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 |
|---|---|---|---|---|
3,360,912 | <p>Why if we have strictly increasing, continuous and onto function its inverse must be continuous? could anyone explain this for me please?</p>
| Conrad | 298,272 | <p>Roughly speaking the sum behaves (for large <span class="math-container">$m,n$</span>) like <span class="math-container">$\sum_{m,n \ne 0}\frac{1}{m^2+n^2}$</span> and that is divergent since the sum say in m for fixed <span class="math-container">$n$</span> is about <span class="math-container">$\frac{1}{n}$</span>... |
1,244,564 | <p>For n > 1 Let $F_n = 2^{2^n} + 1$ be a fermat number and b = $2^{2^{n - 2}}$ * ($2^{2^{n - 1}}$ - 1 ).</p>
<p>Then $b^2$ $\equiv$ 2 (mod $F_n$)</p>
<p>I tried to square the original expression I got something ugly that I couldn't simplify further.</p>
<p>I got $b^2$ = $2^{2^{n - 1}}$ * ($2^{2^n}$ - $2 * 2^{2^{n -... | AgentS | 168,854 | <p>You're almost there! Just notice that $2^{2^n}\equiv -1\pmod{2^{2^n}+1}$ :</p>
<p>$$\begin{align}b^2&=2^{2^{n-1}}(2^{2^n}-2*2^{2^{n-1}}+1)\\&=2^{2^{n-1}}(2^{2^n}+1-2*2^{2^{n-1}})\\&\equiv2^{2^{n-1}}(0-2*2^{2^{n-1}})\pmod{2^{2^n}+1}\\&\equiv-2*2^{2^{n-1}}*2^{2^{n-1}}\pmod{2^{2^n}+1}\\&\equiv-2*2^... |
3,413,837 | <p>Jerry the mouse is hungry and according to some confidential information, there is a tempting piece of cheese at the end of one of the three paths after the junction he just found himself!</p>
<p>Fortunately, Tom is standing right there and Jerry hopes he can get some useful information as to which path he must get... | Hemant Agarwal | 645,672 | <p>Firstly go through this question, famously coined the Hardest logic puzzle : <a href="https://youtu.be/LKvjIsyYng8" rel="nofollow noreferrer">https://youtu.be/LKvjIsyYng8</a> . This YouTube link is an excellent video explaining the question and then the answer. After seeing the video, try solving your question agai... |
233,075 | <p>I am trying to solve the following equation in the Natural Numbers, with the condition <span class="math-container">$a\ge1$</span>, <span class="math-container">$b\ge1$</span>, and <span class="math-container">$r\ge3$</span>:</p>
<p><span class="math-container">$$\frac{a(a + 3)(a(r - 5) + (12 - r))}{9}=\frac{b (9 + ... | Michael E2 | 4,999 | <p>Borrowing a fast perfect-square test from <a href="https://mathematica.stackexchange.com/questions/442/fastest-square-number-test/207813#207813">Fastest square number test</a>, and shortening the length of the test case:</p>
<pre><code>(* OP's *)
Table[
If[IntegerQ[
FullSimplify[
Sqrt[3*((4 a (3 + a) ... |
233,075 | <p>I am trying to solve the following equation in the Natural Numbers, with the condition <span class="math-container">$a\ge1$</span>, <span class="math-container">$b\ge1$</span>, and <span class="math-container">$r\ge3$</span>:</p>
<p><span class="math-container">$$\frac{a(a + 3)(a(r - 5) + (12 - r))}{9}=\frac{b (9 + ... | Ulrich Neumann | 53,677 | <p>Try <code>NSolve</code> with restricted parameter range <code>1<= a,b,r <=50 </code></p>
<pre><code>NSolve[{1/9 a (a + 3) (a (r - 5) + 12 - r) ==1/3 b (9 + b (-14 + r) - r) , 50 >= a >= 1, 50 >= b >= 1 ,50 > r >= 1}, {a, b, r}, Integers]
(**{{a -> 3, b -> 6, r -> 24},
{a -> 5, b ... |
157,823 | <p>Check whether function series is convergent (uniformly):</p>
<p>$\displaystyle\sum_{n=1}^{+\infty}\frac{1}{n}\ln \left( \frac{x}{n} \right)$ for $x\in[1;+\infty)$</p>
<p>I don't know how to do that.</p>
| alejopelaez | 1,318 | <p>Fix an $x$ and think about this tail of the series
$$\sum_{n=3x}^{\infty}\frac{1}{n}\ln{\left(\frac{x}{n}\right)} = -\sum_{n=3x}^{\infty}\frac{1}{n}\ln{\left(\frac{n}{x}\right)} \leq -\sum_{n=3x}^{\infty}\frac{1}{n}\ln{3} \leq -\sum_{n=3x}^{\infty}\frac{1}{n} \leq 0$$
Now use what you know about $\sum_{n=1}^{\infty... |
656,458 | <p>If $A$ is an $n\times n$-matrix, $A^H$ is a Hermitian Matrix and $A^S$ is a Skew Hermitian, show $A=A^H+A^S$.</p>
<p>I am having trouble working with these so far and really cannot find many characteristics except the definitions. A Hermitian is made up of reals on the diagonal and is $A^*=A$. It is skew hermitian... | Ben Grossmann | 81,360 | <p>I think this question can be answered in a fairly straightforward fashion using the definitions
$$
A^H = \frac 12 (A + A^*)\\
A^S = \frac 12 (A - A^*)
$$</p>
|
3,137,160 | <blockquote>
<p>Determine which of the following are subspaces of <span class="math-container">$3 \times 3$</span> matrix <span class="math-container">$M$</span>
all <span class="math-container">$3 \times 3$</span> matrices <span class="math-container">$A$</span> such that the trace of <span class="math-container">... | clark | 33,325 | <p>It suffices to show that a group <span class="math-container">$G$</span> such that <span class="math-container">$|G|=p^k$</span> has an element <span class="math-container">$a$</span> of order <span class="math-container">$p$</span>.</p>
<p>We can do this by induction. For <span class="math-container">$k=1$</span> ... |
1,047,263 | <p>I used to do this on my calculators and it never worked! I think it's because you can't multiply any number by itself to get a negative number. Is that even true? I think it is! I've tried it out and it never worked! Look here:$$0.5\cdot0.5=0.25$$$$0\cdot0=0$$$$-1\cdot-1=1$$$$2.1\cdot2.1=4.41$$$$-7.5\cdot-7.5=5... | ConMan | 82,793 | <p>You are (almost) correct!</p>
<p>It is a property of the real numbers that any number squared is positive, i.e. $\forall x \in \mathbb{R}, x^2 \ge 0$. So all positive real numbers have two square roots - a positive one and a negative one (although the square root <em>function</em> is defined by convention to be the... |
225,253 | <p>Simple question, I just cannot find something that explains it right out and to the point without giving a huge confusing explanation. The question that I am struggling with is to determine a limit of a function if it exists.</p>
<blockquote>
<p>Find: <span class="math-container">$$\lim_{x\to2}{f(x)},$$</span></p>
<... | Yoni Rozenshein | 36,650 | <p>The limit from either side as $x\to2$ exists and equals $4$, so the overall limit as $x\to2$ exists and equals $4$.</p>
<p>Limits completely ignore the value of the function at the given point, as the definition of a limit only cares about what happens in a <em>punctured neighborhood</em> of that point.</p>
<p>The... |
466,722 | <blockquote>
<p>Find a function $f$ and a number $a$ such that:
$$
6+\int_{a}^{x}\frac{f(t)}{t^2}\:\mathrm{d}t=2\sqrt{x}
$$
For all $x>0$</p>
</blockquote>
<p>From Fundamental Theorem of Calculus section. Having some trouble with this. Any help?</p>
| dajoker | 72,504 | <p>Hint ::</p>
<p>Just differentiate, get an expression for $f$ and then substitute back to obtain the value of $a$.</p>
<p><em>EDIT</em> : </p>
<p>To differentiate the integral, use the property :
$$\dfrac d{dx}\large\int^{g(x)}_{f(x)}h(t)dt=h(g(x)).g'(x)-h(f(x)).f'(x)$$</p>
|
1,657,664 | <p>Struggling with a homework problem here and can't understand logically which one would be correct (each has different truth tables). I need to express the following statement using quantifiers, variables, and the predicates M(s), C(s), and E(s) </p>
<blockquote>
<p>"No computer science students are engineering st... | ThisIsNotAnId | 24,567 | <p><strong>HINT:</strong></p>
<p>For every student, his/her being a computer science student guarantees that they are not engineering students.</p>
<p><strong>EDIT:</strong></p>
<p>Your second logical statement claims every student is an engineering major <em>and</em> not a computer science major.</p>
