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 |
|---|---|---|---|---|
1,963,885 | <p>Find the sum
$$\sum_{n=1}^{\infty} \dfrac{4n}{n^4+2n^2+9}.$$</p>
<p>By calculator, we can predict that its sum is equal to $\dfrac{5}{6}$ so I think we should use inequalities to prove it. And I found that</p>
<p>$\dfrac{5}{6(n^4+n^2)} < \dfrac{4n}{n^4+2n^2+9}< \dfrac{5}{6(n^2+n)}$ for all $n\ge n_0$, $n_0$ ... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$(n^2)^2+3^2+2n^2=(n^2+3)^2-(2n)^2=(n^2+2n+3)(n^2-2n+3)$$</p>
<p>$$(n^2+2n+3)-(n^2-2n+3)=?$$</p>
<p>Now if $f(m)=m^2-2m+3,$</p>
<p>$f(m+2)=(m+2)^2-2(m+2)+3=m^2+2m+3$</p>
|
3,389,361 | <p>For what n is this rational,
<span class="math-container">$$\frac{\sqrt{n^2+1}} {\sqrt{2}}$$</span> </p>
<p>So far I have found the integers 1,7,41 and I have found some rational solutions to this as well but I'm looking to get a more general sense. </p>
<ul>
<li>So when is this a rational number?</li>
<li>Are the... | C.S. | 95,894 | <p>Well you want to solve for <span class="math-container">$n^2+1=2y^2$</span>, this is a standard type of <a href="https://en.wikipedia.org/wiki/Pell%27s_equation" rel="nofollow noreferrer">Pell's Equation.</a></p>
|
1,258,199 | <p>The question I need help is:</p>
<blockquote>
<p>Prove that $U(\mathbb I_9) \cong \mathbb I_6$ and $U(\mathbb I_{15}) \cong \mathbb I_4 \times \mathbb I_2$.</p>
</blockquote>
<p><img src="https://i.stack.imgur.com/2fi10.jpg" alt="enter image description here"></p>
<p>U() is the group of units in a ring</p>
<p>... | SomeOne | 87,286 | <p>If I understand your notation correctly then:</p>
<p><strong><em>Theorem</em></strong>: $U(\mathbb{I}_n)$ is the set of all integers $r$ such that: $1 \leq r \leq n-1$ and $\gcd(r,n)=1$</p>
<p><strong><em>Proof:</em></strong> (Check if this set satisfies group axiom with multiplication) </p>
|
12,653 | <p>I've Googled and searched mathstackexchange, but cannot find out how to insert a blue URL reference in a question or answer. Can you give me an example that I can edit it to look at the Latex code?</p>
| Davide Cervone | 7,798 | <p>@AlexR's suggestion can be implemented using <code>\label</code> and <code>\eqref</code> macros in order to get links from within an equation to another equation (but only such links). E.g.</p>
<p>$$\sin^2(\theta) + \cos^2(\theta) = 1\tag{1}\label{dpvc-1}$$</p>
<p>and then a link in the text: \eqref{dpvc-1} is du... |
3,855,672 | <p>This is question 7N #2 from Willard's <em>General Topology</em>, on p. 51.</p>
<blockquote>
<p>For any topological space <span class="math-container">$X$</span>, let <span class="math-container">$H(X)$</span> denote the group of
homeomorphisms of <span class="math-container">$X$</span> onto itself, with composition ... | Henno Brandsma | 4,280 | <p>First note that a homeomorphism <span class="math-container">$h$</span> of <span class="math-container">$[0,1]$</span> sends cutpoints to cutpoints and non-cutpoints to non-cutpoints, so <span class="math-container">$h(0)=1$</span> or <span class="math-container">$h(0)=0$</span>. Similarly for <span class="math-cont... |
384,553 | <p>Any ideas how to solve it?
$$\int\frac{x^4+2x+4}{x^4-1}dx$$
Thanks!</p>
| gt6989b | 16,192 | <p><strong>Hint</strong>
$$
\frac{x^4+2x+4}{x^4-1} = 1 + \frac{2x+5}{x^4-1}
= 1 + \frac{2x+5}{(x+1)(x-1)(x^2+1)}
$$
and use partial fractions.</p>
|
211,689 | <p>For a (well-behaved) one-dimensional function $f: [-\pi, \pi] \rightarrow \mathbb{R}$, we can use the Fourier series expansion to write
$$ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n\sin(nx) \right)$$</p>
<p>For a function of two variables, Wikipedia lists the formula</p>
<p>$$f(x,y) = \su... | JohnD | 52,893 | <p>Yes! And these types of expansions occur in a variety of applications, e.g., solving the heat or wave equation on a rectangle with prescribed boundary and initial data.</p>
<p>As a specific example, we can think of the following expansion as a two dimensional Fourier sine series for $f(x,y)$ on $0<x<a$, $0<... |
2,030,841 | <p>Let $a,b \in \mathbb {Z} $ and let $m$ be an integer greater than $2$. I found a counterexample to the equation</p>
<p>$$(a+b)\mathrm {mod} m = a\mathrm {mod} m + b \mathrm {mod} m $$</p>
<p>where $m>2$. But that was only after I thought I had proven that the equation does hold, so I was wondering if someone co... | cubeception | 337,836 | <p>$a\mod m+b\mod m=x+y$<br>
This part is not always true.<br></p>
<p>Suppose $x=m-1$ and $y = m-1$</p>
<p>$x+y = 2m-2$</p>
<p>But<br>
$a\mod m+b\mod m$<br>
would actaully be $m-2$.</p>
|
80,078 | <p>Given $f$:</p>
<p>$$
f(x) = \begin{cases}
\frac1{x} - \frac1{e^x-1} & \text{if } x \neq 0 \\
\frac1{2} & \text{if } x = 0
\end{cases}
$$</p>
<p>I have to find $f'(0)$ using the definition of derivative (i.e., limits). I already know how to differentiate and stuff, b... | Srivatsan | 13,425 | <p>A good strategy in such problems is to massage the problem into recognizable limits. (EDIT: I like this approach mainly because it avoids Taylor expansion and l'Hôpital's rule. This is, however, not the simplest approach.) </p>
<p>We can "simplify" given function as follows:
$$
\begin{eqnarray*}
\frac{\frac{1}{x}... |
3,845,602 | <p>I was reading Dummit and Foote and encountered the following statement: any two elements in <span class="math-container">$S_n$</span> are conjugate if and only if they have the same cycle types.</p>
<p>However, I am able to produce a counter example:</p>
<p>Let <span class="math-container">$(1 2 3)$</span> and <span... | 1123581321 | 482,390 | <p>I think that you believe you found a counter example because you think that</p>
<p><span class="math-container">$(456)(78)$</span> and <span class="math-container">$(132)(456)(78)(123)$</span> are two different cycle types of the same element.</p>
<p>This is not true since <span class="math-container">$(132)(456)(78... |
1,892,376 | <p>The question was: </p>
<blockquote>
<p>From the letters in MAGOOSH, we are going to make three-letter
"words." Any set of three letters counts as a word, and different
arrangements of the same three letters (such as "MAG" and "AGM") count
as different words. How many different three-letter words can be made... | Emilio Novati | 187,568 | <p>In a metric space $(X,d)$ a similitude can be defined as a function $f:X\to X$ such that
$
d(f(x),f(y))=rd(x,y)
$
for $r>0$.
