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 |
|---|---|---|---|---|
4,316,771 | <p>I'm asked to compute <span class="math-container">$$\sum_{k=-3}^{10} 2k^4$$</span>
I looked up for<a href="https://en.m.wikipedia.org/wiki/Bernoulli_number" rel="nofollow noreferrer">Bernoulli Number</a> on Wikipedia and found a general formula for that. But my teacher has asked me to evaluate this by breaking the s... | am301 | 382,813 | <p><span class="math-container">$\sum^n_{k=1} k^m$</span> is a linear function of the powers <span class="math-container">$\{n^m\}_{m=1}^{n+1}$</span>. Therefore
<span class="math-container">$$
\sum^n_{k=1}k^4=An+Bn^2+Cn^3+Dn^4+En^5
$$</span>
We can plug in the numbers <span class="math-container">$n=1-5$</span> to get... |
983,849 | <p>Given</p>
<ul>
<li>$G$ be a finite group</li>
<li>$X$ is a subset of group $G$</li>
<li>$|X| > \frac{|G|}{2}$</li>
</ul>
<p>I noticed that any element in $G$ can be expressed as the product of 2 elements in $X$. Is there a valid way to prove this？</p>
<p>If the third condition was $|X| = \frac{|G|}{2}$ instead... | Matthias Klupsch | 19,700 | <p>Note that for any $g \in G$ the set $g X^{-1}$ has also more than $\frac{|G|}{2}$ elements, hence $gX^{-1} \cap X \neq \emptyset$ so that there are $x,y \in X$ such that $gx^{-1} = y$, i.e. $g = yx$.</p>
|
398,371 | <p>How to calculate $$\lim_{t\rightarrow1^+}\frac{\sin(\pi t)}{\sqrt{1+\cos(\pi t)}}$$? I've tried to use L'Hospital, but then I'll get</p>
<p>$$\lim_{t\rightarrow1^+}\frac{\pi\cos(\pi t)}{\frac{-\pi\sin(\pi t)}{2\sqrt{1+\cos(\pi t)}}}=\lim_{t\rightarrow1^+}\frac{2\pi\cos(\pi t)\sqrt{1+\cos(\pi t)}}{-\pi\sin(\pi t)}$$... | André Nicolas | 6,312 | <p>This is not quite a solution, more of a comment about the assertion that the L'Hospital's Rule calculation
$$\lim_{t\rightarrow1^+}\frac{\pi\cos(\pi t)}{\frac{-\pi\sin(\pi t)}{2\sqrt{1+\cos(\pi t)}}}=\lim_{t\rightarrow1^+}\frac{2\pi\cos(\pi t)\sqrt{1+\cos(\pi t)}}{-\pi\sin(\pi t)}$$
does not get us any further.</p>... |
3,958,142 | <p>The following problem is in Mathematical Methods in physical science CH8 miscellaneous problems
<span class="math-container">$$
(2x-y\sin(2x))dx = (\sin^2x-2y)dy
$$</span>
it isnt an exact equation the only difference is <span class="math-container">$-$</span> sign
because it not an exact equation I tried to rearran... | Alessio K | 702,692 | <p>Recall we say a differential equation of type
<span class="math-container">$$N(x,y)dx+M(x,y)dy=0$$</span> is exact if <span class="math-container">$$\frac{\partial N}{\partial y}=\frac{\partial M}{\partial x}$$</span></p>
<p>So re-writing the above differential equation as
<span class="math-container">$$
(2x-y\sin(2... |
2,834,219 | <p>I'd like to numerically evaluate the following integral using computer software but it has a singularity at $x=1$:</p>
<p>\begin{equation}
\int_1^{\infty} \frac{x}{1-x^4} dx
\end{equation}</p>
<p>I was thinking of a variable transformation, rewriting the expression, or something of a kind. One of my attempts was t... | Community | -1 | <p>The integral is $-\infty$ because of the pole at $1$. To see this, factor</p>
<p>$$1 - x^4 = (1 - x)p(x)$$</p>
<p>where $p(x)$ is a polynomial which doesn't vanish at $1$. Then near $x = 1$ the integrand is well approximated by $$\frac 1 {p(1)} \frac 1 {1 - x}$$</p>
<p>which is non-integrable. </p>
|
4,616,155 | <p>This is a Question from an Analysis 1 exam. The question is as follows: Decide if the functions <span class="math-container">$f: \mathbb{R} \longrightarrow \mathbb{R}$</span> can be written as the difference of two monotonically increasing functions</p>
<p>a) <span class="math-container">$f(x) = \cos(x)$</span></p>
... | PierreCarre | 639,238 | <p>If <span class="math-container">$f$</span> had compact support <span class="math-container">$K$</span>, it would zero outside <span class="math-container">$K$</span>. So, since <span class="math-container">$K$</span> would be bounded, there would be some <span class="math-container">$x_0 > 0$</span> for which <sp... |
266,834 | <p>I know the values for a function v[x,y] on an irregular grid of (x,y) points. Call the table storing all these points xyvtriples. Because of the irregular grid, the Mathematica function Interpolation only works as</p>
<p><code>interpolatedvfunc = Interpolation[xyvtriples, InterpolationOrder -> 1];</code></p>
<p... | E. Chan-López | 53,427 | <p>Try this:</p>
<pre><code>Total[Map[If[Total[Grad[#, Variables[expr]]] === 0, #, Nothing] &, MonomialList[expr]]]
(*-(1/4)*)
</code></pre>
<p>In general, when there are no subscripts, we proceed as Bob Hanlon indicates:</p>
<pre><code>expr/. Thread[Variables[expr] -> 0]
</code></pre>
<p>Another way to do it is... |
10,807 | <p>Over the past few days, the Chrome browser page with Questions is crashing.</p>
<p>It has crashed while typing answers twice and several other times when just viewing and updating the questions listing.</p>
<p>Did something change in the past three days that could be affecting this as I have not seen this behavior... | Antonio Vargas | 5,531 | <p>It seems that the crash can reliably be reproduced using the following method (supposing you have the ability to go into edit mode on an answer of mine).</p>
<ol>
<li><p>Close Chrome completely.</p></li>
<li><p>Open a new Chrome window and go to this answer: <a href="https://math.stackexchange.com/a/498905/5531">ht... |
3,774,400 | <p>Just like the title says, I don't know how to write an example matrix here to look like matrix. If this makes sense, <span class="math-container">$A$</span> can be <span class="math-container">$[1 0 0;0 1 0;0 1 0]$</span> (like in MATLAB syntax) then if we find determinants of <span class="math-container">$A-\lambda... | mathcounterexamples.net | 187,663 | <p><strong>Two answers</strong></p>
<p><em>First one based on a bit of theory</em></p>
<p>Having <span class="math-container">$\det(A - \lambda I)=0$</span> for all <span class="math-container">$\lambda$</span> is impossible (at least if you work in the field of the reals <span class="math-container">$\mathbb R$</span>... |
2,906,797 | <p>I want to express this polynomial as a product of linear factors:</p>
<p>$x^5 + x^3 + 8x^2 + 8$</p>
<p>I noticed that $\pm$i were roots just looking at it, so two factors must be $(x- i)$ and $(x + i)$, but I'm not sure how I would know what the remaining polynomial would be. For real roots, I would usually just d... | dxiv | 291,201 | <p>Alt. hint: you might as well notice that $\,x=-2\,$ is a root by the <a href="https://en.wikipedia.org/wiki/Rational_root_theorem" rel="nofollow noreferrer">rational root theorem</a>. Then if the remaining two roots are $\,a,b\,$, by <a href="https://en.wikipedia.org/wiki/Vieta%27s_formulas" rel="nofollow nor... |
11,922 | <p>Which is correct terminology: "A Cartesian plane" or "The Cartesian plane"? (As in the directions for a section of homework being, "Plot a point on ______ Cartesian plane." In that context, I feel that one of the two should be used consistently, but find there's little agreement on Google for which is appropriate.</... | Anschewski | 199 | <p>I think from a philosophical point of view, both terms may be used. From an educational point of view, I would generally use the term "the" Cartesian plane in school context as every two Cartesian planes are isomorphic and should most of the time be considered the same object. This may reduce cognitive complexity. O... |
653,449 | <p>According to <a href="http://en.wikibooks.org/wiki/Haskell/Category_theory" rel="noreferrer">the Haskell wikibook on Category Theory</a>, the category below is not a valid category due to the addition of the morphism <em>h</em>. The hint says to " think about associativity of the composition operation." Bu... | Thomas Andrews | 7,933 | <p>hint: What is $hgf$? Write it in two different ways.</p>
|
2,420,435 | <p>I'm learning logic for computer science and came across the question:</p>
<blockquote>
<p>If $\ n$ is a real number, $\frac{1}{n}$ is the reciprocal of $\ n$. Prove that all
numbers have a unique reciprocal.</p>
</blockquote>
<p>I came up with the following method, but it seems so simple that I doubt it'll wor... | vadim123 | 73,324 | <p>The way to prove that a certain property is <em>uniquely</em> satisfied, is to assume that two things satisfy and then prove that they are equal.</p>