<p>Your first... |
3,815,640 | <p>what is the most efficient way to calculate the argument of
<span class="math-container">$$
\frac{e^{i5\pi/6}-e^{-i\pi/3}}{e^{i\pi/2}-e^{-i\pi/3}}
$$</span> without calculator ?</p>
<p>i tried to use <span class="math-container">$\arg z_1-\arg z_2$</span> but the argument of <span class="math-container">$e^{i5\pi/6... | Bernard | 202,857 | <p>First factor out <span class="math-container">$\mathrm e^{-\tfrac{i\pi}3}$</span> both in the numerator and denominator:
<span class="math-container">$$\frac{\mathrm e^{\tfrac{5i\pi}6}-\mathrm e^{-\tfrac{i\pi}3}}{\mathrm e^{\tfrac{i\pi}2}-\mathrm e^{-\tfrac{i\pi}3}} =
\frac{\mathrm e^{\tfrac{5i\pi}6+\tfrac{i\pi}3}-1... |
2,232,390 | <p>$X$ is a subset of $\mathbb{R}$. It contains $\sqrt{3}$ and $\sqrt{2}$ and is closed under addition, subtraction and multiplication.</p>
<ol>
<li>Prove that $X$ contains $\sqrt{8}$.</li>
<li>Prove that $X$ contains $1$.</li>
<li>Prove that $X$ contains $\frac{1}{\sqrt{2}+1}$.</li>
<li>Is it true that $X$ is necessa... | Brenton | 226,184 | <p>Why not just compute the cdf and take a derivative:</p>
<p>$P(\frac{X_1}{X_2} < a) = P(X_2 > \frac{1}{a}X_1) = \int_0^\infty P(X_2 > \frac{1}{a}x)f_{X_1}(x)dx = \int_0^\infty e^{-3x/a} 3e^{-3x} dx = \frac{a}{a+1}$</p>
<p>Take a derivative with respect to a and you have the pdf of $f_{\frac{X_1}{X_2}}(a)$<... |
3,202,955 | <blockquote>
<p><strong>Note:</strong> Please do not give a solution; I would prefer guidance to help me complete the question myself. Thank you.</p>
</blockquote>
<hr>
<p>I am having trouble understanding and finding the continuous and residual spectrum. I am working through the following problem:</p>
<p>Let <spa... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$\lambda \in \sigma_r(T_{\alpha})$</span> iff <span class="math-container">$\lambda I-T_{\alpha}$</span> is injective and there is a non zero element <span class="math-container">$y$</span> orthogonal to <span class="math-container">$im(\lambda I-T_{\alpha})$</span> which means <span ... |
13,030 | <p>At work, we were discussing when is it the best time to change to winter tires for bikes and/or cars.</p>
<p>Using <code>WeatherData[]</code> and <code>DateListPlot[]</code>, it was fairly straightforward for me to create the diagram below:</p>
<p><img src="https://i.stack.imgur.com/Y5wNT.png" alt="Mean temperatur... | image_doctor | 776 | <p>Here is a slightly different approach using <code>GatherBy</code>.</p>
<p>Reformat dates to be just {month, day}:</p>
<pre><code>shortDates = {DateString[First@#, { "Month","Day",}], Last@#} & /@ cityTemp;
</code></pre>
<p>Group the values by day and month, count the days on which the temperature is not posi... |
1,449,776 | <p>I have always known that $a^n=a*a*a*.....$(n times)</p>
<p>Then what exactly is the meaning if $a^0$ and why will it be equal to $1$?</p>
<p>I have checked it in the internet but everywhere the solution is based on the principle that $a^m*a^n=a^{m+n}$ and when $n=0$ it will be $a^m$ and clearly $a^0$ is equal to $... | Enrico M. | 266,764 | <p>You can see that in this way:</p>
<p>$$a^0 = a^{m - m}$$</p>
<p>for every value of $m$. Using the properties of powers we have:</p>
<p>$$a^{m-m} = \frac{a^m}{a^m} = 1$$</p>
<p>Because the two terms are identical so they are canceled.
So</p>
<p>$$a^0 = 1$$</p>
|
707,193 | <p>The inequality to solve:
$$\left[\frac{-K^2+13K+44}{14-K}\right] > 0$$</p>
<p>How do I solve this?
I tried this:
$$
-K^2+13K+44 > 0 \quad \text{(multiply both sides by $14-K$)}\\
K^2-13K < 44\\
K(K-13) < 44
$$
Is this correct? Any way to get a more precise $K$ value? Thanks. </p>
| gt6989b | 16,192 | <p><strong>Hints</strong>
You have 3 cases: $14-K < 0, 14-K >0$, and the easy one $14-K = 0$.
In the first two, you end up with a different sign after multiplication. Take the one you used ($14-K>0$).</p>
<p>Then indeed $-K^2+13K+44>0$ but if you factor the left-hand side to get $(K-a)(K-b)<0$ for some ... |
374,881 | <p>I'd like to know how I can recursively (iteratively) compute variance, so that I may calculate the standard deviation of a very large dataset in javascript. The input is a sorted array of positive integers.</p>
| Rose Perrone | 32,062 | <p>I ended up using this incremental approach:</p>
<pre><code>function mean(array) {
var i = -1, j = 0, n = array.length, m = 0;
while (++i < n) if (a = array[i]) m += (a - m) / ++j;
return j ? m : undefined;
}
function variance(array, mean_value) {
if (!mean_value) return undefined;
var i = -1, j = 0, n... |
688,782 | <p>$$a_n=\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right) \cdots
\left(1-\frac{1}{n^2}\right)
$$</p>
<p>I have proved that this sequence is decreasing. However I am trying to figure out how to find its limit. </p>
| Community | -1 | <p>We write</p>
<p>$$a_n=\prod_{k=2}^n\left(1-\frac{1}{k^2}\right)=\prod_{k=2}^n\frac{(k-1)(k+1)}{k^2}=\prod_{k=2}^n\frac{v_k}{v_{k+1}}$$
where
$$v_k=\frac{k-1}{k}$$
so by change of index
$$a_n=\frac{\displaystyle\prod_{k=2}^nv_k}{\displaystyle\prod_{k=2}^nv_{k+1}}=\frac{\displaystyle\prod_{k=2}^nv_k}{\displaystyle\pr... |
1,179,843 | <p>Proving $\sum_{n=1}^\infty \frac{\xi ^n}{n}$ is not uniformly convergent for $\xi \in (0,1)$.</p>
<p>I am trying to do the above. I have attempted to show it is not a cauchy sequence by considering $||\frac{\xi ^n}{n} ||_{\sup}$ but no avail. Any help please</p>
| RRL | 148,510 | <p>The series fails to converge uniformly if you can show there is some $\epsilon_0 > 0$ such that for every $N \in \mathbb{N}$,</p>
<p>$$\sup_{\xi \in (0,1)} \sum_{n=N}^\infty\frac{\xi^n}{n}\geqslant \epsilon_0.$$</p>
<p>Notice that</p>
<p>$$\sup_{\xi \in (0,1)} \sum_{n=N}^\infty\frac{\xi^n}{n}\geqslant \sup_{\x... |
4,589,921 | <p>Let <span class="math-container">$X$</span> be a set and <span class="math-container">$d : X \times X \to \mathbb R$</span> be a function satisfying the following conditions:</p>
<p><span class="math-container">$d(x, x) = 0$</span> whatever <span class="math-container">$x \in X$</span>;</p>
<p><span class="math-cont... | Thomas | 89,516 | <p><span class="math-container">$0=d(x,x)\le d(x,y)+d(y,x)=d(x,y)+d(x,y)=2d(x,y)$</span></p>
<p>where all properties 1,2 and 3 have been used.</p>
|
4,589,921 | <p>Let <span class="math-container">$X$</span> be a set and <span class="math-container">$d : X \times X \to \mathbb R$</span> be a function satisfying the following conditions:</p>
<p><span class="math-container">$d(x, x) = 0$</span> whatever <span class="math-container">$x \in X$</span>;</p>
<p><span class="math-cont... | latrys | 1,126,389 | <p>You don't say how you establish
<span class="math-container">$$
d(x,y)≤d(x,y)+d(x,y)
$$</span>
but once you <em>do</em> have this, the following steps are valid. In general, not all quantities <span class="math-container">$r \in \mathbb{R}$</span> satisfy <span class="math-container">$r \le r + r$</span>; for exampl... |
3,929,703 | <p>so, we know how to solve if the question was only <span class="math-container">$5$</span> different rings in <span class="math-container">$4$</span> different fingers, which is <span class="math-container">$4^5$</span>. but what if internal order of rings within the finger matters, is this counted in this answer or ... | saulspatz | 235,128 | <p>There are <span class="math-container">$5!=120$</span> ways to arrange the rings in order. Divide them into <span class="math-container">$4$</span> groups and place the first group, in order, on the first finger, the second group on the second finger, and so on.</p>
