of $2$, it isn't exactly $2$.<br>
Why is that?</p>
| Simon S | 21,495 | <p>Alternatively,
$$\cot(x+y) = \frac{\cos(x+y)}{\sin(x+y)} = \frac{\cos x \cos y - \sin x \sin y}{\sin x \cos y + \cos x \sin y} = \frac{\cot x \cot y - 1}{\cot x + \cot y}$$</p>
<p>Now if $x = 37^\circ$, $y = 8^\circ$, then $\cot(x + y) = \cot 45^\circ = 1$ and</p>
<p>$$\begin{align} \ \ \ \ 1 & = \frac{\cot ... |
945,489 | <p>Let $A \subset X$ be a subspace of $X$. Recall that a <strong>retraction</strong> of $X$ onto $A$ is a continuous map $r: X \to A$ such that $r(a) = a$ for every $a \in A$.</p>
<p>Let $X = \bf R$ endowed with the standard topology, and let $A$ be the closed interval $[0,1]$ endowed with the subspace topology. Clear... | Stefan Hamcke | 41,672 | <p>There exists a retraction $r:\Bbb R→I$ where $I=[0,1]$
$$r(x)=\begin{cases}
0,&\text{if }x\le0\\
x,&\text{if }0\le x\le1 \\
1,&\text{if }x\ge1
\end{cases}$$
We could also write $r=\min(1,\max(0,x))$.</p>
|
1,531,154 | <p>A question from my calculus book states,</p>
<blockquote>
<p>Which points on the graph $y=4-x^2$ are the closest to the point (0,2)?</p>
</blockquote>
<p>Using some of my notes, I have a formula as follows (not sure what it's actually called):</p>
<p>$$d=\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$</p>
<p>So I plugged... | Patrick Feltes | 195,257 | <p>This is a classic optimization problem where you are trying to minimize the distance from the point $(0, 2)$. Given the function $$f(x) = 4 - x^2$$, we can represent all of the points on that curve as, $(x, 4 - x^2)$. Now, we can use the distance formula: $$d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$. When we plug in $(x, ... |
57,914 | <p>I have the following code to show a red area defined by inequalities:</p>
<pre><code>ClearAll["Global`*"];
p = Reduce[y <= 3/10 x + 18 && y > x^2/8, {x, y}]
r = RegionPlot[p, {x, -15, 18}, {y, -5, 25},
GridLines -> {Table[i, {i, -15, 18}], Table[j, {j, -5, 25}]},
PlotStyle -> Directive[{... | b.gates.you.know.what | 134 | <p>You can do :</p>
<pre><code>p = ImplicitRegion[y <= 3/10 x + 18 && y > x^2/8, {x, y}]
points = Reduce[Element[{x, y}, p], {x, y}, Integers]
pp = Cases[points, x == xx_ && y == yy_ -> {xx, yy}]
pp // Length
(* 286 *)
Show[RegionPlot[p], ListPlot[pp]]
</code></pre>
<p><img src="https://... |
1,736,405 | <p>I was told to consider the degrees but I'm not sure how the degrees of the polynomial so can help me here. </p>
| Mathmo123 | 154,802 | <p>I'm guessing the intend solution was this:</p>
<ul>
<li>Clearly $\mathbb Q[\sqrt{p},\sqrt[3]p]\subset \mathbb Q[\sqrt[6]p]$.</li>
<li>One can check that $\mathbb Q[\sqrt[6]p]$ is a degree $6$ extension of $\mathbb Q$.</li>
<li>By considering the subfields $\mathbb Q[\sqrt{p}]$, $\mathbb Q[\sqrt[3]{p}]$ of $\mathbb ... |
250,434 | <p>I'm stuck on a couple of practice problems relating to PIDs, they are paraphrased below:</p>
<p>Given a PID $R$ with $a$ and $b$ in $R$ and gcd$(a,b)=1$ I need to show that:</p>
<p>1) There are elements $s$ and $t$ of $R$ such that $sa+tb=1$</p>
<p>2) The $R$-module $R/(a) \bigoplus R/(b)$ is isomorphic to the $R... | Henry | 6,460 | <p>In terms of upper and lower cases, I suspect you saw $$f_{X+Y}(z)=\int_0^z f_X(x)f_Y(z-x)dx$$ as the convolution of two non-negative random variables. </p>
<p><em>With Y Uniform on $[0,2]$:</em></p>
<p>Taking account of where $f_X$ and $f_Y$ are non-zero, I would write $$f_{X+Y}(z)=\int_{\max(0,z-2)}^{\min(1,z)}... |
2,963,255 | <p>I'm very confused by studying a book on linear algebra. The task couldn't be more simple:</p>
<p>Line P is described by the following two points:</p>
<p>P1=(1,0,1) and P2=(1,1,1)</p>
<p>Line Q is described by the following two points:</p>
<p>Q1=(0,1,0) and Q2=(1,2,1)</p>
<p>We need to find the point of intersec... | Dr. Sonnhard Graubner | 175,066 | <p>I got the System
<span class="math-container">$$1=t$$</span>
<span class="math-container">$$s=1+t$$</span>
<span class="math-container">$$1=t$$</span>
so we get</p>
<p><span class="math-container">$$s=2$$</span> and our intersection point is given by <span class="math-container">$$[1;2;1]$$</span></p>
|
2,963,255 | <p>I'm very confused by studying a book on linear algebra. The task couldn't be more simple:</p>
<p>Line P is described by the following two points:</p>
<p>P1=(1,0,1) and P2=(1,1,1)</p>
<p>Line Q is described by the following two points:</p>
<p>Q1=(0,1,0) and Q2=(1,2,1)</p>
<p>We need to find the point of intersec... | mfl | 148,513 | <blockquote>
<p>Any point on line P can be described by the following equation: <span class="math-container">$(1,0,1)+x(1,1,1)$</span>.</p>
</blockquote>
<p>This is not correct because the line contains the points <span class="math-container">$(1,0,1)$</span> and <span class="math-container">$(1,1,1).$</span> Note t... |
26,313 | <p>In page 21 of <em>A Problem seminar</em>, D. J. Newman presents a novel way (at least for me) to determine the expectation of a discrete random variable. He refers to this expression as the <strong>failure probability formula</strong>. His formula goes like this</p>
<p>$f_{0}+f_{1}+f_{2}+\ldots$</p>
<p>where $f_{n... | José Figueroa-O'Farrill | 394 | <p>The answer to the question in the title is affirmative. In the <a href="http://arxiv.org/abs/hep-th/9607161v1" rel="nofollow">Dictionary of Lie superalgebras</a>, there is an entry on the Weyl group of a classical Lie superalgebra. It is generated by reflections associated to the simple even roots, hence it is the... |
294,756 | <p>$$\nabla\cdot(u\times v)=v\cdot(\nabla\times u)-u\cdot(\nabla\times v)$$</p>
<p>How can we go about proving the above vector calculus identity component wise?</p>
| Community | -1 | <p>There's a straightforward algorithm for this. Allow me to demonstrate on</p>
<p>$$ 0.1\overline{37} = 0.1373737373737\ldots$$</p>
<p>The first step is to split off the non-repeating part:</p>
<p>$$ 0.1\overline{37} = 0.1 + 0.0\overline{37} $$</p>
<p>then you pull out the 'common factor' from the repeated part</p... |
294,756 | <p>$$\nabla\cdot(u\times v)=v\cdot(\nabla\times u)-u\cdot(\nabla\times v)$$</p>
<p>How can we go about proving the above vector calculus identity component wise?</p>
| Ross Millikan | 1,827 | <p>You can compute it. Multiply the decimal by $10^n$, where $n$ is the number of digits in the repeat of the decimal and subtract. Example: given $ x=5.181818\ldots \\100x=518.1818\ldots,\\99x=513,\\x=\frac{513}{99}$</p>
|
51,552 | <p>I am not sure if what I want to do is possible in <em>Mathematica</em>. I've provided something simple below to convey what I want to do.