If $X$ is a vector space and the metric is derived from an inner product (so that the notion of angle can be defined) than similarities preserve the angles.</p>
|
2,743,099 | <p>So I have two points lets say <code>A(x1,y1)</code> and <code>B(x2,y2)</code>. I want to find a point <code>C</code> (there will be two points) in which if you connect the points you will have an equilateral triangle. I know that if I draw a circle from each point with radius of equal to <code>AB</code> I will find ... | achille hui | 59,379 | <p>Comment turned to answer per request.</p>
<p>Instead of drawing circles and find the intersection, one can rotate $B$ with respect to $A$ for $\pm 60^\circ$ to get $C(x_3,y_3)$. In matrix notation, the formula is:</p>
<p>$$\begin{bmatrix}x_3\\ y_3\end{bmatrix}
= \begin{bmatrix}x_1\\ y_1\end{bmatrix} +
\begin{bm... |
229,328 | <p>Physicists routinely wrote all 3 Pauli spin matrices as a vector. </p>
<p>$$ \sigma_1 = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0\end{array} \right) \hspace{0.25in}
\sigma_2 = \left( \begin{array}{cc} 0 & -i \\ i & 0\end{array} \right)\hspace{0.25in}
\sigma_3 = \left( \begin{array}{cc} 1 & 0 \\ 0... | Carlo Beenakker | 11,260 | <p>The Pauli spin vector <span class="math-container">$\vec\sigma$</span> relates a two-component spinor <span class="math-container">$s={\alpha\choose \beta}$</span> to its corresponding three-component vector <span class="math-container">$\vec{v}=(a,b,c)$</span>. In Dirac bra-ket notation the relation is written as
<... |
229,328 | <p>Physicists routinely wrote all 3 Pauli spin matrices as a vector. </p>
<p>$$ \sigma_1 = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0\end{array} \right) \hspace{0.25in}
\sigma_2 = \left( \begin{array}{cc} 0 & -i \\ i & 0\end{array} \right)\hspace{0.25in}
\sigma_3 = \left( \begin{array}{cc} 1 & 0 \\ 0... | Dmitry Vaintrob | 7,108 | <p>The Pauli spin vector encodes a two-complex-dimensional unitary representation of the Lie algebra $\bf{su}_2$. A basis of the Lie algebra maps to the matrix algebra $\bf{gl}_2(\mathbb{C}),$ giving a $3\times 2\times 2$ tensor. In 3 dimensions, physicists will sometimes blur the distinction between the basis for the ... |
1,221,224 | <p>I've came across an interesting situation in my code and I want to make sure I'm not missing anything, so I'll try to explain it as simple as possible.</p>
<p>Imagine 4 buckets and 4 ping-pong balls, in how many variations can you put all balls in any of the buckets? The order of balls does not matter, all balls mu... | Rellek | 228,621 | <p>This is not an easy question. This has already been answered on here, and a general form of this question would be the number of ways to order N indistinct objects into K indistinct containers. </p>
<p>This type of problem is defined by a recurrence relation since there is no closed form of this situation. I would ... |
19,067 | <p>I'm trying to modify <code>Plus</code> but am running into trouble with it being <code>Listable</code>:</p>
<pre><code>ClearAll[f, g, h]
Attributes[f] = {Listable};
h /: f[x_h, l_List] := 0
h /: g[x_h, l_List] := 0
f[h[1], {1, 1}] (* {f[h[1], 1], f[h[1], 1]}, not OK I want 0 *)
g[h[1], {1,1}] (* 0 as expected ... | Alexei Boulbitch | 788 | <p>One thing that you could do is to rescale your equation:</p>
<pre><code>Clear[x, ξ, eq1, eq2, A];
eq1 = a x^(17/6) - b x^2 - 1 == 0
eq1 /. x -> A*ξ // PowerExpand
</code></pre>
<p>The result is:</p>
<blockquote>
<p>-1 - A^2 b ξ^2 + a A^(17/6) ξ^(17/6) == 0</p>
</blockquote>
<p>Choose A such that the factor ... |
2,446,000 | <p>I' ve tried with $x^2 = {[x]+1\over 2}$ so $x$ is a square root of half integer. And know? What to do with that?</p>
| Khosrotash | 104,171 | <p>we know $0\leq x-\lfloor x \rfloor <1 $
$$\quad{2x^2-\lfloor x \rfloor-1=0 \to \lfloor x \rfloor=2x^2-1
\\so\\0\leq x-(2x^2-1) <1\to \\
\begin{cases}0\leq x-(2x^2-1) \to & -(x-1)(2x+1)\geq 0 & (*)\\ x-(2x^2-1) <1 \to & x(1-2x)<0 & (**)\end{cases}
\\\begin{cases} (*) \to & x\in[-\frac1... |
140,117 | <p>Does anyone know how one can plot contour lines on a meshed surface? For example I would like to be able to plot contour lines indicating lines of equal distance from a given point such as data coming from (<a href="https://mathematica.stackexchange.com/questions/129207/how-to-estimate-geodesics-on-discrete-surfaces... | george2079 | 2,079 | <p>straightforward draw a line where each contour cuts each triangle:</p>
<pre><code>conttri[tri_, v_] := Module[{s},
s = Select[ Subsets[tri, {2}] ,
Ordering[ Append[MinMax@data[[#]], v]] == {1, 3, 2} & ];
Line[(Interpolation[Transpose[{data[[#]], vertices[[#]]}],
InterpolationOrder -> 1]... |
4,409,499 | <p>Suppose that <span class="math-container">$f:[0,1]\rightarrow [0,1]$</span> is a continuous function such that <span class="math-container">$f(f(x))=x$</span> for all <span class="math-container">$x\in [0,1]$</span>.</p>
<p>We know f is one to one and onto. Morover, it has a fixed point.</p>
<p>If we assume further ... | marco trevi | 170,887 | <p>I believe every function that passes through <span class="math-container">$(0,1)$</span> and <span class="math-container">$(1,0)$</span> and is symmetric along the bisectrix <span class="math-container">$y=x$</span> is a suitable example, for instance
<span class="math-container">$$f_k(x)=\frac{1-x}{1+kx}$$</span></... |
4,409,499 | <p>Suppose that <span class="math-container">$f:[0,1]\rightarrow [0,1]$</span> is a continuous function such that <span class="math-container">$f(f(x))=x$</span> for all <span class="math-container">$x\in [0,1]$</span>.</p>
<p>We know f is one to one and onto. Morover, it has a fixed point.</p>
<p>If we assume further ... | Mercy King | 23,304 | <p>For <span class="math-container">$a \in (0.1)$</span>, define <span class="math-container">$f_a: [0,1] \to [0,1]$</span> by</p>
<p><span class="math-container">$$
f_a(x)= \begin{cases}
-mx+1 & \mbox{ for } 0 \le x\le a\cr
-\frac{x-1}{m} &\mbox{ for } a< x \le 1
\end{cases},
$$</span>
where <span class="ma... |
3,313,145 | <p>For each number I want to get the array of number ascending cumulative</p>
<p>For example :</p>
<p><span class="math-container">$$100 \% = 45 \% + 55 \%$$</span></p>
<p><span class="math-container">$$100 \% = 23.\overline{33} \% + 33.\overline{33} \% + 43.\overline{33} \%$$</span></p>
<p><span class="math-contai... | Henry | 6,460 | <p>Yo seem to want to get <span class="math-container">$n$</span> numbers in arithmetic progression such that their sum is <span class="math-container">$100$</span>, so mean <span class="math-container">$\frac{100}{n}$</span>, and each of the <span class="math-container">$n-1$</span> steps is <span class="math-containe... |
2,644,610 | <p>How is it possible to prove that:
$$
|e^{ia}-e^{ib}|=2\sin\frac{|a-b|}{2}\leq|a-b|?
$$</p>
<p>Specifically, I'm looking for an analytic technique to show that the equality $|e^{ia}-e^{ib}|=2\sin\frac{|a-b|}{2}$ is correct.</p>
| Arthur | 15,500 | <p>Draw a unit circle in the complex plane, and the points $e^{ia}$ and $e^{ib}$. The inequality $|e^{ia}-e^{ib}|\leq |a-b|$ is true because the first term is the distance between $e^{ia}$ and $e^{ib}$ along a straight line, while $|a-b|$ is the distance between the same two points along the arc of the unit circle, pos... |
3,172,149 | <p>Let <span class="math-container">$B^H(t)$</span> be a fractional Brownian motion with Hurst parameter <span class="math-container">$H\in (0,1)$</span>. We define fractional Gaussian noise as <span class="math-container">$X(t)=B^H(t+1)-B^H(t)$</span>. We know the fBm has covariance <span class="math-container">$R(s,t... | corey979 | 321,982 | <p>By expansion and direct substitutions, the expression</p>
<p><span class="math-container">$$E\left[ ( X(t)-X(s) )( X(v)-X(u) ) \right]$$</span></p>
<p>yields</p>
<p><span class="math-container">$$\frac{1}{2} \left(-\left| s-u\right| ^{2 H}+\left| s-v\right| ^{2
H}+\left| t-u\right|^{2 H}-\left| t-v\right| ^{2 H}\... |
2,396,287 | <p>The product has only positive factors so it has zero as lower bound. Also the product is decreasing as all its factors are less than one. In conclusion the series must have a limit. I also compute the first 150 values of the product and I got around 0.297. I believe that the product converges very, very, slowly to z... | Simply Beautiful Art | 272,831 | <p>Hint:</p>
<p>Since</p>
<p>$$\ln(1+x)\sim_0x$$</p>
<p>It follows from logarithmic rules that</p>
<p>$$\frac{\ln(2n+1)}{\ln(2n)}=1+\frac{\ln(1+1/2n)}{\ln(2n)}\sim_\infty1+\frac1{2n\ln(2n)}$$</p>
<p>Reciprocate the product, then take the log of it, and with this, we find</p>
<p>$$-\ln\prod_{k=1}^\infty\frac{\ln(2... |
2,396,287 | <p>The product has only positive factors so it has zero as lower bound. Also the product is decreasing as all its factors are less than one. In conclusion the series must have a limit. I also compute the first 150 values of the product and I got around 0.297. I believe that the product converges very, very, slowly to z... | motoras | 472,463 | <p>I just read <a href="https://math.stackexchange.com/questions/1453665/how-to-calculate-the-this-limit-lim-n-rightarrow-infty-prod-i-1">this question</a> and basically we can use the same approach.</p>
<p>If we denote
$$P_n=\prod_{i=1}^n \frac{ln(2i)}{ln(2i+1)} \ and \ Q_n=\prod_{i=1}^n \frac{ln(2i+1)}{ln(2i+2)} \ t... |
29,143 | <p>In what context should I use $=$ and $\equiv$?</p>
<p>What is the precise difference?</p>
<p>Thanks!</p>
<p>(I wasn't sure what to tag this with, any suggestions?)</p>
| NebulousReveal | 2,548 | <p>It seems that $a \equiv b$ means that $a$ is equivalent to $b$ with respect to some equivalence relation $R$.</p>
|
2,964,512 | <p>A metric <span class="math-container">$d(x,y)$</span> takes two points from some domain <span class="math-container">$X$</span> and returns a non-negative real number. It is the distance between two points.</p>
<p>A norm <span class="math-container">$n(x)$</span> takes only one point from <span class="math-containe... | David C. Ullrich | 248,223 | <p>Of course one big difference is the context: We can talk about a metric on any set, while the notion of a norm only makes sense for a (real or complex) vector space.</p>
<p>So say <span class="math-container">$V $</span> is a vector space. If <span class="math-container">$||.||$</span> is a norm on <span class="mat... |
2,311,583 | <p>$$
\int_{0}^{1}\sqrt{\,1 + x^{4}\,}\,\,\mathrm{d}x
$$ </p>
<p>I used substitution of tanx=z but it was not fruitful. Then i used $ (x-1/x)= z$ and
$(x)^2-1/(x)^2=z $ but no helpful expression was derived.