<p>We have five rings to place on <span class="ma... |
181,855 | <p>In the latest <a href="http://what-if.xkcd.com/113/" rel="noreferrer">what-if</a> Randall Munroe ask for the smallest number of geodesics that intersect all regions of a map. The following shows that five paths of satellites suffice to cover the 50 states of the USA:
<img src="https://i.stack.imgur.com/gyfYt.png" al... | David Eppstein | 440 | <p>It is known to be NP-hard to cover regions (or even just points) with a minimum number of lines. For the Euclidean plane, see Megiddo, Nimrod and Tamir, Arie (1982), "On the complexity of locating linear facilities in the plane", <em>Oper. Res. Lett.</em> 1 (5): 194–197, <a href="https://doi.org/10.1016/01... |
3,106,550 | <p>Determine the values of a for which the following system of linear equations has no solutions, a unique solution, or infinitely many solutions. You can select 'always', 'never', '<span class="math-container">$a = $</span>', or '<span class="math-container">$a \neq$</span>', then specify a value or comma-separated li... | Bernard | 202,857 | <p>The criterion to have solutions is is that the matrix of coefficients of the linear system and the augmented matrix have the same rank. This common rank is then the <code>codimension</code> of the affine space of solutions. </p>
<p>So let's perform row reduction on the augmented matrix:
<span class="math-container"... |
1,580,270 | <p>Consider the groups $G = \{0,1,2\} = \mathbb Z_3$ and $H = \{a,b,c\}$
given by the following multiplication tables:</p>
<p><a href="https://i.stack.imgur.com/hXgBb.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hXgBb.jpg" alt="enter image description here"></a></p>
<p>The first one isn't really... | Lubin | 17,760 | <p>First, find the identity in each. In your first example, it’s “$0$”, while in the second it’s “$b$”. How to see this? It’s the element whose row and column matches the labeling rows. Now see whether you can match some nonidentity element in the first example to a nonidentity element in the second so that you force a... |
659,254 | <p>Say $X_1, X_2, \ldots, X_n$ are independent and identically distributed uniform random variables on the interval $(0,1)$.</p>
<p>What is the product distribution of two of such random variables, e.g.,
$Z_2 = X_1 \cdot X_2$?</p>
<p>What if there are 3; $Z_3 = X_1 \cdot X_2 \cdot X_3$?</p>
<p>What if there are $n$ ... | heropup | 118,193 | <p>We can at least work out the distribution of two IID ${\rm Uniform}(0,1)$ variables $X_1, X_2$: Let $Z_2 = X_1 X_2$. Then the CDF is $$\begin{align*} F_{Z_2}(z) &= \Pr[Z_2 \le z] = \int_{x=0}^1 \Pr[X_2 \le z/x] f_{X_1}(x) \, dx \\ &= \int_{x=0}^z \, dx + \int_{x=z}^1 \frac{z}{x} \, dx \\ &= z - z \log ... |
3,276,984 | <p>Why we divide the small difference 'd√x' by the difference in area 'dx' Where we normally divide the difference in area by the small difference ?<a href="https://i.stack.imgur.com/LDKB7.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LDKB7.jpg" alt="enter image description here"></a></p>
<p>My h... | Fareed Abi Farraj | 584,389 | <p>When talking about derivatives here, we will be studying the variation of a certain function(<span class="math-container">$x^2$</span>, <span class="math-container">$\sqrt{x}$</span>, ...) with respect to the variable <span class="math-container">$x$</span>. In other words, we will be studying how the function chang... |
2,698,357 | <p><strong>Draw a graph on $10$ vertices with no more than $20$ edges that contains no independent set of size $3$.</strong></p>
<p>So I was trying to draw the above graph but I was kind of stuck. What I basically did was draw a bipartite graph with $5$ vertices on each side and then $20$ edges randomly connecting bet... | Robert Z | 299,698 | <p>Take $G=K_5+ K_5$ (the disjoint union of two 5-cliques). Then $G$ has $2\cdot 5=10$ vertices and $2\cdot \binom{5}{2}=20$ edges and any independent set of $G$ has cardinality less than $3$ because it has no more than one vertex from each $K_5$. </p>
<p>P.S. I don't think that by selecting $20$ edges in $K_{5,5}$ w... |
1,023,830 | <p><img src="https://i.stack.imgur.com/X5aCr.png" alt="AB,AC,BC and h are known"></p>
<p><strong>AB,AC,BC</strong> and <strong>h</strong> are known and its a isosceles triangle
<strong>how to find angle BAC?</strong></p>
| Mark Fantini | 88,052 | <p>One of the properties of the Laplace transform is that</p>
<p>$$\mathcal{L} \left( \frac{f(t)}{t} \right) = \int_s^{+ \infty} F(s) \, ds.$$</p>
<p>This means that</p>
<p>$$\mathcal{L} \left( \frac{\cosh(at)}{at} \right) = \frac{1}{a} \int_s^{+ \infty} \mathcal{L}(\cosh(at)) \, ds.$$</p>
<p>Take from here.</p>
|
1,023,830 | <p><img src="https://i.stack.imgur.com/X5aCr.png" alt="AB,AC,BC and h are known"></p>
<p><strong>AB,AC,BC</strong> and <strong>h</strong> are known and its a isosceles triangle
<strong>how to find angle BAC?</strong></p>
| graydad | 166,967 | <p>By definition,$$\mathcal{L}\left\{\frac{\cosh(at)}{at}\right\}=\int_0^\infty e^{-st}\frac{\cosh(at)}{at}dt \\ = \frac{1}{2a}\int_0^\infty e^{-st}\frac{e^{at}+e^{-at}}{t}dt \\ = \frac{1}{2a}\int_0^\infty \frac{e^{t(a-s)}+e^{-t(a+s)}}{t}dt \\=\frac{1}{2a} \int_0^\infty \frac{e^{t(a-s)}}{t}dt+ \frac{1}{2a} \int_0^\i... |
28,955 | <p>I need to crack a stream cipher with a repeating key.</p>
<p>The length of the key is definitely 16. Each key can be any of the characters numbered 32-126 in ASCII.</p>
<p>The algorithm goes like this:</p>
<p>Let's say you have a plain text:</p>
<p>"Welcome to Q&A for people studying math at any level and pr... | Seth | 229,250 | <p>The other answers can work but they make this a much harder problem than it is.</p>
<p>The first step it to determine the key length with IC or the Chi Test, but you seem to have done and determined the key to be 16 bytes.</p>
<p>After that, take the first byte and one every 16 after that. If the message is standa... |
1,134,510 | <p>Regarding My Background I have covered stuff like </p>
<p>1.Single Variable Calculus</p>
<p>2.Multivariable Calculus (Multiple Integration,Vector Calculus etc) (Thomas Finney)</p>
<p>3.Basic Linear Algebra Course (Containing Vector spaces,Linear Transformation)</p>
<p>4.Ordinary Differential Equation</p>
<p>5.R... | Tamim Addari | 83,311 | <p>Well , the answer is </p>
<p>$$(26*25)*(\frac{7!}{3!*4!})$$</p>
<p>Now , why would your answer be wrong ? cause you are not considering that you can you can choose only 2 at a time , then the rearrangement depends on only what number you choosed for the first time . after you choose 2 letters , it is the problem ... |
1,830,989 | <p>so while playing around with circles and triangles I found 2-3 limits to calculate the value of $ \pi $ using the <em>sin, cos and tan</em> functions, I am not posting the formula for obvious reasons.<br>
My question is that is there another infinite series or another way to define the trig functions when the value ... | Andrei | 331,661 | <p>For $x$ in radians
$$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}+...=\sum_{i=0}^\infty\frac{(-1)^i x^{2i+1}}{(2i+1)!}$$
To transform the angle $x$ into degrees, use $y=x\cdot180/\pi$. So for the angle in degrees:
$$\sin(y)=\sum_{i=0}^\infty\frac{(-1)^i (180 y/\pi)^{2i+1}}{(2i+1)!}$$</p>
|
130,914 | <p>I dont know how to proceed with solving $$\sum_{i=1}^{n}i^{k}(n+1-i).$$ Please give advise.</p>
| andreimf | 28,948 | <p>$H_n^{(k)}=\sum_{i=1}^n i^{-k}$ is by definition the $n$-th harmonic number of order $k$. Thus,
$$
\sum_{i=1}^n i^{k} (n+1-i) = (n+1) H_n^{(-k)}- H_n^{(-k-1)}.
