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 |
|---|---|---|---|---|
2,779,458 | <p>I was working on the following definite integral:</p>
<p>$\displaystyle\int^{\infty}_{-\infty}{xe^{-x^{2}}}dx$</p>
<p>I decided to use $u$-substitution to make the integral simpler, by letting $u = -x^{2} \Rightarrow du = - 2x\hspace{1 mm}dx$.</p>
<p>The lower bound then becomes $-(-\infty)^{2}=-\infty$, while th... | Mohammad Riazi-Kermani | 514,496 | <p>Your answer is correct but the method is sloppy. </p>
<p>Note that $$ \displaystyle\int^{\infty}_{-\infty}{xe^{-x^{2}}}dx$$ is defined as $$ \displaystyle\int^{0}_{-\infty}{xe^{-x^{2}}}dx +\displaystyle\int^{\infty}_{0}{xe^{-x^{2}}}dx $$ </p>
<p>Now you apply your substitution and get the same result. </p>
|
38,721 | <p>I encountered a really strange problem with <code>BarLegend</code>.
Say you have a user-defined color function:</p>
<pre><code>cf[z_]:=RGBColor[RandomReal[z, 3]]
</code></pre>
<p>When you want to use it in <code>BarLegend</code> as below:</p>
<pre><code>BarLegend[{cf, {1, 10}}]
</code></pre>
<p>Mathematica will ... | ybeltukov | 4,678 | <p>If you look into the source code</p>
<pre><code>ClearAttributes[BarLegend, ReadProtected]
?? BarLegend
</code></pre>
<p>And into nested functions (<code>BarLegend,Charting`iBarLegend</code> $\to$ <code>Legending`LegendDump`iColorBandLegend</code> $\to$ <code>Legending`LegendDump`parseColorBand</code>) you can find... |
2,523,266 | <p>Find the value of $$\sum_{r=1} ^\infty \frac{r^3+(r^2+1)^2}{(r^4+r^2+1)(r^2+r)}$$
I tried converting the general term into partial fractions but the $$(r^2+1)^2$$ in the numerator is creating problems to do the sum. Please can somebody explain how can I compute the infinite sum</p>
| lhf | 589 | <p><em>Hint for</em> Q3: $1+\sqrt2$ is a unit. If $u$ is a unit, so is $u^n$ for all $n$.</p>
|
2,004,745 | <blockquote>
<p>Find point on the curve $x^2+2y^2=6$ whose distance from the line $x+y-7=0$ is, minimum</p>
</blockquote>
<p>$\bf{My\; Try::}$ Let $(x,y)$ be any point on the curve $x^2+2y^2=6\;,$ Then we have to</p>
<p>minimize $\displaystyle \left|\frac{x+y-7}{\sqrt{2}}\right|$</p>
<p>Using $\bf{Cauchy\; Schwarz... | Robert Z | 299,698 | <p>Hint. Use <a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation" rel="nofollow noreferrer">Stirling approximation</a>:
$$n!\sim\sqrt{2\pi n} \cdot\frac{n^n}{e^n}.$$
which implies that
$$(2n)!\sim2\sqrt{\pi n} \cdot\frac{(2n)^{2n}}{e^{2n}}.$$</p>
|
1,078,493 | <p>How can I prove that for each $n\in\Bbb Z^+$ $$\frac{f_{2n}}{f_{2n-1}}\leq\frac{1+\sqrt{5}}{2}$$ where each $f_i$ is a term of the Fibonacci sequence. </p>
<p>Any help is really appreciated</p>
| Martin Sleziak | 8,297 | <p>To me it seems reasonable to try to prove somewhat stronger claim by induction. (It happens quite often that trying to prove stronger statement might make inductive proof easier.)</p>
<blockquote>
<p>For each $n$ the inequalities $$\frac{F_{2n}}{F_{2n-1}}\le\frac{1+\sqrt5}2 \qquad\text{and}\qquad \frac{F_{2n+1}}{... |
4,257,947 | <p>Edit: Reframed the problem to find an actual answer! Though I may well be talking to myself at this point. Answer below the original problem.</p>
<hr />
<p>The original context of this question is a ridiculous, brainless mobile game. But the geometric question that arises seems quite interesting--I suspect it has al... | Ivan Kaznacheyeu | 955,514 | <p>Let <span class="math-container">$A=(a/2;b/2)$</span>, <span class="math-container">$B=(-a/2;b/2)$</span>, <span class="math-container">$C=(-a/2;-b/2)$</span>, <span class="math-container">$D=(a/2;-b/2$</span>.</p>
<p>Consider rectangle <span class="math-container">$(x,y;x+dx,y+dy)$</span>, taking for simplicity (be... |
4,412,423 | <p>Hello I was trying to prove</p>
<p><span class="math-container">$$\sum_{0 \leq k \leq n}^{} \binom{n-k}{k} = \operatorname{Fib}_{n+1}$$</span>
which the approach I made was</p>
<ol>
<li>Prove that it holds for <span class="math-container">$n=0$</span>, <span class="math-container">$n=1$</span>.</li>
<li>Then use the... | Kajelad | 354,840 | <p>The notation <span class="math-container">$R(X,Y)=[\nabla_X,\nabla_Y]$</span> is somewhat vague. In particular, <span class="math-container">$\nabla_X$</span> refers to a family of differential operators (since covariant derivatives can be applied to all kinds of objects), and the definition of the Riemann tensor ha... |
4,412,423 | <p>Hello I was trying to prove</p>
<p><span class="math-container">$$\sum_{0 \leq k \leq n}^{} \binom{n-k}{k} = \operatorname{Fib}_{n+1}$$</span>
which the approach I made was</p>
<ol>
<li>Prove that it holds for <span class="math-container">$n=0$</span>, <span class="math-container">$n=1$</span>.</li>
<li>Then use the... | J.G. | 56,861 | <p>Since you're not even convinced it has to be a true tensor, we'll work throughout with covariant derivatives, which by definition map tensors to tensors. In fact, it's generally a good idea to use covariant derivatives as much as you can early in a calculation like this; that way you'll need use Christoffel symbols ... |
2,424,061 | <p>A question asks "Is $f(x)=x^2$ an onto function or not?" Here the domain and codomain are not mentioned specifically, so what codomain should I consider?</p>
| Ross Millikan | 1,827 | <p>In a case like this, I would assume the domain and codomain are intended to be the real numbers unless you are taking a complex analysis class.</p>
|
1,684,045 | <p>Let $0<2x<\pi$. If $$\sum_{n=-2}^{\infty}\cos^n x=8,$$ then please find $x.$ </p>
<p>I tried $\sum_{n=-2}^{\infty}\cos^n x=\frac{1}{1-\cos x}+\frac{1}{\cos x}+\frac{1}{\cos^2 x}=\frac{1+\cos x}{\cos^2 x \sin^2 x}=8.$ But I cant find $x.$</p>
| Claudeh5 | 113,848 | <p>Substituting y to cos(x), we have $\sum_{n=-2}^\infty y^n=-\frac{1}{y^2(y-1)}=8.$</p>
<p>The solutions are $1/2$, $\frac{1-\sqrt{5}}{4}$, $\frac{1+\sqrt{5}}{4}$</p>
<p>Then we have $\cos(x)=1/2$ so $x = \pi/3+2k\pi$ or $x=-\pi/3+2k\pi$ ($k \in \mathbb{Z}$)</p>
<p>the two other solutions give four values for x.</p... |
51,301 | <p>With <code>n = 17</code> I would like to get </p>
<blockquote>
<p>{4, 1}</p>
</blockquote>
<p>and with <code>n = 999</code></p>
<blockquote>
<p>{31, 6, 1, 1}</p>
</blockquote>
<p>so that, for example,</p>
<pre><code>Total[#^2 & /@ {31, 6, 1, 1}]
</code></pre>
<p>gives</p>
<blockquote>
<p>999</p>
</b... | ciao | 11,467 | <pre><code>Module[{n = 1, results},
While[(results = Sqrt[IntegerPartitions[#, {n}, Range@Floor@Sqrt[#]^2]]) == {},
n++]; results] &[999]
(*
{{31, 6, 1, 1}, {31, 5, 3, 2}, {30, 9, 3, 3}, {30, 7, 7, 1}, {30, 7,
5, 5}, {29, 11, 6, 1}, {29, 10, 7, 3}, {27, 15, 6, 3}, {27, 14, 7,
5}, {27, 13, 10, 1}, {2... |
51,301 | <p>With <code>n = 17</code> I would like to get </p>
<blockquote>
<p>{4, 1}</p>
</blockquote>
<p>and with <code>n = 999</code></p>
<blockquote>
<p>{31, 6, 1, 1}</p>
</blockquote>
<p>so that, for example,</p>
<pre><code>Total[#^2 & /@ {31, 6, 1, 1}]
</code></pre>
<p>gives</p>
<blockquote>
<p>999</p>
</b... | halirutan | 187 | <p>Here is another one:</p>
<pre><code>squarePartition[n_] := squarePartition[n, 1, Table[k^2, {k, 1, Floor[Sqrt[n]]}]];