Basically what I want to happen is that when the <code>Locator</code> crosses over the blue line, the <code>Disk[]</code> will turn red and stay red until the <code>Locator</code... | kglr | 125 | <p>Partial answer: this does not handle the going-around-the-curve bit.</p>
<pre><code> g1 = Graphics[{Blue, Table[Circle[{0, 0}, i], {i, 3}]}, ImageSize -> 20];
g2 = Graphics[{Red, Table[Circle[{0, 0}, i], {i, 3}]}, ImageSize -> 20];
DynamicModule[{pt = {0.1, 1}},
Plot[x^3, {x, 0, 1}, PlotRange -> {{... |
927,480 | <p>Consider a univariate function $f(x)$. I know the graphical intuition behind why $f'(x)=0$ at the extrema of $f$. But how do you prove it mathematically? </p>
<p>I start with the assumption of $x^*$ being a minimum (the maximum case can be proved likewise), then
$f(x^*+h) \geq f(x^*)$ where $h\in \mathcal{N}(x)$, $... | cuppajoeman | 571,643 | <p>Let <span class="math-container">$n \in \mathbb{N}$</span>, suppose <span class="math-container">$n$</span> requires <span class="math-container">$d$</span> digits in it's base 2 representation, we'd like to show that <span class="math-container">$d = \lfloor \log_2(n) \rfloor+1$</span>.</p>
<p>At most we have <spa... |
3,005,589 | <p>Let <span class="math-container">$R$</span> be a <em>commutative</em> ring with <span class="math-container">$1_R$</span>.</p>
<p>I found the following theorem as an equivalent definition of <span class="math-container">$R-$</span>modules.</p>
<blockquote>
<p><strong>Theorem.</strong> An abelian group <span clas... | egreg | 62,967 | <p>An abelian group <span class="math-container">$M$</span> can be given the structure of <span class="math-container">$R$</span>-module if and only if there exists a ring homomorphism
<span class="math-container">$$
\varphi\colon R\to\operatorname{End}_{\mathbb{Z}}(M)
$$</span>
If <span class="math-container">$M$</spa... |
282,541 | <p>Are there any special (nontrivial) properties of $\mathbb{R}^3$ that distinguish it from any other $\mathbb{R}^n$? If there are, what are some of the important ones?</p>
| aschepler | 2,236 | <p>Not quite a unique property, but close:</p>
<p>A non-trivial vector <a href="http://en.wikipedia.org/wiki/Seven-dimensional_cross_product#Definition">cross product</a> can be defined only in $\mathbb{R}^3$ and $\mathbb{R}^7$.</p>
|
2,895,481 | <p>Consider a one-dimension $10$-step random walk with step size $1$, namely $S_n=\sum_{n=1}^{10} X_n$ and $X_i = 1$ or $-1$ for all $i\in\{1,2,...,10\}$. Further we require $S_i \geq 0$ for any $i$. And we have $S_{10} = 4$. How many paths satisfy this condition?</p>
<p>Actually it is a counting problem. It troubles ... | lulu | 252,071 | <p>Method I (brute force): Clearly we must have seven $+1's$ and three $-1's$. Also clearly, $X_1=1$. If the third $+1$ occurs after the third $-1$ then we violate positivity, so the third $+1$ can only occur at $X_3,X_4,X_5$. Easy to count each case. </p>
<p>If it occurs at $X_3$ all subsequent paths are good, ... |
578,938 | <p>Induction step: We assume that P(k) is true and then we need to show that P(k+1) is true as well.</p>
<p>If k is arbitrary and we assume it's correct, then how come one can't say, </p>
<p>j = (k+1)</p>
<p>and assume p(j) is true because j is arbitrary just like k and it has the same form of p(k).</p>
<p>That log... | Bombyx mori | 32,240 | <p>I think you can check Curtis Mcmullen's notes on a detailed explanation of normal approximation. I read this about 2 years ago and do not recall all details. Here is the link:</p>
<p><a href="http://www.math.harvard.edu/~ctm/home/text/class/harvard/154/11/html/home/course/course.pdf" rel="nofollow">http://www.math.... |
4,318,497 | <p>I'm having trouble finding the line integral of this problem. I have been given a vector field</p>
<p><span class="math-container">$F=(2x\sin(\pi y)-e^z,\pi x^2\cos(\pi y)-3e^z,-xe^z)$</span></p>
<p>Where the curve <span class="math-container">$C$</span> intercepts between <span class="math-container">$z=\ln(1+x)$</... | Math Lover | 801,574 | <p>The given vector field is <span class="math-container">$ \vec F=(2x\sin(\pi y)-e^z,\pi x^2\cos(\pi y)-3e^z,-xe^z)$</span>.</p>
<p><span class="math-container">$r(t)= (t, t, \ln(1+t)), 0 \leq t \leq 1$</span>. As called out in the other answer you have a mistake in the z-component.</p>
<p>You are correct that the vec... |
2,269,874 | <p>I have been learning about the Fibonacci Sequence in school and came across the interesting property: $$\gcd(F_m, F_n) = F_{gcd(m, n)}$$ where $F_m$ and $F_n$ are the $mth$ and $nth$ number from the Fibonacci sequence.</p>
<p>Apparently, this is the defining property of a Strong Divisibility Sequence. I've been sea... | Steven Alexis Gregory | 75,410 | <p>Every Jaccard Similarity is going to look like</p>
<p>$$\dfrac{n_1 + n_2 + n_3 + \cdots + n_N}{d_1 + d_2 + d_3 + \cdots + d_N}$$
where $2n_i \le d_i$ for each $i$.</p>
<p>So $$d_1 + d_2 + d_3 + \cdots + d_N \ge 2(n_1 + n_2 + n_3 + \cdots + n_N)$$.</p>
<p>Hence
$$\dfrac{n_1 + n_2 + n_3 + \cdots + n_N}{d_1 + d_2 + ... |
3,493,871 | <p>The problem is the following:</p>
<p>Let <span class="math-container">$\mathcal{F}$</span> be a family of distinct proper subsets of {1,2,...,n}. Suppose that for every <span class="math-container">$1\leq i\neq j\leq n$</span> there is a unique member of <span class="math-container">$\mathcal{F}$</span> that contai... | Community | -1 | <p><strong>A parital solution:</strong></p>
<p>Let <span class="math-container">$n \in \mathbb{N}$</span> and let <span class="math-container">$\mathcal{F}$</span> such that <span class="math-container">$\forall i,j \in [n]\exists S\in \mathcal{F}:i,j\in S$</span> and that <span class="math-container">$S$</span> is un... |
1,182,429 | <p>Take for instance the following problem. You have two beakers of the same height. One has tick marks that break it into thirds. The other has tick marks that separate it into fourths. The water levels are 1/3 and 1/4 respectively. If I did not know about the concept of LCDs, how would I figure out how much water the... | Wishwas | 19,992 | <p>You are looking for numbers say $x$ and $y$ such that: $$x \times \left(\frac{1}{3} + \frac{1}{4}\right)$$ i.e. $$\frac{x}{3} + \frac{x}{4} = y,$$ where $y$ is an integer. Assuming you do not know about LCM, you will try numbers $x = 1, 2, 3, \ldots$ and $x = 12$ will be the first number for which you will get an in... |
3,449,808 | <p>I am having a tough time solving this problem because I cannot seem to picture this problem. I don't know how the picture in this problem is supposed to look like and therefore I don't know where to start.</p>
<p><a href="https://i.stack.imgur.com/W6LdN.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur... | John Hughes | 114,036 | <p><span class="math-container">$x = 266; y = 2$</span> gives a solution, so evidently your current solution (to the most weirdly edited problem ever!) is wrong. </p>
|
4,157,630 | <p>Compute the factor group: <span class="math-container">$\mathbb{Z} \times \mathbb{Z} \big / \langle (1,2) \rangle$</span></p>
<p>I don't understand why the answer is <span class="math-container">$\mathbb{Z}$</span>.