$$
I don't think it can be simplified further, at least considerably and for any $n$ and $k$. Is this what you meant by "solving"?</p>
|
130,914 | <p>I dont know how to proceed with solving $$\sum_{i=1}^{n}i^{k}(n+1-i).$$ Please give advise.</p>
| pravin | 29,006 | <p>This is a prefix sum of natural numbers. The solution to this is</p>
<p>$$\binom{n+k+1}{k+2}$$</p>
|
2,952,028 | <p>The question asks: Find the values of k for which the line</p>
<p><span class="math-container">$y=2x-k$</span> is tangent to the circle with equation <span class="math-container">$x^2+y^2=5$</span></p>
<p>So I started by substituting,</p>
<p><span class="math-container">$x^2+(2x-k)^2=5$</span></p>
<p><span class... | Michael Rozenberg | 190,319 | <p>The distance from the point <span class="math-container">$(0,0)$</span> to the line <span class="math-container">$2x-y-k=0$</span> should be equal to the radius of the circle.</p>
<p>Thus, <span class="math-container">$$\frac{|2\cdot0-0-k|}{\sqrt{2^2+1^2}}=\sqrt5$$</span> or
<span class="math-container">$$|k|=5.$$<... |
3,787,111 | <p>Sets can have minimal and least elements, and they are two different things, for example:
given the set <span class="math-container">$A=\{\{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{1,3\},\{1,2,3\}\}$</span> and the subset relation, this set has <span class="math-container">$\{1\}$</span> as a minimal element but not as a l... | Charlie Vanaret | 741,916 | <p>You can use a very simple "interval analysis" evaluation approach:</p>
<p>For <span class="math-container">$X = [-2, 3]$</span>, we have <span class="math-container">$f(X) = [-2, 3]^2 + 3 = [0, 9] + 3$</span> (since <span class="math-container">$x \mapsto x^2$</span> maps to nonnegative numbers).</p>
|
3,787,111 | <p>Sets can have minimal and least elements, and they are two different things, for example:
given the set <span class="math-container">$A=\{\{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{1,3\},\{1,2,3\}\}$</span> and the subset relation, this set has <span class="math-container">$\{1\}$</span> as a minimal element but not as a l... | Arthur | 15,500 | <p>The answer is A because the end points of the domain don't necessarily correspond to the end points of the range. Consider <span class="math-container">$f(0)$</span> compared to <span class="math-container">$f(-2)$</span> or <span class="math-container">$f(3)$</span>.</p>
|
152,582 | <p>I have tried to use <code>OpenRead</code> for my application as it appears to be less of a burden on the memory.</p>
<p>I <code>OpenRead</code> a .csv file to extract coordinates in the form {x,y} in the following way</p>
<pre><code>ClearAll[f];
f = OpenRead["mathfile.csv"];
g = ReadList[f, String, RecordL... | BRS | 51,362 | <p>I got some good performance in the following manner. My data was originally an <code>XLSX</code>. I saved it as a MSDOS txt file. Unfortunately, the {x,y} data had long runs punctuated by a text saying 'Segment 1:' e.t.c.. This was stopping the <code>ReadList</code> from working..So after removing the text and white... |
367,669 | <p><img src="https://i.stack.imgur.com/zQFyC.jpg" alt="enter image description here"></p>
<p>This is probably a very simple questions but I am not clear on Möbius transformations and how to solve this problem. I'd appreciate if somebody can point me towards a method to do these sort of questions or a webpage that expl... | Davide Giraudo | 9,849 | <p>Consider the sequence
$$0,0,1,\color{green}{0,1,2},\color{red}{0,1,2,3},0,1,2,3,4, 0,1,2,3,4,5,\dots$$</p>
|
2,316,362 | <p>If $G$ is a Lie Group, a representation of $G$ is a pair $(\rho,V)$ where $V$ is a vector space and $\rho : G\to GL(V)$ is a group homomorphism.</p>
<p>Similarly, if $\mathfrak{g}$ is a Lie Algebra, a representation of $\mathfrak{g}$ is a Lie Algebra homomorphism $\rho : \mathfrak{g}\to \mathfrak{gl}(V)$ to the Lie... | José Carlos Santos | 446,262 | <p>Each finite-dimensional linear representation of a Lie group <span class="math-container">$G$</span> on a vector space <span class="math-container">$V$</span> induces a representation of its Lie algebra <span class="math-container">$\mathfrak g$</span> on <span class="math-container">$V$</span>. If <span class="math... |
1,098,253 | <p>I have got some trouble with proving that for $x\neq 0$:
$$
\frac{\arctan x}{x }< 1
$$
I tried doing something like $x = \tan t$ and playing with this with no success.</p>
| GEdgar | 442 | <p>idm's solution written differently... </p>
<p>For $x \ge 0$,
$$
\arctan x = \int_0^x \frac{dt}{1+t^2}
$$
(in some texts, this may even be the definition)</p>
<p>For $t>0$,
$$
\frac{1}{1+t^2} < 1
$$
so we conclude
$$
\arctan x \le \int_0^x 1\,dt = x
$$
with strict inequality except for $x=0$. </p>
<p>Simil... |
1,098,253 | <p>I have got some trouble with proving that for $x\neq 0$:
$$
\frac{\arctan x}{x }< 1
$$
I tried doing something like $x = \tan t$ and playing with this with no success.</p>
| copper.hat | 27,978 | <p>Note that $\tan 0 = 0$, $\tan' 0 = 1$ and $\tan'x = {1 \over \cos^2 x} > 1$ for $0<|x|< {\pi \over 2}$.</p>
<p>Then $\arctan 0 = 0$; the inverse function theorem gives
$\arctan' 0 = 1$ and $0<\arctan' x < 1$ for $x \neq 0$.</p>
<p>Since $\arctan x = \arctan 0 + \arctan'( \xi ) x$ for some $ \xi \in... |
2,929,025 | <p>Bill gave exams for the entrance at some specific gymnasium. <span class="math-container">$602$</span> students took part, which were classified, after the exams, in an ascending order, and the first <span class="math-container">$108$</span> students will be taken, which will accept to enter. Every student that has... | Phil H | 554,494 | <p>You have to take into consideration the number of ways you can choose <span class="math-container">$5$</span> from <span class="math-container">$112$</span> and the probability of remaining chosen <span class="math-container">$(.98)$</span>. Also its the probability of at least <span class="math-container">$5$</span... |
1,023,463 | <p>I know $x \sin(1/x)$ is not Lipschitz on $[0,1]$, but some experimentation makes me conjecture that it is $1/2$-Holder. What is a good way to prove this?</p>
| DuFong | 193,997 | <p>Check this paper and the first reference paper. <a href="https://arxiv.org/pdf/1407.6871.pdf" rel="nofollow noreferrer">https://arxiv.org/pdf/1407.6871.pdf</a></p>
|
3,348,178 | <p>I have three vectors in 3d that originate at a point. If I look at them along a line perpendicular to a plane that intersects two of them, how do I find the angles between those two vectors and the third one?</p>
<p>Clarification because this is frickin difficult to explain:</p>
<p><a href="https://i.stack.imgur.c... | FFjet | 597,771 | <p><strong>You are wrong at the first step:</strong>
<span class="math-container">$$
x=y \nLeftrightarrow \sqrt{x^2} =\sqrt{y^2}
$$</span></p>
|
300,944 | <p>Show that there are no intergers $x$ and $y$ such that</p>
<p>$P(x,y)=x^2-5y^2=2$</p>
<p>Hint from professor:</p>
<p>Consider the equation in a convenient $\mod (n)$ so that you end up with a polynomial in a single variable. Then proceed as solving number of congruence.</p>
<hr>
<p>Im not sure how to approach t... | pritam | 33,736 | <p>Note that, every perfect square is either $0$ or $1$ modulo $4$. Now check all the cases to see that $x^2-5y^2$ is never equal to $2$ modulo $4$.</p>
|
1,599,890 | <blockquote>
<p>Let $a_n$ be the number of those permutation $\sigma $ on $\{1,2,...,n\}$ such that $\sigma $ is a product of exactly two disjoint cycles. Then find $a_4$ and $a_5$.</p>
</blockquote>
<p>Calculating $a_4$: Possible cases which can happen are $(12)(34),(13)(24),(14)(23)$, any cycle of the form $(123)... | joriki | 6,622 | <p>From the possible answers, it appears that the question intended to ask how many permutations have exactly two cycles in their cycle decomposition, including the $1$-cycles.</p>
<p>If the smaller cycle has $k$ elements, the greater has $n-k$, and $j$ particular elements can form $(j-1)!$ different $j$-cycles. Thus ... |
298,912 | <p>I was reading some basic information from Wiki about category theory and honestly speaking I have a very weak knowledge about it. As it sounds interesting, I will go into the theory to learn more if it is actually useful in practice.</p>
<p>My question is to know if category theory has some applications in practice... | Godot | 38,875 | <p>Category theory is a good and powerful language capable of expressing various concepts of purely algebraical nature.</p>
<p>But it is a terrible tool for actually solving problems.</p>
<p>To convince yourself that the last statement is true try to think about a proof of a theorem from another branch of mathematics... |
298,912 | <p>I was reading some basic information from Wiki about category theory and honestly speaking I have a very weak knowledge about it. As it sounds interesting, I will go into the theory to learn more if it is actually useful in practice.</p>
<p>My question is to know if category theory has some applications in practice... | Community | -1 | <p>The state-of-the-art real world dimensionality reduction technique UMAP is based among other things on category theory:</p>
<p><a href="https://arxiv.org/abs/1802.03426" rel="nofollow noreferrer">https://arxiv.org/abs/1802.03426</a></p>
<p><a href="https://github.com/lmcinnes/umap" rel="nofollow noreferrer">https:... |
41,836 | <p>Nakayama's lemma is as follows:</p>
<blockquote>