<p>In this case, suppose that $p,q$ are both reciprocals of $n$. Then, $qn=1=pn$. We divide both sides by $n$ (which needs to be nonzero) to conclude that $p=q$.</p... |
20,463 | <p>I'm new here and I don't know how things work here and don't have any knowledge about commenting or answering questions. How do I get to know if my question has been commented or answered?</p>
| k1.M | 132,351 | <p>Another way for this problem is to install the software of stackexchange for your device (phone or tablet with android operating system or IOS) and active notifications for this software to send you push notifications.</p>
|
411,875 | <p>This is an exam question I encountered while studying for my exam for our topology course:</p>
<blockquote>
<p>Give two continuous maps from $S^1$ to $S^1$ which are not homotopic. (Of course, provide a proof as well.)</p>
</blockquote>
<p>The only continuous maps from $S^1$ to $S^1$ I can think of are rotations... | Community | -1 | <p>Recall that for any map $f : S^1 \to S^1$ that we have the induced map $f_\ast : H_1(S^1) \to H_1(S^1)$ which is a map between two infinite cyclic groups. Any such map is given by multiplication by $n \in \Bbb{Z}$, so we define the <strong><em>degree of $f$</em></strong> to this integer $n$.</p>
<p>If you think of ... |
411,875 | <p>This is an exam question I encountered while studying for my exam for our topology course:</p>
<blockquote>
<p>Give two continuous maps from $S^1$ to $S^1$ which are not homotopic. (Of course, provide a proof as well.)</p>
</blockquote>
<p>The only continuous maps from $S^1$ to $S^1$ I can think of are rotations... | kahen | 1,269 | <p>If you think of $S^1$ as the unit circle in $\mathbb C$, each homotopy class of loops in $S^1$ with base point $1$ is represented by the map $f_n(z) = z^n$ where $n$ can be any integer, and it's not too hard to show that $f_m$ and $f_n$ are in the same homotopy class if and only if $m=n$.</p>
|
268,461 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/228080/operatornameimfz-leq-operatornamerefz-then-f-is-constant">$|\operatorname{Im}f(z)|\leq |\operatorname{Re}f(z)|$ then $f$ is constant</a> </p>
</blockquote>
<blockquote>
<p>Let $f\colon\mathbb C \t... | Amr | 29,267 | <p>$$X\subseteq F\cup F^c$$
Since $F\subseteq\cup\Omega$ (Here $\cup\Omega$ is the union of all elements of $\Omega$), therefore:
$$X\subseteq\cup\Omega\cup F^c$$
Thus:
$$K\subseteq\cup\Omega\cup F^c$$</p>
|
3,733,428 | <p>I would like to understand why <span class="math-container">$\mathbb{R}^2\setminus \mathbb{Q}^2$</span> endowed with the subspace topology is not a topological manifold. It seems to me it is Hausdorff and second countable. So I am wondering. Why is it not locally Euclidian?</p>
| Alex Kruckman | 7,062 | <p>Because it has lots of holes!</p>
<p>One way to make this precise is to show that <span class="math-container">$\mathbb{R}^2\setminus \mathbb{Q}^2$</span> is not locally compact. In fact, no point has a compact neighborhood.</p>
<p>Let <span class="math-container">$p\in \mathbb{R}^2\setminus\mathbb{Q}^2$</span> and ... |
1,232,143 | <p>If
$$f(x)=\sum_{n=0}^\infty\frac{x^n}{n!}, x\in\mathbb{R}$$
and
$$
g(x) = 1 + \int_0^x f(t) \,dt
$$</p>
<p>prove that $g(x)=f(x)$ for all $x\in\mathbb{R}$
and prove that $f$ is differentiable on $\mathbb{R}$ as well as show that $f'(x)=f(x)$ for all $x\in\mathbb{R}$.</p>
<p>I know that $f$ is continuous on $\math... | Peter | 82,961 | <p>$$f(x)=\sum_{n=0}^\infty \frac{x^n}{n!}=e^x$$</p>
<p>$$g(x)=1+\int_0^x f(t) dt=1+e^x-1=e^x$$</p>
<p>So, we have $f(x)=g(x)$ and $f'(x)=f(x)$</p>
|
835,639 | <p>These days I have read many descriptions of a noncooperative game like the one below.</p>
<p>A noncooperative game is a game in which players are unable to make enforceable contracts outside of the rules/description of such a game.</p>
<p>As a graduate student majoring in math, I wonder if there is any mathematica... | Asinomás | 33,907 | <p>You can define a cooperative game with n players $A=a_1,a_2,\dots a_n$ to be a a function from the power set of $A$ to $\mathbb R$. The idea is each alliance between players has a value, this value can be viewed as a cost or as a reward.</p>
<p>non-cooperative games are usually harder and have many definitions, you... |
2,973,825 | <p>when you are checking to see if a sum of say <span class="math-container">$k^2$</span> from <span class="math-container">$k=1$</span> to to <span class="math-container">$k=n$</span> is equal to a sum of <span class="math-container">$(k+1)^2$</span> from <span class="math-container">$k=0$</span> to <span class="math-... | nonuser | 463,553 | <p>Let <span class="math-container">$S$</span> be a set of all cards.</p>
<p>So we have a set <span class="math-container">$A$</span> of all ordered <span class="math-container">$4$</span>-couples of the set <span class="math-container">$S$</span> and a set <span class="math-container">$B$</span> of all <span class="... |
2,973,825 | <p>when you are checking to see if a sum of say <span class="math-container">$k^2$</span> from <span class="math-container">$k=1$</span> to to <span class="math-container">$k=n$</span> is equal to a sum of <span class="math-container">$(k+1)^2$</span> from <span class="math-container">$k=0$</span> to <span class="math-... | Misha Lavrov | 383,078 | <p>There is in fact a solution which uses your idea: that some suit will be repeated twice.</p>
<p>Say that among the <span class="math-container">$13$</span> ranks within a suit, ordered <span class="math-container">$A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K$</span>, each rank "beats" the next six ranks, wrapping around... |
4,594,043 | <p>For <span class="math-container">$|x|<1,$</span> we have
<span class="math-container">$$
\begin{aligned}
& \frac{1}{1-x}=\sum_{k=0}^{\infty} x^k \quad \Rightarrow \quad \ln (1-x)=-\sum_{k=0}^{\infty} \frac{x^{k+1}}{k+1}
\end{aligned}
$$</span></p>
<hr />
<p><span class="math-container">$$
\begin{aligned}
\int... | Lai | 732,917 | <p>For <strong>natural number</strong> <span class="math-container">$n$</span>, I can use differentiation of the integral
<span class="math-container">$$
\begin{aligned}
I(a) & =\int_0^1 x^a \ln (1-x) d x \\
& =-\int_0^1 x^a \sum_{k=0}^{\infty} \frac{1}{k+1} x^{k+1} d x \\
& =-\sum_{k=0}^{\infty} \frac{1}{... |
2,287,878 | <p>Given the following polynomial: $$P(x)=(x^2+x+1)^{100}$$ How do I find : $$\sum_{k=1}^{200} \frac{1}{1+x_k} $$ Is there a general solution for this type of problem cause I saw they tend to ask the same thing for $\sum_{k=1}^{200} \frac{1}{x_k}$? Also how do I find the coefficient of $a_1$ and the remainder for $$P(x... | dxiv | 291,201 | <blockquote>
<p>Is there a general solution for this type of problem</p>
</blockquote>
<p>Consider for example $P(x)=(x^2+x+1)^n\,$.</p>
<blockquote>
<p>$\sum_{k=1}^{2n} \frac{1}{x_k}$</p>
</blockquote>
<p>Let $y_k = \frac{1}{x_k} \ne 0\,$ then $P(\frac{1}{y_k})=P(x_k)=0\,$, so $y_k$ are the roots of: $$Q(y)=y^{... |
3,757,972 | <p>If <span class="math-container">$\frac{ab} {a+b} = y$</span>, where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are greater than zero, why is <span class="math-container">$y$</span> always smaller than the smallest number substituted?</p>
<p>Say <span class="math-container">$a... | B. Goddard | 362,009 | <p>Another way to think about it: Assuming <span class="math-container">$0<a\leq b$</span>, divide the top and bottom of your fraction by <span class="math-container">$b$</span> to get</p>
<p><span class="math-container">$$\frac{a}{\frac{a}{b}+1}.$$</span></p>
<p><span class="math-container">$a$</span> is the smal... |
757,702 | <p>I am in my pre-academic year. We recently studied the Remainder sentence (at least that's what I think it translates) which states that any polynomial can be written as <span class="math-container">$P = Q\cdot L + R$</span></p>
<p>I am unable to solve the following:</p>
<blockquote>
<p>Show that <span class="mat... | Lai | 732,917 | <p>I am going to prove it by modulo. In modulo <span class="math-container">$$
x^{2}+x+1 \equiv 0\left(\bmod x^{2}+x+1\right) \Rightarrow x+1\equiv -x^2\textrm{ and } x^3 \equiv 1 \left(\bmod x^{2}+x+1\right).
