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 |
|---|---|---|---|---|
3,967,118 | <p>How is it possible to define <span class="math-container">$\lim_{x \rightarrow a} f(x)$</span> if <span class="math-container">$f(x)$</span> is undefined at <span class="math-container">$a$</span>? There is an infinite amount of <span class="math-container">$x$</span>-values on either side of <span class="math-conta... | Ziqi Fan | 851,072 | <p>The limit of <span class="math-container">$f$</span> at a point <span class="math-container">$a$</span> is defined as
<span class="math-container">\begin{equation}
\lim_{x\to a}f\left(x\right) = L \longleftrightarrow \forall \epsilon > 0, \exists \delta > 0, \forall x, 0 < \lvert x-a \rvert < \delta \lon... |
261,156 | <p>Well, I am trying to write a code that makes the number:</p>
<p><span class="math-container">$$123456\dots n\tag1$$</span></p>
<p>So, when <span class="math-container">$n=10$</span> we get:</p>
<p><span class="math-container">$$12345678910$$</span></p>
<p>And when <span class="math-container">$n=15$</span> we get:</... | murray | 148 | <pre><code>FromDigits@Flatten[IntegerDigits /@ Range[15]]
123456789101112131415
</code></pre>
<p>A function to do it:</p>
<pre><code>numberFromRange[n_] := FromDigits@Flatten[IntegerDigits /@ Range@n]
</code></pre>
|
261,156 | <p>Well, I am trying to write a code that makes the number:</p>
<p><span class="math-container">$$123456\dots n\tag1$$</span></p>
<p>So, when <span class="math-container">$n=10$</span> we get:</p>
<p><span class="math-container">$$12345678910$$</span></p>
<p>And when <span class="math-container">$n=15$</span> we get:</... | flinty | 72,682 | <p>Without using <code>IntegerDigits</code> or string processing:</p>
<pre><code>f[x_] := Last@NestWhile[#[[1]]+{1,#[[2]]*10^IntegerLength@#[[1]]}&,{1,0},#[[1]]<=x&]
</code></pre>
<pre><code>f[105]
(* 1234567891011121314151617181920212223242526272829303132333435363738394
0414243444546474849505152535455565758... |
261,156 | <p>Well, I am trying to write a code that makes the number:</p>
<p><span class="math-container">$$123456\dots n\tag1$$</span></p>
<p>So, when <span class="math-container">$n=10$</span> we get:</p>
<p><span class="math-container">$$12345678910$$</span></p>
<p>And when <span class="math-container">$n=15$</span> we get:</... | user1066 | 106 | <pre><code>ToExpression@StringJoin@Array[IntegerString,15]
(* 123456789101112131415 *)
</code></pre>
|
261,156 | <p>Well, I am trying to write a code that makes the number:</p>
<p><span class="math-container">$$123456\dots n\tag1$$</span></p>
<p>So, when <span class="math-container">$n=10$</span> we get:</p>
<p><span class="math-container">$$12345678910$$</span></p>
<p>And when <span class="math-container">$n=15$</span> we get:</... | kglr | 125 | <p>Timings for all the methods (<a href="https://mathematica.stackexchange.com/a/261159/125">g1 : murray</a>, <a href="https://mathematica.stackexchange.com/a/261167/125">g2 : flinty</a>, <a href="https://mathematica.stackexchange.com/a/261188/125">g3/g4 : AccidentalFourierTransform</a>, <a href="https://mathematica.st... |
2,801,294 | <p>I cam across an interesting claim that:</p>
<p>$\mathcal N (\mu_f, \sigma_f^2) \; \mathcal N (\mu_g, \sigma_g^2) = \mathcal N \left(\frac{\mu_f\sigma_g^2+\mu_g\sigma_f^2}{\sigma_f^2+\sigma_g^2}, \frac {\sigma_f^2\sigma_g^2}{\sigma_f^2+\sigma_g^2}\right)$</p>
<p>In trying to understand it I consulted Bromiley:</p>
... | Bernd Wechner | 563,137 | <p>We have an insight into a solution to this confusion from two observations:</p>
<ol>
<li><p>The <span class="math-container">$\mathcal N(\mu, \sigma)$</span> notation is a loose convention. I find no clear definition of it anywhere. A good reference is the Wikipedia mention here:
<a href="https://en.wikipedia.org/wi... |
1,341,896 | <p>given three lines $\ell_1,\ell_2, \ell_3 $ which intersect in one point $P$. How can one construct a triangle such that the given lines become its angle bisectors?</p>
<p>So far I tried to find conditions on how the three given lines have to intersect such that the desired triangle exists. There are altogether six ... | Wojowu | 127,263 | <p>Let $I$ be the point of intersection of the lines (renamed it so that it's clear that we want it to be the incenter) and let $A,B,C$ be the points we want to construct. Let $\alpha,\beta,\gamma$ be angles of the triangle, and let $\delta,\epsilon,\zeta$ be angles $BIC,CIA,AIB$ respectively (see the picture).</p>
<p... |
3,355,215 | <blockquote>
<p>For <span class="math-container">$x$</span> and <span class="math-container">$k$</span> real numbers, for what values of <span class="math-container">$k$</span> will the graphs of <span class="math-container">$f(x)=-2\sqrt{x+1}$</span> and <span class="math-container">$g(x)=\sqrt{x-2}+k$</span> inters... | Alessandro Cigna | 641,678 | <p>You want <span class="math-container">$-2\sqrt{x+1}=\sqrt{x-2}+k\; (*)$</span> has at least one <span class="math-container">$x$</span>-solution.</p>
<p>If we call <span class="math-container">$h(x)=-2\sqrt{x+1}-\sqrt{x-2}$</span> then we have that <span class="math-container">$h$</span> is defined only for <span c... |
911,584 | <p><em>Disclaimer: This thread is a record of thoughts.</em></p>
<p><strong>Discussion</strong>
Given a compact set.</p>
<blockquote>
<p>Do mere neighborhood covers admit finite subcovers?
$$C\subseteq\bigcup_{i\in I}N_i\implies C\subseteq N_1\cup\ldots N_n$$
<em>(The idea is that neighborhoods are in some sens... | gar | 138,850 | <p>It can be solved using ordinary generating functions:</p>
<p>Since there are three groups of numbers of our interest, take three formal variables for the ogf.</p>
<p>Let $x$ indicate 1 or 2, $z$ indicate a 6, and $y$ the other three numbers of the dice.</p>
<p>Hence, the required ogf is:
\begin{align*}
\left(2\, ... |
231,317 | <p>The task I'm faced with is to implement a poly-time algorithm that finds a nontrivial factor of a Carmichael number. Many resources on the web state that this is easy, however without further explanation why.</p>
<p>Furthermore, since Miller-Rabin exits when a nontrivial square root of 1 is found, this can be used ... | Peter | 82,961 | <p>A Carmichael number $N$ is a probable prime to every base $a$ coprime to $N$, but there is a base $a$ with $1<a<N$ not too big and coprime to $N$ (if it is NOT coprime to $N$, $gcd(a,N)$ is a non-trivial factor), such that $N$ is not strong probable prime to base $a$. $a$ is called a witness.</p>
<p>For such ... |
2,700,301 | <blockquote>
<p>Suppose we throw 4 dice consecutively. What is the probability
that the sum of the dice is <span class="math-container">$24?$</span></p>
</blockquote>
<p><strong>My attempt</strong>:</p>
<p>The experiment we consider is throwing 4 dices and looking at the numbers we obtain, i.e. <span class="math-... | Siong Thye Goh | 306,553 | <p>Yes, your approach is correct. </p>
<p>In general, for other target sum, we can use generating function to solve the problem. </p>
|
2,700,301 | <blockquote>
<p>Suppose we throw 4 dice consecutively. What is the probability
that the sum of the dice is <span class="math-container">$24?$</span></p>
</blockquote>
<p><strong>My attempt</strong>:</p>
<p>The experiment we consider is throwing 4 dices and looking at the numbers we obtain, i.e. <span class="math-... | Mohammad Riazi-Kermani | 514,496 | <p>Yes, your approach is correct.</p>
<p>The only way to get a sum of $24$ is having four $6$.</p>
<p>The probability of four independent event where each has the probability of $1/6$ is $ 1/6^4$</p>
|