Isn't <span class="math-container">$\mathbb{Z} \times \mathbb{Z} \big / \langle (1,2) \rangle \cong... | user26857 | 121,097 | <p>The subgroup generated by <span class="math-container">$(1,2)$</span> can be "moved" by an automorphism of <span class="math-container">$\mathbb Z\times\mathbb Z$</span> into a simpler one: the subgroup generated by <span class="math-container">$(1,0)$</span>. Why is this simpler? Since it can be written a... |
3,073,189 | <p>Evaluate </p>
<p><span class="math-container">$$
\int_0^{+\infty} \frac{1}{(1+x^2)(1+x^{\phi})}\, dx \, \, \, (\phi>0)
$$</span></p>
<p>See my answer below for a solution using a nice substitution.</p>
| Frank W | 552,735 | <p>Another way would be to make the substitution <span class="math-container">$x\mapsto\tan x$</span> so that<span class="math-container">$$\begin{align*}\mathfrak{I} & =\int\limits_0^{\pi/2}\frac {\mathrm dx}{1+\tan^nx}\\ & =\int\limits_0^{\pi/2}\mathrm dx-\int\limits_0^{\pi/2}\mathrm dx\,\frac {\tan^n x}{1+\t... |
1,026,807 | <p>Let $R$ be a ring with unity and assume that $R$ has no nonzero zero-divisors. Let $a,b\in R$, and assume that $ab=1$. Show that $ba=1$, and therefore $a,b$ are units. </p>
<p>I think this question boils down to showing that $R$ is communitive under multiplication (of the ring R), but I don't know how to show it gi... | Raúl Alegre | 185,587 | <p>First of all, note that $a\neq 0$ (otherwise $ab=0$).</p>
<p>$$ab = 1\ \Rightarrow\ ab -1 = 0 \ \Rightarrow\ aba-a=0\ \Rightarrow\ a(ba-1) = 0 \ \Rightarrow\ ba -1 = 0 \ \Rightarrow\ ba = 1$$</p>
<p>Since $a(ba-1) = 0$ and $a\neq 0$, the other term must be $0$ (i.e. $ba-1=0$). Otherwise, $a$ and $ba-1$ are nonzero... |
1,751,764 | <p>I'm doing some studies in mathematical methods for physics and I came across something that I don't really understand. I have only been using the $\epsilon_{ijk}$ when I cross some vectors or operators etc... But what does it mean when you write something like this, with Poisson brackets we have that</p>
<p>$$
\{L_... | LouisB | 192,634 | <p>By convention, the repeated index, $c$, on the right hand side is summed over the 3 spatial dimensions. The equation is a short way of writing 9 equations, which can be arranged in 3 groups of 3 equations.</p>
<p>Consider when $a=b=1$. Then $\epsilon_{11c}=0$ for all 3 values of $c$. Thus the equation contains t... |
26,107 | <p>Is there a <em>Mathematica</em>-to-$\LaTeX$ converter, so that I can place <em>Mathematica</em> code, including things like <code>A // MatrixForm</code>, just as they are in the console straight into a $\LaTeX$ document?</p>
| SEngstrom | 6,453 | <p><code>TeXForm</code> is indeed your friend. It even gives you nicely formatted code:</p>
<pre><code>Table[RandomInteger[10], {3}, {4}] // TeXForm
</code></pre>
<p>gives</p>
<pre><code>(*
\left(
\begin{array}{cccc}
9 & 5 & 10 & 9 \\
6 & 10 & 3 & 9 \\
9 & 5 & 9 & ... |
203,836 | <p>Following the wonderful methodologies provided in <a href="https://mathematica.stackexchange.com/q/203517/52181">this post</a>, I have learned how to process microscopic images of the types shown below in order to analyze them. When it comes to analysing the size distribution of the particles, the images always have... | Niki Estner | 242 | <p>First of all, I would exclude the scale from the "particle search algorithm", like this:</p>
<pre><code>imgWithScale = Import["https://i.stack.imgur.com/ryzmV.jpg"];
img = ImageTake[imgWithScale, 280]
</code></pre>
<p>(followed by the same steps you used above).</p>
<p>Then I would extract the nm/pixel scale:</p>... |
171,038 | <p>After some computations I end up with the following expression:</p>
<pre><code>Sqrt[Exp[-z/(4*Sqrt[2])]] Sqrt[Exp[z/(4*Sqrt[2])]]
</code></pre>
<p>where $z$ is actually complex.</p>
<p>For reasons I don't understand, <em>Mathematica</em> won't simplify this to $1$.</p>
<p>According to the <a href="http://referen... | Ulrich Neumann | 53,677 | <p><strong>PowerExpand only shows one solution</strong></p>
<pre><code>PowerExpand[Sqrt[Exp[-z/(4*Sqrt[2])]] Sqrt[Exp[z/(4*Sqrt[2])]],Assumptions -> Element[z, Reals]]
(* 1*)
PowerExpand[Sqrt[Exp[-z/(4*Sqrt[2])]] Sqrt[Exp[z/(4*Sqrt[2])]],Assumptions -> Element[z, Complexes]]
(*E^(I \[Pi] (Floor[1/2 - Im[z]/(8 ... |
2,800,441 | <p>Anyone can give me the path for this, could not figure out which theory/method to use for this...
$$\lim_{n\to\infty}\left(\frac{1}{1\cdot4}+\frac{1}{4\cdot7} + \frac{1}{7\cdot10}+.....+ \frac {1}{(3n-2)\cdot(3n+1)}\right)=?$$</p>
| Community | -1 | <p>If you break the terms in this way $$\frac{1}{(3n-2)(3n+1)}=\frac{1/3}{3n-2}-\frac{1/3}{3n+1}$$</p>
<p>many will cancel.</p>
|
2,348,293 | <p>Sorry, I feel like this should be simple, but I'm stumped and I've searched everywhere. Is it possible to calculate the length of a photographed object in the attached scenario if you know the width and assume it is a perfect rectangle?</p>
<p><img src="https://i.stack.imgur.com/FlSFm.png" alt="pic of flat rectang... | Edward Porcella | 403,946 | <p>I take $l$ and $w$ to be the measured length and width of the unphotographed rectangle. The length (height) of the photo-rectangle (trapezoid) is the perpendicular distance between the lower and upper bases of the trapezoid. Multiply that by the ratio of the measured lower base to the measured upper base to get the ... |
723,563 | <p>In the <strong>AM-GM Inequality</strong>, how do I conclude that $G(a)= a_1^\frac{1}{n},...,a_n^\frac{1}{n}$ must obtain its maximum when $a_1=...=a_n = U(a)$ and ($U(a)= \frac{a_1+...a_n}{n}$ is the arithmetic mean)? Since if the $a_i$'s are not equal, then $G$ can be increased by the following procedure described ... | orion | 137,195 | <p>The question is basically just a sieve, similar to looking for primes.</p>
<p>The period that visits all combinations of said numbers is long ($2\times 3\times 5 \times 7=210$) so whatever you do, it won't be much quicker than the brute method.</p>
<p>For instance, starting with 2 and 5, you have the ones that are... |
3,784,223 | <p>Let <span class="math-container">$g : \mathbb{R} \to \mathbb{R}$</span> be a differentiable function such that <span class="math-container">$g(0) = g^\prime(0) = 0$</span> and <span class="math-container">$g^{\prime\prime}(0)$</span> exists and is positive. Prove that there exists <span class="math-container">$x >... | zhw. | 228,045 | <p>Observe that for small <span class="math-container">$x>0,$</span></p>
<p><span class="math-container">$$\frac{g'(x)}{x}=\frac{g'(x)-g'(0)}{x-0}>g''(0)/2 > 0.$$</span></p>
<p>It follows that <span class="math-container">$g'(x) >0$</span> in some interval <span class="math-container">$(0,a].$</span> Thus <... |
1,942,110 | <p>Let $x,y,z>0$,prove or disprove
$$\sqrt[4]{\dfrac{(xy+yz+xz)(x^2+y^2+z^2)}{9}}\ge\sqrt[3]{\dfrac{(x+y)(y+z)(z+x)}{8}}$$</p>
<p>I tried many times ,use
$$x^2+y^2+z^2\ge\dfrac{1}{3}(x+y+z)^2$$
and kown $$9(x+y)(y+z)(z+x)\ge 8(x+y+z)(xy+yz+xz)$$</p>
| f10w | 31,498 | <p>This inequality is nice. And without any advanced techniques/theorems, it is also hard.</p>
<p>The technique that Michael used in his solution was developed more than ten years ago (credit goes to Nguyen Anh Cuong who was, to the best of my knowledge, the first to introduce the idea, under the name "ABC method"). T... |
4,027,927 | <p>I'm trying to find derivative of <span class="math-container">$\frac{\cos t-\sin t}{\cos t+\sin t}$</span> in a different way: there is a trick to find derivative of the form <span class="math-container">$\frac{ax+b}{cx+d}$</span>:
<span class="math-container">$$\left(\frac{ax+b}{cx+d}\right)'=\frac{ad-bc}{(cx+d)^2... | Ben Grossmann | 81,360 | <p>Another approach: we can rewrite your function as
<span class="math-container">$$
\frac{\cos t - \sin t}{\cos t + \sin t} = \frac{\cos(t)\sin(\pi/4) - \sin(t)\cos(\pi/4)}{\cos(t)\cos(\pi/4) + \sin(t) \sin(\pi/4)}
= \frac{\sin(\pi/4 - t)}{\cos(\pi/4 - t)} = \tan(\pi/4 - t).