$$</span>
In modulo <span class="math-container">$x^{2}+x+1$</span>, we have
<span class="math-container">$$
\... |
786,329 | <p>$$f(x,y)=\begin{cases}
\frac{xy}{x^2+y^2}, \text{ if } x^2+y^2\neq 0
\\
0, \text{ if } x^2+y^2=0
\end{cases}$$</p>
<p>Is it continuous at $(0,0)$?</p>
| dcs24 | 36,711 | <p>No, to disprove show sequential continuity doesn't hold. Look along the sequences $\{ x = 1/n , y = 1/n \}$ and $\{ x = 0 , y = 0 \}$ (the constant sequence)</p>
|
11,290 | <p>I have this code to produce an interactive visualization of a tangent plane to a function:</p>
<pre><code>Clear[f]
f[x_, y_] := x^3 + 2*y^3
Manipulate[
Show[
Plot3D[f[x, y], {x, -1, 1}, {y, -1, 1}, PlotStyle -> Opacity[0.8]],
Plot3D[f[point[[1]], point[[2]]] +
Limit[(f[point[[1]] + h, point[[2]]] - f[po... | halirutan | 187 | <p>The limits can be calculated analytically one time and then used inside your <code>Manipulate</code>. You can use <code>With</code> to place the final expression where you need them:</p>
<pre><code>f[x_, y_] := x^3 + 2*y^3;
With[{l1 = Limit[(f[px + h, py] - f[px, py])/h, h -> 0],
l2 = Limit[(f[px, py + h] - f... |
4,244,983 | <p>In maths it is assumed that every statement is either true (t) or false (f). When proving theorems such as: For all vector spaces <span class="math-container">$V$</span> over a field <span class="math-container">$K$</span> and <span class="math-container">$\lambda \in K$</span> it holds that <span class="math-contai... | Dan | 852,483 | <p>Lets take another viewpoint.
Assume that you have a certain mathematical theory. Some axioms regarding this theory (e.g. in the theory of sets you would probably take the ZFC axioms... etc. and of course with logical axioms, equalitarian axioms and so on). In this case you have a "statement" of the form &q... |
2,187,571 | <p>Consider $e^{it}$ with $t$ real. I can rewrite $t$ as $t=2\pi \tau$:</p>
<p>$ e^{i2\pi\tau} = (e^{i2\pi})^\tau = (1)^\tau = 1 $ for arbitrary $\tau$ or $t$. Where is the mistake in this calculation ? My guess that $(e^x)^y = e^{x\cdot y}$ only applies for real $x$ and $y$, but I don't see why this should be the cas... | Community | -1 | <p>You just (re)discovered that $a^{bc}=(a^b)^c$ may not hold in the complex.</p>
<hr>
<p>Actually, $a^b$ is not so well defined. By logarithms,</p>
<p>$$a^b=\exp(b \log a)=\exp(b(\log|a|+i\angle a+i2k\pi))$$ where the term $i2k\pi$ introduces an indeterminacy.</p>
<hr>
<p>Back to your example,</p>
<p>$$\log e^{i... |
2,495,176 | <p>For how many positive values of $n$ are both $\frac n3$ and $3n$ four-digit integers?</p>
<p>Any help is greatly appreciated. I think the smallest n value is 3000 and the largest n value is 3333. Does this make sense?</p>
| Soph | 232,073 | <p>The question is asking how many values of n satisfy </p>
<ul>
<li>$n \in \mathbb{N}$</li>
<li>$3n \in [1000, 9999]$ </li>
<li>$\frac n3 \in [1000, 9999]$</li>
</ul>
<p>So find the <em>min</em> value of $n$ satisfying the lower bound:</p>
<p>$$ \frac n3 \ge 1000 $$</p>
<p>And the <em>max</em> value of $n$ satisfy... |
3,756,932 | <p>Overall, I think I understand the Monty Hall problem, but there's one particular part that I either don't understand, or I don't agree with, and I'm hoping for an intuitive explanation why I'm wrong or confirmation that I'm right.</p>
<p>The variant that I'm working on is the one with 3 doors. Prize is uniformly loc... | YJT | 731,237 | <p>The main question is why the host opens door #2. If he opens it because he knows it is empty, then indeed it sends a signal about door #1. But if he opens it because he wants to open door #2 and doesn't know it is empty, then the other two doors have equal probabilities and it doesn't matter.</p>
<p>In the latter ca... |
1,431,969 | <p>Just like in the title:</p>
<blockquote>
<p>How to compute the CDF of $X\cdot Y$ if $X,Y$ are independent random variables, uniformly distributed over $(-1,1)$?</p>
</blockquote>
<p>I tried using the next formula:
the density of $X\cdot Y$ is the integral of
$f(u/v)\cdot g(v)$ where $f$ is the density of $X$ a... | Subhasish Basak | 266,148 | <p>Hint : Try jacobian transformation with $u=xy$ and $v=y$ then integrate on v between(-1,1).</p>
|
220,946 | <p>I am a newcomer to Mathematica and I am trying to solve this differential equation:</p>
<pre><code>s = NDSolve[{y'[x] == -4/3*y[x]/x + 4 a/(3 x^2) + 4/(9 x^3) a b -
8/(9 x^3) c, y[0.01] == 20}, y, {x, 0.01, 10}]
</code></pre>
<p>I am getting the error message "Encountered non-numerical value for a derivativ... | SonerAlbayrak | 49,198 | <p>EDIT: The method below only works for upper triangular matrices, I need to modify it to work for all matrices!</p>
<p>One quick way is as follows:</p>
<pre><code>ClearAll[getDiagonal];
getDiagonal[list_List] := getDiagonal[list, {}];
getDiagonal[{}, a_] := a;
getDiagonal[list_List, results_List] := With[{
chosen... |
1,621,302 | <blockquote>
<p>Assume $a,n\in\mathbb{N}$ such that $\gcd{(a,n)}=1$. We say $n$ is prime if $a^{n-1}\equiv 1\mod{n}$ and $a^x\not\equiv1\mod{n}$ for any divisor $x$ of $n-1$.</p>
</blockquote>
<p>I am presented with the following proof but there are jumps within it that I don't understand.</p>
<hr>
<p><em>Proof</e... | Community | -1 | <ul>
<li><p>For <span class="math-container">$1,2,3$</span> :</p>
<p>Let <span class="math-container">$n-1=sd+r$</span> where <span class="math-container">$0 \leq r <d$</span> .Then as you noted it follows that :</p>
</li>
</ul>
<p><span class="math-container">$$1 \equiv a^{n-1} \equiv \left (a^d \right )^s \cdot a^... |
1,573,341 | <blockquote>
<p>Prove that the roots of the equation $z^n=(z+3)^n$ are collinear</p>
</blockquote>
<p>Taking modulus on both sides, $$\vert z-0\vert =\vert z-3\vert$$</p>
<p>This represents the perpendicular bisector of line joining $x=0$ and $x=3$</p>
<p>That was easy. But I tried to solve it using algebra:</p>
... | DeepSea | 101,504 | <p><strong>Hint</strong>:the roots have the same real part which is $-3/2$.</p>
|
4,010,160 | <h2>The Problem</h2>
<p>Let <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span> be a continuous i.i.d. random variables. By definition, we know that...</p>
<ol>
<li>Independence: <span class="math-container">$\mathbb P(X_1 \in A \ \cap \ X_2 \in B)=\mathbb P(X_1 \in A) \ \mathbb P(X... | zero2infinity | 862,690 | <p>No, Suppose <span class="math-container">$A=[-1,1], X=\mathbb{R}, B=(-2,2)$</span> and <span class="math-container">$Y=[-3,3]$</span>. Then this gives a counter example</p>
|
4,010,160 | <h2>The Problem</h2>
<p>Let <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span> be a continuous i.i.d. random variables. By definition, we know that...</p>
<ol>
<li>Independence: <span class="math-container">$\mathbb P(X_1 \in A \ \cap \ X_2 \in B)=\mathbb P(X_1 \in A) \ \mathbb P(X... | Noah Schweber | 28,111 | <p>The topological version of Schroeder-Bernstein <strong>fails badly</strong>. For example, consider <span class="math-container">$(0,1)$</span> and <span class="math-container">$[0,1]$</span> with their usual topologies, which are of course non-homeomorphic:</p>
<ul>
<li><p><span class="math-container">$(0,1)$</span>... |
4,010,160 | <h2>The Problem</h2>
<p>Let <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span> be a continuous i.i.d. random variables. By definition, we know that...</p>
<ol>
<li>Independence: <span class="math-container">$\mathbb P(X_1 \in A \ \cap \ X_2 \in B)=\mathbb P(X_1 \in A) \ \mathbb P(X... | SUNIL PASUPULATI | 594,846 | <p>For example, consider <span class="math-container">$X=[0,1]$</span> and <span class="math-container">$Y=(0,1)$</span>. <span class="math-container">$X$</span> is not homeomorphic to <span class="math-container">$Y.$</span> but <span class="math-container">$X$</span> is homeomorphic to <span class="math-container">$[... |
