qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,260,648 | <p>It's an example in Thomas's Calculus</p>
<p>Q. Prove that <span class="math-container">$\lim_{x \to 2} f(x)=4$</span></p>
<p>if
<span class="math-container">$f(x)=x^2, x\neq 2,
f(x)=1, x=2$</span></p>
<p>In the book they have proved it for <span class="math-container">$\varepsilon>4$</span> even when </p>
<... | alan23273850 | 397,319 | <p>Recall the <span class="math-container">$(\epsilon,\delta)$</span>-definition. It says for <strong>every</strong> <span class="math-container">$\epsilon>0$</span>, you must find a range of <span class="math-container">$(x_0-\delta,x_0+\delta)$</span> such that all <span class="math-container">$x$</span> in this r... |
971,648 | <p>How does this $||x-x'||$ expand to the equation below? </p>
<p>$\|x-x'\|^2 = (x^T)x + (x')^T x' - 2x^T x'$</p>
| John Hughes | 114,036 | <p>Once you know that $\|u\|^2 = \langle u, u \rangle = u^tu $ is the inner product, it's pretty straightforward: apply the distributive law a few times, using $u = x - x'$.</p>
|
1,799,710 | <p>The PDE $$\frac{\partial ^{2}u}{\partial x^{2}}+2\frac{\partial^{2}u}{\partial x\partial y}+\frac{\partial^{2}u}{\partial y^{2}}=x$$ Has</p>
<p>$1.$ Only one particular integral.</p>
<p>$2.$ a particular integral which is linear in x and .</p>
<p>$3.$a particular integral which is a quadratic polynomial is x and ... | Community | -1 | <p>I would like to do it in a systematic way. It is instructive to see how your question fits in the following steps.</p>
<h2>Step 1</h2>
<p>Let $V$ be a $n$-dimensional vector space over $\mathbb{F}$ (say, $\mathbb{R}$). Consider the following two ordered bases of $V$
$$
\alpha=(v_1,\cdots,v_n),\quad \beta=(w_1,\cdo... |
15,351 | <p>For example, in MATLAB, a panel is available where one can see straightaway which variables are used and their dimension sizes. Is such a feature available in <em>Mathematica</em>? I really find it hard to scroll up and down to see where things are in <em>Mathematica</em>; I just want to see at a glance what's been ... | ssch | 1,517 | <p>An ugly hack, look at all things in <code>Global</code> context, keep in table if <code>Dimensions</code> didn't return <code>{}</code></p>
<pre><code>Grid[Select[{#, Dimensions[ToExpression@#]} & /@
Names["Global`*"], #[[2]] != {} &], Alignment -> Left]
</code></pre>
<p><img src="https://i.stack.im... |
3,880,630 | <blockquote>
<p>In right <span class="math-container">$\Delta ABC$</span>, <span class="math-container">$\angle C = 90^\circ$</span>. <span class="math-container">$E$</span> is on <span class="math-container">$BC$</span> such that <span class="math-container">$AC = BE$</span>. <span class="math-container">$D$</span> is... | Michael Rozenberg | 190,319 | <p>Now, let <span class="math-container">$x=2\sqrt2a$</span>, <span class="math-container">$y=2\sqrt2b$</span> and <span class="math-container">$z=3\sqrt2c$</span>.</p>
<p>Thus, we need to prove that:
<span class="math-container">$$64(2a+b)^3(2a+c)^3(b+c)^2\geq729(2a+b+c)^4a^2bc.$$</span>
Also, let <span class="math-co... |
2,668,468 | <p>So I've got the region $R$ like in the image below, and need to find the double integral $$\iint\limits_{R}\frac{1}{4}\sqrt{2x^2+2y^2}dA$$ over that region, given $a=8$ and $c=5$.</p>
<p><a href="https://i.stack.imgur.com/ahhZp.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ahhZp.png" alt="enter... | epi163sqrt | 132,007 | <p>We can also strictly follow @dxiv. Using the differential operator $D_z$ we apply the operator $zD_z$ to a series and get
\begin{align*}
\left(zD_z\right)\left(\sum_{n=0}^\infty a_nz^n\right)=z\sum_{n=0}^\infty na_nz^{n-1}=\sum_{n=0}^\infty na_n z^n\tag{1}
\end{align*}</p>
<blockquote>
<p>We obtain according to (... |
250,426 | <p>I want to factorize any quadratic expressions into two complex-valued linear expressions.</p>
<p>My effort below</p>
<pre><code>a := 1;(*needed*)
p := 2;(*needed*)
q := 3;(*needed*)
f[x_] := a (x - p)^2 + q;(*needed*)
AA := Coefficient[f[x], x^2];
BB := Coefficient[f[x], x];
CC := f[0];
DD = BB^2 - 4 AA CC;
EE = Tim... | Roman | 26,598 | <pre><code>sol = SolveAlways[(I*x + a) (I*x + b) == e (x + I*c) (x + I*d), x]
(* {{a -> -c, b -> -d, e -> -1}, {a -> -d, b -> -c, e -> -1}} *)
e (x + I*c) (x + I*d) /. sol
(* {-(I c + x) (I d + x), -(I c + x) (I d + x)} *)
</code></pre>
|
12,544 | <blockquote>