squarePartition[n_, l_, squares_] :=
Module[{result = IntegerPartitions[n, {l}, squares]},
Sqrt[First[result]] /; result =!= {}
];
squarePartition[n_, l_, squares_] := squarePartition[n, l ... |
51,301 | <p>With <code>n = 17</code> I would like to get </p>
<blockquote>
<p>{4, 1}</p>
</blockquote>
<p>and with <code>n = 999</code></p>
<blockquote>
<p>{31, 6, 1, 1}</p>
</blockquote>
<p>so that, for example,</p>
<pre><code>Total[#^2 & /@ {31, 6, 1, 1}]
</code></pre>
<p>gives</p>
<blockquote>
<p>999</p>
</b... | Basheer Algohi | 13,548 | <pre><code>k = 1;
m[1] = 999;
While[m[k] =!= 0, n[k] = Floor@Sqrt[m[k]]; k = k + 1;
m[k] = m[k - 1] - (Floor@Sqrt[m[k - 1]])^2];
sol = n[#] & /@ Range[1, k - 1]
(* {31, 6, 1, 1} *)
</code></pre>
|
2,824,996 | <p>I'm investigating iterations using Newton's method for $f(x)=x^5-x+1$.</p>
<p>I'm getting unusual results though. I've found that some starting values will result with a "triple oscillation" of results where iterations will loop between values of approximately $1.0003$, $0.7504$ and $-0.0871$. </p>
<p>However the ... | egreg | 62,967 | <p>Let's first suppose $x>0$. Then we can transform the equation into
$$
x=100\log x
$$
and we can consider the function $f(x)=x-100\log x$. Clearly
$$
\lim_{x\to0}f(x)=\lim_{x\to\infty}f(x)=\infty
$$
and we have
$$
f'(x)=1-100/x
$$
which only vanishes at $x=100$. Therefore we know that $x=100$ is a point of minimum... |
1,566,097 | <p>Prove that the function defined by </p>
<p>$$f(x)=\sin\left(\frac{1}{x}\right)$$</p>
<p>is not uniformly continuous on the interval $(0,1)$.</p>
| Pedro | 23,350 | <p>A uniformly continuous function sends cofinal sequences to cofinal sequences, that is, if $|x_n-y_n|\to 0$ and $f$ is uniformly continuous, then $|f(x_n)-f(y_n)|\to 0$. You should be able to prove this using the definition of uniformly continuous function. This doesn't happen with $\sin(x^{-1})$ over $(0,1)$, for th... |
4,451,700 | <blockquote>
<p>Calculate the line integral
<span class="math-container">$$
\int_{L} \frac{-y \,d x+x \,d y}{x^{2}+y^{2}}
$$</span>
where <span class="math-container">$L$</span> is the line segment from <span class="math-container">$(1,0)$</span> to <span class="math-container">$(0,1)$</span> parametrized by
<span clas... | Community | -1 | <p>Consider the quarter arc: <span class="math-container">$\gamma:[0,\pi/2]\to \mathbf{R}^2$</span> with
<span class="math-container">$$
\gamma(t) = (\cos(t),\sin(t))
$$</span>
with the direction from <span class="math-container">$\gamma(0)$</span> to <span class="math-container">$\gamma(1)$</span>. Then <span class="m... |
1,631 | <p>What's a good way to format a citation to a MathOverflow post in a paper?
Here is a first draft (with hyperlink, and assuming it's reference number 16, say), but maybe people can think of improvements:</p>
<blockquote>
<p>[16] Pietro Majer, answer of April 8, 2014, 17:26pm HST to <em>MathOverflow</em> question <... | François G. Dorais | 2,000 | <p>Users of the <a href="http://www.ctan.org/pkg/biblatex" rel="nofollow noreferrer">biblatex</a> package can use an @online entry type. From the <a href="http://ctan.math.washington.edu/tex-archive/macros/latex/contrib/biblatex/doc/biblatex.pdf" rel="nofollow noreferrer">biblatex documentation</a>:</p>
<blockquote>
... |
44,740 | <p>This question is closely related to MO <a href="https://mathoverflow.net/questions/4347/ffxexpx-1-and-other-functions-just-in-the-middle-between-linear-and-expo">f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential.</a> Consider $e^{e^x-1}$, this is the generating function of the ... | Gottfried Helms | 7,710 | <p><em>A late rereading of the question, related to question 2 ...</em> </p>
<p>Here I provide example-data for the Nörlund-summation of $g^{0.5}(1)$ - a function whose power series has convergence radius zero. </p>
<p>I document the index of the coefficients, the coefficients of the formal power series, the... |
886,337 | <p>I need to find $$\int x\sqrt{(a^2 - x^2)}dx$$</p>
<p>I tried putting $x=a cos(t)$ but I ended up getting a very complicated expression, so any tips?</p>
| Rene Schipperus | 149,912 | <p>Let $y=a^2-x^2$ then $dy=-2xdx$ so
$$-\frac{1}{2}\int \sqrt{a^2-x^2} (-2x)dx=-\frac{1}{2}\int \sqrt{y} dy
=-\frac{1}{3}y^\frac{3}{2}= =-\frac{1}{3}(a^2-x^2)^\frac{3}{2}$$</p>
|
1,552,992 | <p>Let $G$ be abelian. Let $n \in \mathbb{Z}^+$ such that the order of $G$ and $n$ are relatively prime. Show that the function $\varphi : G \rightarrow G$ defined by $\varphi (a) = a^n $ is an automorphism for $a \in G$.</p>
<p>I'm stuck in proving 1-1-ness of the function. Let $a,b$ in $G$ such that $\varphi (x) = \... | rwols | 92,883 | <p>I assume you have shown that $\phi$ is a homomorphism.</p>
<p>Showing that a homomorphism is injective is equivalent to showing that its kernel is trivial.</p>
<p>Let $a \in \ker \phi$. We want to show that $a=1$. Let $g$ be the order of $G$ and let $k$ be the order of $a$. By Lagrange's theorem, $k \mid g$. Since... |
4,583,108 | <p>Let <span class="math-container">$M=\langle S\mid R\rangle$</span> be a presentation for a monoid. <span class="math-container">$R=\{a_1=b_1,...\}$</span> is a collection of equalities, and <span class="math-container">$M$</span> is defined as the set of equivalence classes under the the reflexive and transitive clo... | Mark Saving | 798,694 | <p>Consider some such <span class="math-container">$X, R$</span>, and consider some monoid/group <span class="math-container">$G$</span> and some inclusion function <span class="math-container">$i : X \to G$</span>. We say <span class="math-container">$i$</span> comports with <span class="math-container">$R$</span> if ... |
597,638 | <p>I need to evaluate this integral using calculus of residues: </p>
<p>$$\int_0^\infty\frac{\ln(x)}{1+x^4}\mathrm{d}x$$</p>
<p>I know I need to consider $\displaystyle \int_0^\infty$$\frac{\ln(z)}{1+z^4}\mathrm{d}z$.<br>
Then the integrand has singularities at at $-e^{i\pi/4}$, $e^{i\pi/4}$, $-e^{3i\pi/4}$, and $e^{... | xpaul | 66,420 | <p>From the link <a href="https://math.stackexchange.com/questions/269081/calculating-the-integral-int-0-infty-frac-ln-x1xn-using-complex-an/887450#887450">Calculating the integral $\int_0^{\infty}{\frac{\ln x}{1+x^n}}$ using complex analysis</a>, we have for $n=4$
$$ \int_0^\infty\frac{\ln(x)}{1+x^4}\mathrm{d}x=-\frac... |
533,194 | <p>Do you know how to solve this problem? I have two sets and need to solve $A \cup X = B. $</p>
<p>Thanks a lot for your help.</p>
| Did | 6,179 | <p>If $A\cup X=B$ has some solution $X$ then $A\subseteq B$. </p>
<p>If $A\subseteq B$ then $A\cup X=B$ if and only if there exists $C\subseteq A$ such that $X=C\cup(B\setminus A)$.</p>
|
1,514,028 | <p>It's pretty obvious to me as I see the plot of the two functions but how can I prove it with some algebra?</p>
| Michael Hardy | 11,667 | <p>Use the mean value theorem:</p>
<p>There is some number $c$ between $x$ and $0$ for which
$$
\sin' c = \frac{\sin x - \sin 0}{x - 0}.
$$
Since $|\sin'c|=|\cos c|\le 1$, we have
$$
\left|\frac{\sin x} x \right| \le 1,
$$
so
$$
|\sin x| \le |x|.