20,802 | <p>Look at the following example:</p>
<p>Which picture has four apples?</p>
<p>A<a href="https://i.stack.imgur.com/Tpm46.png" rel="noreferrer"><img src="https://i.stack.imgur.com/Tpm46.png" alt="enter image description here" /></a></p>
<hr />
<p>B <a href="https://i.stack.imgur.com/AOv29.png" rel="noreferrer"><img src=... | BCLC | 5,987 | <p>It is wrong exactly for the reasons you have pointed out. There shouldn't be an issue with including the word 'exactly' as in 'exactly 4 apples'.</p>
|
3,580,258 | <p>Hi: The definition I'll use is this: Let <span class="math-container">$F$</span> be an abelian group and <span class="math-container">$X$</span> a subset of <span class="math-container">$F$</span>. Then <span class="math-container">$F$</span> is a free abelian group on <span class="math-container">$X$</span> if for ... | Tsemo Aristide | 280,301 | <p>Consider <span class="math-container">$G=\mathbb{Z}/5\mathbb{Z}$</span> and <span class="math-container">$[1]\in G$</span>, let <span class="math-container">$X=\{[1]\}$</span> <span class="math-container">$F=<X>=G$</span>, let <span class="math-container">$f:X\rightarrow\mathbb{Z}$</span> defined by <span clas... |
326,600 | <p>The intervals are: </p>
<p>$I_1=\{ x\in\mathbb{R}:a\leq x\leq b\}$ with $-1<a<b<1$.</p>
<p>$I_2=\{ x\in\mathbb{R}:-1< x< 1\}$.</p>
<p>I've shown $f_n$ is pointwise convergent to $0$ on $I_1$ and $I_2$. But what is the difference for showing uniform convergence on these two intervals? I have a few o... | Community | -1 | <p>The sequence of function $(f_n)$ is unifom convergent to $f$ in the interval $I$ if:
$$\sup_{x\in I}|f_n(x)-f(x)|=0.$$</p>
<ol>
<li><p>We have
$$ \forall x\in I_1,|f_n(x)-0|\leq \max(|a|^n,|b|^n)\to0,$$
so the sequence $(f_n)$ is uniform convergent to the zero function on $I_1$.</p></li>
<li><p>The sequence $(f_n)... |
1,039,487 | <p>Suppose $a$, $b$ are real numbers such that $a+b=12$ and both roots of the equation $x^2+ax+b=0$ are integers. </p>
<p>Determine all possible values of $a$. </p>
<p>I don't know how to go about doing this without long, messy casework. I tries $(x-s)(x-r)=x^2+ax+b$ and got $-r-s=a$ and $rs=b$, but was unable to fi... | Varun Iyer | 118,690 | <p>Since $a + b = 12$</p>
<p>$$-(r+s) + rs = 12$$</p>
<p>Using Simon's favorite factoring trick:</p>
<p>$$(r-1)(s-1) - 1 = 12 \iff (r-1)(s-1) = 13$$</p>
<p>Only two possible solutions arise:</p>
<p>$r -1 = 1$ and $s -1 = 13$, therefore $r = 2$ and $s = 14$</p>
<p>$r -1 = -1$ and $s -1 = -13$, therefore $r = 0$ an... |
363,881 | <p>The problem gives the curl of a vector field, and tells us to calculate the line integral over $C$ where $C$ is the intersection of $x^2 + y^2 = 1$ and $z= y^2$. I know I should use Stokes Theorem, but how do I find $dS$? </p>
<p>I did $z = \frac{1}{2}(y^2 + 1 - x^2)$ and calculated $dS$ as $\langle-dz/dx,-dz/dy,1\... | Shuhao Cao | 7,200 | <p>I drew a figure illustrating how these two surfaces intersecting with each other<img src="https://i.stack.imgur.com/r0IbW.png" alt="."> </p>
<p>The intersecting curve $C$ is that black line, and let's assume $C$'s direction is rotating counter-clockwise if you look from above. The surface $S$ is part of the $z=y^2$... |
2,194,762 | <p>I have to find all possible values of <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span>, and <span class="math-container">$d$</span> for which the <span class="math-container">$2 \times 2$</span> matrix
<span class="math-container">$\begin{pmatr... | Matthew C | 25,959 | <p>Since you already know semisimple is equivalent to diagonalizable (it is, in the case of complex matrices), then you can consider the possible Jordan forms for 2x2 matrices. You want to avoid the situation of a Jordan block. That means the two remaining situations are: you have a multiple of the identity matrix, or ... |
2,194,762 | <p>I have to find all possible values of <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span>, and <span class="math-container">$d$</span> for which the <span class="math-container">$2 \times 2$</span> matrix
<span class="math-container">$\begin{pmatr... | Community | -1 | <p>Let <span class="math-container">$A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in M_n(\mathbb{C})$</span> (we work over <span class="math-container">$\mathbb{C}$</span>).</p>
<p>If <span class="math-container">$A$</span> is not diagonal, then </p>
<p><span class="math-container">$A$</span> is diagonalizable IFF ... |
79,782 | <p>Calculate $17^{14} \pmod{71}$</p>
<p>By Fermat's little theorem:<br>
$17^{70} \equiv 1 \pmod{71}$<br>
$17^{14} \equiv 17^{(70\cdot\frac{14}{70})}\pmod{71}$</p>
<p>And then I don't really know what to do from this point on. In another example, the terms were small enough that I could just simplify down to an answer... | Bill Dubuque | 242 | <p>Here is the computation using an <a href="http://wwwhomes.uni-bielefeld.de/achim/addition_chain.html" rel="nofollow noreferrer">addition chain.</a></p>
<p>$\qquad 17^2\: \equiv\ 5$</p>
<p>$\qquad 17^3\: \equiv\ 17\cdot 17^2\:\equiv\ 5\cdot 17\ \equiv\ 14$</p>
<p>$\qquad 17^5\: \equiv\ 17^2\cdot 17^3\: \equiv\ 5\c... |
79,782 | <p>Calculate $17^{14} \pmod{71}$</p>
<p>By Fermat's little theorem:<br>
$17^{70} \equiv 1 \pmod{71}$<br>
$17^{14} \equiv 17^{(70\cdot\frac{14}{70})}\pmod{71}$</p>
<p>And then I don't really know what to do from this point on. In another example, the terms were small enough that I could just simplify down to an answer... | Angel Moreno | 327,493 | <p>In this curious problem, the subgroup of order 5 of <span class="math-container">$(Z_{71}^*, \cdot)$</span> is formed by these 5 numbers (5 equivalence classes modulo 71):</p>
<p><span class="math-container">$$ \{1, 5, 25, -17, -14\} $$</span></p>
<p>(When multiplying two, another one comes out of the same subgrou... |
4,523,363 | <p>A set <a href="https://en.m.wikipedia.org/wiki/Lattice_(order)#:%7E:text=A%20lattice%20is%20an%20abstract,greatest%20lower%20bound%20or%20meet" rel="nofollow noreferrer">lattice</a> is a lattice where the meet is the intersection, the join is the union, and the partial order is <span class="math-container">$\subsete... | freakish | 340,986 | <p>Well, not every lattice is isomorphic to <span class="math-container">$P(A)$</span>. For example the <span class="math-container">$P(A)$</span> lattice has <span class="math-container">$2^{|A|}$</span> elements, and there are lattices for any order, e.g.</p>
<p><span class="math-container">$$K_n=\big\{\{1\},\{1,2\},... |
2,203,770 | <p>If I am asked to find all values of $z$ such that $z^3=-8i$, what is the best method to go about that?</p>
<p>I have the following formula:
$$z^{\frac{1}{n}}=r^\frac{1}{n}\left[\cos\left(\frac{\theta}{n}+\frac{2\pi k}{n}\right)+i\sin\left(\frac{\theta}{n}+\frac{2\pi k}{n}\right)\right]$$</p>
<p>for $k=0,\pm1, \pm2... | JMP | 210,189 | <p>You are claiming $\theta=\pi$, whereas it doesn't - actually $\theta=\dfrac{3\pi}{2}$, which gives the solution you list.</p>
<p>For $-8$ use $\theta=\pi$, and for $8i$ use $\theta=\dfrac{\pi}{2}$.</p>
|
1,232,023 | <p>For me, intuitively, integral $2\pi y~dx$ make more sense. I know intuition can not be proof, but by far, most part of math I've learned does match with my intuition. So, I think this one should 'make sense' as well. Probably I didn't understand the way surface area is measured. It will be great if any one could tel... | David K | 139,123 | <p>Here are two ways to measure the surface area of an open cone of base radius $r$ and height $h$ (just the area of the curved surface of the cone; we are not including the area of the flat circular base of the cone).</p>