I also used property $\int_0^a f(a-x)=\int_0^a f(x) $
Please help me out</p>
| Robert Israel | 8,508 | <p>Another non-elementary answer, from Maple, is
$$ \int_0^1 \sqrt{1+x^4}\; dx = \frac{\sqrt {2}+{\it EllipticK} \left( 1/\sqrt {2} \right)}{3} $$</p>
|
1,336,419 | <p>What are the three final numbers of $2003^{2003}$ and $2003^{2003^{2003}}$? </p>
<p>Do I use the Chinese Remainder Theorem here, and if so, how?</p>
| user26486 | 107,671 | <p>$\varphi(1000)=\left(2^3-2^2\right)\left(5^3-5^2\right)=400$, so by <a href="https://en.wikipedia.org/wiki/Euler's_theorem" rel="nofollow">Euler's theorem</a> (since $(3,1000)=1$):</p>
<p>$$2003^{2003}\equiv 3^{2003\pmod{\! 400}}\equiv 3^3\equiv 27\pmod{\! 1000}$$</p>
<p>You can shorten Jack's answer using <a ... |
26,651 | <p>Hi, everybody. I'm recently reading W.Bruns and J.Herzog's famous book-Cohen-Macaulay Rings. I personally believe that it would be perfect if the authors provide for readers more concrete examples. After reading the first two sections of this book, I have two questions.</p>
<ol>
<li><p>Given a non-negative integer ... | Victor Protsak | 5,740 | <p>For me personally, the whole theory started to take shape (and make sense) once I learned about the graded case and understood connections with combinatorics.</p>
<p>For a graded (sometimes called $*$-local) ring, a basic technique for establishing the Cohen-Macaulay property is "Gröbner degeneration": using a Gröb... |
256,298 | <p>I am looking at an example problem in my text:</p>
<p>"Determine whether these system specifications are consistent:</p>
<p>'The diagnostic message is stored in the buffer or it is re-transmitted.'</p>
<p>'The diagnostic message is not stored in the buffer.'</p>
<p>'If the diagnostic message is stored in the buf... | Brian M. Scott | 12,042 | <p>Let $b$ stand for <em>the diagnostic message is stored in the buffer</em>, and let $r$ stand for <em>the diagnostic message is re-transmitted</em>. The three statements can then be abbreviated $b\lor r$, $\lnot b$, and $b\to r$. The question is then whether there is an assignment of truth values to $b$ and $r$ that ... |
1,541,623 | <p>So I'm just getting the grasp of set theory and I have this question.</p>
<blockquote>
<p>Let $|A| = m$ and $|B| = n$. What is the cardinality of the set $A \times B
$?</p>
</blockquote>
<p>I put $\{1,1\}$ as the answer however I wasn't totally sure what the two vertical bars between set $A$ and set $B$ mean. If... | user236182 | 236,182 | <p>$|A|$ denotes the cardinality of $A$. Then $|A\times B|=|A|\cdot |B|=mn$ (see <a href="https://en.wikipedia.org/wiki/Cartesian_product#Cardinality" rel="nofollow">Wikipedia</a>).</p>
|
1,986,333 | <p>Given that in a finite field $K$ the equation $x^2 = 1$ has zero or two solutions depending if $1 \neq -1$, is it true that $x^2 = a$ has at most two roots for a given $a \in K$?</p>
| lisyarus | 135,314 | <p>It follows from a more general statement: in any integral domain, a polynomial of degree $n$ has at most $n$ roots. Note that fields are, by definition, integral domains.</p>
<p>You can easily find a proof of this fact, for example, <a href="https://math.stackexchange.com/questions/7990/roots-of-a-polynomial-in-an-... |
2,192,345 | <p>How can I calculate the integral of $f(z) = e^{-z}$ over the surface of a sphere with radius $R$?
I tried using cylindrical and spherical systems, both gave an unsolvable integral, suspecting there's a way to change the order of the variables. </p>
| Dr. Sonnhard Graubner | 175,066 | <p>HINT: we have $$f_x=3x^2+6x-15$$ or you have made a typo
solving $$3x^2+3y^2-15=6xy+3y^2-15$$ we get
$$3x^2-6xy=0$$ and this can factorized into $$3x(x-2y)=0$$</p>
|
2,192,345 | <p>How can I calculate the integral of $f(z) = e^{-z}$ over the surface of a sphere with radius $R$?
I tried using cylindrical and spherical systems, both gave an unsolvable integral, suspecting there's a way to change the order of the variables. </p>
| Karine Larouche | 426,740 | <p>You obtained $x=2y$ from $3x^2 = 6xy$. But $3x^2 = 6xy$ is also true if $x=0$. So you have to solve for $x=2y$ as you did, but also for $x=0$ which will lead you to $y = \pm \sqrt 5$.</p>
|
786,596 | <p>So I'm trying to solve this practice exam question, </p>
<blockquote>
<p>Let $G$ be a planar graph with at least two edges and does not contain $K_{3}$ as a subgraph. Prove that $|E|\leq 2|V|-4$.</p>
</blockquote>
<p>Now I started doing this by induction, but it seems to me like the base-case is a counter-examp... | Shobhit | 79,894 | <p>HINT:</p>
<p>$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+... +y^{n-1})$</p>
|
78,617 | <p>I'm sorry I'm French so the subject may not be properly translated, but here's my try:</p>
<p>A goat lives in a rectangular place. She's tied to the point P. The length of the row is 8 meters. The problem is that she can eat flowers: it's the shaded area. The farmer doesn't want the goat to eat the flowers, so he h... | NoChance | 15,180 | <p>Thinking like Mr. Goat :), </p>
<p><strong>Approach A:</strong></p>
<p>the only way he can reach the flowers is by:</p>
<p>(A) going around the hut (since it is a solid structure, and assuming he can't jump it) following the blue line until he reaches point F. This is 3+4=7 M.</p>
<p>(B) And from that point he c... |
93,383 | <blockquote>
<p>A hotel can accommodate 50 customers, experiences show that $0.1$ of those who make a reservation will not show up. Suppose that the hotel accepts 55 reservations. Calculate the probability that the hotel will be able to accommodate all of the customers that show up. </p>
</blockquote>
<p>I only ... | Dilip Sarwate | 15,941 | <p>For a very similar problem (with $105$ passengers holding
reservations on a $100$-seat flight and $0.1$ probability
of not showing up), see
<a href="http://courses.engr.illinois.edu/ece313/spring05/homework/PS04.pdf" rel="nofollow">here</a>
and the complete solution
is <a href="http://courses.engr.illinois.edu/ec... |
3,793,268 | <p>How to prove that <span class="math-container">$x(t) = \cos{(\frac{\pi}{8}\cdot t^2)}$</span> aperiodic?</p>
<p>My process was as follows:</p>
<p><span class="math-container">$x(t+T)= \cos{(\frac{\pi(t+T)^2}{8})}$</span>.</p>
<p>So, <span class="math-container">$T^2 + 2tT -16=0$</span> which seems periodic to me...<... | Bastien Tourand | 815,760 | <p>Suppose that <span class="math-container">$x$</span> periodic.