$$
There is no need to assume $x$ is in $[-1,0]$ to get this conclusion.... |
2,881,871 | <p>Is it true that if $N/M$ is a normal subgroup of $G/M$ then $N$ is a normal subgroup of $G$ itself?</p>
<p>I would think that not necessarily because I would expect that under conjugation we could get a subgroup of $G$ isomorphic to $N$ but maybe not $N$ it self but a subgroup identical from $N$ except its elements... | Caleb Stanford | 68,107 | <p>Yes, this appears to be true. Consider the map $f: G \to (G/M)/(N/M)$ that sends $g$ to its equivalence class mod $M$, and then sends the result to its equivalence class mod $N/M$. That is, this is just the projection map from quotienting by $M$ and then by $(N/M)$.</p>
<p>We argue that $N = \ker f$: thus, $N$ is n... |
13,877 | <p><a href="http://my.unix-center.net/~martian/math%20album/%E6%95%B0%E5%AD%A6%E4%B8%93%E8%BE%91%202%20-%20%E5%86%B0%E5%B2%9B%E7%81%AB%E5%B1%B1/%E7%9B%B4%E5%B8%83%E7%BD%97%E9%99%80%E7%9A%84%E5%B7%A8%E5%B2%A9%20%28On%20twin%20prime%29.html" rel="nofollow">a sequence on twin primes</a> In the diagram, How do the stripes ... | Bill Dubuque | 242 | <p><strong>HINT</strong> $\ \ $ Show the sequence of Current values has form $\rm\ 1,\ 1,\ 2,\ \cdots\ a,\ b,\ a+b$</p>
|
3,048,011 | <p>I am learning linear algebra and new to it. I can not solve this problem. I think it has a trick that I don't know.</p>
<blockquote>
<p>for T(a linear map), <span class="math-container">$T:V\rightarrow V$</span> and every <span class="math-container">$v$</span> in <span class="math-container">$V$</span> the <span... | Tsemo Aristide | 280,301 | <p>Suppose <span class="math-container">$a_1T+...+a_kT^k=0$</span>. Let <span class="math-container">$x\neq 0$</span>, <span class="math-container">$a_1T(x)+...+a_kT^k(x)=0$</span>, this implies that <span class="math-container">$a_1=..=a_k=0$</span> since <span class="math-container">$T(x),...T^k(x)$</span> are linear... |
1,977,081 | <p>You play a game using a standard six-sided die. You start with 0 points. Before every roll, you decide whether you want to continue the game or end it and keep your points. After each roll, if you rolled 6, then you lose everything and the game ends. Otherwise, add the score from the die to your total points and con... | Community | -1 | <p>I'm going to tackle the problem of trying to maximize the <em>average</em> score of a strategy, with the disclaimer that this is often not what you <em>actually</em> care about when playing actual games.</p>
<p>Often, the trick to the math is to try and express the result algebraically in terms of simpler (and hope... |
1,977,081 | <p>You play a game using a standard six-sided die. You start with 0 points. Before every roll, you decide whether you want to continue the game or end it and keep your points. After each roll, if you rolled 6, then you lose everything and the game ends. Otherwise, add the score from the die to your total points and con... | Chris Johns | 333,864 | <p>The mathematics of this is actually very simple, every time you roll you have a 1 in 6 chance of losing regardless of previous outcomes so the optimal strategy depends on information not covered by the question and ultimately depends on your ability to judge when your score is adequate. </p>
<p>There are two obvio... |
2,989,607 | <blockquote>
<p>Christina is standing in a rectangular garden. Her distances from the corners of the garden are <span class="math-container">$6$</span> meters, <span class="math-container">$7$</span> meters, <span class="math-container">$9$</span> meters, and <span class="math-container">$d$</span> meters, where <spa... | Batominovski | 72,152 | <p>Let <span class="math-container">$P$</span> be a point inside a rectangle <span class="math-container">$XYZW$</span> such that <span class="math-container">$\{PX,PY,PZ\}=\{6,7,9\}$</span> and <span class="math-container">$PW=d$</span> is an integer. Suppose that the projections of <span class="math-container">$P$</... |
811,490 | <p>Two points A and B are on different sides of a line. Find a point Y on the line such that the absolute value of the difference from Y to A and Y to B is maximal.</p>
<p>My thoughts are as follows. Let's say that point Y is on a line but is not collinear with A and B. Then I could draw a triangle by joining the 3... | fahrbach | 39,901 | <p><strong>Hint:</strong> Find the function that yields the difference in the distances from $A$ to $Y$ and $B$ to $Y$ using the distance formula. This function will be of one variable since we know $A$, $B$, and the equation for the line. Then find the minimum or maximum of this function since we're looking for the gr... |
4,023,679 | <p>So my understanding is that a delta function <span class="math-container">$\delta(t)$</span> has the value of <span class="math-container">$\infty$</span> at <span class="math-container">$t = 0$</span> and <span class="math-container">$\delta(t) = 0$</span> everywhere else. Essentially, it's just one big spike on a ... | José Carlos Santos | 446,262 | <p>Since<span class="math-container">$$\lim_{n\to\infty}\frac{\left|\frac z{(z-n)n}\right|}{\frac1{n^2}}=\lim_{n\to\infty}\left|\frac{n^2z}{nz-n^2}\right|=|z|,$$</span>then, since the series <span class="math-container">$\sum_{n=1}^\infty\frac1{n^2}$</span> converges, your series converges absolutely.</p>
|
3,424,899 | <p>Let be vetors <span class="math-container">$(1, 2, 3,−1)$</span> and <span class="math-container">$(3, 6, 9,−3)$</span> a solutions of some system of linear equations with factors from <span class="math-container">$R$</span>. Prove that the vector <span class="math-container">$(0, 0, 0, 0)$</span> is also a solutio... | M.P | 441,344 | <p>Your approach is all correct. Note that your sum can be rewritten as <span class="math-container">$$\frac{1}{2}\sum_{n=0}^\infty i^n ((-1)^n+1)z^n.$$</span></p>
<p>For an odd index value <span class="math-container">$n$</span>, the term equals <strong>zero</strong>, and for even - it equals <span class="math-contai... |
2,069,201 | <p>let $a,b,c\ge 0$ and such $a+b+c=3$ show that
$$(a^3+1)(b^3+1)(c^3+1)\ge 8$$</p>
<p><strong>My research</strong>:It seem use Holder inequality,so
$$(a^3+1)(b^3+1)(c^3+1)\ge (abc+1)^3$$
Use AM-GM
$$abc\le\dfrac{(a+b+c)^3}{27}=1$$
what? then I think this method is wrong</p>
| David Bowman | 366,588 | <p>Let $f(x,y,z) = (x^3 +1)(y^3 +1)(z^3 +1)$ subject to the constraint $g(x,y,z) = x + y + z - 3 =0$ and $ x,y,z \ge 0$. </p>
<p>Then using lagrange multipliers we have</p>
<ul>
<li>$f_x = 3x^2(y^3 +1)(z^3 +1)= g_x= \lambda$</li>
<li>$f_y = 3y^2(x^3 +1)(z^3 +1)= g_y = \lambda$</li>
<li>$f_z = 3z^2(x^3 +1)(y^3 +1)=g_z... |
2,069,201 | <p>let $a,b,c\ge 0$ and such $a+b+c=3$ show that
$$(a^3+1)(b^3+1)(c^3+1)\ge 8$$</p>
<p><strong>My research</strong>:It seem use Holder inequality,so
$$(a^3+1)(b^3+1)(c^3+1)\ge (abc+1)^3$$
Use AM-GM
$$abc\le\dfrac{(a+b+c)^3}{27}=1$$
what? then I think this method is wrong</p>
| Guy Fsone | 385,707 | <p>BY Jensen inequality with $\phi\left(t\right)=\ln(t^3+1)$</p>
<p>$$\phi\left(\frac13\sum_{i=1}^{3}x_i\right)\le \frac13\sum_{i=1}^{3}\phi\left(x_i\right) $$</p>
<p>since $a+b+c= 3$ we get
$$\ln(2)=\phi\left(1\right)\le \frac13\left(\ln(a^3+1)+\ln(b^3+1)+\ln(c^3+1)\right) $$</p>
<p>hence $$\ln8= 3\ln(2)\le \ln \... |
2,483,254 | <p>I have to compute the limit</p>
<p>$$
\lim\limits_{N \to +\infty} \sqrt{N+1} \log \left(1+\frac{x}{N+1}\right)
$$
where $x \ge 0$ is fixed. I tried to see the previous as</p>
<p>$$
\log \lim\limits_{N \to +\infty} \left(1+\frac{x}{N+1}\right)^{\sqrt{N+1}}
$$
and to change variable, but it doesn't work. Intuitively... | avz2611 | 142,634 | <p>hint: use taylor expansion for the Log term</p>
|
879,051 | <p>How do topologists prove continuity of a function with the usual topology at the endpoints of a closed interval? For instance, how would a topologist prove continuity for $f(x)=x^2$ on the closed interval $[0,1]$ at the point $x = 1$ using open sets (not using calculus)?</p>
| Conifold | 152,568 | <p>Same way you would do it on an open interval. You have to show that inverse images of open sets are open, for $f(x)=x$ the inverse image of any set is the set itself, so it's true. It is only slightly harder for $f(x)=x^2$, but the inverse image of $(a,1]$ say, is $(\sqrt{a},1]$ and both are open (it's enough to con... |
3,237,995 | <p>Let <span class="math-container">$\Bbb L$</span> be a positive integer and define <span class="math-container">$\Bbb Z_L$</span> and <span class="math-container">$\Bbb Z^*_L$</span> as <span class="math-container">$$\Bbb Z_L=\{ 0,1,...,L-1\},\qquad\Bbb Z^*_L=\{i\in \Bbb Z_L\mid \gcd(i,L)=1\}$$</span> where <span cla... | Bernard | 202,857 | <p>This is true because of <em>Bézout's identity</em>: if <span class="math-container">$\gcd(i,L)=1$</span>, there exist intergers <span class="math-container">$u,v$</span> such that