<p>The first way is, you cut the surface of the cone along a straight line from the apex to the ... |
1,773,303 | <p>Let $k$ be a field and $L=k(\sqrt{a_1},\sqrt{a_2},\cdots,\sqrt{a_n})$ with $a_i\in k$. Suppose $[L:k]=2^n$. Let $a=\sqrt{a_1}+\sqrt{a_2}+\cdots\sqrt{a_n}$. Show that $L=k(a)$.</p>
| H. Potter | 289,192 | <p>Since you know that being Hausdorff is an "increasing property" (that means if $\mathcal{T}'\subseteq \mathcal{T}$ and $\mathcal{T}'$ is Hausdorff, then $\mathcal{T}$ is Hausdorff), you know that the only possible topology is the discrete topology. Now your goal is just to show that any Hausdorff topology on a finit... |
3,474,847 | <p><strong>Prop:</strong> The union of two open sets is open. </p>
<p><strong>Pf:</strong> Let <span class="math-container">$(X,d)$</span> be a metric space, and <span class="math-container">$U$</span> and <span class="math-container">$V$</span> are non empty and open sets such that <span class="math-container">$U\sub... | Ben Grossmann | 81,360 | <p><em>Note: since the asker has been updating their proof since originally posting the question, this answer may no longer address the proof in its current form.</em></p>
<hr>
<p>Your proof could be substantially improved. Some criticisms below.</p>
<blockquote>
<p>Let <span class="math-container">$(X,d)$</span>... |
3,606,255 | <p>I want to ask you something about the following quadratic inequality: <span class="math-container">$$9x^2+12x+4\le 0.$$</span> Let us find the "=0" points. Here we have <span class="math-container">$9x^2+12x+4=9(x+\frac{2}{3})^2\ge0$</span> (we can factor this by calculating the discriminant <span class="math-contai... | Michael Rozenberg | 190,319 | <p><span class="math-container">$$(3x+2)^2\leq0$$</span> gives <span class="math-container">$$3x+2=0$$</span> or <span class="math-container">$$x=-\frac{2}{3}.$$</span>
Also, if you want to use <span class="math-container">$\Delta$</span>, so for <span class="math-container">$\Delta=0$</span> we have the following form... |
1,351,881 | <p>This argument comes up once every while on Lambda the Ultimate. I want to know where the flaw is.</p>
<p><em>Take a countable number of TMs all generating different bitstreams. Construct a Cantor TM which runs every nth TM upto the nth bit and outputs the reversed bit.</em></p>
<p><em>Now I have established there ... | Asaf Karagila | 622 | <p>There are uncountably many Turing machines, if you allow them to use oracles (which are real numbers, really).</p>
<p>But if you disallow that, then the only way that this process gives back a Turing machine is if the enumeration itself is computable. So what you have shown is that the set of Turing machines is not... |
1,435,665 | <p>Matrix $\mathbf{D}$ is a full rank diagonal matrix. $\mathbf{ADA}^T=\mathbf{D}$.</p>
<p>Can we conclude that $\mathbf{A}^2=\mathbf{I}_n$? (edited from $\mathbf{A}=\pm\mathbf{I}$)</p>
<p>It's almost sure that some one has asked this question before, but I cannot find proper keywords to search for that post.</p>
| Rory Daulton | 161,807 | <p>The equation $y^2+xy-3x=5$ becomes, after differentiation using the chain rule and the product rule,</p>
<p>$$2y\frac{dy}{dt}+y\frac{dx}{dt}+x\frac{dy}{dt}-3\frac{dx}{dt}=0$$</p>
<p>Can you continue from there?</p>
|
1,435,665 | <p>Matrix $\mathbf{D}$ is a full rank diagonal matrix. $\mathbf{ADA}^T=\mathbf{D}$.</p>
<p>Can we conclude that $\mathbf{A}^2=\mathbf{I}_n$? (edited from $\mathbf{A}=\pm\mathbf{I}$)</p>
<p>It's almost sure that some one has asked this question before, but I cannot find proper keywords to search for that post.</p>
| Brevan Ellefsen | 269,764 | <p>Doesn't $\frac{d}{dt} (y^2 +xy - 3x = 5) \Rightarrow 2y\frac{dy}{dt} + y\frac{dx}{dt} + x\frac{dy}{dt} - 3\frac{dx}{dt} $
??</p>
|
2,736,295 | <p>Let $f(x,y) = xy^2$ and the domain $D = \lbrace (x,y)| x,y\geq0, x^2 + y^2 \leq 3 \rbrace$</p>
<p>$f_x(x,y) = y^2$ and $f_y(x,y) = 2xy$ </p>
<p>Therefore, the critical points should be $\lbrace (x,y)| y = 0, \sqrt{3} \geq x \geq 0 \rbrace$. </p>
<p>The determinant of the Hessian is
$$\det(Hf(x,y))=
\begin{vmatri... | quasi | 400,434 | <p>Critical points are <em>candidates</em> for the max and min values.
<p>
Boundary points are also candidates.
<p>
But for the specified domain $D$, the only possible critical points are the points where $y=0$, and those points are on the boundary, hence are not actually critical points.
<p>
It follows that the maxim... |
3,019,882 | <p>Í already have found a proof About the statment which I did not understand, I will write it down, so maybe someone can explain me the part that I did not understand. But if there are easier ways to prove the statement I would be delighted to know. </p>
<p><span class="math-container">$\alpha:=\sqrt{2}+\sqrt{3}\noti... | Joel Cohen | 10,553 | <p>We know that any element of <span class="math-container">$\mathbb{Q}[\sqrt{2}]$</span> can be <strong>uniquely</strong> written as <span class="math-container">$a + b \sqrt{2}$</span> with <span class="math-container">$a, b \in \mathbb{Q}$</span> (the unicity is important). So, if <span class="math-container">$\alph... |
316,016 | <p>Could you recommend any approachable books/papers/texts about matroids (maybe a chapter from somewhere)? The ideal reference would contain multiple examples, present some intuitions and keep formalism to a necessary minimum.</p>
<p>I would appreciate any hints or appropriate sources.</p>
| Tara Fife | 215,095 | <p>There is a great introduction to matroids that is about 40 pages, free, and sounds more like what you are looking for: <em>What is a matroid?</em> by James Oxley, <a href="https://www.math.lsu.edu/~oxley/survey4.pdf" rel="nofollow">available from the author's webpage</a>.</p>
|
2,606,999 | <p>$f: X \to Y$ prove that if $A \subseteq X$, then A is a subset of the pre-image of the image of $A$, which is shown by these symbols respectively: $f^{-1}[f[A]]$</p>
<p>This is my proof:</p>
<p>If $A \nsubseteq f^{-1}[f[A]]$, then there exists "$x$" as an element of $A$, such that $f^{-1}[f[A]] \neq x$
which is a ... | String | 94,971 | <p>It suffices to note that for any $x\in A$ we have some image $f(x)$. This implies
$$
x\in f^{-1}(f(x))\subseteq f^{-1}(f(A))
$$
To sum up, we have
$$
x\in A\implies x\in f^{-1}(f(A))
$$
which in turn means $A\subseteq f^{-1}(f(A))$.</p>
|
335,258 | <p>Find the domain of the function:
$$f(x)= \sqrt{x^2 - 4x - 45}$$</p>
<p>I'm just guessing here; how about if I square everything and then put it in the graphing calculator?
Thanks,
Lauri</p>
| Lauri | 67,552 | <p>Well, I just happened to run into a mathematician yesterday (in person) and he just pulled the polynomial out of the square root and added the "greater than or equal to"symbol.Because, a square root number can't be a negative number,but it can be zero. Then we just factored it and it factored out to (x+5)(x-9), whi... |
167,440 | <blockquote>
<p>Let $\Bbb F$ be a field of characteristic $p\gt 0$ and $f(x)=x^{p^n}-c \in\Bbb F[x]$ where $n$ is a positive integer. If $c \notin \{a^p:a\in \Bbb F \}$, show that $f$ is irreducible in $\Bbb F[x]$.</p>
</blockquote>
<p>I recently started studying Field theory, so I don't know How to approach this pr... | Community | -1 | <p>Go up to a splitting field $E$ so that in there $f(x)$ splits completely. Now a root of $f(x)$ is $\sqrt[p^n]{c}$ and so we can write $f(x)$ over $E$ as </p>
<p>$$f(x) = x^{p^n} - (\sqrt[p^n]{c})^{p^n} = (x - \sqrt[p^n]{c})^{p^n}$$</p>
<p>by the schoolboy binomial theorem. Now recall $f(x) \in \Bbb{F}[x]$ so that ... |