<p>Can <span class="math-container">$n!$</span> be a perfect square when <span class="math-container">$n$</span> is an integer greater than <span class="math-container">$1$</span>?</p>
</blockquote>
<p>Clearly, when <span class="math-container">$n$</span> is prime, <span class="math-container">$n!$</span> ... | Dave Marain | 840,798 | <p>√n ≤ n/2 for n ≥ 4. Thus if p is a prime such that n/2 < p ≤ n, we have
√n < p → n < p² so p² cannot be a factor of n if n ≥ 4.</p>
|
1,915,560 | <p><a href="https://math.dartmouth.edu/archive/m105f13/public_html/m105f13notes1.pdf" rel="nofollow">These notes on harmonic sum</a> present the following inequality:</p>
<p>$$\frac{1}{n} < \int_{n-1}^n \frac{dt}{t}$$ for $n≥2$.</p>
<p>How can this be shown? By induction and by evaluation the integral?</p>
| Bernard | 202,857 | <p>This is a direct consequence of the <em>Mean Value theorem</em> for definite integrals:
$$\int_n^{n+1}\frac1t\,\mathrm d\mkern 1mu t= \frac1c(n+1-n)\quad\text{for some }c\in [n,n+1].$$</p>
|
49,630 | <p>Most programmers (including me) are painfully aware of quadratic behavior resulting from a loop that internally performs 1, 2, 3, 4, 5 and so on operations per iteration,</p>
<p>$$\sum_{i=1}^n i = \frac{n \left(n+1\right)}{2} $$</p>
<p>It’s very easy to derive, e.g. considering the double of the sum like $(1... | Geoff Robinson | 13,147 | <p>These "power sum" polynomials are known as Bernoulli polynomials, and have been studied for centuries, and there is a vast literature about them. There are many inductive formulae relating them. As you observed, the right thing to do is to consider a polynomial $f_{k}(x)$ with $f_{k}(0) = 0$
and $f_{k}(x+1) - f_{k}(... |
3,005,922 | <p>Suppose <span class="math-container">$f:\Bbb R^n\to \Bbb R^m$</span> is a linear map. Show that <span class="math-container">$Df(a)=f(a)$</span>.</p>
<p>Tried using limit definition:</p>
<p><span class="math-container">$$\lim\limits_{h \to 0}\frac{\Vert f(a+h)-f(a)-f(a)h\Vert}{\Vert h\Vert}$$</span><span class="ma... | Jacky Chong | 369,395 | <p><strong>Hint:</strong> Let <span class="math-container">$f(x) = Ax$</span>. Let
<span class="math-container">\begin{align}
\frac{\|f(a+h)-f(a)-Ah\|}{\|h\|} = \frac{\|A(a+h)-Aa-Ah\|}{\|h\|} = 0.
\end{align}</span></p>
|
3,005,922 | <p>Suppose <span class="math-container">$f:\Bbb R^n\to \Bbb R^m$</span> is a linear map. Show that <span class="math-container">$Df(a)=f(a)$</span>.</p>
<p>Tried using limit definition:</p>
<p><span class="math-container">$$\lim\limits_{h \to 0}\frac{\Vert f(a+h)-f(a)-f(a)h\Vert}{\Vert h\Vert}$$</span><span class="ma... | Community | -1 | <p>As @WillM. says in the comments, what you are trying to show is not true and in fact doesn't even make sense. <span class="math-container">$f(a)$</span> is an element of <span class="math-container">$\mathbb{R}^n$</span> whereas <span class="math-container">$Df(a)$</span> is a linear map from <span class="math-conta... |
2,605,626 | <p>$\sin(t)$ is continuous on $[0,x]$ and $\frac{1}{1+t}$ is continuous on $[0,x]$ so $\frac{\sin(t)}{1+t}$ is continuous on $[0,x]$ so the function is integrable.</p>
<p>How do I proceed? What partition should I consider ? </p>
<p>Edit : We haven't done any properties of the integral so far except the basic definiti... | Jack D'Aurizio | 44,121 | <p>For any $a>0$ and any $x>0$ we have
$$ e^{ax}-\cos(x)-a\sin(x) \geq (1-\cos x)+a(x-\sin x) \geq 0 $$
such that
$$ \int_{0}^{x}\sin(t)e^{-at}\,dt \geq 0 $$
ensures the non-negativity of the Laplace transform of $\mathbb{1}_{(0,x)}(u)\sin(u)$. The inverse Laplace transform of $\frac{1}{1+u}$ is also non-negativ... |
2,001,449 | <p>Where in the analytic hierarchy is the theory of all true sentences in ZFC? In higher-order ZFC? In ZFC plus large cardinal axioms?</p>
<p>Edit: It seems that this is ill-defined. Why is this ill-defined for ZFC, but true for weaker theories like Peano arithmetic and higher-order arithmetic?</p>
| Noah Schweber | 28,111 | <p>It depends what you mean by "true sentences of ZFC."</p>
<p>If you mean the set of true sentences in the <em>language</em> of set theory - that is, the theory of the ambient model of set theory $V$ - then this isn't in the analytic hierarchy at all. This is because in ZFC we can define the true theory of $V_{\omega... |
3,369,669 | <blockquote>