Then its derivative wrt <span class="math-container">$t$</span> is also.</p>
<p>However <span class="math-container">$x'(t)=-\frac{1}{4} \pi x \sin(\frac{\pi}{8}x^2)$</span>, which is obviously aperiodic because of the factor <span class="math-container"... |
195,176 | <p>Find the values of the real constants $c$ and $d$ such that</p>
<p>$$\lim_{x\to 0}\frac{\sqrt{c+dx}-\sqrt{3}}{x}=\sqrt{3}$$</p>
<p>I really have no clue how to even get started.</p>
| Michael Hardy | 11,667 | <p>$$\lim_{x\to 0}\frac{\sqrt{c+dx}-\sqrt{3}}{x}=\sqrt{3}$$</p>
<p>Since the denominator goes to $0$, the limit cannot exist unless the numerator also goes to $0$. The numerator is $\sqrt{c+dx}-\sqrt{3}$, so that would have to go to $0$ as $x$ goes to $0$. But it goes to $\sqrt{c+d\cdot0} - \sqrt{3}$. Hence $c+d\cd... |
981,949 | <blockquote>
<p>Find the derivative and evaluate at $f\;'(2):$ $$\log_4(2x^2+1)$$ </p>
</blockquote>
<p>$\log_4(2x^2+1)=y$<br>
$4^y=2x^2+1$ </p>
<p>$4^y\ln4 \times y\;'=4x$<br>
$y\;'=\dfrac{4x}{4^y\ln4}\implies \dfrac{4x}{(2x^2+1)\ln4}$ </p>
<p>What am I doing wrong? I evaluated at $2$ and got $1.154$</p>
| Paul | 17,980 | <p>Notice that $(\log_ax)'=\frac{1}{x\ln a}$. SO </p>
<p>$$(\log_4(2x^2+1))\big|_{x=2}=\frac{1}{(2x^2+1)\ln 4} \times 4x\big|_{x=2}=\frac{4}{9\ln 2}$$ </p>
|
981,949 | <blockquote>
<p>Find the derivative and evaluate at $f\;'(2):$ $$\log_4(2x^2+1)$$ </p>
</blockquote>
<p>$\log_4(2x^2+1)=y$<br>
$4^y=2x^2+1$ </p>
<p>$4^y\ln4 \times y\;'=4x$<br>
$y\;'=\dfrac{4x}{4^y\ln4}\implies \dfrac{4x}{(2x^2+1)\ln4}$ </p>
<p>What am I doing wrong? I evaluated at $2$ and got $1.154$</p>
| Claude Leibovici | 82,404 | <p>You result $$y'=\dfrac{4x}{(2x^2+1)\ln4}$$ is perfectly correct. So, if $x=2$, $y'=\frac{8}{9 \log (4)}=0.641198$</p>
|
58,306 | <p>Let $X$ be a topological space, and let $\mathscr{F}, \mathscr{G}$ be sheaves of sets on $X$. It is well-known that a morphism $\varphi : \mathscr{F} \to \mathscr{G}$ is epic (in the category of sheaves on $X$) if and only if the induced map of stalks $\varphi_P : \mathscr{F}_P \to \mathscr{G}_P$ is surjective for e... | Georges Elencwajg | 3,217 | <p>Take $X=\mathbb R$ for your topological space, the constant sheaf $\underline {\mathbb Z} $ for $\mathcal F$ and for $\mathcal G$ the direct sum of two skyscraper sheaves with fibers $\mathbb Z$ at two distinct points $P,Q\in \mathbb R$, that is $\mathcal G=\mathbb Z^P \oplus \mathbb Z^Q$.<br>
The natural rest... |
1,234,500 | <p>I am a student in 12th grade and am fond of mathematics. I enjoy reading mathematics but when it comes to problems I just get completely stuck. Its not that I don't understand the problem but often don't know how to go about tackling it. When I see the solution, often I understand it perfectly but arriving at that s... | lab bhattacharjee | 33,337 | <p>$$m(x-2)=|(x+3)(x+1)|\ge0$$</p>
<p>If $m=0,$ there are two real solutions</p>
<p>Else $m(x-2)=|(x+3)(x+1)|=0$ has no solution</p>
<p>So, $$m(x-2)=|(x+3)(x+1)|>0$$</p>
<p>Now $|(x+3)(x+1)|=-(x+3)(x+1)$ if $-3\le x\le-1$</p>
<p>$=+(x+3)(x+1)$ otherwise</p>
<p>If $m>0,x-2>0\iff x>2\implies m(x-2)=x^2+... |
3,225,176 | <p>Prove that the number A is not primary</p>
<p>Such that : </p>
<p><span class="math-container">$A=\frac{2^{4n+2}+1}{5}$</span> </p>
<p><span class="math-container">$n≥2$</span></p>
<p>n=2 then <span class="math-container">$A=205$</span></p>
<p>Please I need some ideas to approach it</p>
| Community | -1 | <p>First, the numerator is divisible by <span class="math-container">$5$</span>. This follows since <span class="math-container">$4^{2n+1}+1=(4+1)(4^{2n}-4^{2n-1}+\dots-4+1)$</span>. </p>
<p>Second, @marty cohen has shown that the numerator is the product of factors each greater than five.</p>
<p>It follows that ... |
3,132,009 | <p>This question is a <strong>cross post</strong> from <a href="https://mathoverflow.net/questions/333204/reference-request-introduction-to-finsler-manifolds-from-the-metric-geometry-po">MathOverflow</a>. I have requested the migration of the question, but unfortunately it is not possible after two months of posting.</... | Dante Grevino | 616,680 | <p>I have found the following references:</p>
<ol>
<li><p>An introductory textbook by A. Papadopoulos about the Busemann's approach: <a href="https://books.google.com.ar/books/about/Metric_Spaces_Convexity_and_Nonpositive.html?id=JrwzXZB0YrIC&redir_esc=y" rel="nofollow noreferrer">Metric Spaces, convexity and non-... |
2,298,971 | <p>Using Wolframalpha, the limit is $\sqrt{e}$. Now because we already know $\lim\limits_{n \rightarrow \infty}(1+\frac{1}{n})^n = e$ </p>
<p>the term has to be ~$(1+\frac{1}{n})^{\frac{n}{2}}$. My attempt to transform the equation fails though, because im not sure how to get rid of the $-\frac{1}{2}$:</p>
<p>$(1+\fr... | lab bhattacharjee | 33,337 | <p>Try $$\lim_{n\to\infty}(\left(1+\dfrac1{2n+1}\right)^n=\left(\lim_{n\to\infty}\left(1+\dfrac1{2n+1}\right)^{2n+1}\right)^{\lim_{n\to\infty}\frac n{2n+1}}$$</p>
<p>Now $\lim_{n\to\infty}\dfrac n{2n+1}=\lim_{n\to\infty}\dfrac1{2+\frac1n}=?$</p>
<p>Alternatively, $$\lim_{m\to0}(1+m)^{\frac{m}{2}-\frac{1}{2}}=\dfrac{\... |
11,618 | <p>I'm teaching a preparatory course on mathematics at a university. The content is mostly calculus, manipulating expressions and solving equations and inequalities. I show a couple of simple derivations/proofs and ask the students to occasionally prove some simple equality, so the course is by no means rigorous. Most ... | blmayer | 6,547 | <p>I would say to first sketch the curve $4^x$ and it's derivative $4^x \ln 4$, and the same for $3^x$ and $2^x$, then it will be easy to see that the case $e^x$ is a particular case. For the first two functions the derivative graph is a little higher than the function, but with $2^x$ the derivative is lower, so there ... |
1,786,514 | <p>Let $S$ be a set containing $n$ elements and we select two subsets: $A$ and $B$ at random then the probability that $A \cup B$ = S and $A \cap B = \varnothing $ is?</p>
<p>My attempt</p>
<p>Total number of cases= $3^n$ as each element in set $S$ has three option: Go to $A$ or $B$ or to neither of $A$ or $B$</p>
<... | Tom Collinge | 98,230 | <p>Pick any subset $A$, and there is only one subset $B$, namely $S \setminus A$ which satisfies $A \cup B = S$ and $A \cap B = \emptyset $.</p>
<p>There are $2^n$ subsets to choose from so the probability of selecting such a pair is $1/2^n$.</p>
<p>(Or, $1/(2^n - 1)$ if one constrains that $A \ne B$).</p>
|
2,728,317 | <p>As I know when you move to "bigger" number systems (such as from complex to quaternions) you lose some properties (e.g. moving from complex to quaternions requires loss of commutativity), but does it hold when you move for example from naturals to integers or from reals to complex and what properties do you lose?</p... | TheGeekGreek | 359,887 | <p>The most important property you loose when moving from real to complex numbers is definitely the notion of an <em>order</em>, i.e. $\mathbb{R}$ is an ordered field whereas $\mathbb{C}$ is not. This follows from the following proposition (<em>Abstract Algebra</em> by <em>P.A. Grillet</em>):</p>
<blockquote>
<p>A f... |
3,767,935 | <p>Prove or disprove that if <span class="math-container">$$\prod\limits_{x=2}^{\infty} f(x)=0$$</span> and <span class="math-container">$f(x)\neq0$</span> for any <span class="math-container">$x\geq0$</span> then <span class="math-container">$$\prod\limits_{x=2}^{\infty} f(x\varphi)=0$$</span> for any constant <span c... | Community | -1 | <p>Not really: you can consider your favourite function <span class="math-container">$f$</span> such that <span class="math-container">$f(n)=1-\frac1n$</span> for all odd <span class="math-container">$n\in \Bbb N$</span> and <span class="math-container">$f(n)=1-2^{-n-1}$</span> for all even <span class="math-container"... |
1,579,579 | <p>I am struggling with showing that for algebraic number $\alpha$, the ring generated by $\mathbb{Q}[\alpha]$ is a field. I understand that to do this, I will have to show that any $r+s\alpha, r,s\in \mathbb{Q}$ has an inverse in $\mathbb{Q}[\alpha]$. I'm lost on how to go about doing this, though. Help? </p>
| basket | 294,706 | <p>The ring $\mathbb{Q} [\alpha]$ is the image of $f: \mathbb{Q}[x] \to \mathbb{C}$ where $f: p(x) \mapsto p(\alpha)$ ($f$ is evaluation at $\alpha$.) Since $\alpha$ is algebraic it has a minimal polynomial $q_{\alpha}(x)$. Show the kernel of $f$ is the principal ideal generated by $q_{\alpha}(x)$. Since $q_{\alpha}(x)... |
1,137,930 | <p>Please, help me to understand the mathematics behind the following formula of CPI. Why do we calculate CPI the way it's done on the pic? The formula reminds me the expected value from stochastic, but do we have a random value here? </p>