$$</span>
By the chain rule, we find the de... |
2,410,940 | <p>The question states that if $f (x)$ is a polynomial such that $x-1|f(x^n)$ prove that $f(x^n)$ is divisible by $x^n-1$</p>
<p>This is how I proceeded since$x-1|f(x^n)$ </p>
<p>$f(1)=0$ </p>
<p>$\frac {f (x^n)-f(1)}{x^n-1}=g(x)$ </p>
<p>since $f(1)=0$ </p>
<p>$\frac{f(x^n)}{x^n-1}=g(x)$
hence $x^n-1|f(x^n)$</p>... | José Carlos Santos | 446,262 | <p>Your proof is wrong because you did not prove that $g(x)$ is a polynomial.</p>
<p>Let $z\in\mathbb C$ be such that $z^n=1$. Then $f(z^n)=f(1)=0$. So, $z$ is a root of $f(x^n)$. Therefore, $f(x^n)$ is a multiple of$$(x-z_1)(x-z_2)\ldots(x-z_n),\tag{1}$$where $z_1,\ldots,z_n$ are the $n^\text{th}$ roots of unity. But... |
204,405 | <p>By the 2-shift map I mean the map $T:\{0,1\}^\mathbb{Z}\to \{0,1\}^\mathbb{Z}$ that shifts the sequence leftwise. By a root I mean an homeomorphism $\psi:\{0,1\}^\mathbb{Z}\to\{0,1\}^\mathbb{Z}$ that commutes with $T$ and such that $\psi^2=T$.</p>
<p>I read somewhere that it does not have one, but could not find a ... | YCor | 14,094 | <p><b>(05/05/2015)</b> If $T$ is a permutation of any set $X$ with a (finite) odd number of 2-cycles, then $T$ is not a square in the group of permutations of $X$ (because if $T=U^2$ then $T$ commutes with $U$ --this assumption in the question is thus redundant--, hence $U$ preserves the union $J$ of 2-cycles of $T$ an... |
1,947,915 | <p>Can I ask what is the Laplace transform of $\sqrt[3]{t}$ using the Gamma function?
This was my initial answer. Note that there is a theorem, $\Gamma(\frac{1}{3})=3\Gamma(\frac{4}{3})$. $$\int^{\infty}_{0}\exp^{-\beta} \beta^{-\frac{2}{3}}=3\int^{\infty}_{0}\exp^{-\beta} \beta^{\frac{4}{3}}$$ And was able to get the ... | Alex | 232,656 | <p>The Gamma function can be defined as</p>
<p>$$\Gamma(z)=\int_0^\infty\text{d}x\,e^{-x}x^{z-1}$$</p>
<p>The Laplace transform you want to compute is</p>
<p>$$\mathcal{L}\left\{t^{1/3}\right\}=\int_0^\infty\text{d}t\,e^{-ts}t^{1/3}$$</p>
<p>Performing the change of variables $ts=y$ you have</p>
<p>$$\begin{align}... |
717,882 | <p>The $x^{2/2}$ can be represented by these ways:
$$\begin{align}
x^{2\over2}=\sqrt{x^2} = |x|\\
\end{align}
$$
And<br>
$$\begin{align}
x^{2\over2}=x^{1} = x\\
\end{align}
$$
Which one is correct?
And what is the domain of $x^{2 \over 2}$?</p>
| Blah | 6,721 | <p>The second one, <em>by definition</em> the symbol
$$
x^\dfrac{m}{n} \qquad\qquad m,n\in\mathbb{Z}, n\neq 0
$$
is <strong>not</strong> defined if $x<0$, it should be considered meaningless.</p>
|
1,522,359 | <blockquote>
<p>Let $W_0, W_1, W_2, \dots$ be random variables on a probability space
$(\Omega, \mathscr{F}, \mathbb{P})$ where</p>
<p>$$\sum_{k=0}^{\infty}P(|W_k|>k) <\infty$$</p>
<p>Prove that $$\limsup \frac{|W_k|}{k} \le 1 \ \text{a.s.} $$</p>
</blockquote>
<hr>
<p>I initially thought the conc... | Andreas Blass | 48,510 | <p>The difference between the two formulas in the title is that the first one makes sense and the second one is meaningless. The second one pretends to form the lim sup of some statements (not of numbers), and there's no such thing.</p>
|
17,669 | <p>I just inherited two slide rules from my grandfather-in-law, one wood with smooth action despite nearly a century without
use.<sup>1</sup> (I used a K+E slide rule myself as an undergraduate in the 1970's.) It struck me that the conversion of multiplication/division to addition/subtraction via the
logarithmically ru... | nyamal0 | 13,269 | <p>A few thoughts:</p>
<ul>
<li><p>Joe Pasquale from UCSD has done similar things:
<a href="https://cseweb.ucsd.edu/~pasquale/SlideRuleTalkLasVegas14.pdf" rel="noreferrer">https://cseweb.ucsd.edu/~pasquale/SlideRuleTalkLasVegas14.pdf</a> and
<a href="https://cseweb.ucsd.edu/~pasquale/FreshmanSeminarF03/" rel="norefe... |
3,230,619 | <p>Hello i cant figure out how to evaluate this integral and find a volume for a triple integral which is given by <span class="math-container">$z=x^2+y^2 \text{ and } z = x+y$</span> this figure for what i found is paraboloid intersected by a straight plane. I have tried to find the intersection of the two figures but... | DinosaurEgg | 535,606 | <p>In the notation set above, it is easy to show that, since the rectangular pulses never overlap due to the ordering prescribed:</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty}f_{a_j b_j}(x)f_{a_k b_k}(x)dx=(b_j-a_j)\delta_{jk}$$</span></p>
<p>and the basis of functions chosen is orthogonal. Now, we su... |
3,230,619 | <p>Hello i cant figure out how to evaluate this integral and find a volume for a triple integral which is given by <span class="math-container">$z=x^2+y^2 \text{ and } z = x+y$</span> this figure for what i found is paraboloid intersected by a straight plane. I have tried to find the intersection of the two figures but... | Behnam Esmayli | 283,487 | <p>See the short nice proof in the lecture notes of Piotr Hajlasz at </p>
<p><a href="http://www.pitt.edu/~hajlasz/Notatki/Harmonic%20Analysis2.pdf" rel="nofollow noreferrer">http://www.pitt.edu/~hajlasz/Notatki/Harmonic%20Analysis2.pdf</a></p>
<p>Theorem 2.31</p>
|
1,194,181 | <p>Let $G:\mathbb{R}^n \to\mathbb{R}^n$ be transformation such that $G(x):=Ax+b$ where $A\in\mathcal{M}_{nxn}(\mathbb{R})$ and $b\in\mathbb{R}^n$ such that $det(A-I)\neq0$ .</p>