132,053 | <p>I have an expression $e^{x-a}(e^{bx}+c)$. I just want it multiplied throughout to get $e^{(b+1)x-a}+ce^{x-a}$. To achieve this I have tried <code>Expand</code>, <code>Simplify</code>, <code>Collect</code> (using pattern <code>Exp[q_*x]</code>). The best output I could get is</p>
<pre><code>Expand[Exp[x-a]*(Exp[b*x]... | BoLe | 6,555 | <p><code>MapAt</code> with specified position:</p>
<pre><code>tmp = Exp[x - a] (Exp[b x] + c) // Expand;
pos = Position[tmp, p_Plus /; ! FreeQ[p, x]];
MapAt[Collect[#, x] &, tmp, pos]
</code></pre>
<blockquote>
<p><code>c E^(-a + x) + E^(-a + (1 + b) x)</code></p>
</blockquote>
|
2,145,098 | <p>This was part of a three part question where I was supposed to prove two sets have equal cardinality by finding bijections. I've created a bijection $f: \Bbb Z \Rightarrow 2\Bbb Z$ by $f(x)=2x$. I've created a bijection $g: (0,1) \Rightarrow (4,50)$ by $g(x)=46x+4$. I think those are both correct. My last question i... | Forever Mozart | 21,137 | <p>Do you know that the rationals are countable? Let </p>
<p>$\{q_i:i\in\omega\}$ enumerate $\mathbb Q\cap (0,1)$, and </p>
<p>$\{p_i:i\in\omega\}$ enumerate $\mathbb Q \cap [0,1]$.</p>
<p>Map $q_i\to p_i$ and let every irrational number map to itself.</p>
|
46,726 | <p>In many proofs I see that some variable is "fixed" and/or "arbitrary". Sometimes I see only one of them and I miss a clear guideline for it. Could somebody point me to a reliable source (best a well-known standard book) which explains, when and how to use both in proofs?</p>
<p>EDIT: A little add-on to the question... | Eric Naslund | 6,075 | <p>Rather then arbitrary and fixed, lets look at something very similar where the difference is more pronounced. There is a very important distinction between <strong>uniform</strong> and <strong>fixed</strong>. A perfect example is convergence of functions.</p>
<p>Let <span class="math-container">$f_n$</span> be a s... |
1,651,452 | <p>I am trying to solve a modular arithmetic system and I got to the point where I have to solve $22t \equiv 9 \pmod{7}$ for $t$. I researched on the internet and found that there are many ways to solve this, such as using linear diophantine equations, the euclidean algorithm or using inverses.</p>
<p>Can someone show... | John Frederick Chionglo | 190,989 | <p>A solution t = 2 - 7y where y is from the set of Integers.
<a href="https://i.stack.imgur.com/FH2In.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/FH2In.jpg" alt="enter image description here"></a></p>
<h2>Notes (Chionglo, 2015)</h2>
<ol>
<li>A linear system of congruencies is equivalent to a l... |
1,518,258 | <p>I haven't been able to come up with a counterexample so far. </p>
| AnotherPerson | 185,237 | <p>$a=b=2$. $a$ divides $2!$, $b$ divides $3!$, but their product $4$ does not divide $3!$</p>
|
376,530 | <p>Let <span class="math-container">$G$</span> be a graph which does not contain a simple cycle <span class="math-container">$v_1\ldots v_k$</span> and two "crossing" chords <span class="math-container">$v_iv_j$</span> and <span class="math-container">$v_pv_q$</span>, <span class="math-container">$i<p<j... | David Wood | 25,980 | <p>Assume <span class="math-container">$n=2k+2$</span> is even. Let <span class="math-container">$G$</span> be the graph obtained from a matching <span class="math-container">$v_1w_1,\dots,v_kw_k$</span> by adding two vertices <span class="math-container">$x$</span> and <span class="math-container">$y$</span> both adja... |
2,168,554 | <p>Assuming you are playing roulette.</p>
<p>The probabilities to win or to lose are:</p>
<p>\begin{align}
P(X=\mathrm{win})&=\frac{18}{37}\\
P(X=\mathrm{lose})&=\frac{19}{37}
\end{align}</p>
<p>Initially 1$ is used. Everytime you lose, you double up the stake. If you win once, you stop playing. If required ... | Breh | 458,959 | <p>I wrote this assuming the payoff is double the bet, as in Roulette when betting on spaces with these odds. So a \$1 bet yields \$2, or a \$1 net. When repeated, one is starting with a \$1 loss, and doubles their bet to \$2. If they win, they receive \$4, but have invested \$3 in the series, meaning the net gain is \... |
611,198 | <p>A corollary to the Intermediate Value Theorem is that if $f(x)$ is a continuous real-valued function on an interval $I$, then the set $f(I)$ is also an interval or a single point.</p>
<p>Is the converse true? Suppose $f(x)$ is defined on an interval $I$ and that $f(I)$ is an interval. Is $f(x)$ continuous on $I$? <... | Euler....IS_ALIVE | 38,265 | <p>The conclusion is not true even if $f$ is one to one. Consider the function on $[-1,1]$ defined by $f(x) = x$ for $x \not=-1,0,1 $, and $f(-1) = 0, f(0) = 1, f(1)=-1$. Then this function is not continuous, but is one to one and the range $f(I)$ is the interval $[-1,1]$.</p>
|
3,555,687 | <blockquote>
<p>There are 3 straight lines and two circles on a plane. They divide the plane into regions. Find the greatest possible number of regions.</p>
</blockquote>
<p><a href="https://i.stack.imgur.com/YbUAU.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/YbUAU.png" alt="enter image descrip... | WimC | 25,313 | <p>The number <span class="math-container">$21$</span> for a general configuration with maximal number of intersections follows from Euler’s formula. There are <span class="math-container">$18$</span> vertices (<span class="math-container">$3$</span> from line/line, <span class="math-container">$12$</span> from line/ci... |
29,443 | <p>I'm looking at a <a href="http://en.wikipedia.org/wiki/Partition_function_%28statistical_mechanics%29#Relation_to_thermodynamic_variables" rel="nofollow noreferrer">specific</a> derivation on wikipedia relevant to statistical mechanics and I don't understand a step.</p>
<p>$$ Z = \sum_s{e^{-\beta E_s}} $$</p>
<p>$... | Ehsan M. Kermani | 8,346 | <p>Have you seen the link of the definition of variance or expected value in wiki, <a href="http://en.wikipedia.org/wiki/Variance" rel="nofollow">here</a>?</p>
|
348,674 | <p>I am a beginner at matrices and I am trying to find out whether or not the linear transformation defined by the matrix $A$ is onto, and also whether it is 1-1.</p>
<p>Here is the matrix $A$: </p>
<p>$$\begin{bmatrix}
1 & 3 & 4 & -1 & 2\\
2 & 6 & 6 & 0 & -3\\
3 &... | DonAntonio | 31,254 | <p>Seen as an operator, your matrix is a map from $\,5-$dimensional space to a $\,4-$dimensional one, so $\;1-1\;$ it certainly cannot be. </p>
<p>Now, if its rank is $\,4\,$ then it is onto (why?), so you really have to reduce it and check this...</p>
|
3,902,501 | <p>Over <span class="math-container">$\mathbb R$</span>, the only linear maps are those of the form <span class="math-container">$ax$</span>.</p>
<p>If we discuss rational functions over <span class="math-container">$\mathbb R$</span>, this extra structure would allow us to describe a wider variety of linear maps.</p>
... | Ethan | 207,264 | <p>Using partial fractions and the fact that <span class="math-container">$\mathbb{R}[X]$</span> is a PID, it's possible to write every element of <span class="math-container">$\mathbb{R}(X)$</span> <em>uniquely</em> as a sum <span class="math-container">$\sum_{p<\mathbb{R}[X],n\in\mathbb{N}} \frac{f_{p,n}}{p^n}$</s... |
3,097,816 | <p>Show that the product of four consecutive odd integers is 16 less than a square.</p>
<p>For the first part I first did
<span class="math-container">$n=p(p+2)(p+4)(p+6)
=(p^2+6p)(p^2+6p+8).$</span>