<span class="math-container">$$ui+vL=1,$$</span>
hence, denoting <span class="math-container">$[\,.]$</span> the congruence classes modul... |
4,458,244 | <p>To factorize <span class="math-container">$xy\left(x+y\right)+yz\left(y-z\right)-xz\left(x+z\right)$</span> I used the fact that <span class="math-container">$x=-y$</span> and <span class="math-container">$y=z$</span> and <span class="math-container">$x=-z$</span> make the expression zero. Hence it factorize to <spa... | User | 1,044,391 | <p>You can observe that, if <span class="math-container">$x=-y$</span> or (not "and") <span class="math-container">$x=-z$</span> or (not "and") <span class="math-container">$y=z$</span> , then the expression equals to zero. Thus, we have</p>
<p><span class="math-container">$$xy(x+y)+yz(y-z)-xz(x+z)=... |
3,051,737 | <ul>
<li>Show that If <span class="math-container">$X$</span> is an any set, then <span class="math-container">$X\subset X\cup\left\{ X\right\}$</span></li>
</ul>
<p>Proof. Let <span class="math-container">$t\in X$</span>. We must show <span class="math-container">$t\in X\cup\left\{ X\right\}$</span>, that is we need ... | YoungMath | 570,404 | <p>Apparently, an answer on ground level of set theory is needed here. Since the inclusion <span class="math-container">$X \subseteq X \cup \{X\}$</span> is already discussed in full detail, I want to make a remark about the case <span class="math-container">$X \neq X \cup \{X \}$</span>.</p>
<p>The <em>Axiom of Regul... |
477,834 | <p>What is the transformation such that a general cubic polynomial to be made a cube,</p>
<p>$$ax^3+bx^2+cx+d = y^3\tag{1}$$</p>
<p>can be transformed to Weierstrass form,</p>
<p>$$x^3+Ax+B = t^2\tag{2}$$</p>
<p>(The special case $b = c = 0$ is easier.) Given an initial rational point, I know a method how to find s... | Jyrki Lahtonen | 11,619 | <p>The point $P=(x,y)=(-1,0)$ is a point of inflection in the sense that the line $x=-1$ makes 3-fold contact there. This makes it a prime candidate to be moved to the point at infinity. Recalling that in Weierstrass form the two coordinates have poles of orders $2$ and $3$ respectively at the point of infinity (follow... |
4,022,066 | <p>I don't quite understand how <span class="math-container">$\dfrac1 x$</span> is unbounded while <span class="math-container">$\dfrac{x}{1+x^2}$</span> is not.</p>
| Alan | 175,602 | <p>Look at where <span class="math-container">$\dfrac 1 x$</span> goes to infinity, what causes that unbounded behavior? Notice that that is avoided in the second function.</p>
|
4,516,512 | <p>The finite subgroups of <span class="math-container">$\mathrm{GL}_2(\mathbb{C})$</span> are pretty much classified in several expository papers in the web as well as posts on this site. However, I would like to know that how does one show that the Heisenberg group <span class="math-container">$H_3$</span> cannot be ... | spin | 12,623 | <p>Here is another way:
If <span class="math-container">$\varphi: H_3 \rightarrow GL_2(\mathbb{C})$</span> is faithful, it must be irreducible. (Otherwise the representation would decomposable into a direct sum of two <span class="math-container">$1$</span>-dimensional representations, in which case <span class="math-c... |
2,678,895 | <p>I was wondering how to map the set $\mathbb{Z}^+$ to the sequence $1, 2, 3, 1, 2, 3, \ldots$. I thought it would be easy, but I was only able to obtain an answer through trial and error.</p>
<p>For a function $f \colon \mathbb{Z}^+ \rightarrow \mathbb{Z}$, we have that</p>
<p>$f(x) = x \bmod 3$ gives the numbers $... | r.e.s. | 16,397 | <p>$f(x)=\left\lfloor 10^x\cdot {123\over999}\right\rfloor\mod 10$.</p>
<p>More generally, with $b>n>1$,
$$f(x)=\left\lfloor b^x\cdot {(123\ldots n)_b\over b^n-1}\right\rfloor\mod b.
$$</p>
|
2,678,895 | <p>I was wondering how to map the set $\mathbb{Z}^+$ to the sequence $1, 2, 3, 1, 2, 3, \ldots$. I thought it would be easy, but I was only able to obtain an answer through trial and error.</p>
<p>For a function $f \colon \mathbb{Z}^+ \rightarrow \mathbb{Z}$, we have that</p>
<p>$f(x) = x \bmod 3$ gives the numbers $... | Acccumulation | 476,070 | <p>In group theory, the modulus function is often taken to be a function from <strong>Z</strong> to <strong>Z</strong>/n<strong>Z</strong>. That is, the output is not an integer, but the equivalence class of integers that have the same remainder when divided by n. E.g. 1->{...-5,-2,1,4,7...}. In computer science, on th... |
2,590,359 | <p>I have to find a distance between a plane represented with a triangle and a point in $3D$ space. I searched other questions but the plane is usually represented by a plane equation or a normal, so I am having trouble finding this distance when the plane is represented by three points.</p>
| user | 505,767 | <p><strong>HINT</strong></p>
<p>You can find the normal vector by <strong>cross product</strong> and then apply the formula for the distance between plane and point.</p>
|
3,615,874 | <p>I'm trying to solve <span class="math-container">$\frac{d}{dx}e^{x}$</span>: I know <span class="math-container">$\ln(e^{x})=x$</span>, so <span class="math-container">$$(A)\frac{d}{dx}\ln(e^{x})=\frac{d}{dx}x=1$$</span>I can also apply the chain rule : <span class="math-container">$$(B)\frac{d}{dx}\ln(e^{x})=\frac{... | Community | -1 | <p><span class="math-container">$$|x^4\sin^2(1/x)|<|x^4|=x^4<\varepsilon$$</span></p>
|
3,615,874 | <p>I'm trying to solve <span class="math-container">$\frac{d}{dx}e^{x}$</span>: I know <span class="math-container">$\ln(e^{x})=x$</span>, so <span class="math-container">$$(A)\frac{d}{dx}\ln(e^{x})=\frac{d}{dx}x=1$$</span>I can also apply the chain rule : <span class="math-container">$$(B)\frac{d}{dx}\ln(e^{x})=\frac{... | Manjoy Das | 568,656 | <p>Let <span class="math-container">$\ f(x)$</span>= <span class="math-container">$\ x^{4} sin^{2}(\frac {1}{x})$</span>. Then <span class="math-container">$$\ |f(x)-0|=|x^{4} sin^{2}(\frac {1}{x})|$$</span> <span class="math-container">$$\leq |x^{4}|(since \ |sin^{2}(\frac {1}{x})| \leq 1)$$</span> <span class="math-... |
83,507 | <p>Is there any shorter / more handy / more flexible way to get directory tree as a nested <code>Rules/Associations</code> list?</p>
<p>I tried to incorporate <code>FileNames</code> + <code>GatherBy</code> and <code>GroupBy</code>, but nothing was as short and handy as old fashioned:</p>
<pre><code>ClearAll @ filesIn... | C. E. | 731 | <p>Leonid Shifrin's <a href="https://mathematica.stackexchange.com/a/447/731">implementation of a trie</a> can be used for this purpose. It is available both for lists and for associations. For associations:</p>
<pre><code>ClearAll[makeTreeAssoc];
makeTreeAssoc[wrds : {__String}] := Association@makeTreeAssoc[FileNameS... |
83,507 | <p>Is there any shorter / more handy / more flexible way to get directory tree as a nested <code>Rules/Associations</code> list?</p>
<p>I tried to incorporate <code>FileNames</code> + <code>GatherBy</code> and <code>GroupBy</code>, but nothing was as short and handy as old fashioned:</p>
<pre><code>ClearAll @ filesIn... | alancalvitti | 801 | <p>Try this recursive query which is linear-time in input (timing study <a href="https://mathematica.stackexchange.com/questions/69942/10-0-2-breaks-a-recursive-trie-query">here</a>)</p>
<pre><code>trie = Query[GroupBy@First, If[First@# == {}, {}, trie@#] &, Rest];
</code></pre>
<p>Variations on the above are pos... |
1,835,519 | <p>If $V$ is a subspace of an Hilbert space $H$, I know that the orthogonal of $V$, $V$$^o$, is ($V$closed)$^o$, even if $V$ is not closed.<br>
Does this mean that $V$ is always dense in $V$$^o$? </p>
<p>Thanks!</p>
| tomi | 215,986 | <p>I guess you want a function (polynomial) for trhe radius of the vase at different points so that you can work out the surface area to determine the rate of evaporation...?</p>
<p>You can do this with geogebra. Look at this <a href="http://mrhonner.com/archives/3923" rel="nofollow">http://mrhonner.com/archives/3923<... |
2,525,055 | <p>How can the differential of $\mathbf r\cdot\mathbf r$ be written as $2\mathbf r\cdot d\mathbf r$? That is,</p>
<p>$$d(\mathbf r\cdot\mathbf r)=2\mathbf r\cdot d\mathbf r$$</p>
<p>How? Here, $\mathbf r$ is a vector.</p>
| AlkaKadri | 258,038 | <p>The first thing to realize is that you can differentiate vector functions. In single variable calculus, we are interested in functions that have $1$ "input" and $1$ output. I see that now you're likely in a multivariate calculus course, although you're still only interested in functions that have $1$ input, function... |
1,503,369 | <p>Let <span class="math-container">$n$</span> be a Von Nemuan natural number, i.e. the set of all natural numbers less then <span class="math-container">$n$</span>.