<p>Let <span class="math-container">$A$</span> be a ring, and <span class="math-container">$\frak{a}$</span> an ideal such that <span class="math-container">$\frak{a}$</span> is contained in every maximal ideal. Let <span class="math-container">$M$</span> be a finitel... | Arrow | 69,037 | <p>I think the following proof is valid and avoids both determinants and maximal ideals. The cost is induction over all <span class="math-container">$A$</span>-modules generated by <span class="math-container">$n$</span> elements.</p>
<p><strong>Nakayama.</strong> Let <span class="math-container">$J$</span> be the Jaco... |
148,809 | <p>Let $M$ be a complete Riemannian manifold, does there exists a positive non-constant harmonic function $f \in L^1(M)$? Who can answer me or give me a counter example? Thank you very much!</p>
| Neal | 20,569 | <p>No. </p>
<p>The second result on Google gives <a href="http://intlpress.com/JDG/archive/1984/20-2-447.pdf" rel="nofollow">http://intlpress.com/JDG/archive/1984/20-2-447.pdf</a>, "Uniqueness of $L^1$ solutions for the Laplace equation and heat equation on Riemannian manifolds" by Peter Li, J Diff Geo 20 (1984) 447-... |
4,613,449 | <p>I first calculated it using the substitution method and got the result <span class="math-container">$\frac 1 2\ln^2x+C$</span>, <span class="math-container">$C\in \Bbb R$</span>, but I am getting a wrong result attempting to use per parts on the same problem. Is there something I am missing? Is there a reason per pa... | Henry | 6,460 | <p>Ignoring the indefinite integration constant temporarily,</p>
<p>if you are attempting to apply <span class="math-container">$\int u\, dv = uv-\int v \, du$</span> here,</p>
<p>then with <span class="math-container">$u=\log_e(x)$</span> and <span class="math-container">$dv=\frac1x\, dx$</span>, and thus <span class... |
4,613,449 | <p>I first calculated it using the substitution method and got the result <span class="math-container">$\frac 1 2\ln^2x+C$</span>, <span class="math-container">$C\in \Bbb R$</span>, but I am getting a wrong result attempting to use per parts on the same problem. Is there something I am missing? Is there a reason per pa... | Abezhiko | 1,133,926 | <p>Integration by parts brings :
<span class="math-container">$$
I := \int\frac{1}{x}\,\ln(x)\,\mathrm{d}x = \ln^2(x) - \int\ln(x)\,\frac{1}{x}\,\mathrm{d}x
$$</span>
hence <span class="math-container">$2I = \ln^2(x)$</span> and the desired result.</p>
|
179,659 | <p>I want to assign each cell in my current notebook a tag so that I can rerun cells with specific tags later in the notebook <a href="https://mathematica.stackexchange.com/a/71333/44420">(Using a method similar to this answer)</a>.</p>
<p>My questions are</p>
<ul>
<li>How do I assign cells in my current notebook tag... | Carl Woll | 45,431 | <p>You can use the menu item <code>Cell | CellTags</code> to do this. If you want to inspect and modify <a href="http://reference.wolfram.com/language/ref/CellTags" rel="noreferrer"><code>CellTags</code></a> programmatically, you can use <a href="http://reference.wolfram.com/language/ref/CurrentValue" rel="noreferrer">... |
234,409 | <p>I'm trying to obtain the coordinates of the border of the continents. I need this information to be ordered such that when I do, for example,</p>
<pre><code>ListLinePlot[data]
</code></pre>
<p>It does not yield a messed up image, as happens for disordered points. Initially I was trying by highlighting points on imag... | creidhne | 41,569 | <p>Although <code>ListLinePlot</code> can plot coordinates for geographic data, I recommend that you avoid this method and use <code>GeoGraphics</code> instead. Some of the advantages are:</p>
<ul>
<li>selecting map features, e.g, islands</li>
<li>using map projections (with the <code>GeoProjection</code> option)</li>
... |
4,516,356 | <p>In Ch. 20 of Spivak's <em>Calculus</em>, he shows that the remainder terms for <span class="math-container">$\arctan$</span> and <span class="math-container">$\log{(1+x)}$</span> become large with the order of the Taylor polynomial used to approximate these functions. Thus these approximations</p>
<blockquote>
<p>ar... | Community | -1 | <p>It is easy to show, using differentiation the formula</p>
<p><span class="math-container">$arctanx+arctan(\dfrac{1}{x})=\dfrac{\pi}{2}$</span>. Hence, knowing the value</p>
<p>for <span class="math-container">$0<x<1$</span> we obtain the value of <span class="math-container">$arctanx$</span> for <span class="m... |
216,803 | <p>Let $X$ be a nonempty set. Find the topology $\tau$ on $X$ satisfying one of the following conditions:</p>
<ul>
<li><p>$(X, \tau)$ has the largest number of compact subsets.</p></li>
<li><p>$(X, \tau)$ has the least number of compact subsets.</p></li>
<li><p>$(X, \tau)$ has the largest number of connected subsets.<... | user123123 | 45,121 | <p>Try thinking about when $X$ is given the discrete topology, and see if any of them become trivial. For example, if $X$ is discrete, then every subset is dense. Then you should be able to use arguments involving the discrete and trivial topologies to answer the other questions.</p>
|
216,803 | <p>Let $X$ be a nonempty set. Find the topology $\tau$ on $X$ satisfying one of the following conditions:</p>
<ul>
<li><p>$(X, \tau)$ has the largest number of compact subsets.</p></li>
<li><p>$(X, \tau)$ has the least number of compact subsets.</p></li>
<li><p>$(X, \tau)$ has the largest number of connected subsets.<... | joriki | 6,622 | <p>In each case, the property in question has a definite behaviour with respect to <a href="http://en.wikipedia.org/wiki/Comparison_of_topologies" rel="nofollow">comparison of topologies</a>, which you can determine by going through its definition and checking whether its conditions are easier or harder to fulfil when ... |
1,851,084 | <p>I have to solve the following problem:
find the matrix $A \in M_{n \times n}(\mathbb{R})$ such that:
$$A^2+A=I$$ and $\det(A)=1$.
How many of these matrices can be found when $n$ is given?
Thanks in advance.</p>
| boaz | 83,796 | <p>Note that once you have one matrix that satisfies $A^2+A=I$, you have infinity many, since for every invertible matrix $P$ we get
$$
(P^{-1}AP)^2+(P^{-1}AP)=I\qquad\text{and}\qquad \det(P^{-1}AP)=1
$$
Now, if $4\mid N$, then choose the matrix
$$
A=diag(\phi,-1-\phi,\ldots,\phi,-1-\phi)
$$
where $\phi=\frac{\sqrt{5}-... |
1,356,932 | <blockquote>
<p><strong>Problem.</strong> Let $h\in C(\mathbb{R})$ be a continuous function, and let
$\Phi:\Omega:=[0,1]^{2}\rightarrow\mathbb{R}^{2}$ be the map defined
by \begin{align*}
\Phi(x_{1},x_{2}):=\left(x_{1}+h(x_{1}+x_{2}),x_{2}-h(x_{1}+x_{2})\right) \tag{1}
\end{align*}
What is the (Lebesgue) meas... | Batominovski | 72,152 | <p>Since $\int_0^1\,g(x)\,\text{d}x=0$ and $\int_0^1\,x\,g(x)\,\text{d}x=0$, we have
$$\int_0^1\,x^2\,g(x)\,\text{d}x=\int_0^1\,\left(x^2-x+\frac{1}{6}\right)\,g(x)\,\text{d}x\leq\int_0^1\,\left|x^2-x+\frac{1}{6}\right|\,\big|g(x)\big|\,\text{d}x\,.$$
By the Cauchy-Schwarz Inequality,
$$\int_0^1\,x^2\,g(x)\,\text{d}x\l... |
548,902 | <p>I need to find the limit:
$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\left[ {{{(a + {1 \over n})}^2} + {{(a + {2 \over n})}^2} + ... + {{(a + {{n - 1} \over n})}^2}} \right]$ </p>
<p>any ideas here? I've tried to use "squeeze theorem" but with no luck.. </p>
| Luiz Cordeiro | 58,818 | <p>Maybe you're not familiar with the definition of differentiability in higher dimensions: see <a href="http://en.wikipedia.org/wiki/Differentiable_function#Differentiability_in_higher_dimensions" rel="nofollow">Wikipedia</a>, for example. The idea is this: in one-variable calculus, the derivative of a function at a p... |
3,104,051 | <p>I have the work of the proof done, but at the end after showing
<span class="math-container">$3^{2(n+1)} - 1=9(3^{2n} - 1)+8$</span>
I make the statement that since <span class="math-container">$9(3^{2n} - 1)$</span> is a multiple of 8 and 8 is a multiple of 8 then <span class="math-container">$3^{2n} - 1$</span> is... | nonuser | 463,553 | <p>Induction hypothesis: <span class="math-container">$$8\mid 9^n-1$$</span>so <span class="math-container">$$ 8\mid 9(9^n-1) = 9^{n+1}-9$$</span></p>
<p>but then <span class="math-container">$$8\mid (9^{n+1}-9) +8 = 9^{n+1}-1$$</span></p>
|
3,390,979 | <blockquote>
<p>Let <span class="math-container">$\sum_i^na_i=n$</span>, <span class="math-container">$a_i>0$</span>. Then prove that <span class="math-container">$$ \sum_{i=1}^n\left(\frac{a_i^3+1}{a_i^2+1}\right)^4\geq n $$</span></p>
</blockquote>
<p>I have tried AM-GM, Cauchy-Schwarz, Rearrangement etc. but n... | Michael Rozenberg | 190,319 | <p>You need to use another queue: </p>
<p>By Rearrangement, AM-GM and C-S we obtain: <span class="math-container">$$\sum_{i=1}^n\left(\frac{a_i^3+1}{a_i^2+1}\right)^4\geq \ \sum_{i=1}^n\left(\frac{a_i+1}{2}\right)^4\geq\sum_{i=1}^na_i^2=\frac{1}{n}\sum_{i=1}^n1^2\sum_{i=1}^na_i^2\geq\frac{1}{n}\left(\sum_{i=1}^na_i\ri... |
4,552,955 | <p>I'm solving a probability problem, and I've ended up with this sum:</p>
<p><span class="math-container">$$\sum\limits_{k=0}^{n-a-b}\binom{n-a-b}{k}(a+k-1)!(n-a-k)!$$</span></p>
<p>WolframAlpha says I should get the answer <span class="math-container">$\frac{n!}{a\binom{a+b}{a}}$</span>, but I don't see how to get th... | epi163sqrt | 132,007 | <blockquote>
<p>We obtain
<span class="math-container">\begin{align*}
\color{blue}{\sum_{k=0}^{n-a-b}}&\color{blue}{\binom{n-a-b}{k}(a+k-1)!(n-a-k)!}\\
&=(n-1)!\sum_{k=0}^{n-a-b}\binom{n-a-b}{k}\binom{n-1}{n-a-k}^{-1}\tag{1}\\
&=n!\sum_{k=0}^{n-a-b}\binom{n-a-b}{k}\int_{0}^1z^{n-a-k}(1-z)^{a+k-1}\,dz\tag{2}... |
2,150,832 | <p>I don't understand this equation $\int_0^t ds \int_0^{t'} ds' \delta(s-s')= \min(t,t')$.