I know that you are supposed to rearrange this to give an equation in the form <span class="math-container">$(ap^2+bp+c... | Keith Backman | 29,783 | <p>The trick is to start with the middle number, call it <span class="math-container">$k$</span>. Then the four consecutive odd integers are <span class="math-container">$k-3,k-1,k+1,k+3$</span>.</p>
<p>Multiply the outer two: <span class="math-container">$(k-3)(k+3)=k^2-9$</span></p>
<p>Multiply the inner two: <span... |
1,536,904 | <blockquote>
<p>A father send his $3$ sons to sell watermelons.The first son took $10$
watermelons,the second $20$ and the third $30$ watermelons.The father
gave order to sell all in the same price and collect and the same
number of money.How is this possible?</p>
</blockquote>
<p>Any ideas for this puzzle?</p... | mvw | 86,776 | <p>We can model this as:
$$
10 p = 20 p = 30 p \Rightarrow
$$
Subtracting $10 p$ from both sides of the first equation gives
$$
0 = 10 p \Rightarrow \\
p = 0
$$</p>
|
47,495 | <p>Firstly, let me divulge. I've been doing a lot of research on the summation of two coprime numbers and unfortunately have failed to come up with the properties I'm seeking; it is my hope that someone here might be of some help.</p>
<p>Let $(j, k)\in \mathbb{N}^2$ be coprime.</p>
<p>Can $\Omega(j + k)$ or $\omega(j... | Alastair Irving | 8,084 | <p>Suppose $p,q$ are prime. Thus $\omega,\Omega,\lambda$ are known constants at $p,q$ so we want to know if $\omega,\Omega(p+q)$ can be written as functions in $p,q$. To answer this you need to define what is meant by a function, as clearly $\omega(p+q)$ is a function in $p,q$.</p>
<p>Presumably you mean a function... |
1,615,732 | <p>Consider $CW$-complex $X$ obtained from $S^1\vee S^1$ by glueing two $2$-cells by the loops $a^5b^{-3}$ and $b^3(ab)^{-2}$. As we can see in Hatcher (p. 142), abelianisation of $\pi_1(X)$ is trivial, so we have $\widetilde H_i(X)=0$ for all $i$. And if $\pi_2(X)=0$, we have that $X$ is $K(\pi,1)$.</p>
<p>My questio... | Lee Mosher | 26,501 | <p>Letting $\widetilde X$ be the universal cover of $X$ we have
$$\pi_2(X) \approx \pi_2(\widetilde X) \approx H_2(\widetilde X,\mathbb{Z})
$$
The first isomorphism comes from the long exact sequence of homotopy groups of a fibration, using discreteness of the fiber of the universal covering map. The second isomorphism... |
3,702,309 | <p>The question reads,
Find all differentiable functions <span class="math-container">$f$</span> such that <span class="math-container">$$f(x+y)=f(x)+f(y)+x^2y$$</span> for all <span class="math-container">$x,y \in \mathbb{R} $</span>. The function <span class="math-container">$f$</span> also satisfies <span class="ma... | Alex Ravsky | 71,850 | <p>The problem conditions are implicitly self-contradictory. If we accept them and shall follow them then we can rigorously prove a lot of claims about the function <span class="math-container">$f$</span> and we can believe that everything is OK, until we obtain an explicit contradiction.</p>
<p>I <a href="http://ceads... |
3,702,309 | <p>The question reads,
Find all differentiable functions <span class="math-container">$f$</span> such that <span class="math-container">$$f(x+y)=f(x)+f(y)+x^2y$$</span> for all <span class="math-container">$x,y \in \mathbb{R} $</span>. The function <span class="math-container">$f$</span> also satisfies <span class="ma... | Alex Peter | 808,129 | <p>Your expression is not a representation of a function on as wide domain as <span class="math-container">$\mathbb{R}$</span>, but it does not mean that it is totally meaningless.</p>
<p>Replace <span class="math-container">$y=x$</span> and you have got</p>
<p><span class="math-container">$$f(2x)=2f(x)+x^3$$</span></p... |
2,465,705 | <p>In differential geometry, you are often asked to reparameterize a curve using arc-length. I understand the process of how to do this, but I don't understand what we are reparameterizing from.</p>
<p>What is the curve originally parameterized by (before we REparameterize it by arc-length)?</p>
| Community | -1 | <p>It is parameterized by the parameter that appears in the parametric equations given to you !</p>
<p>It is often the case that there is a natural way to define a curve, be it from its geometric definition or from a convenient coordinate system, and the parameter stands out.</p>
<p>In the case of trajectories, it ca... |
402,970 | <p>I had an exercise in my text book:</p>
<blockquote>
<p>Find $\frac{\mathrm d}{\mathrm dx}(6)$ by first principles.</p>
</blockquote>
<p>The answer that they gave was as follows:</p>
<p>$$\lim_{h\to 0} \frac{6-6}{h} = 0$$</p>
<p>However surely that answer would be undefined if we let $h$ tend towards $0$, as fo... | Mikasa | 8,581 | <p>The answer is right. We know that if the following limit $$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$ exists, then it is denoted by $f'(x)$. In the case $f$ is a constant function, the numerator of previous fraction in limit is <strong>absolutely</strong> zero but the variable $h$ in the denominator <strong>is going</stro... |
1,851,698 | <p>While playing with the results of defining a new operation, I came across a number of interesting properties with little literature surrounding it; the link to my original post is here: <a href="https://math.stackexchange.com/questions/1785715/finding-properties-of-operation-defined-by-x%E2%8A%95y-frac1-frac1x-frac1... | wyboo | 373,047 | <p>I found this number in an unrelated context.
Consider the differential equation:</p>
<p><span class="math-container">$$f'(x) = f(f(x))$$</span></p>
<p>The function that is the solution to this has the property that its derivative is the same as composing the function with itself. Assume the function takes the form o... |
226,249 | <p>I am struggling to convert a base64 string to a list of UnsignedInteger16 values.
I have limited experience with mathematica, so please excuse me if this should be obvious.</p>
<p>I read the base64 string from an XML file and ultimately into a variable base64String.
This is a long string with 7057 UnsignedInteger16 ... | alex | 20,155 | <p>There doesn't seem to be a built-in method within LineLegend or any other legend function to allow multiple lines. It seems that the fastest way forward is to have this created via Graphics, and then introduced in the plot as an inset using the Epilog option</p>
<pre><code>sep = 0.05; (* separation between the two l... |
1,948,634 | <blockquote>
<p>Is it possible to find $6$ integers $a_1,a_2,\ldots,a_6 \geq 2$ such that $$a_1+a_1a_2+a_1a_2a_3+a_1a_2a_3a_4+a_1a_2a_3a_4a_5+a_1a_2a_3a_4a_5a_6 = 248?$$</p>
</blockquote>
<p>I was wondering how we could establish the existence of such numbers. Is there a way to do it without finding the actual $6$ n... | trang1618 | 370,434 | <p>Note that:$$a_1(1+a_2(1+a_3(1+a_4(1+a_5(1+a_6))))) = 248 = 2^3(31)$$
$a_1$ has to be 2, because $1+a_2(1+a_3(1+a_4(1+a_5(1+a_6))))\geq 63.$ And so now we're solving for:
$$a_2(1+a_3(1+a_4(1+a_5(1+a_6)))) = 123.$$ </p>
<p>Following the same kind of argument, keeping in mind the constraint $a_i\geq2$, we arrive to t... |
238,673 | <p><a href="http://vihart.com/doodling/" rel="nofollow">Vi Hart's doodling videos</a> and a 4 year old son interested in mazes has made me wonder:</p>
<p>What are some interesting mathematical "doodling" diversions/games that satisfy the following criteria:</p>
<p>1) They are "solitaire" games, i.e. require only one ... | Karolis Juodelė | 30,701 | <p>You could draw an <a href="http://en.wikipedia.org/wiki/Elementary_cellular_automaton" rel="nofollow">elementary cellular automaton</a> with random initial row.</p>
|
973,985 | <p>I don't know where to start. </p>
<p>I know that the sum of the angles is less than or equal to 180.