Why it holds that <span class="math-container">$\mathscr{P}^-(n)\overset{def}{=}\{u:u\subseteq \{0,...,n-1\}\}$</span>
<span class="math-container">$=\mat... | Crostul | 160,300 | <p>For all $i$, let $B_i$ be a basis of $U_i$. Since $U = \sum_i U_i$, you have that
$\cup_i B_i =B$ generates $U$. Moreover, by your assumption, you have
$$|\cup_i B_i| \le \sum_i |B_i| = \sum_i \dim U_i = \dim U \le |\cup_i B_i|$$
hence equality holds. This means that $B$ is a basis of $U$, and that all the $B_i$s ar... |
2,485,325 | <p>Let $a$ and $b$ be two positive integers such that gcd$(a,b)=1$. Prove that there exists $(x,y)\in\mathbb{Z}\times \mathbb{Z}$ satisfying </p>
<p>(1) $ax-by=1$</p>
<p>(2) $0<x<b$ and $0<y<a$.</p>
<p>I know (1) is by the fact that gcd$(a,b)=1$ but I don't know how to prove (2). Any hint? </p>
| G Tony Jacobs | 92,129 | <p>First of all, if $a$ or $b$ equals $1$, then the claim is false, so suppose they are both greater than $1$.</p>
<p>Suppose $(x_0,y_0)$ is one solution. Then we also have that $(x_0+kb,y_0+ka)$ is a solution for any integer $k$. (Do you see why?)</p>
<p>Choose the value of $k$ that makes $x=x_0+kb$ as small as poss... |
1,349,720 | <p>Say we have two irreducible polynomials in $Q [x] $. We call them $f, g$. Say one of the roots of $f $ is $a$. Is it possible that $g$ satisfies a root of the form $a^n$ for some natural number $n $? </p>
<p>Thanks </p>
| Khadija Mbarki | 247,868 | <p>Take $f(x)=x-a$ and for a fixed naturel number $n,$ $g(x)=x-a^n.$ The two polynomials are irreducibles in $Q[x]$ and the root of $f$ is $a$ and for $g$ is $a^n.$ </p>
|
4,529,873 | <p>I am currently dealing with the two terms "Gateaux-derivation" and "Frechet-derivation" and would like to know if there is a Frechet differentiable function <span class="math-container">$f$</span> and Gateaux differentiable function <span class="math-container">$g$</span>, so that <span class="ma... | FuncAna09 | 588,212 | <p>Hello @grew you are right. Here are the calculations for the Gateaux derivative of the function</p>
<p><span class="math-container">$$f(x,y)=
\begin{cases} \exp\left(-\frac{(y-x^2)^2}{x^6}\right)&\text{for}~x\neq 0,\\ 0&\text{o.w.}\end{cases}$$</span>
in the point <span class="math-container">$(x_0,y_0)=(0,0... |
3,118,410 | <p>I've been trying to use the change of base property but I'm not having much luck. Can anyone give me any ideas on how I should approach this problem? The answer is 49/5</p>
<p>Thanks.</p>
| twnly | 587,142 | <p>Using change-of-bases, we have <span class="math-container">$\log_x(5x) \cdot \log_7(x)= \frac{\log (5x)}{\log (x)} \cdot \frac{\log (x)}{\log (7)} = \frac{\log (5x)}{\log 7} = \log_7(5x)= 2$</span>, where the base of the intermediate <span class="math-container">$\log$</span>s do not really matter.</p>
|
4,179,661 | <p><strong>Question:</strong> Prove that <span class="math-container">$p(x^q-1) + q(x^{-p}-1) \geq 0$</span> for <span class="math-container">$x > 0$</span> and <span class="math-container">$p,q \in \mathbb{Z}^+$</span>.</p>
<p><strong>My attempts:</strong> I initially thought to make the expression <span class="mat... | Q9y5 | 891,345 | <p>Continue with what you have tried. For integers, <span class="math-container">$q<p$</span> implies <span class="math-container">$q\leq p-1$</span>, therefore <span class="math-container">$q\equiv 1\left(\text{mod }p-1\right)$</span> means either <span class="math-container">$q=1$</span> or <span class="math-conta... |
750,468 | <p>Show that if $f: U \to \mathbb{C}$ is analytic on an open set $U$ in $\mathbb{C}$, then the Cauchy Riemann equations for $f$ hold in $U$. </p>
<p>My question is, how do the Cauchy-Riemann equations differ on an open set $U$ versus a function $f$ on $\mathbb{C}$? Not sure how to prove this.</p>
| drhab | 75,923 | <p>This is not a real answer but more an alternative proof. If it is useless in this context then please let me know. In that case I will delete the answer.</p>
<p>If $g\circ f=id$ then $g$ is surjective since $g\left(f\left(a\right)\right)=a$
for each $a\in A$. So if $g$ is injective as well then it is bijective
(i.e... |
918,306 | <blockquote>
<p>A set $X$ is said to be denumerable if there is a bijection $\mathbb{Z}^+\rightarrow X$</p>
</blockquote>
<p>Statement to prove:</p>
<blockquote>
<p>The of rationals $\mathbb{Q}$ is denumerable.</p>
</blockquote>
<p>Proof:</p>
<blockquote>
<p>Define $f:\mathbb{Q} \rightarrow \mathbb{Z}\times \... | Lee Mosher | 26,501 | <p>Question 1: The image of $f$ is not $\mathbb{Z} \times \mathbb{Z}^+$ because $(4,2) \in \mathbb{Z} \times \mathbb{Z}^+$ is not in the image. The image of $f$ is an infinite subset of $\mathbb{Z} \times \mathbb{Z}^+$ because the domain is infinite and $f$ is injective.</p>
<p>Question 2: Yes and yes.</p>
|
234,153 | <p>Prove: we always have at least one x>0 is a $x^3+bx^2+cx-d^2=0$ 's root (b, c, d are real numbers and $d≠0$)</p>
| GregRos | 47,775 | <p>Let,
$$p(x) = x^3 + bx^2 + cx - d^2$$
Then we have, $p(0)= -d^2 < 0$
We know that there exists $x > 0$ so that:
$$x^3 > bx^2 + cx - d^2$$
For any set of constants $b, c, d$. Let's call this pont $r$. Thus, we know that $p(0) < 0$ and $p(r)>0$ and that $r > 0$. </p>
<p>You can continue the rest.</p... |
20,817 | <p>I am referencing to this question on mathunderflow:<br>
<a href="https://math.stackexchange.com/questions/223238/what-is-the-average-of-rolling-two-dice-and-only-taking-the-value-of-the-higher">What is the average of rolling two dice and only taking the value of the higher dice roll?</a></p>
<p>Now I tried:</p>