<p>How would you prove G has a unique fixed point $p\in\mathbb{R}^{n}$ ? </p>
| Marm | 159,661 | <p><strong>Hint:</strong></p>
<p>Rewrite: $Ax+b=x$ to $(A-I)x=-b$ and use that $det(A-I)\neq 0$.</p>
<p>What does $det(B)\neq 0$ says about the associated linear map?</p>
|
3,119,573 | <blockquote>
<p>Let <span class="math-container">$R$</span> be the equivalence relation on the real numbers given by
<span class="math-container">$$R = \{(x, y) \in \Bbb R^2: (x−y)(x+y) = 0 \} $$</span>
What are the equivalence classes of <span class="math-container">$R$</span>?</p>
</blockquote>
<p>So I wrote t... | Robert Lewis | 67,071 | <p>The condition which defines <span class="math-container">$R$</span>,</p>
<p><span class="math-container">$(x - y)(x + y) = 0, \tag 1$</span></p>
<p>is clearly equivalent to</p>
<p><span class="math-container">$x^2 - y^2 = 0 \equiv y^2 = x^2 \equiv y = \pm x. \tag 2$</span></p>
<p>Is <span class="math-container">... |
1,212,185 | <p>I have the following equation to solve, I think is just the case of founding a suitable change of variables, but I couldn't think of anything: $$y'' + (y')^2 = 2e^{(-y)}.$$</p>
<p>Any suggestions?</p>
| Mike | 17,976 | <p>$$u=e^y,u'=e^yy',u''=e^yy'^2+e^yy''$$</p>
<p>So the equation reduces to</p>
<p>$$u''=2,u=x^2+ax+b$$
$$e^y=x^2+ax+b$$</p>
|
296,737 | <p>Im trying to show that the ring of polynomials in one variable over the complex numbers is not isomorphic to the ring over $\mathbb C$ with two variables $x$ and $y$ modulo $\langle x^2-y^3\rangle$. I've shown previously that if the relationship $p^2=q^3$ holds for some $p$ and $q$ in one variable, there exists $r$ ... | Mariano Suárez-Álvarez | 274 | <p>If $I$ is a maximal ideal in $\def\CC{\mathbb C}\CC[X]$, then there is an $\alpha\in\CC$ such that $I=(X-\alpha)$, and using this it is easy to see that $\dim_\CC I/I^2=1$.</p>
<p>On the other hand, the ideal $J=(X,Y)\subset A=\CC[X,Y]/(X^2-y^3)$ is maximal and $J/J^2$ is a vector space of dimension $2$.</p>
<p>It... |
296,737 | <p>Im trying to show that the ring of polynomials in one variable over the complex numbers is not isomorphic to the ring over $\mathbb C$ with two variables $x$ and $y$ modulo $\langle x^2-y^3\rangle$. I've shown previously that if the relationship $p^2=q^3$ holds for some $p$ and $q$ in one variable, there exists $r$ ... | Mariano Suárez-Álvarez | 274 | <p>Yet another argument, the most laborious:</p>
<p>One can show with some calculation that the automorphism group of the $\mathbb C$-algebra $\mathbb C[X,Y]/(X^2-Y^3)$ is isomorphic to $\mathbb C^\times$, which is a nice abelian group.</p>
<p>On the other hand, the automorphism group of $\mathbb C[T]$ is the group $... |
1,997,050 | <p>Because strange conjectures have the tendency to enter my mind, I've become convinced that the only powers of $10$ that take the form $\frac{k(k+1)}{2}$ for $k\in \mathbb{Z}^{+}$ (i.e. is triangular) are $10^{0}$ and $10^{1}$. However, I'm having a difficult time making progress towards how to prove or disprove this... | Ross Millikan | 1,827 | <p>Because $k$ and $k+1$ are coprime, one of them must be the power of $2$ and the other must be the power of $5$. The exponent on the $2$ has to be one larger than the exponent on the $5$ because of the two in the denominator. But once $k \gt 1, 5^k \gt 2^{k+1}+1$, so no other powers of $10$ are triangular.</p>
|
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Kevin McGerty | 1,878 | <p>You can prove it using only basic facts about continuity/compactness and the same estimate which makes the winding number/fundamental group of $S^1$ proofs work: first check that if $p(z)=z^n + a_{n-1}z^{n-1} + \ldots + a_0$ is a polynomial then $|p(z)|$ tends to infinity as $|z|$ tends to infinity (this is the "lea... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Gian Maria Dall'Ara | 1,049 | <p>I add another proof separately since it is rather different.</p>
<p>It is based on the observation that an irreducible polynomial of degree $n$ on a field $F$ gives rise to a field structure on $F^n$, compatible with the vector addition (just quotient out polynomials with coefficients in $F$ by the maximal ideal ge... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Dan Ramras | 4,042 | <p>When I was a freshman, I was asked to prove the fundamental theorem of algebra on the final exam for multivariable calculus (I'm completely serious: I think the problem just stated the FTA and asked us to give a proof.)</p>
<p>I didn't succeed, but what I was supposed to do (I think) was apply the Gauss-Bonnet Theo... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Lucas Culler | 5,499 | <p>As in one of the previous posts, consider the projective space $CP^n$ of nonzero all polynomials</p>
<p>$$ c_nT^n + c_{n-1} T^{n-1} + \cdots + c_1 T + c_0 $$
considered up to nonzero scalar multiple. We'll show directly that any such polynomial admits a factorization into linear factors.</p>
<p>Consider the map $... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Johannes Ebert | 9,928 | <p>Here is another short proof, showing that no finite extension of $\mathbb{C}$ exists. If $A$ were such an extension of dimension $d$,
then the projective space $\mathbb{P} A \cong \mathbb{CP}^{d-1}$
were a compact commutative Lie group, hence a torus.