but how do i prove this.</p>
| Hagen von Eitzen | 39,174 | <p>Let the sum be $x$. Cut an arbitrary triangle into two triangles. The sum of the smaller triangles is $2x$ but also $x+180^\circ$.</p>
|
2,240,756 | <p>I tried rewriting $(1+x+x^2)^\frac{1}{x}$ as $e^{\frac{1}{x}\ln(1+x+x^2)}$ and then computing the taylor series of $\frac{1}{x}$ and $\ln(1+x+x^2)$ but I'm still not getting the correct answer..</p>
| Masacroso | 173,262 | <p>HINT: a workaround following your way to see the problem is this: you want to write the Taylor series around $x=a$ of $$f(x):=\frac{\ln (1+x+x^2)}x$$</p>
<p>in terms of $$\frac1{1-z}=\sum_{k=0}^\infty z^k,\quad |z|<1\tag{1}$$</p>
<p>and</p>
<p>$$\ln(1+z)=\sum_{k=1}^\infty(-1)^{k+1}\frac{z^k}k,\quad|z|<1\tag... |
735,563 | <p>For both sets C and D, provide a proof that C ∪ (D − /C) = C</p>
<p>"/C" is a set's complement of C</p>
| Ellya | 135,305 | <p>It may help to know that the diagonal entries of $D$ are the eigenvalues of $A$, and the rows of $P$ are the corresponding eigenvectors of $A$, so calculate the eigenvalues and eigenvectors, then $A^5=(P^{-1}D^5P)$</p>
|
3,672,839 | <p>What is the derivative of <span class="math-container">$f(x)=|x|^\frac{3}{2},\forall x\in \mathbb{R}$</span>.</p>
<p>When <span class="math-container">$x>0$</span> it is fine but the problem is when <span class="math-container">$x\leq 0$</span>.I was trying by the defintion of derivative but what is <span class=... | Asaf Karagila | 622 | <p>I think that the first encounter I had was in linear algebra I or II. After proving that every finite dimensional vector space has a basis, some people wondered what happens if we omit "finite dimensional", and the answer is positive with Zorn's lemma being mentioned.</p>
<p>But maybe this doesn't count, since ther... |
2,220,105 | <p>Hatcher makes the following definition in exercise 7 on page 258. </p>
<p>For a map $f: M \rightarrow N $ between connected, closed, orientable $n$-manifolds with fundamental classes $[M]$ and $[N]$, the <em>degree</em> of $f$ is defined to be the integer $d$ such that $f_{*}[M]=d[N]$ (so the sign of the degree dep... | Bram28 | 256,001 | <p>The claim I will prove is that any string of $n$ numbers can be sorted <em>in any way I want</em> in at most $2n-3$ inversions.</p>
<p>Proof by weak induction over $n$:</p>
<p>Base: $n =2$. Then I have two numbers, which are either already in the order I want, which takes no inversions, or must be switched, which ... |
4,930 | <p>This is not an urgent question, but something I've been curious about for quite some time.</p>
<p>Consider a Boolean function in <em>n</em> inputs: the truth table for this function has 2<sup><em>n</em></sup> rows.</p>
<p>There are uses of such functions in, for example, computer graphics: the so-called ROP3s (ter... | Jon Awbrey | 1,636 | <p>I doubt if there's anything like the best of all possible formal languages for boolean expressions, but there are many ways of coming up with calculi that are more efficient than most of the ones currently in common use.</p>
<p>You might enjoy exploring the possibilities of using <a href="http://mywikibiz.com/Minim... |
4,930 | <p>This is not an urgent question, but something I've been curious about for quite some time.</p>
<p>Consider a Boolean function in <em>n</em> inputs: the truth table for this function has 2<sup><em>n</em></sup> rows.</p>
<p>There are uses of such functions in, for example, computer graphics: the so-called ROP3s (ter... | Jon Awbrey | 1,636 | <p>By way of clarifying the question to the point where it might get a definite answer, or at least a parametric set of answers, I think it would help to draw the distinction between the language-fixed and the language-variable sides of the problem a little more sharply. Rhubbarb alluded to this distinction in asking ... |
1,363,882 | <p>I am aware that the area under the curve of $\frac{1}{x}$ is infinite yet the area under the curve of $\frac{1}{x^2}$ is finite. </p>
<p>Calculus and series wise, I understand what is going on, but I can't seem to get a good geometric intuition of the problem.
Both curves can be shown to converge to $0$ (the curves... | Ad van Straeten | 573,311 | <p>I made drawings of both functions.</p>
<p>With respect to 1/x^2 I was very strict: I calculated the area under the straight line connecting ( 1,1) and (2,1/4) and so on. This gives two series that are both convergent by comparison with the series 1, 1/2, 1/4 ....The area of this series is larger than the area we ar... |
2,867,718 | <p>How do I integrate $\sin\theta + \sin\theta \tan^2\theta$ ?</p>
<p>First thing,
I have been studying maths for business for approximately 3 months now. Since then, I studied algebra and then I started studying calculus. Yet, my friend stopped me there, and asked me to study Fourier series as we'll need it for our ... | Tyler6 | 411,359 | <p>What happens in the video is that the $\sin(x)$ is factored out, and so it becomes $\sin(x)(1+\tan^2(x))$. This is simply factoring, and by expanding you can see it is equivalent to $\sin(x)+\sin(x) \tan^2(x)$. </p>
<p>As you said, the Greek letter $\theta$ is just a common variable, which is often used in trigonom... |
2,867,718 | <p>How do I integrate $\sin\theta + \sin\theta \tan^2\theta$ ?</p>
<p>First thing,
I have been studying maths for business for approximately 3 months now. Since then, I studied algebra and then I started studying calculus. Yet, my friend stopped me there, and asked me to study Fourier series as we'll need it for our ... | the_candyman | 51,370 | <p>This is a step-by-step solution using only <strong>basic facts</strong>.</p>
<blockquote>
<p>$$\tan(\theta) = \frac{\sin\theta}{\cos\theta}$$</p>
</blockquote>
<p>Then:</p>
<p>$$\int\sin\theta (1 + \tan^2\theta)d\theta = \int\sin\theta \left(1 + \frac{\sin^2\theta}{\cos^2\theta}\right)d\theta = \int\sin\theta \... |
1,318,552 | <blockquote>
<p>Let $f$ be a twice differentiable function on $\left[0,1\right]$ satisfying $f\left(0\right)=f\left(1\right)=0$. Additionally $\left|f''\left(x\right)\right|\leq1$ in $\left(0,1\right)$. Prove that $$\left|f'\left(x\right)\right|\le\frac{1}{2},\quad\forall x\in\left[0,1\right]$$</p>
</blockquote>
<p>... | wdacda | 124,843 | <p>This is an equivalent statement:</p>
<p>Let $g:[0,1] \to \mathbb R$ be a continuous function on $[0,1]$. Assume that $g$ is differentiable on $(0,1)$,<br>
$$
\int_0^1 g(x) dx = 0 \qquad \text{and} \qquad \forall x \in (0,1) \quad \bigl| g'(x) \bigr| \leq 1. \tag{1}\label{1}
$$
Then $|g(x)| \leq 1/2$ for all $x \... |
2,551,368 | <p>If I have a liquid that has a specific volume of 26.9 inches cubed per pound, and that liquid sells in the form of a gallon weighing 15.2 lbs, it would be correct to say it's total volume is 15.2*26.9, correct? So that would be 408.88 inches cubed. So if I had a volume of 1.157 inches cubed to fill with that liquid,... | DanielWainfleet | 254,665 | <p>Only if $X=\{0\}.$ If $X\ne \{0\}$ let $p\in X$ and let $\overline {B(p,1)}$ be a closed unit ball. Let $0\ne q\in X.$ Let $q'=q/\|q\|.$ Let $n\in \Bbb N.$ Then $$F(n)= \{B(p+q'(4j+1)/4n\; , 1/4n) : j\in \{0,...,n-1\}\}$$ is a family of pair-wise disjoint sets (by the Triangle Inequality) with $n$ members. Members ... |