<p... | Seth Chandler | 5,775 | <p>You could create the TransformedDistribution described by bill s or you could use Mathematica's built in <code>OrderDistribution</code> command, which addresses precisely the sort of problem you have posed. </p>
<pre><code>Mean[OrderDistribution[{DiscreteUniformDistribution[{1, 6}], 2}, 2]]//N
(* 4.47222 *)
</code>... |
3,519,261 | <p>Here is the question.</p>
<blockquote>
<p>If <span class="math-container">$s_1=0$</span>, prove that <span class="math-container">$s_{n+1}=s_n+\frac{1}{2}(x-s_n^2)$</span> is monotone and converges if <span class="math-container">$0<x<1$</span>. </p>
</blockquote>
<p>I tried this question by induction, bu... | peterwhy | 89,922 | <p><span class="math-container">$$\begin{align*}\sum_{i=1}^3(2+i)^2 &= \sum_{j=3}^5 j^2&&(j=2+i)\\
&= \sum_{j=1}^5 j^2 - \sum_{j=1}^2 j^2\\
&= \sum_{j=1}^5 j^2 - 5
\end{align*}$$</span></p>
|
3,519,261 | <p>Here is the question.</p>
<blockquote>
<p>If <span class="math-container">$s_1=0$</span>, prove that <span class="math-container">$s_{n+1}=s_n+\frac{1}{2}(x-s_n^2)$</span> is monotone and converges if <span class="math-container">$0<x<1$</span>. </p>
</blockquote>
<p>I tried this question by induction, bu... | machine_1 | 363,728 | <p>No, you can't use the original formula directly. If you change what is to be squared, then here is how the formula transforms:</p>
<p><span class="math-container">$SUM((a + i)^2) = i(i+1)(2i+1)/6 + a(i^2 + i) + a^2(i)$</span></p>
<p>In other words:</p>
<p><span class="math-container">$f_n = f_o + a(i^2 + i) + a^2... |
2,717,786 | <p>Problem: I am reading through Rudin's text on functional analysis and came across the proof for the following theorem: Suppose $X, Y$ are Banach spaces, and $T$ is a bounded linear operator from $X$ to $Y$. Then $T$ is compact $\iff$ $T^*$ is compact. In his proof of the theorem (picture below)<a href="https://i.sta... | Tsemo Aristide | 280,301 | <p>$\|T^{\ast}y_{n_i}^* - T^{\ast}y_{n_{j}}^* \| = \sup_{x \in U} |((T^* (y_{n_i})-T^*(y_{n_j}))(x)|=Sup_{x\in U}|\langle y_{n_i}-y_{n_j},T(x)\rangle|$ adjunction formula.</p>
|
582,804 | <p>I want to draw phase plane diagram of the following differential equation </p>
<p>$$\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 10 y = 0.$$</p>
<p>Please check if my approach is correct. I have some doubts about it. They are in <strong>bold</strong>.</p>
<p>First of all I am changing it to a system of differential equat... | JohnD | 52,893 | <p>What you have looks good. Some comments to each of your questions:</p>
<p>Q1/Q2: Your eigenvalues/eigenvectors yield the general solution which you wrote and this tells the tale: you have a complex conjugate pair of eigenvalues (so trajectories <em>spiral</em>) and their real part is positive (so they spiral <em>ou... |
975,398 | <p>I want to know about the convergence or divergence of the following series:</p>
<p>$$\sum \sin (a_n) $$</p>
<p>where </p>
<p>$$a_n=\frac{(-1)^n}{n}$$</p>
<p>The tests that I tried were inconclusive. Is it possible to know?</p>
<p>Or, more generally, does this series converge for all conditionally convergent ser... | Some Math Student | 181,130 | <p>For $x$ close to $0$, we have $\frac{|x|}{2}\leq |\sin{x}|\leq |x|$. Does that give you any ideas?</p>
<p>Edit: can be done way simpler, just noticed. $\sin(-x)=-\sin(x)$, and the sequence $\frac{1}{n}$ has the limit $0$. Now, if I were to mention a guy called Leibniz, we are almost done.</p>
|
2,918,287 | <p>We have
\begin{align}
U_{0} &= 1 &&\text{and} & V_{0} &= 2 \\
U_{n+1} &= \frac{U_{n}+V_{n}}{2} &&\text{and} & V_{n+1} &= \sqrt{U_{n+1}V_n}
\end{align}
How to prove two sequence have a common limit?</p>
<p>I found $(U_n)$ is increasing and $(V_n)$ is decreasing but I don't kn... | Somos | 438,089 | <p>Since <span class="math-container">$U_n$</span> is increasing and <span class="math-container">$V_n$</span> is decreasing, all you have to show is that the difference <span class="math-container">$V_n-U_n$</span> has limit <span class="math-container">$0$</span>. But, the limit exists since both sequences are monoto... |
1,917,852 | <p>Suppose $(\Omega, \mathcal{F}, \mu)$ is a probability space and $f$ is integrable function. Then, is it true that $\mu \{ \omega: |f(w)|=\alpha\}=0$ for all but countably many $\alpha$ ? \</p>
| H. H. Rugh | 355,946 | <p>It doesn't depend on integrability of $f$. Measurability suffices.
Let
$$ A_n = \{ a\in [0,+\infty] : \mu(|f|=a)>1/n \}$$
Then the cardinality of $A_n$ is at most $n$.
And $\cup_{n\geq 1} A_n$ is the set of $a$'s for which $\mu(|f|=a)$ is strictly positive.</p>
<p>Remark: The result also holds for any $\sigma$... |
824,255 | <p>Suppose I have a polynomial (of any order) and I'm not able to calculate the roots. Is there a way to get at least some information about the roots such as how many of them are complex, negative or positive? For example, I can safely identify the behavior (and therefore roots' character) of $f(x)=ax+b$ or even a qua... | mm-aops | 81,587 | <p>do you know that $\sin$, $x$ and $1/x$ are continuously differentiable everywhere (but $0$ in the last case)? if so just use the fact that a composition and a product of $C^1$ functions is $C^1$</p>
<p>edit: to be clear - $C^1$ means continuously differentiable</p>
|
1,723,707 | <p>I have proven that Martingales have orthogonal increments. </p>
<p>From this I need to show that $\operatorname{Cov}[M(t),M(s)]$ relies only on $\min\{s,t\}$.</p>
<p>I used the expected value definition of Covariance and eliminated one piece using the orthogonal increments bit. Also, I introduced conditioning on $... | John Dawkins | 189,130 | <p>You are on the right track. Assuming $s<t$, by conditioning,
$$
E[M(s)M(t)]=E[M(s)E[M(t)|\mathcal F_s]=E[M(s)^2].