2,493,481 | <p>I'm currently studying calculus of variations. I couldn't find a rigorous definition of a functional on this site.</p>
<ol>
<li>What is the general definition of a functional?</li>
<li>Why for calculus of variations in physics, I must to use for a functional <em>a convex function</em> for the space of the admissib... | Community | -1 | <blockquote>
<p>What is a functional?</p>
</blockquote>
<p>As it is well written in <a href="https://en.wikipedia.org/wiki/Functional_(mathematics)" rel="nofollow noreferrer">Wikipedia</a>, a functional simply refers to a mapping from a space $X$ into the real numbers. (Technically one might refer to other <a href="... |
1,822,179 | <p>Pointwise limit of the sequence of functions </p>
<p>$$h_n(x)=\begin{cases}1,&\text{if }x\ge \frac1n\\nx,&\text{if }x\in[0,\frac1n)\end{cases}$$</p>
<p>The trouble with this question is that I think that $h(0)$ is not defined but Im not completely sure. We can see that for any $n\in\Bbb N$ we have that $h_... | Community | -1 | <p>Assuming that by $h$ you mean the pointwise limit of $(h_n)$: no, $h(0)$ is defined. In fact, $h(0) :=\lim_{n \to \infty} h_n(0) = 0$. </p>
<p>What is confusing you is that $h$ is not continuous. We have for $x > 0$, there exists $k \in \Bbb N$ such that $x > 1/k$, so $h_n(x) = 1$ for all $n \ge k$, thus $h_n... |
33,646 | <h1>Preamble</h1>
<p>I am a novice SE user, a toddler. In this post I want to criticize some moderator actions, which seems risky and futile, regarding unwritten policies on the meta; however, just for the record, I write this post so that unbiased readers find it helpful.</p>
<p>Please note that I am not throwing a ta... | hardmath | 3,111 | <p>It is reasonable and desirable that you can come here to Meta to ask about what you perceive to be "unwritten policies," and it is true (as noted in comments) that StackExchange Communities vary as to policy priorities. You seem motivated to help the Math.SE Community, and I appreciate your effort to gain... |
353,920 | <p>I am studying a problem where a quadratic matrix equation emerges. The equation is as follow (all capital letters are n by n matrices)</p>
<p><span class="math-container">$(I-X^{\prime}L)X=D$</span></p>
<p>where <span class="math-container">$L$</span> and <span class="math-container">$D$</span> are both symmetric ... | Piyush Grover | 30,684 | <p>A good resource is : Abou-Kandil, Hisham, Gerhard Freiling, Vlad Ionescu, and Gerhard Jank. Matrix Riccati equations in control and systems theory. Birkhäuser, 2012.</p>
|
400,715 | <p>Consider the metric space $(\mathbb{Q},d)$ where $\mathbb{Q}$ denotes the rational numbers and $d(x,y)=|x-y|$. Let $$E:=\{x \in\mathbb{Q}:x>0, 2<x^2<3\}$$</p>
<p>Is $E$ closed and bounded in $\mathbb{Q}?$ Is it compact? Justify your answers.</p>
| Brian M. Scott | 12,042 | <p>HINT: You should have no trouble deciding whether $E$ is bounded in $\Bbb Q$. To decide whether it’s closed in $\Bbb Q$, ask yourself whether $\Bbb Q\setminus E$ is open in $\Bbb Q$: can you find an open set $U$ in $\Bbb R$ such that $U\cap\Bbb Q=\Bbb Q\setminus E$? Remember, $\sqrt2$ and $\sqrt3$ are irrational.</p... |
400,715 | <p>Consider the metric space $(\mathbb{Q},d)$ where $\mathbb{Q}$ denotes the rational numbers and $d(x,y)=|x-y|$. Let $$E:=\{x \in\mathbb{Q}:x>0, 2<x^2<3\}$$</p>
<p>Is $E$ closed and bounded in $\mathbb{Q}?$ Is it compact? Justify your answers.</p>
| Dimitris | 37,229 | <p><strong>Hint</strong></p>
<p>About boundness when do you call a set unbounded? What is your intuition about this set? Can a $x\in E$ be too much large?</p>
<p>Regarding compactness, a characterization of compact metric spaces is: </p>
<blockquote>
<p>$(X,d)$ is compact iff every sequence has a convergent subseq... |
26,192 | <p>I have a list of rules that represents a list of parameters to be applied to a circuit model (Wolfram SystemModeler model): </p>
<pre><code>sk = {
{R1 -> 10080., R2 -> 10080., C1 -> 1.*10^-7, C2 -> 9.8419*10^-8},
{R1 -> 10820., R2 -> 4984.51, R3 -> 10000., R4 -> 10000., C1 -> 1.*10^-7,
C... | Michael E2 | 4,999 | <p>Here's a version where the counting is done by structuring the data with <code>GatherBy</code> and using <code>MapIndexed</code>.</p>
<pre><code>lengthToNames = {6 -> "sallenKey", 4 -> "sallenKeyUnityGain"};
gatheredSk = GatherBy[sk, Length];
positionToNames = MapIndexed[
With[{basename = Length[#[[1]]] /.... |
2,439,202 | <p>i want to solve this question but i don't know what way i use :
$$\lim_{n\to \infty}a_n= \lim_{n\to\infty} \frac{\binom n 1} {n} +\frac {\binom n 2} {n^2} + \cdots + \frac {\binom n n}{n^n} $$</p>
| Angina Seng | 436,618 | <p>This sum is
$$-1+\sum_{k=0}^n\binom{n}{k}\frac1{n^k}=
-1+\left(1+\frac1n\right)^n.$$
This has a well-known limit...</p>
|
262,500 | <p>What is the Green’s function of the boundary value problem
$$
\frac{\mathrm d^2 y}{\mathrm d x^2}-\frac{1}{x}\frac{\mathrm dy}{\mathrm dx}=1,\quad y(0)=y(1)=0,
$$</p>
<p>this boundary problem is not self adjoint, so please help me how to solve it.</p>
| Community | -1 | <p>Denoting $y'(x) = v(x)$, we have that $$v'(x) - \dfrac{v(x)}{x} = 1$$
Let $v(x) = x g(x)$. Then we get that $$xg'(x) + g(x) - g(x) = 1\implies g'(x) = \dfrac1x$$
Hence, we have $$g(x) = \log(x) + c_1 \implies v(x) = x \log(x) + c_1 x$$
Hence, we now need to solve for
$$y'(x) = x \log(x) + c_1 x$$
$$y(x) = \int x \lo... |
2,404,258 | <p>I'm asked to prove that </p>
<h2>$|z-i||z+i|=2$</h2>
<p>defines an ellipse in the plane.</p>
<p>I have tried replacing $z = x+iy $ in the previous equation and brute forcing the result to no avail. Considering that $ |z-i||z+i| = |z^2+1| $ eases the algebra a bit but didn't help me that much.</p>
<p>Edit: I kno... | José Carlos Santos | 446,262 | <p>This expression <em>doesn't</em> define an ellipse. It's the set of those points of the plane such that the <em>products</em> of its distances to $i$ and to $-i$ is equal to $2$. If we were talking about the <em>sum</em> of the distances, then, yes, it would be an ellipse.</p>
<p>In order to prove that it is not an... |
1,810,582 | <p>Statement 1: </p>
<blockquote>
<p>If $f :A\mapsto B$ & if $X,Y$ are subsets of $B$, then the inverse image of union of two sets $X$ & $Y$ is the union of inverse images of $X$ & $Y$.</p>
</blockquote>
<p>Statement 2:</p>
<blockquote>
<p>Let $f:A\mapsto B$ be a one-to-one correspondence from $... | Jay | 608,546 | <p>This is what I have but I also want to add my messy work after this:</p>
<p>Let <span class="math-container">$x\in X$</span> be such that <span class="math-container">$x\in G\subset X, G$</span> open. There is a countable dense subset <span class="math-container">$E$</span> of <span class="math-container">$X$</span... |
7,247 | <p>Let $f(x) \in L^p(\mathbb{R})$ and $K \in C^m(\mathbb{R})$. Can I then say that $(f \ast K) (x) = \int_{\mathbb{R}} f(t) K(x-t) dt$ is in $C^m$? </p>
<p>I know that this is true if $K$ has compact support, but I was wondering if it is possible to have a stronger result (perhaps $K$ vanishing at $\infty$?). </p>
| Willie Wong | 1,543 | <p>No. If $K$ does not have compact support, then $f*K$ need not even be bounded. Consider the function $K = 1 / \log( 3 + |x|)$, and $f(x) = (1 + |x|)^{-1/p - \epsilon}$. For $\epsilon > 0$. Then $f\in L^p$, but the convolution integral does not converge anywhere, even when $K$ vanishes at $\infty$. </p>
<p>What you ... |
1,364,417 | <p>Find a real number k such that the limit $$\lim_{n\to\infty}\ \left(\frac{1^4 + 2^4 + 3^4 +....+ n^4}{n^k}\right)$$ has as positive value.