1,412,091 | <p>The typewriter sequence is an example of a sequence which converges to zero in measure but does not converge to zero a.e.</p>
<p>Could someone explain why it does not converge to zero a.e.?</p>
<blockquote>
<p><span class="math-container">$f_n(x) = \mathbb 1_{\left[\frac{n-2^k}{2^k}, \frac{n-2^k+1}{2^k}\right]} \tex... | RozaTh | 386,136 | <p>I drew the first 63 functions in the sequence to help me understand its convergences. It might help others as well to understand answers given above:</p>
<p><a href="https://i.stack.imgur.com/2D209.png" rel="noreferrer"><img src="https://i.stack.imgur.com/2D209.png" alt="The typewriter sequence"></a></p>
<p>Unfort... |
1,412,091 | <p>The typewriter sequence is an example of a sequence which converges to zero in measure but does not converge to zero a.e.</p>
<p>Could someone explain why it does not converge to zero a.e.?</p>
<blockquote>
<p><span class="math-container">$f_n(x) = \mathbb 1_{\left[\frac{n-2^k}{2^k}, \frac{n-2^k+1}{2^k}\right]} \tex... | robjohn | 13,854 | <p>I wrote this sequence of functions as a counterexample for <a href="https://math.stackexchange.com/questions/3735925/how-to-prove-that-f-n-to-0">a question</a> that was deleted.</p>
<p><span class="math-container">$$
f_n(x)=\left[\frac km\le x\lt\frac{k+1}m\right]
$$</span>
where <span class="math-container">$[\dots... |
3,839,244 | <p>Is <span class="math-container">$\{3\}$</span> a subset of <span class="math-container">$\{\{1\},\{1,2\},\{1,2,3\}\}$</span>?</p>
<p>If the set contained <span class="math-container">$\{3\}$</span> plain and simply I would know but does the element <span class="math-container">$\{1,2,3\}$</span> include <span class=... | User203940 | 333,294 | <p>Let <span class="math-container">$X$</span> be a set. We say <span class="math-container">$Y \subseteq X$</span> (<span class="math-container">$Y$</span> is a subset of <span class="math-container">$X$</span>) if, for all <span class="math-container">$x \in Y$</span>, we have <span class="math-container">$x \in X$</... |
1,935,320 | <p>If $a^2-b^2=2$ then what is the least possible value of: \begin{vmatrix} 1+a^2-b^2 & 2ab &-2b\\ 2ab & 1-a^2+b^2&2a\\2b&-2a&1-a^2-b^2 \end{vmatrix}</p>
<p>I tried to express the determinant as a product of two determinants but could not do so. Seeing no way out, I tried expanding it but that ... | Ivan Neretin | 269,518 | <p>Why, the general answer is just as simple: consider a $120^\circ$ rotation around any axis.</p>
<p>In 2D, the possibilities are limited to Jacky Chong's answer, its transposition, and $I$.</p>
<p>In 3D, the general answer is $$\left(\begin{array}{c|c|c}
{3\over2}x^2-{1\over2} & {3\over2}xy+{\sqrt3\over2}z&... |
765,738 | <p>I am trying to prove the following inequality:</p>
<p>$$(\sqrt{a} - \sqrt{b})^2 \leq \frac{1}{4}(a-b)(\ln(a)-\ln(b))$$</p>
<p>for all $a>0, b>0$.</p>
<p>Does anyone know how to prove it?</p>
<p>Thanks a lot in advance!</p>
| medicu | 65,848 | <p>Inequality that is another form of a well-known inequalities:</p>
<p>For $0<y<x$
$$ \sqrt{xy}<\frac{x-y}{\ln x-\ln y} < \frac{x+y}{2}. (1)$$
For $a>b$ inequality of enunciation transform so:
$$(\sqrt{a} - \sqrt{b})^2 < \frac{1}{4}(a-b)(\ln a-\ln b)<=>$$
$$(\sqrt{a} - \sqrt{b})^2 < \frac{1... |
599,656 | <p>If the objects of a category are algebraic structures in their own right, this often places additional structure on the homsets. Is there somewhere I can learn more about this general idea?</p>
<hr>
<p><strong>Example 1.</strong> Let $\cal M$ denote the category of magmas and <strong>functions</strong>, not necess... | Flowers | 93,842 | <p>I cannot comment, but you are obviously mistaken in your calculation. Your quadratic formula that is. Here, <a href="http://www.wolframalpha.com/input/?i=x%5E2%2B104x%E2%88%92896%3D0" rel="nofollow">WA does not make mistakes</a></p>
<p>EDIT:
$$\frac{-104\pm\sqrt{104^2+4\cdot 896}}{2}\\
=\frac{-104\pm\sqrt{14400}}{2... |
1,593,007 | <p>maybe this is a stupid question but I have the following expression:</p>
<p>$ 10^{-18}(e^{50,9702078⋅0,75}) = 10^{-18}(4⋅10^{16}) $</p>
<p>How would I go about simplifying the big exponent on the left to what's on the right? With the use of a calculator.</p>
<p>Thanks a lot! </p>
| Yuri Negometyanov | 297,350 | <p>$$yy' = x+1$$
$$y^2 = (x+1)^2+C.$$
$$y = \pm\sqrt{(x+1)^2+C},\quad(x+1)^2 + C \geq 0.$$
Or:
$$y = \pm\sqrt{(x+1)^2+C},$$
$$
x\in
\begin{cases}
(-\infty, +\infty),\text{ if }C \geq -1\\[3pt]
\left(-\infty. -\sqrt{-C}-1\right)\, \cup (\sqrt{-C}-1, +\infty),\text{ if }C <-1.
\end{cases}
$$</p>
|
1,865,868 | <p>the problem goes like that "in urn $A$ white balls, $B$ black balls. we take out without returning 5 balls. (we assume $A,B\gt4$) what would be the probability that at the 5th ball removal, there was a white ball while we know that at the 3rd was a black ball".</p>
<p>What I did is I build a conditional probability... | Claude Leibovici | 82,404 | <p>You could even solve the problem using basic calculus since minimizing the distance is the same as minimizing the square of the distance. </p>
<p>The constraints being $$x+y+z=2\qquad , \qquad 12x+3y+3z=12$$ take advantage of their linearity and solve these two equations for $x$ and $z$ as functions of $y$. Tou wil... |
1,993,034 | <p>if I have say $x_4$ = free, what value goes in the $x_4$ position of the parametric form, is it $1$ or $0$ or can it be any value since it's free?</p>
| rosaqq | 384,490 | <p>You just say $x_4 = t$, $t\in \Bbb R$</p>
<p>Example:
$$\mathbf v = [x_1, x_2, x_3, 0] + t[0, 0, 0, 1],\quad t\in \Bbb R$$
$x_1$, $x_2$, and $x_3$ are set values while $t$ is free.</p>
|
102,402 | <p>I wanted to make a test bank of graphs of linear equations for my algebra classes. I want the $y$-intercept of each graph to be an integer no less than $-10$ and no greater than $10$. Generally, you want these graphs to be small, so i've decided on a $20 \times 20$ grid (10 units from the origin). Additionally, i ... | 2'5 9'2 | 11,123 | <p>If you extended by $11$ in all directions (instead of $10$), then your grid is a picture of $\mathbb{F}_{23}^2$. (It's nicer to work with the field $\mathbb{F}_{23}$ than the non-field $\mathbb{F}_{21}$.) Your question translates to counting how many affine lines are there in this vector space. Lines in this space w... |
1,910,085 | <p>For all integers $n \ge 0$, prove that the value $4^n + 1$ is not divisible by 3.</p>
<p>I need to use Proof by Induction to solve this problem. The base case is obviously 0, so I solved $4^0 + 1 = 2$. 2 is not divisible by 3.</p>
<p>I just need help proving the inductive step. I was trying to use proof by contrad... | parsiad | 64,601 | <p><strong>Hint:</strong> Suppose $3$ does not divide $4^{n}+1$ for some $n\in\{0,1,\ldots\}$.
If $3$ divides $4^{n+1}+1$,
$$
0\equiv4^{n+1}+1\equiv4^{n}4+1\equiv \text{ } ?\text{ (mod }3\text{)}.