$$
So,
$$
cov[M(s),M(t)]=E[M(s)M(t)]-E[M(s)]E[M(t)]=E[M(s)^2]-(E[M(s)])^2=var[M(s)],
$$
this depends on $t$ only through the fact that $t\in(s,\infty)$.</p>
|
3,428,083 | <p>I'm reading a book, and it defines the following definition: </p>
<p>Let <span class="math-container">$f:X \to [-\infty,+\infty]$</span> be a function , where <span class="math-container">$(X,d)$</span> is some metric space.</p>
<p>Define <span class="math-container">$\lim\inf\limits_{y\to x}f(y) :=\inf \{\lim\inf... | Asaf Karagila | 622 | <p>If the sequence is decreasing, it has to be finite, since a well-ordering does not have any infinitely decreasing sequences. But that means that we have to stop somewhere, and if you take "decreasing" to mean "continue to decrease as long as you can" that means that the last element of the sequence must be the minim... |
2,155,201 | <p>By a real ODE I mean an ordinary differential equation with only real coefficients and the resulting function is a function of a real argument. If such a solution exists, can you give an example?</p>
<p>Edit: To add to this, is it still possible if the initial conditions must also be real?</p>
| Community | -1 | <p>Consider $y(x)=(1-x^2)^{3/2}$.</p>
<p>Then $$y'(x)=-\frac322x(1-x^2)^{1/2}=-3x\sqrt[3]{y(x)},$$ with the initial condition $y(0)=1$.</p>
<p>But $y(\sqrt2)=i.$</p>
|
311,261 | <p>I would like to fit a formula $ax^b + cx^d+ e$ to a set of points. I have two questions.</p>
<ol>
<li>If my data were perfect, how many points do I need in the worst case to get $a,b,c,d,e$ exactly?</li>
<li>If my data aren't perfect, but the underlying unknown function is really something like $a\sqrt{x} + cx^{0.... | vonbrand | 43,946 | <p>You have 5 unknowns, so 5 points are enough to set up a set of 5 equations in the 5 unknowns. Solving it will require numerical methods.</p>
<p>A "good method" would be to (ab)use least squares: You can define for your $n$ data points $(x_k, y_k)$:
$$
f(a, b, c, d, e) = \sum_{1 \le k \le n} (y_k - a x_k^b + c x_k^d... |
2,458,259 | <p>Are there any sequences s.t. $\{a_{n}\}^{\infty}_{n=1}$ and $\{b_{n}\}^{\infty}_{n=1}, b_{n}\neq0, \forall n$ which diverge, but $\{\frac{a_{n}}{b_{n}}\}^{\infty}_{n=1}$ converges to $0$.</p>
<p>I have tried to plug in a bunch of examples, but failing to come up with one. </p>
| Zach Boyd | 60,023 | <p>Hint: pick the numerator so that it goes to infinity and then make the pick the denominator so that it goes to infinity faster. Does that help? </p>
|
249,508 | <p>Where did these symbols come from? Like Pi, Fee and this weird E/sideways M and the triangle.</p>
| ctype.h | 50,159 | <p>They are Greek letters. The first two you mentioned are Pi ($\pi$) and Phi ($\phi$), respectively (note the spelling). What you refer to as "weird E/sideways M" is the capital letter Sigma ($\Sigma$), and the one that looks like a triangle is the capital letter Delta ($\Delta$). Here is a table of Greek letters and ... |
249,508 | <p>Where did these symbols come from? Like Pi, Fee and this weird E/sideways M and the triangle.</p>
| MrWolvwxyz | 44,786 | <p>They are mostly Greek letters that have been adopted because scientists and mathematicians ran out of letters. It also probably has to do with the first real break throughs happening in Greek culture eg. Euclid, Pythagoras, Aristotle.</p>
|
249,508 | <p>Where did these symbols come from? Like Pi, Fee and this weird E/sideways M and the triangle.</p>
| eccfcco | 51,565 | <p>One other thing that I would like to add, is that $\Sigma$ is the symbol used in summations. If you don't understand that, just look it up on Google.</p>
|
1,779,314 | <p><strong>harmonic numbers</strong>
$H_n=\sum_{i=1}^{n}\frac{1}{i}$</p>
<p><strong>Euler's constant</strong></p>
<p>$\gamma=\lim_{n\to \infty} [H_n-\ln(n)]$</p>
<p><strong>Factorial</strong></p>
<p>$n!=n(n-1)(n-2)\cdots2\cdot1$; valid for all non-negative integers</p>
<p>Show that,</p>
<p>$$(s+1)\gamma-\int_0^1\... | almagest | 172,006 | <p>Assuming baskets are indistinguishable (as well as apples etc), there are surprisingly few arrangements possible. For apples the possible splits between baskets are: (5,2,0), (5,1,1), (4,3,0), (4,2,1), (3,3,1), (3,2,2).</p>
<p>For type (5,2,0) the possible splits of lemons are: (0,3,2), (0,2,3), (0,1,4), (0,0,5) (a... |
637,099 | <p>I'm having trouble verifying my proof, would appreciate some input on this one.</p>
<p>Let $(X,d)$ be a metric space with $E\subset X$.</p>
<p>Suppose $E$ is closed in $X$, which means that $E=\overline{E}$.
By defintion of the <em>boundary</em> of $E$, denoted $\partial E$ we have that</p>
<p>$\partial E=\overli... | copper.hat | 27,978 | <p>Take $f_n(x) = \sqrt{n(n+1)} 1_{[{1 \over n+1}, {1 \over n}]}(x)+{1 \over n}$.</p>
<p>(Note that the second condition follows from the first by Hölder, since the
exponential term is bounded and
$\|fg\|_1 \le \|f\|_1 \|g\|_\infty$.)</p>
|
2,033,470 | <blockquote>
<p>Assume that an integer <span class="math-container">$a$</span> has a multiplicative inverse modulo an integer <span class="math-container">$n$</span>. Prove that this inverse is unique modulo <span class="math-container">$n$</span>.</p>
</blockquote>
<p>I was given a hint that proving this Lemma:
<sp... | Stefan4024 | 67,746 | <p>For the first part note as $(n,a) = 1$, then there exist $x,y \in \mathbb{Z}$, s.t. $nx + ay = 1$. Multiply both sides by $b$ and you will get $nxb + (ab)y = b$. The LHS is obviously divisible by $n$, so then the RHS must be too. Hence $n \mid b$.</p>
<p>This thing proves that the inverse exist. To note that it's u... |
2,033,470 | <blockquote>
<p>Assume that an integer <span class="math-container">$a$</span> has a multiplicative inverse modulo an integer <span class="math-container">$n$</span>. Prove that this inverse is unique modulo <span class="math-container">$n$</span>.</p>
</blockquote>
<p>I was given a hint that proving this Lemma:
<sp... | JSchoone | 393,635 | <p>What you have shown already is a technique that, using the Extended Euclidean Algorithm will give you the inverse (if it exists). Note also that you can determine if an element $a \mod n$ has an inverse by checking that they have no common divisors (that is, $\gcd(a,n)=1$).</p>
<p>Now to answer your question. Suppo... |
1,501,988 | <p>How do I solve the following equation?
$$\lvert{x}\rvert- \lvert{2+x}\rvert= x$$</p>
<p>I was thinking about dividing it into 4 cases: plus plus, plus minus, minus plus and minus minus. What is the best way to solve this?</p>
| juantheron | 14,311 | <p>$$|x|-|x+2|=\left\{\begin{matrix}
-x+(x+2)& \;\;, x\leq -2 \\\\
-x-(x+2)&\;\;, -2<x<0 \\\\
x-(x+2) &\;\; ,x\geq 0 &
\end{matrix}\right.$$</p>
<p>$\bullet\; $ If $x\leq -2\;,$ Then equation convert into $-x+x+2=x\Rightarrow x=2\;\;(\bf{False})$</p>
<p>$\bullet\; $ If $-2<x< 0\;,$... |
105,577 | <p>The analytic rank of the Mordell elliptic curve $y^2=x^3-86069^5$ indicates that it has rank 2. However, deriving a set of generators, and hence the regulator, is proving to be a little bit of an intractable problem.</p>
<p>I'm posting this question in the hope that someone may have investigated this curve already ... | Kevin Acres | 25,232 | <p>Tom A. Fisher has kindly identified the second generator for this curve, at a height of $1715.46805605884712533431816634$, and with a regulator of $1435241.11022534407264592039437$.</p>
<p>Many thanks to Tom for this impressive workout of his 12 descent code.</p>
<p>P2:= E![6409818948420000148009253515371033320765... |
1,822,314 | <p>I think of two numbers, $x$ and $y$.
When I add them together I get $5$ and when I find the difference I get $13$. What numbers did I think of?</p>
<p>I need to know how to write this down in simultaneous equation form</p>
<p>many thanks</p>
| didgocks | 190,918 | <p>Two Numbers:</p>
<p>$$x,y$$</p>
<p>The Sum is:</p>
<p>$$x+y=5$$</p>
<p>The difference is:</p>
<p>$$x-y=13$$</p>
<p>Now, let's solve these for $x$</p>
<p>$x+y=5$ and $ x-y=13$</p>
<p>$ x=5-y $ and $ x = 13+y$</p>
<p>Since these two equations have $x$ in common,</p>
<p>$$5-y = 13+y$$ </p>
<p>Solve for y,</p... |
1,002,203 | <p>Show that $\int_0^{\infty}\dfrac{\ln x}{1+x^2}\hspace{1mm}dx=0$</p>
<p>The problem I am facing is with the hint.</p>
<p>The hint says, use the substitution $u = 1/x$, it makes no sense to me.</p>
<p>Why would we use the substitution u = 1/x, its makes the integral strange.</p>
<p>Can anyone explain</p>
| Community | -1 | <p><strong>Hint</strong></p>
<p>Change the variable <span class="math-container">$u=1/x$</span> for the integral</p>
<p><span class="math-container">$$\int_0^1\frac{\ln x}{1+x^2}dx$$</span></p>
|