If I am not mistaken every even $k$ can be the answer. But the answer is 5.</p>
| GohP.iHan | 151,481 | <p>Do you know Riemann Sum?</p>
<p>$\begin{eqnarray}
&& \lim_{n\to\infty} \left( \frac{1^4+2^4+3^4+\ldots+n^4 }{n^k } \right) \\
&=& \lim_{n\to\infty}n^{4-k} \left( \left(\frac1n\right)^4 +\left(\frac2n\right)^4 + \left(\frac3n\right)^4 + \ldots + \left(\frac nn\right)^4 \right) \\
\end{eqnarray} $</p>... |
2,354,383 | <p>Why doesn't a previous event affect the probability of (say) a coin showing tails?</p>
<p>Let's say I have a <strong>fair</strong> and <strong>unbiased</strong> coin with two sides, <em>heads</em> and <em>tails</em>.</p>
<p>For the first time I toss it up the probabilities of both events are equal to $\frac{1}{2}$... | paw88789 | 147,810 | <p>Your question mentions a 'fair and unbiased coin.' That very concept means a coin that on any given flip has a $0.5$ probability of heads and a $0.5$ of tails on any given flip, $\underline{\rm{independently}}$ of what has happened on any other flip.</p>
<p>To understand the statement 'It isn't that future flips co... |
316,374 | <p>If <span class="math-container">$a_n$</span> satisfies the linear recurrence relation <span class="math-container">$a_n = \sum_{i=1}^k c_i a_{n-i}$</span> for some constants <span class="math-container">$c_i$</span>, then is there an easy way to find a linear recurrence relation for <span class="math-container">$b_n... | Qiaochu Yuan | 290 | <p>Yes. Take the <a href="https://en.wikipedia.org/wiki/Companion_matrix" rel="noreferrer">companion matrix</a> <span class="math-container">$M$</span> of the <a href="https://en.wikipedia.org/wiki/Characteristic_polynomial" rel="noreferrer">characteristic polynomial</a> of your original recurrence. Then the squared re... |
3,102,218 | <p>Given a fraction:</p>
<p><span class="math-container">$$\frac{a}{b}$$</span></p>
<p>I now add a number <span class="math-container">$n$</span> to both numerator and denominator in the following fashion:</p>
<p><span class="math-container">$$\frac{a+n}{b+n}$$</span></p>
<p>The basic property is that the second fr... | fleablood | 280,126 | <p>Intuition?</p>
<p>For me the intuition is this: The absolute difference in size becomes less significant when we are comparing big things than when we are comparing small thing. e.g. If one person ways <span class="math-container">$100$</span> lbs more than another that is significant. If one elephant is <span ... |
299,795 | <p>I seem to have completely lost my bearing with implicit differentiation. Just a quick question:</p>
<p>Given $y = y(x)$ what is $$\frac{d}{dx} (e^x(x^2 + y^2))$$</p>
<p>I think its the $\frac d{dx}$ confusing me, I don't what effect it has compared to $\frac{dy}{dx}$. Any help will be greatly appreciated.</p>
| amWhy | 9,003 | <p>Note:
$$x^9 + x^8 + x^7 = 0 \;\; \iff \;\;x^7(x^2+x+1)=0.$$</p>
<p>Simply solve the equations $x^7=0$ and $x^2+x+1=0.$</p>
<p>No need for WolframAlpha: simply use the quadratic formula to obtain solutions to the second equation.</p>
<p>You'll get one real and two complex (non-real) solutions.</p>
|
1,721,345 | <p>Found a game related to knight's tour <a href="http://www.emathhelp.net/math-games-and-logic-puzzles/knight-score/" rel="nofollow" title="Knight's Score">Knight's Score</a></p>
<p>The basic idea is to gather as much points as possible, avoiding some cells.</p>
<p>What is the math of this?
Can algorithm be cons... | Hagen von Eitzen | 39,174 | <p>Yes, the game is solvable in the sense that an optimal move sequence can be found. This follows from the fact that there are only finitely many move sequences.</p>
|
897,955 | <p>For a function $f:X\rightarrow Y$ we have $f(U)\subset V\iff U\subset f^{-1}(V)$ proof</p>
<p>I'm following a book and it just uses this, it doesn't say anything about the function, so I've not assumed it's one-to-one. I've spent to much time on this (as in it should be 4 lines for both ways at most) so I hope you ... | drhab | 75,923 | <p>First of all note that by definition of the preimage of $V\subset Y$ under $f:X\rightarrow Y$ we have: $$x\in f^{-1}\left(V\right)\iff f\left(x\right)\in V$$</p>
<hr>
<p>$\Rightarrow$</p>
<p>$x\in U\Rightarrow f\left(x\right)\in f\left(U\right)\subset V\Rightarrow x\in f^{-1}\left(V\right)$. Application of defini... |
1,109,706 | <p>I have just started going over abstract algebra.</p>
<p>One of the question is </p>
<p>$*$ is defined on $\mathbb C$ such that $a*b=|ab|$</p>
<p>I tried to check three axioms : 1) Associativity 2) identity 3) inverse</p>
<p>I found out with long computation that Associativity works.</p>
<p>For identity, $a*e=e*... | k.stm | 42,242 | <p>It’s <em>not</em> a group as it fails to have an identity.</p>
<p><em>Hint</em>. Look at $a = -1$ and the range of the defined operation “$*$”.</p>
|
1,109,706 | <p>I have just started going over abstract algebra.</p>
<p>One of the question is </p>
<p>$*$ is defined on $\mathbb C$ such that $a*b=|ab|$</p>
<p>I tried to check three axioms : 1) Associativity 2) identity 3) inverse</p>
<p>I found out with long computation that Associativity works.</p>
<p>For identity, $a*e=e*... | MJD | 25,554 | <p>There is a theorem that if $\langle G, \ast\rangle$ is a group, then each function $f_a$ defined by $$f_a(x) = a\ast x$$ must be a bijection. (That is, the left coset $aG$ must be equal to $G$.) For example, the function $f_3(x) = 3 + x$ is a bijection of the real numbers.</p>
<p>This is not hard to show. To see ... |
489,528 | <p>I need to check for convergence here. Why is every example completely different and nothing which I've learned from the examples before helps me in the next? Hate it!</p>
<p>$$ \sum_{n\geq1} \frac{n^{n-1}}{n!} $$</p>
<p>so I tried the ratio test and I got</p>
<p>$$ \frac{1}{1+\frac{1}{n}} $$
which converges to $... | marty cohen | 13,079 | <p>Let's look at the more general function
$\sum_{n\geq1} x^n\frac{n^{n-1}}{n!}$</p>
<p>The ratio of consecutive terms is</p>
<p>$\begin{align}
x\frac{\frac{(n+1)^n}{(n+1)!}}{\frac{n^{n-1}}{n!}}
&=x\frac{(n+1)^n}{(n+1)!}\frac{n!}{n^{n-1}}\\
&=x\frac{(n+1)^n}{(n+1)n^{n-1}}\\
&=x\frac{(n+1)^{n-1}}{n^{n-1}}\... |
2,260,975 | <p>Problem: Let $\mu : \mathcal{B}(\mathbb{R}^k) \rightarrow \mathbb{R}$ be a nonnegative and finitely additive set function with $\mu(\mathbb{R}^k) < \infty $.
Suppose that
\begin{equation}
\mu(A) = \sup \{ \mu(K) : K \subset A, K \text{ compact}\}
\end{equation}
for each $A \in \mathcal{B}(\mathbb{R}^k)$.
Then $... | Community | -1 | <p>First, $\mu$ is $\sigma$-additive iff it is continuous at $\emptyset$. Suppose that $\mu$ is not $\sigma$-additive, i.e. there exists a decreasing sequence of sets $\{A_n\}$ s.t. $\cap_{n\ge 1}A_n=\emptyset$ but $\inf_n\mu(A_n)=\epsilon>0$. Let $K_n\subset A_n$ compact s.t. $\mu(K_n)\le \mu(A_n)+\epsilon2^{-(n+1)... |
2,900,014 | <p>How would solve for $a$ in this equation without using an approximation ?