There are plenty of ways showing that $\mathbb{CP}^{m}$ is not a... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | trew | 9,514 | <p>There seems to be a new elementary proof using only Bolzano–Weierstraß and an inequality:</p>
<ul>
<li>Oswaldo Rio Branco de Oliveira, <em>The fundamental theorem of algebra: A most elementary proof</em>, American Mathematical Monthly 119:9 (2012) 753-758, doi:<a href="https://doi.org/10.4169/amer.math.monthly.119.0... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Vergilius | 33,927 | <p>In elementary proofs like the one elucidated in <a href="https://mathoverflow.net/a/10670/33927">Kevin McGerthy's answer</a>, there is usually a step where for a given <span class="math-container">$n\in \mathbb{N}_0$</span> and given <span class="math-container">$\alpha\in \mathbb{C}_0$</span> a <span class="math-co... |
1,752,320 | <p>Is $\text{Gal} (\overline{\Bbb Q}/\Bbb Q)$ countable or uncountable? It seems like it should be countable (because the algebraic closure of $\Bbb Q$ is countable and there are countably many permutations of the irrational algebraic numbers, and a countable union of countable sets is countable. However, I've seen ref... | Adam Hughes | 58,831 | <p>Let $I\subseteq \Bbb N$ be any subset and let $K_I=\Bbb Q(\{\sqrt{p_i}\}_{i\in I})$ where $p_i$ is the $i^{th}$ prime. Then there are precisely $2^{\Bbb N}$ in fact the Galois group of the compositum of this extension is exactly isomorphic to</p>
<p>$$\prod_{i\in\Bbb N}\Bbb Z/2\Bbb Z$$</p>
<p>and this of course in... |
1,752,320 | <p>Is $\text{Gal} (\overline{\Bbb Q}/\Bbb Q)$ countable or uncountable? It seems like it should be countable (because the algebraic closure of $\Bbb Q$ is countable and there are countably many permutations of the irrational algebraic numbers, and a countable union of countable sets is countable. However, I've seen ref... | Kostas Kartas | 350,587 | <p>Let me elaborate on Lubin's comment: there are no countably infinite Galois groups. This follows from a general statement about topological spaces: If <span class="math-container">$X$</span> is compact, Hausdorff with no isolated points, then <span class="math-container">$X$</span> is uncountable. For the proof, see... |
2,251,501 | <p>How can I calculate the number of perfect cubes among the first $4000$ positive integers?</p>
<p>Is there any trick to solving such questions?</p>
| Ovi | 64,460 | <p>You could find the largest cube $\le 4000$.</p>
<p>$\sqrt[3]{4000} \approx 15.9$, so there are $15$ perfect cubes among the first $4000$ positive integers.</p>
|
2,384 | <p>I am tutoring several talented students, middle school level and early high school level, in mathematics. I am always looking for new sources from which to draw questions. Can anyone recommend books, web-sites, etc. with a interesting questions?</p>
<p>I know of the ArtofProblemSolving.com. I am currently using the... | deinst | 943 | <p>Check out the <a href="http://blossoms.mit.edu/" rel="nofollow">Blossoms</a> videos. Although aimed more at a classroom setting there may be interesting snippets to grab.</p>
|
2,384 | <p>I am tutoring several talented students, middle school level and early high school level, in mathematics. I am always looking for new sources from which to draw questions. Can anyone recommend books, web-sites, etc. with a interesting questions?</p>
<p>I know of the ArtofProblemSolving.com. I am currently using the... | Dan Christensen | 3,515 | <p>I have developed some educational freeware to teach students how to write simple, though thoroughly rigorous mathematical proofs. The software verifies every line of proof as it is entered. It also provides some important insights into the nature of modern mathematics. One of the worked examples in the tutorial is t... |
1,214,419 | <p>How do I show that any open interval is an half open set and use this to conclude that any open set is also half open?</p>
<p>I am in an introduction to proofs writing class. I have a feeling I need to use the fact that an open set is a union of open intervals but I don't know that I am even going in the right dir... | skyking | 265,767 | <p>You can prove it directly. If $U$ is an open set then for any $c\in U$ there's an open interval $I=(a,b)\subseteq U$ such that $c\in I$. Or put another way $a<c<b$, now both $[c,b)$ and $(a,c]$ are subsets of $I$ and therefore subset of $U$ which means that $U$ is half-open.</p>
<p>Note that your definition w... |
1,452,559 | <p>This is a problem from a linear algebra textbook. Given a finite dimensional inner product space $V$ with orthonormal basis $e_1, \ldots, e_n$, show that if a list of vectors $v_1, \ldots, v_n$ satisfies $\|e_j - v_j\| < \frac{1}{\sqrt{n}}$ for all $j$ in $\{1, \ldots, n\}$, then $v_j$'s form a basis of $V$.</p>
... | levap | 32,262 | <p>Define $u_i := v_i - e_i$ so $v_i = e_i + u_i$ and $||u_i||^2 < \frac{1}{n}$. It is enough to show that $(e_1 + u_1, \ldots, e_n + u_n)$ are linearly independent. Let $a_i \in \mathbb{C}$ be scalars such that $\sum_{i=1}^n a_i (e_i + u_i) = 0$. Then
$$ \sum_{i=1}^n a_i e_i = -\sum_{i=1}^n a_i u_i. $$
Taking norms... |
1,452,559 | <p>This is a problem from a linear algebra textbook. Given a finite dimensional inner product space $V$ with orthonormal basis $e_1, \ldots, e_n$, show that if a list of vectors $v_1, \ldots, v_n$ satisfies $\|e_j - v_j\| < \frac{1}{\sqrt{n}}$ for all $j$ in $\{1, \ldots, n\}$, then $v_j$'s form a basis of $V$.</p>
... | Hanno | 316,749 | <p>This scribble is split in two: The first half answers the OP,
while <span class="math-container">$(2)$</span> offers a general symmetrical example to demonstrate that the <strong><em>assumptions cannot be relaxed from <span class="math-container">$\,$</span>"<span class="math-container">$<$</span>"<span class="ma... |
73,522 | <p>I am learning a little about Inset, but after trying all options, I can't get this to work. I need to inset separately made <code>Graphics2D</code> objects, into one main <code>Graphics2D</code> but have them show up there with same relative sizes they had on their own. I use the Axes of the main Graphics as measur... | Mr.Wizard | 121 | <p>This seems to be what you want:</p>
<pre><code>Graphics[
MapThread[
Inset[##, {0, 0}, #2 - # & @@ First@PlotRange@#] &,
{{g1, g2}, {{0, 0}, {2, 2}}}
],
Axes -> True,
PlotRange -> {{-6, 6}, {-4, 4}}
]
</code></pre>
<p><img src="https://i.stack.imgur.com/7RQdC.png" alt="enter image desc... |
73,522 | <p>I am learning a little about Inset, but after trying all options, I can't get this to work. I need to inset separately made <code>Graphics2D</code> objects, into one main <code>Graphics2D</code> but have them show up there with same relative sizes they had on their own. I use the Axes of the main Graphics as measur... | dofuuu | 25,250 | <pre><code>g1 = Graphics[{LightGray, Disk[{0, 0}, 1]}, AspectRatio -> Automatic];
g2 = Graphics[{Opacity[.5], Blue, Rectangle[{0, 0}, {1, .5}]}, AspectRatio -> Automatic];
Graphics[
{
Inset[g1, {0, 0}, {0, 0}, Scaled[{.35, 1}]],
Inset[g2, {0, 2}, {0, 0}, Scaled[{.35, 1}]]
},
Axes -> True,
PlotRange -> {... |
3,684,467 | <blockquote>
<p><span class="math-container">$\ell^{p}$</span> spaces [edit]</p>
<blockquote>
<p>See also: <span class="math-container">$L^{p}$</span> space and <span class="math-container">$L$</span> -infinity</p>
</blockquote>
<p>For <span class="math-container">$0<p<\infty, \ \ell^{p}$</span> is the subspace o... | PtF | 49,572 | <p>If <span class="math-container">$0<p<1$</span> then the triangular inequality does not hold for <span class="math-container">$\|\cdot\|_p$</span>. For instance, take <span class="math-container">$\ell^{1/2}(\mathbb R)$</span> and consider the elements </p>
<p><span class="math-container">$$x=(1/4, 0, 0, \ldot... |
3,684,467 | <blockquote>
<p><span class="math-container">$\ell^{p}$</span> spaces [edit]</p>
<blockquote>
<p>See also: <span class="math-container">$L^{p}$</span> space and <span class="math-container">$L$</span> -infinity</p>
</blockquote>
<p>For <span class="math-container">$0<p<\infty, \ \ell^{p}$</span> is the subspace o... | Calvin Khor | 80,734 | <p>Question 2: compare <span class="math-container">$d(x,y)$</span> with <span class="math-container">$d(x,0)=d(y,0)$</span>, where <span class="math-container">$x = (1,0,0,\dots)$</span> and <span class="math-container">$y_n = (0,1,0,\dots)$</span>. Note that this is a finite dimensional counterexample i.e. basically... |
1,660,402 | <p>Suppose I transform an integral $$I=\int f(x,y) \, dx \, dy$$ using polar coordinates, setting $x=r\cos\theta$ and $y=r\sin\theta$. We get
$$
\begin{split}
dx &= \cos\theta \, dr - r\sin\theta \, d\theta\\
dy &= dr\,\sin\theta + r \cos \theta \, d\theta.