<p><img src="https://i.stack.imgur.com/djpLX.png" alt="enter image description ... | axiom | 167,868 | <p><code>Not reflexive</code> Correct. Not every element is related to itself.</p>
<p><code>Not irreflexive</code> Correct. Since we have $(2, 2)$. Irreflexivity requires that <em>no</em> element should be related to itself. </p>
<p><code>Not symmetric</code> Correct. We have $(3, 4)$ but not $(4, 3)$. </p>
<p><co... |
2,941,421 | <p>A study is conducted to test the hypothesis that people with glaucoma have higher variability in systolic blood pressure(SBP). The study includes 41 people with glaucoma whose mean SBP is 140 mmHg with a standard deviation of 25 mmHg. If the population standard deviation is 20 mmHg, verify the claim at 1% significan... | Phil H | 554,494 | <p><span class="math-container">$63.691$</span> is "greater" than <span class="math-container">$62.5$</span> not less which means your p value test statistic is <span class="math-container">$> 0.01$</span>. Hence you fail to reject the null and conclude that there is no statistically significant difference in SBP va... |
1,008,253 | <p>The Wronskian for $\sin^2x, \cos^2x$ is</p>
<p>\begin{align}
& \left| \begin{array}{cc} \sin^2 x & \cos^2 x \\ 2\sin x\cos x & -2\cos x\sin x \end{array} \right| \\[8pt]
= {} & -2\sin^2x \cos x \sin x - 2 \cos^2 x \sin x \cos x,
\end{align}
with $x = \frac{π}{6},$ this is $=$
$$
-\sqrt{\frac{3}{2}}... | Ivo Terek | 118,056 | <p>It suffices to show that the Wronskian is not zero for a single value of $x$. We have: $$W(x) = \begin{vmatrix} \sin^2x & \cos^2x \\ 2\sin x \cos x & -2 \sin x \cos x\end{vmatrix} = -2\sin^3x \cos x - 2\sin x \cos^3 x$$
$$W(x) = -2\sin x \cos x = -\sin(2x)$$
Then, $W(\pi/4) = -1 \neq 0$, so the functions are... |
2,460,003 | <p>I need some help showing that these are equivalent. I made a couple attempts to get this right but so far the following work is as far as I've gotten.</p>
<p>Here is the question in its entirety:</p>
<blockquote>
<p>Let n be a natural number. Give a combinatorial proof of the following:
$\binom{2n+2}{n+1} = \b... | jonsno | 310,635 | <p>The question asks for combinatorial proof. <strong>But</strong> going by your way, ie using Pascal's identity it can be done as follows:
$$\binom{2n}{n} + \binom{2n}{ n-1} = \binom{2n+1}{n}$$</p>
<p>And $$\binom{2n}{n} + \binom{2n}{n+1} = \binom{2n+1}{n+1}$$</p>
<p>so adding both the equations,</p>
<p>$$\begin{al... |
789,769 | <p>I was reading some questions about prime numbers posted in latest days and a question came to my mind:</p>
<blockquote>
<p>What is the state of art of the research into prime numbers distribution?</p>
</blockquote>
<p>I read then several other questions (<a href="https://math.stackexchange.com/questions/69628/ri... | O. S. Dawg | 137,416 | <p>For a partial answer see:</p>
<p><a href="http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes" rel="nofollow">http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes</a></p>
<p>Everything there is certainly state of the art.</p>
|
789,769 | <p>I was reading some questions about prime numbers posted in latest days and a question came to my mind:</p>
<blockquote>
<p>What is the state of art of the research into prime numbers distribution?</p>
</blockquote>
<p>I read then several other questions (<a href="https://math.stackexchange.com/questions/69628/ri... | Eric Naslund | 6,075 | <p>Essentially you are asking for a survey of the modern developments in understanding prime numbers, and I believe it would be best to start by looking at some expository papers.</p>
<p>Here is a paper by Granville title <a href="http://www.dms.umontreal.ca/~andrew/PDF/PrinceComp.pdf" rel="nofollow">Analytic Number T... |
356,497 | <p>I usually solve a quadratic equation:</p>
<p>$$ax^2+bx+c=0$$</p>
<p>Through a method I learned in school: For a monic quadratic, you make $x=y-\frac{b}{2}$.</p>
<p>The method is intended for a monic equation but in this case (non-monic equation), I divide all the equation by $a$ to transform it in a monic equatio... | Cameron Buie | 28,900 | <p>Your cubic is already monic, so there's no need to adjust. Furthermore, $$x^3+x^2-12x=x(x^2+x-12)=x(x+4)(x-3)$$ readily, so we can solve your equation without much difficulty.</p>
<p>In answer to your question, though, the reason that the coefficient of $x$ in $1/x$ is $0$ is because there <strong>is no term</stron... |
356,497 | <p>I usually solve a quadratic equation:</p>
<p>$$ax^2+bx+c=0$$</p>
<p>Through a method I learned in school: For a monic quadratic, you make $x=y-\frac{b}{2}$.</p>
<p>The method is intended for a monic equation but in this case (non-monic equation), I divide all the equation by $a$ to transform it in a monic equatio... | Pedro | 23,350 | <p><strong>ADD</strong> With what you've done, you can let $u=1/x$ to move on, however, note that $x=0$ is a solution to the equation, so you can factor $x$ out and move on with a quadratic.</p>
<hr>
<p>Note that the method you know about quadratic equations has an analog for cubics, however, what you do is anihilate... |
820,614 | <p>In a scalene triangle,does there exist three cevians which are equal in length,where length is measured between the corresponding vertex and the intersection point of the cevian with the corresponding side? This is just a question I had in my mind. </p>
| Mark Fischler | 150,362 | <p>Because of the circular symmetry, it is obvious that the center of mass is on the $z$ axis. The $z$ component of the center of mass is
$$
\frac{\int_V z \rho \;\mathrm{d}v}{\int_V \rho \;\mathrm{d}v} = \frac{M_{xy}}{M}
$$
But you mis-calculated both $M$ and $M_{xy}$. </p>
<p>By far the easiest way to do these integ... |
771,607 | <p>Let $G$ be a finite $p$-group and $K$ be a normal subgroup. I want to show that there exists a normal subgroup $N$ of $G$ such that $N \leq K$ and $[K:N]=p$. I tried in this way: from Sylow's theorem, there exists a normal series $G=G_0 \rhd G_1 \rhd \cdots \rhd G_a=\{e\}$ such that $|G_i/G_{i+1}|=p$, then $K=K \cap... | DonAntonio | 31,254 | <p>Hints:</p>
<p>Using the all-powerful theorem that says that a finite $\;p$- group has a non-trivial center, prove by induction on $\;n\;$ that <em>any</em> group of order $\;p^n\;$ has <strong>a normal subgroup</strong> of order $\;p^k\;$ , for all $\;0\le k\le n\;$ .</p>
<p>This solves at once your question...</... |
720,504 | <p>Why is $\displaystyle \lim_{x \to \infty} \ x^{2/x} = 1$ since this is an indeterminate form $\infty^{0}$ and I can't see any manipulation that would suggest this result?</p>
| MCT | 92,774 | <p>$\lim \limits_{x \to \infty} x^{\frac{1}{x}} = 1$</p>
<p>Proof using AM-GM and Sandwich Theorem</p>
<p>$\frac{1 + 1 + 1 + \dots + \sqrt{x} + \sqrt{x}}{x} \geq \sqrt[x]{x} \geq 1$</p>
<p>$\frac{x - 2 + 2\sqrt{x}}{x} \geq \sqrt[x]{x} \geq 1$</p>
<p>$1 - \frac{2}{x} + \frac{2}{\sqrt{x}} \geq \sqrt[x]{x} \geq 1$</p>... |
720,504 | <p>Why is $\displaystyle \lim_{x \to \infty} \ x^{2/x} = 1$ since this is an indeterminate form $\infty^{0}$ and I can't see any manipulation that would suggest this result?</p>
| robjohn | 13,854 | <p>For all $x\in\mathbb{R}$: $1+x\le e^x$. Substitute $x\mapsto\frac xn$ and then raise to the $2/x$ power (assuming $x\gt0$):
$$
\left(\frac xn\right)^{2/x}\le\left(1+\frac xn\right)^{2/x}\le e^{2/n}\tag{1}
$$
Multiplying by $n^{2/x}$ gives the following for any $x$ and $n$ greater than $0$:
$$
x^{2/x}\le e^{2/n}n^{2/... |
924,551 | <p>$$\displaystyle \lim_{x \to \infty}\dfrac{8-\sqrt{x}}{8+\sqrt{x}}$$ </p>
<p>I tried rationalizing the numerator: </p>
<p>$$\lim_{x \to \infty}\dfrac{8-\sqrt{x}}{8+\sqrt{x}} \times \dfrac{(8-\sqrt{x})}{(8-\sqrt{x})}$$ </p>
<p>$$\lim_{x \to \infty}\dfrac{64-16\sqrt{x}+x}{64-x}$$</p>
<p>Is this correct? how do I p... | egreg | 62,967 | <p>First of all get rid of the square root: since squaring is a continuous function and $\lim_{t\to\infty}t^2=\infty$, you have
$$
\lim_{x\to\infty}\frac{8-\sqrt{x}}{8+\sqrt{x}}=
\lim_{t\to\infty}\frac{8-\sqrt{t^2}}{8+\sqrt{t^2}}=
\lim_{t\to\infty}\frac{8-t}{8+t}
$$
Now you have reduced to the limit at infinity of a ra... |
519,516 | <p>I have been reading about uniform spaces and topological groups. There does not look to be a lot of literature on the topic, much less accesible literature, and the books that I have been reading do not mention any examples of uniform spaces, other than metric spaces and topological groups. There is another which I ... | Michael Hardy | 11,667 | <p>It's just a way of looking at the chain rule.</p>
<p>The chain rule is differentiation by substitution.</p>
<p>One can write $$\frac{d}{dx} f(g(x)) = f'(g(x)) g'(x),$$ or one can look at
$$
\frac{d}{dx} f(g(x))
$$
and then do this substitution:
$$
u = g(x),\qquad \frac{du}{dx} = g'(x).