is it possible?</p>
<p>where $x>0$ and $0<a<\infty$</p>
<p>$x=\Sigma _{i=1}^{n} i^a$</p>
<p>for example $120=\Sigma _{i=1}^{6} i^a$ what is $a$ in this equation?</p>
| user | 505,767 | <p>There is not a general closed form but we can use integral estimation</p>
<p>$$\sum_{i=1}^{6} i^a\approx \int_{0.5}^{6.5}x^adx=\left[\frac{x^{a+1}}{a+1}\right]_{0.5}^{6.5}=\frac{6.5^{a+1}}{a+1}-\frac{0.5^{a+1}}{a+1}=120 \implies a\approx 2.175$$</p>
<p>where the value for $a$ needs to be evaluated by numerical met... |
1,400,876 | <p>Given a measurable space $(\Omega, \cal{F})$, $f:(\Omega, \cal F) \to (\Bbb{B},\cal B)$, where $\cal B$ is the Borel $\sigma$-algebra of $\Bbb R$, is said to be $\cal {F}$-measurable if $f^{-1}(B)\in \cal F $ for any $B\in \cal B$.</p>
<p>Any function $f:\Omega\to\Bbb R$ is trivially $2^\Omega$ measurable. A consta... | layman | 131,740 | <ol>
<li><p>For any $\sigma$-algebra $\mathcal{F}$ on $\Omega$, and any $A \in \mathcal{F}$, the indicator function $\Bbb 1_{A}(x) : = \begin{cases} 1 & x \in A \\ 0 & x \not \in A \end{cases}$ is $\mathcal{F}$-measurable. It's a good (and easy) exercise to prove this.</p></li>
<li><p>Also, if $\Omega$ is a to... |
620,370 | <p>I am tackling a problem which asks:</p>
<p>Find the sum of all the multiples of 3 or 5 below 1000.</p>
<p>My reasoning is that since
Since $\left\lfloor\frac{1000}{3}\right\rfloor = 333$ and $\left\lfloor\frac{1000}{5}\right\rfloor = 200$</p>
<p>This sum can be denoted as:
\begin{equation}
\sum\limits_{n=1}^{333... | Jeppe Stig Nielsen | 70,134 | <p>Some numbers, such as $15$ and $30$, are multiples of <em>both</em> $3$ and $5$. These should be counted only once.</p>
<p><strong>Edit:</strong> After comments to the question (not to this answer), I see that another issue is that you include $1000$ as the last term, but $1000$ is not below $1000$.</p>
|
4,295,212 | <p>I have a statics problem in which my approach differs from the suggested solution, so I was wondering if anyone can point out my flawed logic. Below is the question.</p>
<p>Two identical uniform rods <span class="math-container">$ab$</span> and <span class="math-container">$bc$</span>, each of length <span class="m... | Cesareo | 397,348 | <p>Calling the forces indexed by point of application</p>
<p><span class="math-container">$$
\cases{
T_a = T(\cos\beta,\sin\beta)\\
F_a = (0,R_a)\\
F_b = (0,R_b)\\
F_c = (H_c,V_c)\\
F_{g_1} = (0,-W)\\
F_{g_2} = (0,-W)\\
a = (0,0)\\
b = 2l(\sin\alpha,0)\\
c = 2l(\sin\alpha,\cos\alpha)\\
g_1 = \frac 12 (a+c)\\
g_2 = \fra... |
1,990,105 | <p>Presume that $(x_n)$ is a sequence s. t.</p>
<p>$|x_n-x_{n+1}| \le 2^{-n}$ for all $n \in \mathbb N$</p>
<p>Prove that $x_n$ converges.</p>
<p>What I've tried to think: since $2^{-n}$ converges to 0, and the difference between the terms $x_n$ and $x_{n+1}$ is smaller or equal to it, then $x_n$ must be a cauchy s... | ILoveMath | 42,344 | <h3>Hint:</h3>
<p>For <span class="math-container">$m>n$</span>, write</p>
<p><span class="math-container">$$|x_n - x_m| = |x_n + x_{n+1} - x_{n+1} + x_{n+2} - x_{n+2} + ... + -x_{m-n-1} + x_{m-n-1} - x_m|$$</span></p>
<p>now use triangle inequality</p>
|
310,513 | <p>I haven't found this yet and I'm somehow not sure if my idea is correct.</p>
<p>The Problem: Let $X$ be a separable Banach-Space, let $x_k\to x$ weakly and such that for every $\lambda_k \to \lambda$ weakly-* there holds $\lambda_k(x_k)\to \lambda(x)$. Then $x_k\to x$ strongly (in the norm of $X$).</p>
<p>My idea ... | A Blumenthal | 54,337 | <p>I think the reason for your confusion is the condition made on the convergence $x_k \rightarrow x$. The condition that for any $\lambda_k \rightarrow \lambda$ weak* we have $|\lambda_k(x_k) - \lambda(x)| \rightarrow 0$ implies $x_k \rightarrow x$ weakly, by taking $\lambda_k$ to be the constant sequence. You certain... |
2,672,497 | <p>$$\lim _{n\to \infty }\sum _{k=1}^n\frac{1}{n+k+\frac{k}{n^2}}$$ I unsuccessfully tried to find two different Riemann Sums converging to the same value close to the given sum so I could use the Squeeze Theorem. Is there any other way to solve this?</p>
| RRL | 148,510 | <p>Writing this as</p>
<p>$$\lim_{n \to \infty}S_{nn} = \lim_{n \to \infty}\frac{1}{n}\sum _{k=1}^n\frac{1}{1+\frac{k}{n}+\frac{k}{n}\frac{1}{n^2}}
$$</p>
<p>a technique that often works is to evaluate the double limit</p>
<p>$$\lim_{m \to \infty} \lim_{n \to \infty} S_{mn} = \lim_{m \to \infty} \lim_{n \to \infty} ... |
1,018,672 | <blockquote>
<p><span class="math-container">$$\int_0^{\infty} \frac{1}{x^3-1}dx$$</span></p>
</blockquote>
<p>What I did:</p>
<p><span class="math-container">$$\lim_{\epsilon\to0}\int_0^{1-\epsilon} \frac{1}{x^3-1}dx+\lim_{\epsilon\to0}\int_{1+\epsilon}^{\infty} \frac{1}{x^3-1}dx$$</span></p>
<hr />
<p><span class="ma... | Lucian | 93,448 | <ul>
<li>Subdivide $(0,\infty)$ into $(0,1)$ and $(1,\infty)$.</li>
<li>On the latter, let $t=\dfrac1x$.</li>
<li>$a^3-b^3=\big(a-b\big)\Big(a^2+ab+b^2\Big)$.</li>
<li>Complete the square in the denominator.</li>
</ul>
|
1,018,672 | <blockquote>
<p><span class="math-container">$$\int_0^{\infty} \frac{1}{x^3-1}dx$$</span></p>
</blockquote>
<p>What I did:</p>
<p><span class="math-container">$$\lim_{\epsilon\to0}\int_0^{1-\epsilon} \frac{1}{x^3-1}dx+\lim_{\epsilon\to0}\int_{1+\epsilon}^{\infty} \frac{1}{x^3-1}dx$$</span></p>
<hr />
<p><span class="ma... | robjohn | 13,854 | <p><strong>Cauchy Principal Value</strong></p>
<p>The integral, as written, diverges. In the case of an improper integral such as this,
$$
\int_0^\infty\frac1{x^3-1}\mathrm{d}x
=\int_0^1\frac1{x^3-1}\mathrm{d}x+\int_1^\infty\frac1{x^3-1}\mathrm{d}x\tag{1}
$$
however, neither of the integrals on the right converge.</p>... |
1,018,672 | <blockquote>
<p><span class="math-container">$$\int_0^{\infty} \frac{1}{x^3-1}dx$$</span></p>
</blockquote>
<p>What I did:</p>
<p><span class="math-container">$$\lim_{\epsilon\to0}\int_0^{1-\epsilon} \frac{1}{x^3-1}dx+\lim_{\epsilon\to0}\int_{1+\epsilon}^{\infty} \frac{1}{x^3-1}dx$$</span></p>
<hr />
<p><span class="ma... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
1,056,045 | <blockquote>
<p>Show that </p>
<p>$2^{3n}-1$ is divisble by $7$ for all $n$ $\in \mathbb N$</p>
</blockquote>
<p>I'm not really sure how to get started on this problem, but here is what I have done so far:</p>
<p>Base case $n(1)$:</p>
<p>$\frac{2^{3(1)}-1}{7} = \frac{8-1}{7} = \frac{7}{7}$</p>
<p>But not sur... | Crostul | 160,300 | <p>$$2^{3n}-1 = 8^n-1 = (8-1)(8^{n-1} + 8^{n-2} + \dots + 8^2 +8+1) = 7 \cdot (\mbox{something})$$</p>
|
1,056,045 | <blockquote>
<p>Show that </p>
<p>$2^{3n}-1$ is divisble by $7$ for all $n$ $\in \mathbb N$</p>
</blockquote>
<p>I'm not really sure how to get started on this problem, but here is what I have done so far:</p>
<p>Base case $n(1)$:</p>
<p>$\frac{2^{3(1)}-1}{7} = \frac{8-1}{7} = \frac{7}{7}$</p>
<p>But not sur... | Josh B. | 4,308 | <p>Assume $2^{3n}-1$ is divisible by 7. Look at $2^{3(n+1)}-1$ and show that it is divisible by 7.</p>
<p>$2^{3(n+1)}-1 = 2^{3n+3}-1 = 8*2^{3n}-1 = (7+1)*2^{3n}-1 = (7*2^{3n} + 2^{3n})-1 = 7*2^{3n} + (2^{3n}-1)$ which is divisble by 7.</p>
|
1,394,550 | <p>I have tried to prove the following inequality:</p>
<p>$$
\left(1+\frac{\log n}{n}\right)^n \gt\frac{n+1}{2}, \mbox{for}\;n\in\{2,3,\ldots\}
$$</p>
<p>which seems to be correct (confirmed by numerical result).</p>
<p>Can anyone give me some help or hint? Thanks a lot.</p>
| k170 | 161,538 | <p>$$
\left(1+\frac{\log n}{n}\right)^n\gt\frac{n+1}{2}
$$
Let $n=2$, then
$$
\left(1+\frac{\log 2}{2}\right)^2\gt\frac{2+1}{2}
$$
$$
1+\log 2+\frac14(\log 2)^2\gt1+\frac12
$$
Let $n=k+1$, then
$$\left(1+\frac{\log(k+1)}{k+1}\right)^{k+1}\gt\left(1+\frac{\log k}{k}\right)^k+\frac12$$
Applying the induction hypothesis, ... |
3,324,803 | <p><strong>definition.</strong></p>
<p>The phase flow of the differential equation <span class="math-container">$\dot{x}=\vec{v}\ (x)$</span> is the one-parameter diffeomorphism group for which <span class="math-container">$\vec{v}$</span> is the phase velocity vector field, namely,
<span class="math-container">$$
\ve... | sirous | 346,566 | <p><span class="math-container">$$\log_3(x+1)+\log_2 x=5$$</span></p>
<p><span class="math-container">$$\log_3(x+1)=5-\log_2 x$$</span></p>
<p><span class="math-container">$$x+1=3^{5-\log_2 x}$$</span></p>