I tried to work with the property of the dirac delta function that $\int_a^b \delta(x-c)dx = 1$ if $c \in [a,b]$, but I can't see how I can obtain the minimum. Can someone help me? </p>
<p>Thank you in advance!</p>
| Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\n... |
2,964,897 | <blockquote>
<p>Using Leibniz on <span class="math-container">$\sum_{n=1}^\infty \sin(\pi \sqrt{n^2+1})$</span></p>
</blockquote>
<p>So the question actually is how to rewrite <span class="math-container">$\sin(\pi\sqrt{n^2+1})$</span> in the form of <span class="math-container">$(-1)^n\times a_n$</span> so that I c... | Community | -1 | <p><span class="math-container">$$\sin\left(\pi\sqrt{n^2+1}\right)=\sin\left(\pi n\sqrt{1+\frac1{n^2}}\right)\sim\sin\left(\pi n\left(1+\frac1{2n^2}\right)\right)=(-1)^n\sin\left(\frac\pi{2n}\right)\sim(-1)^n\frac{\pi}{2n}.$$</span></p>
|
307,545 | <p>If $\gcd(a,b)=1$, how can I find the values that $\gcd(a+b,a^2+b^2)$ can possibly take? I can't find a way to use any of the elemental divisibility and gcd theorems to find them. </p>
| Math Gems | 75,092 | <p>Put <span class="math-container">$\,\rm (a,b)=1\,$</span> below.</p>
<p><strong>Theorem</strong> <span class="math-container">$\rm\,\ \ (a\!+\!b,\ a^2\!+\!b^2)\, =\, (\color{#c00}{2a^2,\ \ 2ab,\ \ 2b^2},\ a\!+\!b)\, \overset{\rm\color{#c00}E}=\, (\color{#c50}{2(a,b)^2}\!,\ a\!+\!b)$</span></p>
<p><strong>Proof</str... |
307,545 | <p>If $\gcd(a,b)=1$, how can I find the values that $\gcd(a+b,a^2+b^2)$ can possibly take? I can't find a way to use any of the elemental divisibility and gcd theorems to find them. </p>
| N. S. | 9,176 | <p><span class="math-container">$$ \gcd(a+b,a^2+b^2) \mid \gcd((a+b)(a-b), a^2+b^2) = \gcd(a^2-b^2, a^2+b^2) \mid \gcd [ ( a^2+b^2)+ (a^2-b^2) , ( a^2+b^2)+ (a^2-b^2) ]=2 \gcd(a^2,b^2)=2$$</span></p>
<p>Now it is easy to check that both 1 and 2 are possible...</p>
|
7,223 | <p>I want to produce a <em>Mathematica</em> Computable Document in which <code>N</code> appears as a variable in my formulae. But <code>N</code> is a reserved word in the <em>Mathematica</em> language. Is there a way round this other than using a different symbol? It seems a severe limitation if you cannot use <em>Math... | Jens | 245 | <p>The methods suggested by David and István already do the job perfectly, but one could add something to the collection:</p>
<p>As described in the documentation on <a href="http://reference.wolfram.com/mathematica/tutorial/OperatorsWithoutBuiltInMeanings.html">Operators without Built-in Meanings</a>, there are some ... |
7,223 | <p>I want to produce a <em>Mathematica</em> Computable Document in which <code>N</code> appears as a variable in my formulae. But <code>N</code> is a reserved word in the <em>Mathematica</em> language. Is there a way round this other than using a different symbol? It seems a severe limitation if you cannot use <em>Math... | Sjoerd C. de Vries | 57 | <p>To continue with Ajasja's land mine theme, it's not so problematic to use <code>N</code> as a <em>symbol</em> in equations like this:</p>
<p><img src="https://i.stack.imgur.com/snqa9.png" alt="Mathematica graphics"></p>
<p>as long as you keep the following in mind:</p>
<ul>
<li>Don't try to use it in an assignmen... |
7,223 | <p>I want to produce a <em>Mathematica</em> Computable Document in which <code>N</code> appears as a variable in my formulae. But <code>N</code> is a reserved word in the <em>Mathematica</em> language. Is there a way round this other than using a different symbol? It seems a severe limitation if you cannot use <em>Math... | Mr.Wizard | 121 | <p>The question of reassigning build-in Symbols has come up a number of times. There is an approach not yet posted here, however <em>it is not recommended</em>. Nevertheless for the sake of completeness I shall illustrate it.</p>
<p>Because <a href="http://reference.wolfram.com/language/ref/N.html" rel="nofollow nor... |
4,181,442 | <p>I know this is a dumb question but...</p>
<p>Is <span class="math-container">$x ≠ 1,3,5$</span> the same as <span class="math-container">$x$</span> does not belong to {<span class="math-container">$1,3,5$</span>}, for example ?</p>
<hr />
<p>Sorry for the formatting.</p>
<p>Btw, anyone has a link with mathjax's comm... | herb steinberg | 501,262 | <p><span class="math-container">$x\notin (1,3,5)$</span> means that <span class="math-container">$x$</span> is not a member of a set. The set is defined as elements, which could be letters, intervals or anything else. <span class="math-container">$x\ne 1$</span>, etc. means that equality is defined which is no neces... |
160,169 | <p>Consider the following implementation of the complex square root:</p>
<pre><code>f[z_]:=Sqrt[(z - I)/(z + I)]*(z + I);
</code></pre>
<p>This implementation has branch points at $\lambda=\pm i$ and a (vertical) branch cut connecting them.</p>
<p>Then</p>
<pre><code>g[z_]:=Sinc[f[z]];
</code></pre>
<p>(recalling ... | José Antonio Díaz Navas | 1,309 | <p>You can try with:</p>
<pre><code>Series[PowerExpand@g[z], {z, I, 4}]
</code></pre>
<p>$1-\frac{1}{3} i (z-i)-\frac{1}{5} (z-i)^2+\frac{11}{315} i (z-i)^3+\frac{61 (z-i)^4}{5670}+O\left((z-i)^5\right)$</p>
<p>and</p>
<pre><code>SeriesCoefficients[PowerExpand@g[z], {z, I, 4}]
(* 61/5670 *)
</code></pre>
|
631,053 | <p>I have a container of 100 yellow items.</p>
<p>I choose 2 at random and paint each of them blue.</p>
<p>I return the items to the container.</p>
<p>If I repeat this process, on average how many cycles will I make before all 100 items are painted?</p>
<p>It is obviously 50 (100/2) if there is no replacement. But ... | Henry | 6,460 | <p>You could calculate the exact figure for the expected number using recursion and find the mean is about $258.32$ with a standard deviation of about $62.59$.</p>
<p>Alternatively, if you took the balls one at a time, this would be the <a href="http://en.wikipedia.org/wiki/Coupon_collector%27s_problem" rel="nofollow"... |
1,424,198 | <p>My mathematical logic textbook defines $\{x \ | \ \text {_} x \text {_} \ \}$, but I'm not sure what the $\text {_} x \text {_}$ means. </p>
<p>Do the _ just mean 'for any expression involving $x$', or is there something I'm missing?</p>
| Community | -1 | <p>The symbols in question just mean "any expression involving $x$."</p>
|
172,139 | <p>I have a data set, I am trying to join all the data by a line. But I am afraid the plot is not doing it properly. </p>
<p>This is the example (or almost) of my problem:</p>
<pre><code>data = {{0, π}, {π/2, π/2}, {π/2,
3 π/2}, {π, 0}, {π,
2 π}, {3 π/2, π/2}, {3 π/2,
3 π/2}, {2 π, π}};
</code></p... | Hugh | 12,558 | <p>Your data do lie on the boundary of a square. However their order is not going around the square. Here I plot them and number them. </p>
<pre><code>Graphics[{Point[data],
Table[Text[ToString[n], data[[n]], {1, 1}], {n, Length@data}]}]
</code></pre>
<p><img src="https://i.stack.imgur.com/Tq3C4.png" alt="Mathemat... |