$$
Then one writes
$$
\frac{... |
1,107,013 | <p>Suppose that $f$ is a differentiable real function in an open set $E \subset \mathbb{R^n}$, and that $f$ has a local maximum at a point $x \in E$. Prove that $f'(x)=0$</p>
| Hagen von Eitzen | 39,174 | <p>As $x\to+\infty$, the left hand side converges to $1$, hence for $x$ big enough it stays strictly between $\frac12$ and $\frac32$, wheras the right hand side oscillates between $-2$ and $2$. By the IVT, there exist a solution each time the cosine goes up or down.</p>
|
3,109,300 | <p>For the case that <span class="math-container">$m\geq0$</span> I don't need to apply L'Hospital.</p>
<p>Let <span class="math-container">$m<0$</span></p>
<p>We have <span class="math-container">$x^m=\frac{1}{x^{-m}}$</span></p>
<p>We also know that <span class="math-container">$x^{-m}\rightarrow 0$</span> as <... | Alan Zhang | 547,659 | <p>take <span class="math-container">$ln$</span> first, <span class="math-container">$ln(\frac{e^{-1/x}}{x^m})=-\frac{1+mxln(x)}{x}$</span>, use L'Hospital, we get <span class="math-container">$mln(x)+m$</span> goes to <span class="math-container">$-\infty$</span>, thus apply exp again, we get zero. </p>
|
194,123 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/150482/probability-of-a-random-binary-string-containing-a-long-run-of-1s">Probability of a random binary string containing a long run of 1s?</a> </p>
</blockquote>
<p><strong><em>EDIT</strong>: Cocopuffs b... | Cocopuffs | 32,943 | <p><strong>Edit:</strong> This is my (incorrect) attempt at a solution. The induction step is not valid at $L=2$.</p>
<p>Clearly $p(0) = 1$.</p>
<p>Use induction on $L$ and assume that $p(0),...,p(L-1) = 1$.</p>
<p>Case 1: $L$ is odd, $L = 2m-1.$ Then $$1 = p(m) = \frac{1}{2^{m-1}} + \sum_{i=1}^{m-1} \frac{p(m+i)}{2... |
1,599,510 | <p>I understand this proof <a href="http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Ng.pdf" rel="nofollow">http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Ng.pdf</a> (Lemma 2.2) until the point "and hence of $-1 - m^2\mod p$ ". Why is this true, and how does the final line then follow?</p>
| user236182 | 236,182 | <p>It's a classical proof that uses the Pigeonhole Principle. You can prove a more general result with the same method: </p>
<p>If $p$ is an odd prime (it's clear when $p=2$), then for all $a,b,c\in\mathbb Z$ such that $p\nmid a,b$ exist $x,y\in\mathbb Z$ such that $p\mid ax^2+by^2+c$.</p>
<p>Proof: First I'll prove ... |
1,599,510 | <p>I understand this proof <a href="http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Ng.pdf" rel="nofollow">http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Ng.pdf</a> (Lemma 2.2) until the point "and hence of $-1 - m^2\mod p$ ". Why is this true, and how does the final line then follow?</p>
| Hagen von Eitzen | 39,174 | <p>The map $x\mapsto -1-x$ is a bijection of the residues modulo $p$. Hence if $m^2$ runs through $n+1$ distinct residues, then $-1-m^2$ also runs through $n+1$ distinct residues.</p>
<p>After that, the two sets of residues (i.e., the $n+1$ residues of form $l^2\pmod p$ and the $n+1$ residues of form $-1-m^2\pmod p$) ... |
1,639,156 | <p>Use logical quantifiers to write:
"Everybody loves somebody sometimes" (Where U=all people)
I came up with this but not sure how to type symbols in here.</p>
<p>$$\forall x \in U\,: \exists y\in U: x \text{ loves } y.$$</p>
<p>So... upside down A="For all"
Backwards E for "there exists"
curly little e for "belongs... | DanielWainfleet | 254,665 | <p>$\forall Body(1)\; \exists Body(2)\; \exists Time(t)\; (Body(1)\ne Body(2)\land Body(1)\; loves\; Body(2)\; at \;Time(t)).$</p>
|
48,359 | <p>I'm trying to numerically integrate a function which has a vector-valued slow part and a much faster component which is shared by all the components, i.e. an integral of the form
$$
\int_a^b\begin{pmatrix}f(x)\\ g(x) \\ h(x)\end{pmatrix}w(x)\,\text dx.