<p>For integer solutions for x we must have:</p>
<p><span class="math-container">$$5-\log_2 x>0$$</span></p... |
2,564,256 | <p>The definition of general complex log for any non-zero complex number $z$ is
$$Log(z)=\log|z|+i[\arg(z)+2m\pi], m\in \mathbb{Z}$$</p>
<p>With this, if $n\in \mathbb{N}$ then
$Log(z^{1/n})=\frac{1}{n} Log(z)$ holds for all non-zero complex number $z$. </p>
<p>I verified this for $Log(i^{1/2})=\frac12 Log(i)$ succ... | Dylan | 135,643 | <p>You have </p>
<p>$$ i = \exp\left(i\frac{\pi}{2} + i2n\pi\right) $$</p>
<p>Taking the third power yields</p>
<p>$$ i^{1/3} = \exp\left(i\frac{\pi}{6} + i \frac{2n\pi}{3} i\right) $$</p>
<p>Since $|i| = |i^{/3}| = 1$, taking the log yields</p>
<p>$$ \log (i^{1/3}) = i\left(\frac{\pi}{6} + \frac{2n\pi}{3} \right)... |
2,051,785 | <p>$PQRS$ is a convex quadrilateral. The intersection of the two diagonals is $O$. If the areas of triangles $PQS$, $QRP$, and $SPR$ are $1, 2,$ and $3$, respectively. What is the area of triangle $PQO$?</p>
| Jean Marie | 305,862 | <p>With the excellent hint given by @Triskele, one finds $y=2/5$ under the assumption that <strong>such a quadrilateral exists</strong>.</p>
<p>This existence can be established. Moreover, there is no unicity: up to a rotation, there is one degree of freedom. More precisely, taking line PR as the $x$ axis and $O$ as t... |
4,005,738 | <p>I am interested in a method of how to compute the expected number of steps for absorpion in a Markov Chain with only one absorption node and given the starting and pre-final node (before absorption).</p>
<p>So, if the transition probability matrix is given, for example by:</p>
<p><span class="math-container">$P = \b... | Jeroen van der Meer | 874,176 | <p>I'm slightly correcting my comments, and turning it into an answer. Let <span class="math-container">$P$</span> be the Markov chain you're interested in. Write <span class="math-container">$j_i$</span> for the initial state, <span class="math-container">$j_f$</span> for the last state before absorption, and <span cl... |
3,668,702 | <p>As the title says, how should I go about finding the shortest distance between all pairs of nodes (Each node has x and y co-odrinates associated with it) on a graph?</p>
<p>A brute force method is to run shortest path finding algorithms between all the pairs of the points. Is there a better way to approach this pro... | peter.petrov | 116,591 | <p>There is a standard algorithm just for this. </p>
<p><a href="https://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm" rel="nofollow noreferrer">Floyd-Warshall's algorithm</a> </p>
<p>Its execution time is <span class="math-container">$O(n^3)$</span> where n is the number of vertices.<br>
It is easy to imp... |
362 | <p>The Math Stack Exchange chat room has had some heated arguments/suspensions in the past and I was wondering if there were any plans for the newly opened Math Overflow chat room to have its own moderators to provide the required special attention to prevent similar problem situations?</p>
| Tim Post | 35,352 | <p><a href="https://mathoverflow.net/users?tab=moderators">Moderators of Math Overflow</a> <em>are also</em> moderators in chat. That said ...</p>
<p>Math Overflow can certainly have an election like any other site, and the same procedure would apply as on any other site. </p>
<p>Once a year, we check in with the mod... |
1,770,656 | <p>In lecture I was given the following reasoning:</p>
<p>There are 7 options for the first character, 6 options for the last character, and 9! combinations for the 9 characters in between. Then you divide out the repeats -- 4 I's, 4 S's, and 2 P's. So the answer is 7 * 6 * 9! / (4! * 4! * 2!) = 13230</p>
<p>My own r... | Graham Kemp | 135,106 | <p>You were wanting to count words that <em>neither</em> begin <em>nor</em> end with I. However you have found the count of words that do not <em>both</em> begin <em>and</em> end in I.</p>
<p>Let $\Omega$ be the set of all words formed with those letters (distinct permutations of the string), $B$ be the set of ... |
1,771,279 | <p>Let $B=\{x \in \ell_2, ||x|| \leqslant 1\}$ - Hilbert ball $X \subset B$ - open convex connected set in Hilbert ball, $\bar{X}$ - closure of $X$. $F: \bar{X} \to \bar{X}$ - continuous map that holomorphic into each point of $X$. Is it true in general, that $F$ has fixed point?</p>
| Chill2Macht | 327,486 | <p>At first ignoring the holomorphic assumption, in general, no, since the Hilbert Ball is not compact for an infinite-dimensional Hilbert space.</p>
<p>Moreover, just because $X$ is open and connected does not mean that it is convex, and even though $\bar{X}$ is closed, since it is not a priori a subset of a compact ... |
4,101,068 | <p>Can two random variables <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> have the covariance matrix</p>
<p><span class="math-container">$\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}? $</span></p>
<p>The matrix is positive definite if <span class="math-container">$xy<0... | Sandipan Dey | 370,330 | <p>In order to show that the matrix <span class="math-container">$A=\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$</span> can't be a covariance matrix, we need to show that it's not a PSD matrix, i.e., it's sufficient to show an example vector (<span class="math-container">$\exists \mathbb{x} \in \mathbb{R^2}$... |
1,609,945 | <p>Prove that the set of points for which $\arg(z-1)=\arg(z+1) +\pi/4$ is part of the set of points for which $|z-j|=\sqrt 2$ and show which part clearly in a diagram.</p>
<p>Intuition for the geometrical interpretation of this would be really appreciated. </p>
<p>And I was wondering greatest and lowest value for $|z... | Rafael | 303,887 | <p>We will prove that $X=\{|s-t|:s\in S, t\in T\}$ is closed.</p>
<p>Let $(|s_{n}-t_{n}|)$ be a sequence in $X$ such that $lim|s_{n}-t_{n}|=x$. Then $(s_{n})$ is a sequence in $S$ and $(t_{n})$ is a sequence in $T$.
Since $S$ is limited, there is a subsequence $(s_{n_{k}})$ of $(s_{n})$ such that $lim\ s_{n_{k}}=s$; s... |
1,956,855 | <p>I'm doing some math work involving proofs, and one of the definitions is:</p>
<p>|a| = -a when a < 0</p>
<p>Isn't the absolute value of a, positive a no matter what a is in the beginning? Am I looking at this wrong? Could use an explanation.</p>
| adjan | 219,722 | <p>$|a|$ is always positive, but if $a$ is negative at the beginning, the absolute value is <em>equal</em> to $-a$, because negating a negative number results in a positive. This is merely a a property of $a$, not of $|a|$.</p>
|
1,956,855 | <p>I'm doing some math work involving proofs, and one of the definitions is:</p>
<p>|a| = -a when a < 0</p>
<p>Isn't the absolute value of a, positive a no matter what a is in the beginning? Am I looking at this wrong? Could use an explanation.</p>
| Mikhail Tikhonov | 272,803 | <p>If $a<0$ you can consider it as $-|a|$ since $|a|$ is absolute value(like unsigned value).</p>
<p>So $a=-|a|$, and $|a|=-a$</p>
<p>Example: $a=-1$, $|a|=1=-(-1)=-a$</p>
|
4,580,478 | <p>So I'm currently reading about signed measures in "Real analysis" by Folland. In it, he defines a signed measure as follows.</p>
<p><em>Let <span class="math-container">$(X,\mathcal{M})$</span> be a measurable space. Then a signed measure <span class="math-container">$\nu$</span> is a function from <span c... | GEdgar | 442 | <p>Answer for the original question.</p>
<p>Let <span class="math-container">$a_n \in [-\infty,\infty]$</span>, for <span class="math-container">$n=1,2,3,\dots$</span>. Write <span class="math-container">$\mathbb N = \{1,2,3,\dots\}$</span>. Let <span class="math-container">$\mathfrak S$</span> be the set of all bije... |
2,796,618 | <p>I am trying to</p>
<p>i) determine the infimum</p>
<p>ii) show that there's a function for which $\int_{0}^{1} {f'(x)}^2 dx$ is the infimum</p>
<p>iii) show if such function is unique.</p>
<p>I tried out several functions that suit the given condition, but couldn't see how $\int_{0}^{1} {f'(x)}^2 dx$ changes as ... | rapier | 521,292 | <p>You can solve this problem with variational calculus.
Let $y=f(x)$ and $g(x,y,y')=(y')^2=(f'(x))^2$. Then the function $f(x)$ that gives infimum satisfies
\begin{equation}
\frac{\partial g}{\partial y}-\frac{d}{dx}\left(\frac{\partial g}{\partial y'}\right)=0,
\end{equation}
if we use the Euler–Lagrange equation.