4,292,427 | <p><span class="math-container">$$ \frac{d^{2}y}{dt^2}+ 2t \frac{dy}{dt}+ t y=0 ~~ \tag{1} $$</span></p>
<p>At least I know that in this case of ODE can be solved by finding out 2 particular solutions.</p>
<p>As those 2 particular solutions are known, the general solution for this ODE can be written as below form.... | Aleksas Domarkas | 562,074 | <p>Maple:
<span class="math-container">$$y \left( t \right) ={\it \_C1}\,{{\rm e}^{-t/2}}{{ \rm KummerM}\left(1/16,\,1
/2,\,-1/4\, \left( 2\,t-1 \right) ^{2}\right)}+{\it \_C2}\,{{\rm e}^{-
t/2}}{{\rm KummerU}\left(1/16,\,1/2,\,-1/4\, \left( 2\,t-1 \right) ^{2}
\right)}
$$</span></p>
|
3,518,719 | <p>Evaluate <span class="math-container">$$\lim_{n \to \infty} \sqrt[n^2]{2^n+4^{n^2}}$$</span></p>
<p>We know that as <span class="math-container">$n\to \infty$</span> we have <span class="math-container">$2^n<<2^{2n^2}$</span> and therefore the limit is <span class="math-container">$4$</span></p>
<p>In a more... | Narasimham | 95,860 | <p>We need to look at how Frenet-Serret relations are derived. Imagine a circular band or belt in a horizontal plane . All vectors <span class="math-container">$n$</span> normal to the band remain in a plane. and derivative <span class="math-container">$\dot b $</span> of bi-normal that is vertical up rotating in the n... |
3,348,780 | <p>I worked through <span class="math-container">$\int \frac{e^x}{(1-e^x)^2}dx$</span> using u-substitution, but my answer, <span class="math-container">$(1-e^x)^{-1}+C$</span> is incorrect. It should be <span class="math-container">$- \ln|1-e^x|+C$</span></p>
<blockquote>
<p><span class="math-container">$$\int \fra... | Allawonder | 145,126 | <p>Your answer is the correct one, as you can confirm by differentiating both. In general you can always confirm whether you got the right primitive by differentiation.</p>
|
3,077,312 | <p>The proof given in my book (and I came up with as well) is:</p>
<p><a href="https://i.stack.imgur.com/H6eqf.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/H6eqf.png" alt="Proof"></a></p>
<p>However, the part that throws me off is line #3 where they do <span class="math-container">$\Sigma A_{jk}... | Community | -1 | <p>What you have is that the <span class="math-container">$(i,j)$</span>-th element of <span class="math-container">$C^t$</span> is</p>
<p><span class="math-container">$$ \sum_k A_{jk}B_{ki} $$</span></p>
<p>where <span class="math-container">$A_{jk}$</span> is the <span class="math-container">$(j,k)$</span>-th eleme... |
59,932 | <p>So I need to make a plot of some ~2000 data points I have from a spreadsheet. I'm able to import the data just fine, it stores it like so</p>
<pre><code>sundat
</code></pre>
<blockquote>
<p>{{280.,0.082},{280.5,0.099},......{3995.,0.0087},{4000.,0.00868}}</p>
</blockquote>
<p>And I can call individual data sets... | kglr | 125 | <pre><code>sundat = {{280., 0.082}, {280.5, 0.099}, {3995., 0.0087}, {4000., 0.00868}};
conversionfactors = {2., .5};
data2 = Transpose[conversionfactors Transpose[sundat]]
</code></pre>
<p>or</p>
<pre><code>data2 = conversionfactors # & /@ sundat
</code></pre>
<p>both give</p>
<pre><code>(* {{560.,0.041},{561... |
59,932 | <p>So I need to make a plot of some ~2000 data points I have from a spreadsheet. I'm able to import the data just fine, it stores it like so</p>
<pre><code>sundat
</code></pre>
<blockquote>
<p>{{280.,0.082},{280.5,0.099},......{3995.,0.0087},{4000.,0.00868}}</p>
</blockquote>
<p>And I can call individual data sets... | Mr.Wizard | 121 | <p>Although I prefer kguler's double <code>Transpose</code> you could also use <a href="http://reference.wolfram.com/mathematica/ref/Inner.html" rel="nofollow noreferrer"><code>Inner</code></a>:</p>
<pre><code>sundat = {{280., 0.082}, {280.5, 0.099}, {3995., 0.0087}, {4000., 0.00868}};
Inner[Times, sundat, {2., .5}, ... |
2,466,527 | <p>Let $A$ be the matrix of $T:P_2\to P_2$ with respect to basis $B=\{v_1,v_2,v_3\}$. Find $T(v_1)$</p>
<p>$$A=\begin{bmatrix}
1 & 3 & -1 \\
2 & 0 & 5 \\
6 & -2 & 4
\end{bmatrix}$$</p>
<p>$v_1=3x+3x^2$</p>
<p>$v_2=-1+3x+2x^2$</p>
<p>$v_3=3+7x+2x^2$</p>
<hr>
<p>First part of question asks t... | Patrick Stevens | 259,262 | <p>Let $c=ab+a$ to make the numbers nicer; then $x_1 = 2$ and $x_n = c x_{n-1} - 1$.</p>
<p>Then $x_2 = 2c-1, x_3 = 2c^2-c-1, x_4 = 2c^3-c^2-c-1$.</p>
<p>I formulate an inductive hypothesis! $x_i = 2c^{i-1} - c^{i-2} - \dots - c - 1$.</p>
<p>This is easy to prove inductively. You can also simplify it a bit if you li... |
817,934 | <p>How to prove</p>
<p>$$\int\frac{12x\sin^{-1}x}{9x^4+6x^2+1}dx=-\frac{2\sin^{-1}x}{3x^2+1}+\tan^{-1}\left(\frac{2x}{\sqrt{1-x^2}}\right)+C$$</p>
<p>where $\sin^{-1}x$ and $\tan^{-1}x$ are inverse of trig functions. I don't know how to find the integral because of inverse of trig functions. I missed calc class twice... | Pranav Arora | 117,767 | <p>$$\int \frac{12\sin^{-1} x}{(3x^2+1)^2}\,dx=\sin^{-1}x\int \frac{12x}{(3x^2+1)^2}\,dx -\int \left(\frac{1}{\sqrt{1-x^2}}\int \frac{12x}{(3x^2+1)^2}\,dx\right)\,dx$$
To evaluate $\displaystyle \int \frac{12x}{(3x^2+1)^2}$, use the substitution $3x^2+1=u \Rightarrow 6x\,dx=du$ to get: $\frac{-2}{3x^2+1}$, i.e
$$\int \... |
119,696 | <p>Let us suppose i have two graphs for sequences A and B as follows</p>
<pre><code>a1 := {{0.9, 0.086133}, {0.086133, 0.0082432}, {0.0082432,
0.0007889}, {0.0007889, 0.0000755}, {0.0000755,
7.2256*10^-6}, {7.2256*10^-6, 6.9151*10^-7}, {6.9151*10^-7,
6.618*10^-8}, {6.618*10^-8, 6.3336*10^-9}, {6.3336*10... | Sumit | 8,070 | <p>I would suggest creating the legend separately and combine with <code>Grid</code>.</p>
<pre><code>Grid[{{Show[plot1, plot2, PlotRange -> All],
SwatchLegend[{Red, Yellow}, {"plot1", "plot2"}]}}]
</code></pre>
<p><a href="https://i.stack.imgur.com/f0uH1.png" rel="nofollow noreferrer"><img src="https://i.s... |
3,607,430 | <blockquote>
<p>Given that <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span> are the angles of a right-angled triangle, prove that:
<span class="math-container">$$\begin{align}
\sin a\sin b\sin(a-b) &+\sin b\sin c\sin(b-c)+\sin c\sin a\sin(... | Rezha Adrian Tanuharja | 751,970 | <p>Points <span class="math-container">$P,Q,R$</span> are collinear if and only if they satisfy the expression <span class="math-container">$m\vec{P}+(1-m)\vec{Q}=\vec{R}$</span>. Substitute <span class="math-container">$\vec{P}=\{a,0\}, \vec{Q}=\{0,\frac{3}{a}\}, \vec{R}=\{6,-1\}$</span> To obtain the following expres... |
95,965 | <p>Joyal's <a href="http://en.wikipedia.org/wiki/Combinatorial_species" rel="nofollow">combinatorial species</a>, endofunctors in the category of finite sets with bijections $\mathbf B$ have found numerous applications. One generalisation is given by so-called "tensor species" (also "tensorial species", or, "linear sp... | Jan Weidner | 2,837 | <p>In the theory of algebraic operads, the language of "tensor species" is often used,