$$
Because <code>NIntegrate</code> is nicely Listable on its fir... | Szabolcs | 12 | <p>Here's how this can be done:</p>
<pre><code>ClearAll[algebraicQ]
algebraicQ[x_] := Module[{result},
result = Element[x, Algebraics];
result /; MatchQ[result, True | False]]
</code></pre>
<p>The key to these types of problems is usually a special use of <a href="http://reference.wolfram.com/mathematica/ref/Con... |
2,591,621 | <p>Can this equation $x^3-12x=c$ have $2$ different solutions in $[-2,2]$? In $(-\infty,-2]$? In $[2,+\infty)$?</p>
<p>I said:
Let the equation have 2 different solutions, one in $[-2,x_1]$ and one in $[x_1,2]$ and let $f(x)=x^3-12x-c,f(-2)<0$. According to Bolzano's theorem, $f(-2) \cdot f(x_1)<0 \implies f(x_1... | Peter | 82,961 | <p>The derivate of the function $$f(x)=x^3-12x-c$$ is $$f'(x)=3x^2-12=3(x-2)(x+2)$$</p>
<p>Hence the derivate is non-positive in the given interval because $x-2$ is non-positive and $x+2$ is non-negative. We have two isolated roots , namely $2$ and $-2$. Hence , $f(x)$ is strictly decreasing in the interval $[-2,2]$, ... |
2,591,621 | <p>Can this equation $x^3-12x=c$ have $2$ different solutions in $[-2,2]$? In $(-\infty,-2]$? In $[2,+\infty)$?</p>
<p>I said:
Let the equation have 2 different solutions, one in $[-2,x_1]$ and one in $[x_1,2]$ and let $f(x)=x^3-12x-c,f(-2)<0$. According to Bolzano's theorem, $f(-2) \cdot f(x_1)<0 \implies f(x_1... | ajotatxe | 132,456 | <p>Let $f(x)=x^3-12x-c$. Then $f'(x)=3x^2-12$, which vanishes at $=2$ and $x=-2$. This means that $f$ is increasing at $(-\infty,-2]$ and $[2,\infty)$, and decreasing in $[-2,2]$.</p>
<p>There are five cases:</p>
<ol>
<li>$f(-2)<0$. Then $f$ has a single root, and it is in $(2,\infty)$. This is for $c>16$.</li>... |
4,377,390 | <p>The famous German physicist Walter Schottky (1986-1976), in a publication on "thermal agitation of electricity in conductors" in the 1920ies, calculated the integral <span class="math-container">$\int_{0}^{\infty} \frac{1}{(1-x^2)^2+r^2 x^2}\;dx$</span> to be <span class="math-container">$\frac{2\pi}{r^2}$... | Quanto | 686,284 | <p>In the special case of <span class="math-container">$r=2$</span>, the integral satisfies the inequity
<span class="math-container">\begin{align}
& \int_{0}^{\infty} \frac{1}{\left(1-x^{2}\right)^{2}+r^{2} x^{2}} d x
\overset {r=2}=
\int_{0}^{\infty} \frac{1}{\left(1+x^{2}\right)^{2}} d x
< \int_{0}^{\infty... |
3,381,219 | <p><a href="https://i.stack.imgur.com/mM0OF.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/mM0OF.png" alt="3B) in the picture"></a>What is an example of an infinite intersection of infinite sets is infinite?</p>
<p>I know that the intersection of infinite sets does not need to be infinite. However,... | HiMatt | 615,440 | <p>Trivially, you could intersect an infinite set with itself. For instance, consider the set of real numbers <span class="math-container">$\mathbb{R}$</span>. Then
<span class="math-container">$$\bigcap_{i=1}^{\infty} \mathbb{R} = \mathbb{R}.$$</span>
If you want something nontrivial, consider
<span class="math-contai... |
3,117,260 | <p>So according to the commutative property for multiplication:</p>
<p><span class="math-container">$a \times b = b \times a$</span> </p>
<p>However this does not hold for division</p>
<p><span class="math-container">$a \div b \neq b \div a$</span> </p>
<p>Why is it that in the following case:</p>
<p><span class="... | marshal craft | 167,793 | <p>With multiplication and division alone (no addition or subtraction) they are associative with respect to one another, BUT division itself is NOT commutative.</p>
<p>So basically you can do the multiplication or division in either order, BUT you must respect which way you interpret the inputs to the left or right of... |
3,117,260 | <p>So according to the commutative property for multiplication:</p>
<p><span class="math-container">$a \times b = b \times a$</span> </p>
<p>However this does not hold for division</p>
<p><span class="math-container">$a \div b \neq b \div a$</span> </p>
<p>Why is it that in the following case:</p>
<p><span class="... | Marc van Leeuwen | 18,880 | <p>The exact same situation occurs with addition and subtraction instead of multiplication and division. Do you find it hard to justify rewriting say <span class="math-container">$a+b-c$</span> to <span class="math-container">$a-c+b$</span>? And if not, which laws are you using to justify this? I would say associativit... |
2,270,346 | <p>I have a pretty simple question here it looks like but I just can't seem to do it. I'd like to be able to do it the easiest way possible. </p>
<blockquote>
<p>Solve $\dot{x}=y$ and $\dot{y}=x$ for $x(t)$ and $y(t)$.</p>
</blockquote>
<p>I need solve these two equations so I can draw a phase plane portrait and s... | HBR | 396,575 | <p>Yes. You can differentiate both equations wrt. time and plug one into the other and vice-versa. I mean:
$$\dot{x}=y\tag{1}$$
$$\dot{y}=x\tag{2}$$</p>
<p>Differentiating $(1)$ and introducing $(2)$ you have:
$$\ddot{x}=\dot{y}=x$$
doing an analogous procedure with $(2)$
$$\ddot{y}=y$$</p>
|
2,270,346 | <p>I have a pretty simple question here it looks like but I just can't seem to do it. I'd like to be able to do it the easiest way possible. </p>
<blockquote>
<p>Solve $\dot{x}=y$ and $\dot{y}=x$ for $x(t)$ and $y(t)$.</p>
</blockquote>
<p>I need solve these two equations so I can draw a phase plane portrait and s... | Patrick Moloney | 349,293 | <p>I understand it now. So if I had this problem </p>
<p>$$\dot{x}=y \,\,\,\,\,\,\,\,\,\,\,\,(1)$$ $$ \dot{y}=-2x \,\,\,\,\,\,(2)$$</p>
<p>Differentiate $(1)$ </p>
<p>$$\ddot{x}= \dot{y}= -2x \implies \ddot{x}-2x=0$$</p>
<p>This equation has characteristic polynomial $P(\lambda)$</p>
<p>$$P(\lambda) = \lambda^2-2=... |
2,289,668 | <p>I have a problem which states:</p>
<p>Let $f(x)=4x-2$ and $\epsilon > 0$.</p>
<p>I must find a $\delta>0$ s.t. $0<|x-1|<\delta$ implies $|f(x)-2|<\epsilon$.</p>
<p>How can I solve a problem such as this?</p>
| ADA | 235,471 | <p>Let $\epsilon > 0$ then $|f(x)-2|=|4x-2-2|=|4x-4|=4|x-1|$ select $\delta = \epsilon / 4$, then $|f(x)-2| < \epsilon $ whenever $|x-1| < \delta$. </p>
|
2,289,668 | <p>I have a problem which states:</p>
<p>Let $f(x)=4x-2$ and $\epsilon > 0$.</p>
<p>I must find a $\delta>0$ s.t. $0<|x-1|<\delta$ implies $|f(x)-2|<\epsilon$.</p>
<p>How can I solve a problem such as this?</p>
| hamam_Abdallah | 369,188 | <p>$f (x)-2=4 (x-1) $.</p>
<p>We look for $\delta>0$ such that</p>
<p>$$|x-1|<\delta\implies 4|x-1|<\epsilon $$</p>
<p>or
$$|x-1|<\delta\implies |x-1|<\frac {\epsilon}{4} $$</p>
<p>a $\delta\leq \frac {\epsilon}{4} $ works.</p>
|
1,087,107 | <p>In <a href="http://raudys.com/kursas/Options,%20Futures%20and%20Other%20Derivatives%207th%20John%20Hull.pdf" rel="nofollow">Hull (2008, p. 307)</a>, the following equation is found (Eq. 13A.2):</p>
<p>$$E[\max(V-K,0)]=\int_{K}^{\infty} (V-K)g(V)\:dV$$</p>
<p>Where $g(V)$ is the PDF of $V$ and $V,K>0$.</p>
<p>I... | André Nicolas | 6,312 | <p>Let $X$ and $Y$ be (say positive) independent random variables, with continuous distributions, densities $f(x)$ and $g(y)$, and cumulative distribution functions $F(x)$ and $G(y)$. Let $W=\max(X,Y)$. Then $\Pr(W\le w)=\Pr(X\le w)\Pr(Y\le w)=F(w)G(w)$.</p>
<p>Thus the cumulative distribution function of $W$ is $F(w)... |
2,105,653 | <p>Show that $\left|Im(2+ z^{c} -4z^2) \right| \leq 9.5$ When $ \left| z \right| \leq \frac {3}{2}$</p>
<p>$z^c$= compliment of z</p>
<p>$\left|Im(2+ z^{c} -4z^2) \right| \leq \left| 2+z^{c} - 4z^2 \right|$</p>
<p>I have tried to split it up directly i have tried to force complete the square i always get a weird va... | Siong Thye Goh | 306,553 | <p>Credit: @MartinBladt is right that something is wrong with the question.</p>
<p>Let $z = \frac{\sqrt{5}}{2}+i$</p>
<p>$$|z|=\sqrt{\frac54+1}=\frac32$$</p>
<p>$$|\Im(2+z^c+-4z^2)|=|-\Im (z)-4(2) \Re(z)\Im(z)|=1+8\frac{\sqrt{5}}{2}=1+4\sqrt{5}>9.5$$</p>
<p>Edit:</p>
<p>The following statement is not valid: </p... |
156,585 | <p>I am struggling to evaluate the following integral:<br>
$$\int \frac{1}{(1-x^2)^{3/2}} dx$$<br>
I tried a lot to factorize the expression but I didn't reach the solution.