A... |
2,185,434 | <p>I recently read this problem</p>
<blockquote>
<p>A bag contains $500$ beads, each of the same size, but in $5$ different colors. Suppose there are $100$ beads of each color and I am blindfolded. What is the least number of beads I must pick before I can be sure there are $5$ beads of the same color among the bead... | vrugtehagel | 304,329 | <p>You probably read the question wrong. You need $5$ beads of <em>the same</em> color. This means you can get $4$ beads of each color maximum to not-have $5$ beads that are the same color (which is $20$ beads) and then the next one makes sure you have $5$ beads of the same color (that was the $21$st bead).</p>
|
2,185,434 | <p>I recently read this problem</p>
<blockquote>
<p>A bag contains $500$ beads, each of the same size, but in $5$ different colors. Suppose there are $100$ beads of each color and I am blindfolded. What is the least number of beads I must pick before I can be sure there are $5$ beads of the same color among the bead... | The Count | 348,072 | <p>You have $5$ colors, and the worst possible scenario is picking $4$ of each color before you pick a fifth of one color. So you can get $20$ beads before you get the fifth of at least one of the colors. Does this make sense?</p>
|
1,985,849 | <p>I am really stumped by this equation.
$$e^x=x$$
I need to prove that this equation has no real roots. But I have no idea how to start.</p>
<p>If you look at the graphs of $y=e^x$ and $y=x$, you can see that those graphs do not meet anywhere. But I am trying to find an algebraic and rigorous proof. Any help is appre... | Barry Cipra | 86,747 | <p>We have $e^x=x$ if and only if $x=\ln x$, which means, for one thing that there can be no solution with $x\le0$. For $x\gt0$, we have</p>
<p>$$\ln x=\int_1^x{dt\over t}\le\int_1^xdt=x-1\lt x$$</p>
<p>so there is no solution for $x\gt0$ either.</p>
<p>Remark: The key inequality between the two integerals may loo... |
90,548 | <p>Suppose we are handed an algebra $A$ over a field $k$. What should we look at if we want to determine whether $A$ can or cannot be equipped with structure maps to make it a Hopf algebra?</p>
<p>I guess in order to narrow it down a bit, I'll phrase it like this: what are some necessary conditions on an algebra for ... | James | 49,213 | <p>In the book <em>Hopf algebras</em> by Eiichi Abe (1980), there is a theorem on page 57-58 that reads
that </p>
<blockquote>
<p>Given a $k$-linear space $H$, suppose that there are $k$-linear maps $\mu:H\otimes H\rightarrow H$, $\eta:k\rightarrow H$, $\Delta:H\rightarrow H\otimes H$, $\varepsilon:H\rightarrow k$ s... |
782,929 | <p>prove or disprove following matrix is
positive definite matrix ?</p>
<p>$$\begin{bmatrix}
\dfrac{\sin(a_1-a_1)}{a_1-a_1}&\cdots&\dfrac{\sin(a_1-a_n)}{a_1-a_n}\\
\vdots& &\vdots\\
\dfrac{\sin(a_n-a_1)}{a_n-a_1}&\cdots&\dfrac{\sin(a_n-a_n)}{a_n-a_n}
\end{bmatrix}_{n\times n}$$
and where we ... | Adam | 89,966 | <p>Are there any other assumptions you forgot to add? A counter-example which comes to mind is to take $n = 2$ and $a_1 = a_2 = 1$ so that the matrix is just
$$ \left( \begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix} \right), $$
which is positive semidefinite, but not positive definite since one of its eigenvalues is... |
1,595,243 | <p>Let $f:\mathbb R \to \mathbb R$ be a function such that $|f(x)|\le x^2$, for all $x\in \mathbb R$. Then, at $x=0$, is $f$ both continuous and differentiable ?</p>
<p>No idea how to begin. Can someone help?</p>
| user295959 | 295,959 | <p>Note $|f(x)|\leq x^2$ implies $f(0)=0$. Then $$\lim_{x\rightarrow 0}\frac{|f(x)-f(0)|}{|x|}\leq \lim_{x\rightarrow 0}|x|=0.$$</p>
|
2,666,640 | <p>Assume I have a monoid $M$ and I am not guaranteed that all elements have an inverse.</p>
<p>Say I have the property that:</p>
<p>$a^m = a^{m+n}$</p>
<p>Can I claim than it must be that $i=a^n$?</p>
<p>Why or why not?</p>
| vadim123 | 73,324 | <p>The answer is yes, if the monoid is a <a href="https://en.wikipedia.org/wiki/Cancellation_property" rel="nofollow noreferrer">cancellative</a> monoid. </p>
<p>Here's an example of a non-cancellative monoid under multiplication: $$\{[1],[4]\}\subseteq \mathbb{Z}/6\mathbb{Z}$$</p>
<p>$[4][4]=[4]$, but $[4]\neq [1]$... |
2,666,640 | <p>Assume I have a monoid $M$ and I am not guaranteed that all elements have an inverse.</p>
<p>Say I have the property that:</p>
<p>$a^m = a^{m+n}$</p>
<p>Can I claim than it must be that $i=a^n$?</p>
<p>Why or why not?</p>
| lisyarus | 135,314 | <p>In general you cannot simplify expressions in monoids this way. Consider a monoid $M = \{i,x\}$, where $i$ is the neutral element and $x$ satisfies $x^2=x$. Then, $x^m=x^{n+m}$ for $n,m \geq 1$, but $i \neq x^n$.</p>
<p>However, in some monoids you can do this (for example, in groups and submonoids of groups).</p>
|
1,760,148 | <p>If I have a connected metric space $X$, is any ball around a point $x\in X$ also connected?</p>
| Community | -1 | <p>It is easy to see the answer is no. Take the unit circle in the plane, with the metric inherited from the Euclidian metric in the plane. Then remove the "top" point (coordinates $(0,1)$). Then a "ball" (small disk) around a point close to "the top", restricted to the "circle with the top removed," will be disconnect... |
483,131 | <p>How can you prove that if $t$ is less than or equal to $1$, that the probability of the sum of a sequence of uniform random variables being less than or equal to $t$ equals $t^k/k!$ ?</p>
<p>In other words:</p>
<p>Prove if $t \leq 1$, $$P(U_1+U_2 + \dots +U_k \leq t)=\frac{t^k}{k!}$$</p>
<p>My thought is that int... | André Nicolas | 6,312 | <p>We are told that $a-b$ is divisible by all of the $p_i$, where $i$ ranges from $1$ to $n$. We want to show that $a-b$ is divisible by $p_1p_2\cdots p_n$. </p>
<p>We <strong>must</strong> assume that the $p_i$ are <strong>distinct</strong>. If they are not, the result does not necessarily hold. </p>
<p>So we ... |
1,009,868 | <p>I had the question of $0^{-1}$ on a math test and I naturally assumed that this evaluates to zero, but from what I have seen from various sources it is equal to infinity which I do not quite understand. I would sooner believe that this it is just undefined.</p>
| k170 | 161,538 | <p>$$ 0^{-1}=\frac{1}{0}=\mbox{undefined} $$
because
$$ \lnot\exists x\in\mathbb{R}:1=0\times x $$</p>
|
1,009,868 | <p>I had the question of $0^{-1}$ on a math test and I naturally assumed that this evaluates to zero, but from what I have seen from various sources it is equal to infinity which I do not quite understand. I would sooner believe that this it is just undefined.</p>
| Graham Kemp | 135,106 | <p>Strictly speaking we say <span class="math-container">$0^{-1}$</span> is undefined.</p>
<p><span class="math-container">$x^{-1}$</span> is the multiplicative inverse of the number <span class="math-container">$x$</span>. By definition of the multiplicative inverse, the product of a number and its multiplicative in... |
4,373,216 | <p>Find power series solution for <span class="math-container">$y''-xy=0$</span> close to <span class="math-container">$x=0$</span>.</p>
<p><span class="math-container">$\sum_{n=2}^\infty n(n-1)a_nx^{n-2}-x\sum_{n=0}^\infty a_nx^n=0$</span></p>
<p>Then,
<span class="math-container">$a_{n+2}(n+2)(n+1)-a_{n-1}=0 \implies... | 2'5 9'2 | 11,123 | <p>Use this property of integrals:</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty}x^2f(x)\,dx=\int_{-\infty}^{0}x^2f(x)\,dx+\int_{0}^{1}x^2f(x)\,dx+\int_{1}^{2}x^2f(x)\,dx+\int_{2}^{\infty}x^2f(x)\,dx$$</span></p>
|
1,918,674 | <p>For what value $k$ is the following function continuous at $x=2$?
$$f(x) = \begin{cases}
\frac{\sqrt{2x+5}-\sqrt{x+7}}{x-2} & x \neq 2 \\
k & x = 2
\end{cases}$$</p>
<p>I was thinking about multiplying the numerator by it's conjugate, but that makes the denominator very messy, so I don't rlly know what to d... | Mark | 147,256 | <p>You actually <em>should</em> multiply by the conjugate, in this case. For $x \neq 2$, we have
$$\begin{aligned}
f(x) &= \frac{\sqrt{2x+5}-\sqrt{x+7}}{x-2} \cdot \frac{\sqrt{2x+5}+\sqrt{x+7}}{\sqrt{2x+5}+\sqrt{x+7}}\\
&= \frac{(2x+5)-(x+7)}{(x-2)(\sqrt{2x+5}+\sqrt{x+7})}\\
&= \frac{1}{\sqrt{2x+5}+\sqrt{x+... |
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