$$
(I have left out a step which you should fill in to arrive at a contradiction)</p>
|
1,910,085 | <p>For all integers $n \ge 0$, prove that the value $4^n + 1$ is not divisible by 3.</p>
<p>I need to use Proof by Induction to solve this problem. The base case is obviously 0, so I solved $4^0 + 1 = 2$. 2 is not divisible by 3.</p>
<p>I just need help proving the inductive step. I was trying to use proof by contrad... | Community | -1 | <p>Induction step:</p>
<p>$$4^{n+1}+1=3\cdot 4^n+4^n+1\equiv4^n+1\mod 3$$</p>
<hr>
<p>By the way, by recurrence,</p>
<p>$$4^n\equiv1\mod3,$$
$$4^n+1\equiv2\mod3,$$
$$4^n+2\equiv0\mod3.$$</p>
|
1,910,085 | <p>For all integers $n \ge 0$, prove that the value $4^n + 1$ is not divisible by 3.</p>
<p>I need to use Proof by Induction to solve this problem. The base case is obviously 0, so I solved $4^0 + 1 = 2$. 2 is not divisible by 3.</p>
<p>I just need help proving the inductive step. I was trying to use proof by contrad... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\ \ f_n =\, 4^n\!+1\ \Rightarrow\,\ f_{n+1}\!-f_n =\! \overbrace{\color{#c00}3\cdot 4^n}^{\large 4^{\Large n+1}-4^{\Large n} }\! $ so $\,\ \color{#c00}3\mid f_{n+1}\!\iff \color{#c00}3\mid f_n$</p>
<p><strong>Remark</strong> $ $ Said $\,{\rm mod}\ 3\!:\ f_{n+1}\equiv f_n\ $ so by induction ... |
265,509 | <p>Given a (possibly disjoint) region, defined by a discrete set of points, how can I use <code>ListContourPlot[]</code> together with <code>Mesh</code> to highlight a specific area of the plot? For instance, how can I mesh the region where the points are smaller than a certain value?</p>
<p>Here I construct a minimal ... | cvgmt | 72,111 | <pre><code>data = Table[Exp[x^2 - y^2], {x, -1, 1, .01}, {y, -1, 1, .01}];
ListContourPlot[data, Contours -> {1.0}, ContourStyle -> Transparent,
ContourShading -> {Directive[Orange, HatchFilling[-Pi/4, 1, 10]],
Directive[Cyan, HatchFilling[Pi/4, 1, 8]]}]
</code></pre>
<p><a href="https://i.stack.imgur.c... |
686,848 | <p>This is probably an elementary question about fields, but I think it is a little tricky. </p>
<blockquote>
<p>Can we make the integers $\mathbb{Z}$ into a field?</p>
</blockquote>
<p>Let me be more precise. Is it possible to make $\mathbb{Z}$ into a field so that the underlying additive group structure is the us... | Community | -1 | <p><strong>Theorem:</strong> If $F$ is a field and $x \in F$ with $x \neq 0$, then either:</p>
<ul>
<li>there exists $y \in F$ such that $y + y = x$, or</li>
<li>$x + x = 0$.</li>
</ul>
|
3,614,875 | <p><strong>Prop:</strong> Every sequence has a monotone subsequence.</p>
<p><strong>Pf:</strong> Suppose <span class="math-container">$\{a_n\}_{n\in \mathbb{N}}$</span> is a sequence. Choose <span class="math-container">$a_{n_1} \in \{a_1,a_2,...\}$</span>. Further choose smallest possible <span class="math-container"... | Brian M. Scott | 12,042 | <p>Say that a term <span class="math-container">$a_n$</span> is <em>dominant</em> if <span class="math-container">$a_n\ge a_m$</span> whenever <span class="math-container">$n\le m$</span>.</p>
<ul>
<li>Show that if there are infinitely many dominant terms, they form an infinite non-increasing subsequence.</li>
</ul>
... |
2,698,903 | <p>So this has caused some confusion. In one exercise one was asked to prove that</p>
<p>$$\lim_{k \rightarrow \infty} \int_A \cos(kt)dt=0$$</p>
<p>where $A \subset [-\pi, \pi]$ is a measurable set.</p>
<p>My initial idea was to take any $a,b \in A$ and then show that:</p>
<p>$$\lim_{k \rightarrow \infty} \int_a^b ... | Giuseppe Negro | 8,157 | <p>I think that your approach is a good start, but you have to extend your result to general measurable sets. You can do that by approximation. </p>
<p>Let $\mathbb T=[-\pi, \pi)$. </p>
<p><strong>Step 1</strong>. Show that for any open set $O\subset \mathbb T$, it holds that $\lim_{k\to \infty} \int_O \cos(kt)\, dt ... |
1,545,045 | <p>Is it possible to cut a pentagon into two equal pentagons?</p>
<p>Pentagons are not neccesarily convex as otherwise it would be trivially impossible.</p>
<p>I was given this problem at a contest but cannot figure the solution out, can somebody help?</p>
<p>Edit: Buy "cut" I mean "describe as a union of two pentag... | N. F. Taussig | 173,070 | <p>Graham Kemp and user164385 have provided you with nice solutions. Here is an alternative approach.</p>
<p>Line up the $28$ workers in some order, whether alphabetically, by seniority, or some other criterion. The first person in line has $27$ ways of selecting a work partner. The next person in line who is not a... |
109,973 | <p>I have stuck solving this problem of financial mathematics, in this equation:
$$\frac{(1+x)^{8}-1}{x}=11$$</p>
<p>I'm stuck in this eight grade equation:
$$x^{8}+8x^{7}+28x^{6}+56x^{5}+70x^{4}+56x^{3}+28x^{2}+9x-11=0$$</p>
<p>But I cannot continue past this. This quickly lead me to find a general way of solving eq... | Nick Alger | 3,060 | <p>As has been pointed out above, from Galois theory in general there is no algebraic solution to polynomials of degree 5+. </p>
<p>But what about numerical solvers? Consider the polynomial
$$p(x)=a+bx+cx^2+dx^3+x^4$$
and its companion matrix,
$$A(p)=\left[\begin{matrix}0 & 0 & 0 & -a \\ 1 & 0 & 0... |
1,668,487 | <p>$$2^{-1} \equiv 6\mod{11}$$</p>
<p>Sorry for very strange question. I want to understand on which algorithm there is a computation of this expression. Similarly interested in why this expression is equal to two?</p>
<p>$$6^{-1} \equiv 2\mod11$$</p>
| Clément Guérin | 224,918 | <p>You have $2\times 6=12$ which is $1$ mod $11$. Hence, we see that that $2\times 6=1$ mod $11$.</p>
<p>Now the set of congruence numbers modulo $11$ (often noted $\mathbb{Z}/11$) is a ring with addition and multiplication coming from $\mathbb{Z}$. </p>
<p>Saying that $6^{-1}\text{ mod }11=2$ is saying that $6$ mod... |
847,672 | <p>I repeat the Peano Axioms:</p>
<ol>
<li><p>Zero is a number.</p></li>
<li><p>If a is a number, the successor of a is a number.</p></li>
<li><p>zero is not the successor of a number.</p></li>
<li><p>Two numbers of which the successors are equal are themselves equal.</p></li>
<li><p>If a set S of numbers contains zer... | Thomas Andrews | 7,933 | <p>In logic, $0$ is a constant, not a property. That is, we can say "$x=0$" rather than "$x$ is a zero."</p>
<p>Even in second order logic, the statement of induction is $0\in S$. Not, "for all zeros $z, z\in S$."</p>
<p>You can, of course, interpret lots of language in lots of different ways. But mathematics uses "$... |
749,035 | <p>Call a square-free number a 3-prime if it is the product of three primes. Similarly for 2-primes, 4-primes , 5-primes, etc. Are there two consecutive 3-primes with no 2-prime between them?Are there infinitely many?</p>
| Community | -1 | <p>Here's a list to get you started: $$[[230, 231], [255, 258], [285, 286], [429, 430], [434, 435], [609, 610], [645, 646], [741, 742], [805, 806], [902, 903], [969, 970], [986, 987], [1001, 1002], [1022, 1023], [1065, 1066], [1085, 1086], [1105, 1106], [1130, 1131], [1221, 1222], [1245, 1246], [1265, 1266], [1309, 131... |
825,848 | <p>I'm trying to show that the minimal polynomial of a linear transformation $T:V \to V$ over some field $k$ has the same irreducible factors as the characteristic polynomial of $T$. So if $m = {f_1}^{m_1} ... {f_n}^{m_n}$ then $\chi = {f_1}^{d_1} ... {f_n}^{d_n}$ with $f_i$ irreducible and $m_i \le d_i$.</p>
<p>Now I... | Marc van Leeuwen | 18,880 | <p>One can reason using only some field theory, rather than linear algebra, if one prefers. I will assume the Cayley-Hamilton theorem, and the fact that all eigenvalues are roots of the minimal polynomial$~\mu$, which holds because an eigenvector for$~\lambda$ is killed by $P[T]$ (if and) only if $P$ is a multiple of $... |
272,588 | <p>I am trying to do the partition of the list variable called
<strong>test</strong> so that answer will look like the variable called <strong>goal</strong>. The <strong>test</strong> list is actually a list of long data. In this example I have a short version example according to the code below which also includes m... | Syed | 81,355 | <p>Assuming that the number of items in <code>test</code> is not divisible by the total number of elements in each partition, e.g.,</p>
<blockquote>
<pre><code>test = {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u,
v, w, x, y};
</code></pre>
</blockquote>
<p>a possible solution using <code>TakeList<... |
317,981 | <p>Prove the following:</p>
<p>$$\lim_{n \to \infty} \displaystyle \int_0 ^{2\pi} \frac{\sin nx}{x^2 + n^2} dx = 0$$</p>
<p>How would I prove this? I know you have to show your steps, but I'm literally stuck on the first one, so I can't. </p>
| user 1591719 | 32,016 | <p><strong>HINT:</strong> Make use of </p>
<p>$$\left|\dfrac{\sin nx}{x^2 + n^2}\right|\le\dfrac{1}{x^2 + n^2}$$</p>
<p>I think you also want to see <a href="https://math.stackexchange.com/questions/151839/absolute-value-integral-inequality-proof-step">Absolute value integral inequality proof step</a>. </p>
|
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