Please someone help me.</p>
| Listing | 3,123 | <p><strong>Hint</strong>:</p>
<p>Set $x=\sin(t)$, then everything will turn out very well.</p>
<p>This often helps when you have some expresion like $1-x^2$ in your integral.</p>
|
16,080 | <p>I'm having trouble solving problem 12 from Section 1.2 in Hatcher's "Algebraic Topology".</p>
<p>Here's the relevant image for the problem:
<img src="https://i.stack.imgur.com/QNb5W.png" alt="enter image description here"></p>
<p>I'm trying to find $\pi_1(R^3-Z)$, where $Z$ is the graph shown in the first figure. ... | Aaron Mazel-Gee | 2,159 | <p>I remember doing this problem and having similar difficulties, until I realized that it was actually way easier to find $\pi_1(Y)$ than it was to find $\pi_1(\mathbb{R}^3-Z)$. That these are the same is more or less a proof by picture (remember that fundamental group is an invariant of homotopy type). I suggest dr... |
2,906,832 | <p>I want to write $\csc$ and $\tan$ and terms of classical trigonometric functions like $\sin$ and $\cos$. I know about the identity $\sin(x)^2+\cos(x)^2=1$. But I am not sure where to go from here. </p>
| Whoneon | 588,814 | <p>$$ \csc x= \frac{1}{\sin x}$$
or
$$ \csc x= \frac{1}{\pm\sqrt{1-cos^2 x}}$$ </p>
|
1,789,077 | <blockquote>
<p>There is a square <span class="math-container">$Q$</span> consisting of <span class="math-container">$(0,0), (2,0), (0,2), (2,2)$</span>.</p>
<p>A point <span class="math-container">$P$</span> satisfies following condition:</p>
<p>The straight line passing through <span class="math-container">$P$</span>... | user5954246 | 315,298 | <p>using $\sin x+\sin y=2\sin(\frac{x+y}2)\cos(\frac{x-y}2)$ formula we get:</p>
<p>$$\sin3x(\cos x-1)=0$$
$$\implies \sin 3x=0 $$or $$\cos x=1$$</p>
<p>so$$x=n\pi/3$$or$$x=2n\pi$$ where $n\in \mathbb{Z}$. now it is cakewalk to find final equation.</p>
|
3,997,992 | <p>Taken from <a href="https://artofproblemsolving.com/wiki/index.php/1970_Canadian_MO_Problems/Problem_10" rel="nofollow noreferrer">https://artofproblemsolving.com/wiki/index.php/1970_Canadian_MO_Problems/Problem_10</a></p>
<p>Problem <br>
Given the polynomial <span class="math-container">$f(x)=x^n+a_{1}x^{n-1}+a_{2}... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>This is an isosceles triangle, with basis <span class="math-container">$AB$</span>, hence, if <span class="math-container">$I$</span> is the midpoint of <span class="math-container">$[AB]$</span>, its area is equal to <span class="math-container">$\frac12 AB\cdot CI$</span>. Note that <... |
1,871,565 | <p>I am reading about deformation theory. I am treating mostly the algebraic case, but I would like to know a bit about all facets of this field of mathematics, so the geometric case is also of great interest to me. What are good references for the theory of deformations of complex analytic structures on a manifold?</p... | Martin R | 42,969 | <p><em>Hint:</em> Substitute <em>polar coordinates</em>
$$
x = r \cos \varphi, y = r \sin \varphi, dxdy = r dr d\varphi
$$
and use the mean-value property of harmonic functions.</p>
<p>(The boundedness is only needed to ensure that the integral exists.)</p>
|
3,168,381 | <p>Given <span class="math-container">$f(x) = \frac{{4x}}{\sqrt{x}-3}$</span>, what's the domain of <span class="math-container">$g(x) = \frac{{1}}{f(x)}$</span> ?</p>
<p>My textbook includes in the answers <span class="math-container">$x \neq 9$</span>, which I think is erroneous.</p>
| JMoravitz | 179,297 | <p>These "<em>find the domain</em>" style questions are very frequently poorly worded. A better worded question would be:</p>
<p><em>Find the</em> <strong>maximal</strong> <em>subset of <span class="math-container">$\Bbb R$</span> such that the expression is well-defined for all elements in our subset and defines a f... |
4,418,091 | <p>Is <span class="math-container">$\mathbb Q-\mathbb N$</span> dense in <span class="math-container">$\mathbb R$</span>? I believe that the solution is about two relatively prime integers. However, I do not know how to proceed.</p>
| mich95 | 229,072 | <p>You can see that <span class="math-container">$\mathbb{Q}-\mathbb{N}$</span> is dense in <span class="math-container">$\mathbb{Q}$</span>.(If <span class="math-container">$q\in \mathbb{N}$</span>, consider <span class="math-container">$q_n=q+1/(n+1)$</span>.
<span class="math-container">$\mathbb{Q}$</span> is dense ... |
157,301 | <p>Here is the limit I'm trying to find out:</p>
<p><span class="math-container">$$\lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)}$$</span></p>
<p>Since it is an indeterminate form, I simply applied l'Hopital's Rule and I ended up with:</p>
<p><span class="math-container">$$\lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)} = \... | DonAntonio | 31,254 | <p>Another idea using $\,\,\displaystyle{\frac{\sin x}{x}\underset{x\to 0}{\longrightarrow}1\,,\,\cos kx\underset{x\to 0}\longrightarrow 1\,\,(k=\text{a constant})\,\,,\,\sin 2x=2\sin x\cos x}$:</p>
<p>$$\frac{x^3}{\tan^3 2x}=\frac{x^3}{\frac{\sin^32x}{\cos^32x}}=\cos^32x\frac{x^3}{\left(2\sin x\cos x\right)^3}=\frac{... |
454,622 | <p>I am trying to solve a particular probability question. </p>
<p>I have a fair 10-sides die, whose sides are labelled 1 through 10. I am trying to find the probability of rolling a multiple of 5 or an odd number. </p>
<p>I find the probability as: </p>
<p>P(multiple of 5) OR P(odd number)=P(multiple of 5) + P(odd... | André Nicolas | 6,312 | <p>Let $A$ be the event we roll an odd number, and $B$ the event we roll a multiple of $5$, The events $A$ and $B$ <strong>are</strong> independent. For it is clear that the probability of $A\cap B$ is $\frac{1}{10}$, and the product $\Pr(A)\Pr(B)$ is also $\frac{1}{10}$.</p>
<p>However, consider a $9$-sided die. Then... |
60,750 | <p>If we have two finitely generated residually finite groups $G$ and $H$, is there are relation between</p>
<p>the profinite completions $\hat{G},\hat{H}$ and the profinite completion of a semidirect
product $\hat{G \rtimes H}$</p>
<p>(and analogous question for pro-p completions)</p>
| Yiftach Barnea | 5,034 | <p>I convinced myself the Mark's second claim is true. Here is a detailed argument. Let us start by checking whether $\widehat{G} \rtimes \widehat{H}$ actually exists. We assume that <del>both</del> $G$ <del>and $H$ are</del> is finitely generated. Let $\varphi:H \to \textrm{Aut}(G)$ be the map that defines the semidir... |
2,841,102 | <p>I'm asked to find the minimum and maximum values of $f(x, y, z) = x^2+y^2+z^2$ given the constraints $x+2y+z=5$ and $x-y=6$. </p>
<p>I have successfully computed the following:
$x = \frac{57}{11}, y = \frac{-9}{11}, z= \frac{16}{11}$. </p>
<p>I was then able to obtain $f(\frac{57}{11}, \frac{-9}{11}, \frac{16}{11}... | quasi | 400,434 | <p>The two constraints yield the set of points on a line.
<p>
Since a line is unbounded, there is no maximum.</p>
|
4,224,043 | <blockquote>
<p><strong>Question:</strong> Let <span class="math-container">$G$</span> be a matchable graph, and let <span class="math-container">$u$</span> and <span class="math-container">$v$</span> be distinct vertices of <span class="math-container">$G$</span>. Show that <span class="math-container">$G - u - v$</sp... | José Carlos Santos | 446,262 | <p>Suppose that, for some <span class="math-container">$v\in\Bbb R^2\setminus\{(0,0)\}$</span>, there are two distinct numbers <span class="math-container">$a,b\in(0,\infty)$</span> such that <span class="math-container">$av,bv\in\partial C$</span>. I will assume that <span class="math-container">$a<b$</span>. Let <... |
2,992,411 | <p><span class="math-container">\begin{cases}
\frac {dP}{dt} = rP(t)(1-\frac {P(t)}{K}) ,t \geq 0 \\
P(0) = P_o
\end{cases}</span></p>
<p><span class="math-container">$r, K$</span> and <span class="math-container">$P_o$</span> are positive constants.</p>
<p>We say that <span class="math-container">$P(t), t \geq0... | badjohn | 332,763 | <p>Consider the simpler and, naively, smaller set <span class="math-container">$\{0, 1\} ^ \mathbb{N}$</span>. There is a clear near bijection to the interval <span class="math-container">$[0, 1]$</span> by writing the reals in binary. So, this might make it more clear that this set is uncountable and yours also. </... |
17,423 | <p>In most of books on elementary algebra, intermediate algebra and college algebra, the degree of the non-zero polynomial <span class="math-container">$$f(x)=a_nx^n+\cdots a_1x+a_0$$</span> with <span class="math-container">$a_n\neq 0$</span> is defined to be <span class="math-container">$n$</span>. </p>
<p>But I am ... | Benjamin Dickman | 262 | <p>I do not believe that this is a concern that would surface, or be worth surfacing, in the courses named by this question's title. In my reading, the question is analogous to worrying about whether you can ask about e.g. the thousands place of <span class="math-container">$7521$</span>: To do so assumes that the base... |
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