\end{split}
$$</p>
<p>Now the volume element $dx \, d... | Michael Hardy | 11,667 | <p>The volume element $dx\,dy$ can be thought of as an infinitely small change in $x$ with $y$ not changing, times an infinitely small change in $y$ with $x$ not changing -- thus the area of an infinitely small rectangle.</p>
<p>In the above, if we make $\theta$ change while $r$ is fixed we move along a circle centere... |
2,401,252 | <p>Prove the identity
$$\frac{(2n)!}{2^n \cdot n!} = (2n-1)(2n-3)\cdots 3\cdot 1$$
using combinatorial proof.</p>
| EugenR | 473,662 | <p>Ok, I think, I get the LHS: we order $2n$ objects, thus $(2n)!$, we don't care about the order of objects in each pair, thus $2^n$, and we don't about the order of pairs, thus $n!$ I am struggling with the RHS though. Looks like proof by induction, but I don't see it...</p>
<p>EDIT: I think, you could apply recursi... |
1,960,862 | <blockquote>
<p>If $a,b,c>0$, Then prove that $$\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}\geq a+b+c$$</p>
</blockquote>
<p>$\bf{My\; Try::}$ Using Cauchy- Schwarz Inequality</p>
<p>$$\frac{a^4}{abc}+\frac{b^4}{abc}+\frac{c^4}{abc}\geq \frac{a^2+b^2+c^2}{3abc}$$</p>
<p>Now How can i solve after that , Help re... | Christian Blatter | 1,303 | <p>A parametric representation
$$t\mapsto{\bf z}(t)=\bigl(x(t),y(t)\bigr)\qquad(0\leq t\leq T)$$
of a curve $\gamma\subset{\mathbb R}^2$ is <em>regular</em> if it is $C^1$, and if ${\bf z}'(t)=\bigl(x'(t),y'(t)\bigr)\ne{\bf 0}$ for all $t\in[0,T]$. If we are given such a curve $\gamma$ then we have to, and can, <em>pro... |
1,548,314 | <p>Does there exist a continuous function $f:\mathbb R^2\to \mathbb R$ such that $\displaystyle \frac{\partial f}{\partial x}$ <strong>does not exists</strong> but $\displaystyle \frac{\partial^2 f}{\partial x\partial y}$ exists.</p>
<p>I think yes. But I am unable to find an example of such function. Can anyone help ... | Eric S. | 263,514 | <p>See this <a href="https://en.wikipedia.org/wiki/Weierstrass_function" rel="nofollow">Wikipedia page</a> on the Weierstrass function $W_{\alpha}(x)$. It is continuous everywhere, but differentiable nowhere. So let
$$f(x,y)=W_{\alpha}(x)$$
Then $\frac{\partial f}{\partial x}$ does not exist anywhere, and since $\frac{... |
1,828,024 | <p>If each ticket for a lottery has a 1 in 258,890,850 chance... what happens if you buy 10 tickets? </p>
<p>Is it:</p>
<ul>
<li>A 10 in 258,890,850 chance?</li>
<li>A 1 in 258,890,840 chance?</li>
<li>A 1 in 25,889,085.0 chance?</li>
</ul>
<p>What is the best way to represent the chances? This is very simple, but I... | Ross Millikan | 1,827 | <p>A maximal element of $S$ must be a member of $S$. An upper bound need not be, but it has to be comparable to all members of $S$. $10$ is maximal in $S$ because it is not $\propto$ any member of $S$. It is not an upper bound as $12 \not \propto 10$. Neither is $12$ an upper bound as $10 \not \propto 12$ The "in $\... |
563,170 | <p>If $$\lim_ {x\to a}\ f(x)$$ exists and $$\lim_ {x\to a}\ (f(x)g(x))$$ exists, then $$\lim_ {x\to a}\ g(x)$$ exists. Please exdpain why the above statment is true or false.</p>
<p>EDIT: Edited question to be more specific.</p>
| p.s. | 17,433 | <p>Not sure what the intuition is, but it's a straightforward integral to work out.
$$
\begin{aligned}\int_0^\infty \int_{0}^\infty& \int_{0}^\infty [x_1 \le x_2] [x_1 \le x_3] \lambda_1 \lambda_2 \lambda_3 e^{-\lambda_1 x_1 -\lambda_2 x_2-\lambda_3 x_3}dx_3 dx_2 dx_1\\
&=
\int_0^\infty \int_{x_1}^\infty \int_... |
328,263 | <p>Let $\ S\ $ be a non-empty Set, and suppose s$\ \in S $.
Assuming $\ S\ $ is finite, what can we deduce about the relationship between $\ |\mathcal P(S\ \setminus \{s\} )| $ and $\ | \mathcal P(S)|?$</p>
<p>May I know if the question is looking for $\ |\mathcal P(S\ \setminus \{s\} )| =$ $\ | \mathcal P(S)| \over ... | dtldarek | 26,306 | <p>Let $$P_1 = \{ A \subset S \mid s \in A\}$$ and $$P_2 = \{ A \subset S \mid s \notin A \}.$$ Then $\mathcal{P}(S) = P_1 \uplus P_2$ and $\mathcal{P}(S\setminus\{s\}) = P_2$. However, $|P_1| = |P_2|$ by bijection $f : \mathcal{P}(S) \to \mathcal{P}(S)$ (observe that $f$ is its own inverse, that is $f \circ f = \mathr... |
4,098,179 | <p>Often when working on a proof, I get to a computation which appears to be elementary (e.g. requiring only standard algebra and perhaps calculus) but messy. Solving this via pen and paper is tedious and error prone, yet the path to a solution is not always elegant (or, at the least, an elegant path is not always app... | Somos | 438,089 | <p>You asked</p>
<blockquote>
<p>Often when working on a proof, I get to a computation which appears to be elementary (e.g. requiring only standard algebra and perhaps calculus) but messy. Solving this via pen and paper is tedious and error prone, yet the path to a solution is not always elegant (or, at the least, an e... |
2,479,483 | <p>given the matrix equation $Ax=b$ (shown as $[A|b]$ )</p>
<p>$$\left[\begin{array}{ccc|c} 2 & 2 & 0 &2 \\ 0 &k &1 &1 \\ 1 &2 &k&2 \end{array}\right]$$</p>
<p>I've used Gaussian elimination to find that this equals:</p>
<p>$$\left[\begin{array}{ccc|c} 1&1 &0 &1 \\ 0 &... | Przemysław Scherwentke | 72,361 | <p>HINT: Every linear space has a basis.</p>
|
1,838 | <p>I understand what the problem with Gimbal Lock is, such that at the North Pole, all directions are south, there's no concept of east and west. But what I don't understand is why this is such an issue for navigation systems? Surely if you find you're in Gimbal Lock, you can simply move a small amount in any direction... | Josephine Moeller | 950 | <p>I don't imagine that this is a practical issue for navigation any longer, given the advent of GPS technology. However, it is of practical concern in 3-d animation and robotics. To get back to your navigation example, suppose that I have a mechanical gyro mounted in an airplane flying over the North Pole. </p>
<p>If... |
3,934,824 | <p>There are three containers, one container has diamond in it, other containers are empty. Each
container has label as clue whatever is inside the containers. Those labels are:</p>
<hr />
<ul>
<li><strong>Container#1:</strong> The diamond is not present</li>
<li><strong>Container #2:</strong> The diamond is not presen... | subrosar | 602,170 | <p>There are three possibilities. Either the first statement is true, the second is true, or the third is true. If the first is true, then statement 2 and 3 are false, leading to a contradiction. If the third is true, then statement 1 and 2 are false, leading to a contradiction. Therefore statement 2 must be true, mean... |
4,618,382 | <p>An answer to <a href="https://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics">this question</a> claimed that an interesting mathematical mistake is believing that</p>
<blockquote>
<p>If <span class="math-container">$f$</span> is a smooth function with <span class="math-container">$f... | Alex Ortiz | 305,215 | <p>The example in that list is worded in a very bare-bones manner. All it says is</p>
<blockquote>
<p>If <span class="math-container">$f$</span> is a smooth function with <span class="math-container">$df = 0$</span>, then <span class="math-container">$f$</span> is constant.</p>
</blockquote>
<p>This of course isn't tru... |
2,170,833 | <p>A proposition in a lecture script on measure and integration theory reads:</p>
<blockquote>
<p><strong>Proposition 1.1.4</strong> Let $$\mathcal{I} = \{ (a,b]: -\infty \leq a < b < \infty \}\cup\{ (b,\infty) : -\infty < b < \infty \}\cup\{ \emptyset \}\cup\mathbb{R}.$$ Then $$
\epsilon = \left\{ \big... | Cameron Buie | 28,900 | <p>What it means is that the elements of $\epsilon$ are precisely the finite unions of elements of $\mathcal I.$ For example, $(0,1]\cup(2,\infty)$ would be an element of $\epsilon.$</p>
<p>Put another way, given an arbitrary natural number $N,$ and arbitrary sets $I_1,...,I_N\in\mathcal I,$ we have that $\bigcup_{i=1... |
1,712,769 | <p>I found this question in text book. I am looking at the solution but I can't understand it. Can anyone enlighten me. Thanks!</p>
<p>Let function $f : X → Y$ , with $|X| = m$ and $|Y|= n$.</p>
<ol>
<li>How many functions $f$ are possible ?</li>
</ol>
<p>Solution:$n^m$</p>
<ol start="2">
<li>How many one-to-one fu... | drhab | 75,923 | <p>1) For each of the $m$ elements $x\in X$ there are $n$ choices for $f(x)\in Y$. That gives $$n\times n\times\cdots\times n=n^m$$ possibilities.</p>
<p>2) Let $X=\{x_1,\dots,x_m\}$. There are $n$ choices for $f(x_1)$. After that there are $n-1$ choices for $f(x_2)$, et cetera. So we find $$n\times(n-1)\times\cdots\t... |
2,980,446 | <p>I have this problem and I really don't know, how to edit it to get some solution.