111,835 | <p>If $p$ is prime, then $F_{p-\left(\frac{p}{5}\right)}\equiv 0\bmod p$, where $F_j$ is the $j$th Fibonacci number, and $\left(\frac{p}{5}\right)$ is the Jacobi symbol.</p>
<p>Who first proved this? Is there a proof simple enough for an undergraduate number theory course? (We will get to quadratic reciprocity by th... | Qiaochu Yuan | 232 | <p>Here's a proof that only uses a little Galois theory of finite fields (and QR). I don't know if it's any of the proofs referenced by Gerry. Recall that
$$F_n = \frac{\phi^n - \varphi^n}{\phi - \varphi}$$</p>
<p>where $\phi, \varphi$ are the two roots of $x^2 = x + 1$. Crucially, this formula remains valid over $\ma... |
2,154,351 | <p>I've recently (yesterday) started to studying complex analysis on my own. So i came across this question at the end of the first chapter
" Let $G$ be an open set in $ \mathbb{C} $ . Is $\{ z : z \in G \lor \bar z \in G \}$ open?</p>
<p>My attempt :</p>
<p>By the definition a set is open if $ A \subseteq \mathbb{C... | Olod | 1,100 | <p>Geometrically, an open set is a union of open disks. This property ('to be a union of open disks') is preserved under the complex conjugation (for it is the symmetry with respect to the real line): whenever $U \subseteq \mathbf C$ is open, $\overline U$ is also open. Now if $G$ is open, then $\overline G$ is open, t... |
2,154,351 | <p>I've recently (yesterday) started to studying complex analysis on my own. So i came across this question at the end of the first chapter
" Let $G$ be an open set in $ \mathbb{C} $ . Is $\{ z : z \in G \lor \bar z \in G \}$ open?</p>
<p>My attempt :</p>
<p>By the definition a set is open if $ A \subseteq \mathbb{C... | Nosrati | 108,128 | <p>Geometrically it's simple that consider $\Omega=\{ z : z \in G \lor \bar z \in G \}$ is open, but analytically:</p>
<hr>
<p>Assumption: $G$ is open in $\mathbb{C}$, that is for every $z\in G$, there exise $r>0$ such that $\mathcal{D}(z;r)\subset G$.</p>
<p>Now let $\Omega=\{ z : z \in G \lor \bar z \in G \}=G\... |
2,335,043 | <p>Let D be the ring of integers of a number field K. Let I be a non-zero ideal of D. Why D/I is finite?</p>
<p>Would you help me please? Thank you in advance.</p>
| Dando18 | 274,085 | <p>Sometimes we express $0^0 = 1$, then we have $e^{0^0-1}=e^{1-1}=e^0=1$. However, normally $x^x$ is undefined at $x=0$. So we take the limit, $ \lim_{x\to 0} (e^{x^x-1}) $. Substitute $x \mapsto x^x$ (see below),</p>
<p>$$ \lim_{x\to 1} (e^{x-1}) = e^0 = 1 $$</p>
<p>Here I show that $x^x \to 1$ as $x \to 0$</p>
<p... |
338,089 | <p>Denote by <span class="math-container">$Q_n$</span> the <span class="math-container">$n$</span>-dimensional hypercube. A vertex of <span class="math-container">$Q_n$</span> is represented by a vector of <span class="math-container">$n$</span> <span class="math-container">$\{0,1\}$</span>-bits. An edge corresponding ... | Maryam Ajorlou | 97,838 | <p>I offer you to try these books too:</p>
<p>-A Guide to Classical and Modern Model Theory, Written by Annaliza Marcja and Carlo Toffalori. I could find many tangible examples in this book when I was trying to understand Marker's.</p>
<p>-A course in Model Theory, Katrin Tent and Martin Ziegler. Although one can fin... |
338,089 | <p>Denote by <span class="math-container">$Q_n$</span> the <span class="math-container">$n$</span>-dimensional hypercube. A vertex of <span class="math-container">$Q_n$</span> is represented by a vector of <span class="math-container">$n$</span> <span class="math-container">$\{0,1\}$</span>-bits. An edge corresponding ... | Alex Kruckman | 2,126 | <p>The best book for you is probably <em>A Shorter Model Theory</em> by Hodges.</p>
<p>Some comments on your question, though: First, you should be aware that the model theory of finite structures and the model theory of infinite structures have extremely different characters - so much so that finite model theory is e... |
9,719 | <p>I have proved this using Venn diagram but when I am trying to prove this using the rule that <strong><em>"If $ A \subset B \text{ and } B \subset A $ then $ A = B $"</em></strong>, I am having some problems with my understanding of the same,here is how I did so far:</p>
<p>Let $x \in A \cap (B-C) \Rightarrow x \in... | Niloct | 14,802 | <p>Simplyfing the notation of @Marnix answer:</p>
<p>$$
A \cap (B - C) \\
= A \cap B \cap C' \text{ (definition of set complement)}\\
= A \cap B \cap (A' \cup C') \text{ (trick, } A \cap B \cap A' = \emptyset\text{)}\\
= A \cap B \cap (A \cap C)' \text{ (complement of intersection of sets in braces)}\\
= (... |
9,719 | <p>I have proved this using Venn diagram but when I am trying to prove this using the rule that <strong><em>"If $ A \subset B \text{ and } B \subset A $ then $ A = B $"</em></strong>, I am having some problems with my understanding of the same,here is how I did so far:</p>
<p>Let $x \in A \cap (B-C) \Rightarrow x \in... | user2838619 | 162,041 | <p>It is possible to continue your solution$$\begin{align}
A\cap(B-C) & = \{x\space|\space x\in A\space\land(x\in B\space\land x\notin C)\} \\
& = \{x\space|\space (x\in A\space\land x\in B)\space\land x\notin C\} \\
& = \{x\space|\space (x\in A\space\land x\in B)\space\land (x\notin A\lor x\notin C)\} \... |
574,536 | <p>I tried this problem by counting the number of 5-digit numbers and then dividing it by 2 just like we do for three digits. Please help. Thank you.</p>
| qwr | 122,489 | <p>There are $90000$ $5$-digit numbers, exactly half have an even sum of digits (base $10$ has equal numbers of even and odd digits) </p>
|
3,753,788 | <p>In his book Serge Lang provides the following proof that every bounded set that is closed is also compact.</p>
<p>" Let S be a closed and bounded set. This implies that there exists <span class="math-container">$B$</span> s.t. <span class="math-container">$|z|\leq B$</span> where <span class="math-container">$z... | José Carlos Santos | 446,262 | <ol>
<li>The sequence <span class="math-container">$\{x_{n_2}\}$</span> is a subsequence of the sequence <span class="math-container">$\{x_{n_1}\}$</span>. Since <span class="math-container">$\{x_{n_1}\}$</span> converges to <span class="math-container">$a$</span>, any of its subsequences also converges to <span class=... |
2,288,402 | <p>I tried simplifying this first:</p>
<p>$$\frac{\pi}{3}\le \arg(-z+3+i)\le \frac{5\pi}{3} \Leftrightarrow \\
\frac{\pi}{3}\le \arg(-(z+(3-i)))\le \frac{5\pi}{3} \Leftrightarrow \\
\frac{\pi}{3}\le \arg(z+(3-i))+\pi \le \frac{5\pi}{3} \Leftrightarrow \\
???$$</p>
<p>My book does the following:</p>
<p>$$\frac{\pi}{3... | bigfocalchord | 298,824 | <p>Here is how I was taught how to do this (I think it might be a slightly longer method):</p>
<p><span class="math-container">$$ \frac{\pi}{3}\le \arg(-z+3+i)\le \frac{5\pi}{3} $$</span></p>
<p>Split this inequality into two inequalities:</p>
<p><span class="math-container">$$ \frac{\pi}{3} \le \arg(-z+3+i) ~~\text{an... |
585,044 | <blockquote>
<p>Let $R(n)$ be the number of ordered pairs of integers $(x,y)$ with $x^2+y^2 = n$. Using the identity $$(a+b)^2+(a-b)^2 =2(a^2 +b^2),$$
prove that for all integers $n$, $R(2n) = R(n)$.</p>
</blockquote>
<p>I'm not really sure how to start this question. Any help would be appreciated.</p>
| shyamupa | 111,694 | <p>Assume a solution $(x_1,y_1)$, so
$$
x_1^2+y_1^2=n
$$</p>
<p>use the identity to obtain a solution of $x^2+y^2=2n$. Using the fact that this was an arbitrary solution ...</p>
|
3,517,376 | <p>A friend of mine presented a problem I found interesting:</p>
<blockquote>
<p>Compute the following: <span class="math-container">$$\sum_{n=0}^\infty\left(\prod_{k=1}^j(1+x^k)\right)[x^{mn}]$$</span> where <span class="math-container">$P(x)[x^n]$</span> denotes the <span class="math-container">$x^n$</span> coeffi... | Thomas Andrews | 7,933 | <p>I haven't checked this, but it looks correct.</p>
<p>When <span class="math-container">$m\mid j$</span> and <span class="math-container">$\gcd(m,n)=d$</span> then:</p>
<p><span class="math-container">$$\prod_{k=0}^{m-1}\left(1-e^{2\pi ink/m}z\right)=\left(1-z^{m/d}\right)^{d}$$</span></p>
<p>Setting <span class="... |
2,257,601 | <p>Let $$x^4-4x^3+mx^2+nx+1=0$$ have four roots positive real number. Then : what is $n,m \in \Bbb R$</p>
<p>My try </p>
<p>Let four roots be $x=a,b,c,d \in \Bbb R^+$ so, $$x^4-4x^3+mx^2+nx+1=0\\(x-a)(x-b)(x-c)(x-d)=0$$
Next?</p>
| Andreas | 317,854 | <p>A geometric interpretation leads to the solution.</p>
<p>Write $f(x) = x^4 - 4x^3 + mx^2 + nx + 1 = (x-1)^4 - [(6-m)x^2 - (4+n) x]$. </p>
<p>So the roots $f(x) = 0$ are attained at the intersections of two functions
$g(x) = (x-1)^4 $ and $h(x) = (6-m)x^2 - (4+n) x$. From Salahamam_ Fatima's analysis it is already ... |
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