see Chapter 5 of "Algebraic Operads, Jean-Louis Loday & Bruno Vallette, Grundlehren der mathematischen Wissenschaften, Volume 346, Springer-Verlag (2012).</p>
<p>For example one can define an operad very concise as a monoid in ... |
23,192 | <p>The question if there is an upper bound known for Brun's constant was discussed briefly here: <a href="http://gowers.wordpress.com/2009/05/22/what-is-wolfram-alpha-good-for/" rel="nofollow">http://gowers.wordpress.com/2009/05/22/what-is-wolfram-alpha-good-for/</a> but no sure answer was given. </p>
<p>So I thought ... | Michael Lugo | 143 | <p>Crandall and Pomerance, "Prime numbers: a computational perspective" (Google books) says that Brun's constant B, the sum of the reciprocals of the twin primes, is known to be between 1.82 and 2.15.</p>
<p><b>edited to add</b>: I'm aware that this isn't much of a citation. It would be nice if someone who has access ... |
167,013 | <p>I don't understand this behavior: why does <code>Limit[z/(z - a), z -> 0]</code> give zero and not a condition depending on <code>a</code>, provided it has not been defined before? is there a way to make it work properly? (By working properly I mean give the correct result, namely 1 if $a=0$ and 0 otherwise.) </p... | halirutan | 187 | <p>Obviously, with the assumptions Mathematica uses for <code>a</code>, the value of it does not matter for the residue or the limit. Here are two counter-examples:</p>
<pre><code>Residue[Gamma[z] Gamma[z - 1] Gamma[z - a], {z, 0}]
(* -Gamma[-a] + 2 EulerGamma Gamma[-a] -
Gamma[-a] PolyGamma[0, -a] *)
</code></pre>
... |
159,965 | <blockquote>
<p>Find limits of a function $f:\mathbb{R}^3\rightarrow \mathbb{R}$ given by formula $f(x,y,x)=x+y+z$ on set $M=\left\{ (x,y,z)\in\mathbb{R}^3:x^2+y^2\le z\le 1 \right\}$. Does $f$ reaches all its limits?</p>
</blockquote>
<p>To answer the last question I need to know if $M$ is a closed and bounded set.... | H. Kabayakawa | 32,428 | <p>$M$ is like a cone with the vertex in $(0,0,0)$ and the base in the circumference $\{x^2+y^2=1, z=1\}$, then $M$ is a compact set in $\mathbb{R}^3$. The function is lineal with gradient $(1,1,1)$ and $M$ is a convex set because the surface $z=x^2+y^2$ is a revolution paraboloid. The paraboloid has normal $(2x,2y,-1... |
159,965 | <blockquote>
<p>Find limits of a function $f:\mathbb{R}^3\rightarrow \mathbb{R}$ given by formula $f(x,y,x)=x+y+z$ on set $M=\left\{ (x,y,z)\in\mathbb{R}^3:x^2+y^2\le z\le 1 \right\}$. Does $f$ reaches all its limits?</p>
</blockquote>
<p>To answer the last question I need to know if $M$ is a closed and bounded set.... | Christian Blatter | 1,303 | <p>Your set $M$ is defined by inequalities of the form $g_i(x,y,z)\geq0$ with continuous $g_i:{\mathbb R}^3\to{\mathbb R}$, therefore it is closed; and all $(x,y,z)\in M$ satisfy $x^2+y^2\leq 1$ as well as $0\leq z\leq 1$, therefore $M$ is bounded. (In fact $M$ looks like an inverted sugar cone.) It follows that any co... |
3,608,114 | <p>Consider the example where I have a matrix <span class="math-container">$\mathbf{D}$</span> in <span class="math-container">$-1/1$</span> coding with <span class="math-container">$5$</span> columns,</p>
<p><span class="math-container">$$D = \begin{bmatrix}-1&-1&-1&1&1\\1&-1&-1&-1&1\\... | asd.123 | 613,151 | <p>At the comments @Rodrigo de Azevedo said you can apply <a href="https://en.m.wikipedia.org/wiki/Gaussian_elimination" rel="nofollow noreferrer">Gaussian elimination</a> which is quite intuitive method to apply but I personally suggest that if you want to find the linear dependence relation between rows or columns of... |
64,881 | <p>I am having trouble with this problem from my latest homework.</p>
<p>Prove the arithmetic-geometric mean inequality. That is, for two positive real
numbers $x,y$, we have
$$ \sqrt{xy}≤ \frac{x+y}{2} .$$
Furthermore, equality occurs if and only if $x = y$.</p>
<p>Any and all help would be appreciated.</p>
| Bruno Joyal | 12,507 | <p>Since $x$ and $y$ are positive, we can write them as $x=u^2$, $y=v^2$. Then</p>
<p>$$(u-v)^2 \geq 0 \Rightarrow u^2 + v^2 \geq 2uv$$</p>
<p>which is precisely it.</p>
|
69,961 | <p>I want to determine the set of natural numbers that can be expressed as the sum of some non-negative number of 3s and 5s.</p>
<p>$$S=\{3k+5j∣k,j∈\mathbb{N}∪\{0\}\}$$</p>
<p>I want to check whether that would be:
0,3, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and so on.</p>
<p>Meaning that it would include 0, 3, 5,... | André Caldas | 17,092 | <p>First, it will be easier to determine the set
$$I = \{3k+5j | k,j \in \mathbb{Z}\}.$$</p>
<p>This set is an <a href="http://en.wikipedia.org/wiki/Ideal_%28ring_theory%29" rel="nofollow" title="ideal on the Wikipedia">ideal</a> over $\mathbb{Z}$.
That is, if $a,b \in I$ then $a+b \in I$ ($I+I \subset I$),
and if
$a ... |
4,627,334 | <p>To my understanding that a primitive triple <span class="math-container">$x$</span> and <span class="math-container">$y$</span> can be written as <span class="math-container">$x = q^2 - p^2$</span> while <span class="math-container">$y=2pq$</span> for relatively prime opposite parity <span class="math-container">$q ... | Joseph Harrison | 1,131,061 | <p>As you say, we can write <span class="math-container">$x = q^2 - p^2$</span> and <span class="math-container">$y = 2pq$</span> where <span class="math-container">$p, q$</span> are relatively prime with different parity. Now the area of the triangle is half of <span class="math-container">$xy$</span>. If this area is... |
90,940 | <p>It seems known that the category of hypergraphs is a topos.
I am looking for any reference here, or just a statement of this in the literature, but can't find anything. There is one paper </p>
<blockquote>
<p>A category-theoretical approach to hypergraphs,
W. Dörfler and D. A. Waller, ARCHIV DER MATHEMATIK, Vol... | YKY | 11,555 | <p>According to this presentation:</p>
<blockquote>
<p>Will Grilliette and Lucas Rusnak, <em>Natural Generalizations of Graphs Part II: Commas, Topoi, & Homomorphisms</em>,
Discrete Mathematics Seminar, Texas State University (2017) DOI:<a href="https://doi.org/10.13140/RG.2.2.13627.92961" rel="nofollow norefe... |
2,262,011 | <p>This might be a somewhat stupid question, but I've been wondering if it is possible to define some other topology on $\mathrm{Spec} (A)$ other than Zariski topology in a way that it has some interesting properties as well.</p>
<p>First of all, I am new as this is my first encounter with anything close or related to... | Tsemo Aristide | 280,301 | <p>There is a generalization of the notion of topology called a Grothendieck topology. Grothendieck has defined many examples of Grothendieck topolgies in the category of schemes for example the Etale topology which is one notion used in the proof of Weil conjectures.</p>
<p><a href="https://en.wikipedia.org/wiki/Grot... |
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