<span class="math-container">$\lim_{n \to \infty} \sqrt{n} * [\sqrt{n+1} - \sqrt{n}]$</span></p>
<p>So my question Is what to do with <span class="math-container">$\sqrt{n} * [\sqrt{n+1} - \sqrt{n}]$</span>? </p>
| Mohammad Riazi-Kermani | 514,496 | <p>Note that <span class="math-container">$$\sqrt {n+1}-\sqrt n =\frac {1}{\sqrt {n+1}+\sqrt n}$$</span></p>
<p>Thus the desired limit is <span class="math-container">$$\lim _{n\to \infty} \frac {\sqrt n }{\sqrt {n+1}+\sqrt n} = 1/2$$</span></p>
|
173,888 | <p>$$\frac{\mathrm{d}y}{\mathrm{d}x} = f(y)$$
where $f(y)$ is continuous on $|y-a|\leq \epsilon$,and $f(y)=0$ iff $y=a$.</p>
<p>To Proof : For the initial value point on $y=a$,the equation has local unique solution iff $\left|\int_a^{a+\epsilon}\frac{\mathrm{d}y}{f(y)}\right|= \infty$ </p>
<p>How to proof Initial val... | Vstal | 813,692 | <p>I don't quite understand the counterexample Robert Israel gives. This equation obviously has an infinite number of solutions.I'd like to give a proof, though it's too late for that.</p>
<p><span class="math-container">${\rm{Proof:}}$</span></p>
<p>Assume that <span class="math-container">$(x_0,y_0)\in\left\{(x,y):0\... |
239,622 | <p>For what values of $1\le p \le \infty$ does $f(x,y)=\frac{1}{1+|x|+|y|}$ with $(x,y) \in \mathbb{R}^2$ belong to $L^p(\mathbb{R}^2)$?</p>
<p>Using Wolfram Alpha I've found that the answer should be $p > 2$, but I don't know how to begin proving it. I've thought about doing a change of variable, after considering... | Did | 6,179 | <p>For a conceptual proof, (1.) do not focus on the exact values of the integrals involved and (2.) turn to polar coordinates. </p>
<p>All norms on $\mathbb R^2$ are equivalent and, if $(x,y)=(r\cos\theta,r\sin\theta)$, then $r$ is the $\ell^2$ norm of $(x,y)$ while $|x|+|y|$ is its $\ell^1$ norm hence $ar\leqslant|x|... |
234,137 | <p>I have a long list consists of float numbers with some of them duplicated as shown below.</p>
<p><a href="https://i.stack.imgur.com/Qwz8I.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Qwz8I.png" alt="list output" /></a></p>
<p>I want to divide this list into bins;</p>
<ol>
<li>with equal bin wid... | Gravifer | 72,025 | <p>I think the best way would be to use <code>HistogramList</code>:</p>
<p><a href="https://reference.wolfram.com/language/ref/HistogramList.html" rel="nofollow noreferrer">HistogramList—Wolfram Language Reference</a></p>
<p>various sorts of bin specifications are supported.</p>
|
339,585 | <p>Show that the Function $$g(x):=\begin{cases} \left| x \right| \sin{( \cot{x} )} & \text{for }x\notin \left\{ 0,\frac{1}{42} \right\}, \\ 0 & \text{for }x=0, \\ 10^{42} & \text{for }x=\frac{1}{42} \end{cases}$$ </p>
<p>is not continuous, but in the Point $\xi=0$ is continuous. You can use that $\sin,\co... | Berci | 41,488 | <p><strong>Hint:</strong> What you have to prove is
$$\lim_{x\to\frac1{42}}|x|\sin(\cot x) \ne 10^{42} \\
\lim_{x\to 0}|x|\sin(\cot x) = 0\,.
$$</p>
|
339,585 | <p>Show that the Function $$g(x):=\begin{cases} \left| x \right| \sin{( \cot{x} )} & \text{for }x\notin \left\{ 0,\frac{1}{42} \right\}, \\ 0 & \text{for }x=0, \\ 10^{42} & \text{for }x=\frac{1}{42} \end{cases}$$ </p>
<p>is not continuous, but in the Point $\xi=0$ is continuous. You can use that $\sin,\co... | Michael Hardy | 11,667 | <p>To show continuity at $x=0$, you're being asked to show
$$
\lim_{x\to0} |x|\sin(\cot x) = \text{the given value} = 0.
$$
Since $\sin(\text{anything})$ is between $1$ and $-1$ (inclusive), you've got $|x|\sin(\text{something})$ between $|x|$ and $-|x|$, and those both approach $0$ as $x\to 0$, so that's all you need ... |
3,813,434 | <p>I am trying to prove the following :</p>
<blockquote>
<p>Let <span class="math-container">$R$</span> be a commutative ring with unity, <span class="math-container">$a,b\in R$</span>; suppose <span class="math-container">$(a)+(b)$</span> is principal. Show that <span class="math-container">$(a)\cap(b)$</span> is prin... | Aphelli | 556,825 | <p>Write <span class="math-container">$a=\alpha d$</span>, <span class="math-container">$b=\beta d$</span>, <span class="math-container">$d(u\alpha+v\beta-1)=0$</span>, <span class="math-container">$\mu=\alpha\beta d$</span>.</p>
<p>Let <span class="math-container">$c \in (a) \cap (b)$</span>. Then <span class="math-co... |
3,649,613 | <p>Let <span class="math-container">$\phi: U \rightarrow \mathbb{R}^m$</span>, <span class="math-container">$U\subseteq \mathbb{R}^n$</span> open, such that <span class="math-container">$\phi(U)$</span> is open and <span class="math-container">$\phi$</span> is a homeomorphism onto its image.</p>
<blockquote>
<p>Let ... | Hanul Jeon | 53,976 | <p>You are almost done. Let <span class="math-container">$\mathcal{A}$</span> be an open cover of <span class="math-container">$K$</span>. Then <span class="math-container">$\{U\cap W\mid W\in\mathcal{A}\}$</span> is an open cover of <span class="math-container">$K$</span> which consists of open subsets of <span class=... |
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