<p>let <span class="math-container">$V$</span> be a real finite dimensional vector space and <span class="math-container">$f,g$</span> are nonzero linear functional on <span class="math-container">$V$</span> real vector space,Assume that <span class="math-container">$Ker(f)\subset Ker(g)$</span> ... | Olórin | 187,521 | <p>The kernel of a non-zero linear form on an <span class="math-container">$n$</span>-dimensional vector space is of dimension <span class="math-container">$n-1$</span> by the rank theorem. Hence both your kernels have the same dimension. As one kernel is included in the other they are equal. So the quotient zero.</p>
|
2,745,623 | <blockquote>
<p>Maximize the generic bivariate quadratic form constrained to the unit circle.</p>
<p>$$\begin{array}{ll} \text{maximize} & f(x_1, x_2) := ax_1^2 + 2bx_1 x_2 + cx_2^2\\ \text{subject to} & g(x_1, x_2) := x_1^2 + x_2^2 - 1 = 0\end{array}$$</p>
</blockquote>
<p>Using the standard Lagrange M... | Maziar Sanjabi | 479,764 | <p>Multiply the Lagrange conditions as follows:
\begin{align}
x_1 \times (a x_1 + b x_2 = \lambda x_1)\\
x_2 \times (b x_1 + c x_2 = \lambda x_2)\\
\end{align}
and sum them up. Now use the fact that $(x_1,x_2)$ is on the unit circle and you would easily get: $f(x_1,x_2) = a x_1^2 + 2bx_1x_2 + cx_2^2 = \lambda$.</p>
|
8,193 | <p><strong>NB. Some answers appear to be for a question I did not ask, namely, "Why is standardized testing bad?" Indeed, these answers tend to underscore the premise of my actual question, which can be found above.</strong></p>
<p>As a foreigner who has spent some time in the US, it seems to me that in the U... | Benjamin Dickman | 262 | <p>One possible reference (mentioned here specifically for its introduction) is:</p>
<blockquote>
<p>van den Heuvel-Panhuizen, M., & Becker, J. (2003). Towards a didactic model for assessment design in mathematics education. In <em>Second international handbook of mathematics education</em> (pp. 689-716). Spring... |
8,193 | <p><strong>NB. Some answers appear to be for a question I did not ask, namely, "Why is standardized testing bad?" Indeed, these answers tend to underscore the premise of my actual question, which can be found above.</strong></p>
<p>As a foreigner who has spent some time in the US, it seems to me that in the U... | Daniel R. Collins | 5,563 | <p>A late answer, because I think none of the existing responses have yet to address the theme of "why particularly in the U.S.?" (laying aside, momentarily, the other solid criticisms of standardized tests in general). </p>
<p>Recall that the U.S. has an almost unique <a href="https://en.wikipedia.org/wiki/Federal_go... |
3,942,070 | <p>Prove that <span class="math-container">$\mathbb{Z}_8$</span> is not an internal direct product of two proper subgroups.</p>
<p>I'm really struggling to get the wheels turning on this one. I know what it means for group to be an internal product of two subgroups, I'm just not sure how to get started in showing that ... | Moosh | 837,933 | <p>I am a bit rusty with group theory, so there may be another way of doing this, but <span class="math-container">$\mathbb{Z}_4$</span> only has so many subgroups, in fact it only has 2 proper subgroups so there are only 4 possibilities for you to check. I dont remember if there is an easy way to find the size of a pr... |
3,057,874 | <blockquote>
<p>The following formula shall be proved by induction:
<span class="math-container">$$F(m+n) = F(m-1) \cdot F(n) + F(m) \cdot F(n+1)$$</span>
Where <span class="math-container">$F(i), i \in \mathbb{N}_0$</span> is the Fibonacci sequence defined as:
<span class="math-container">$F(0) = 0$</span>, <s... | Bram28 | 256,001 | <p>Here is a proof in the same spirit as RobertZ's.</p>
<p>First, let's relate the Fibonacci numbers to the following problem: </p>
<blockquote>
<p>Suppose that you want to go up some flight of stairs and at every step you can take either one or two stairs: in how many ways can you get up the stairs? </p>
</blockq... |
4,650,606 | <p>Let <span class="math-container">$X$</span> be a complex Banach space, and let <span class="math-container">$P$</span> be a bounded linear operator acting on the dual <span class="math-container">$X^{*}$</span> such that that <span class="math-container">$P^2=P$</span>.
I research for a bounded linear operator <span... | coudy | 716,791 | <p>Note that <span class="math-container">$Q$</span> must be a projector: <span class="math-container">${Q^2}^*={Q^*}^2 = P^2 = P = Q^*$</span>, hence <span class="math-container">$Q^2=Q$</span>. The space <span class="math-container">$X$</span> must decompose into the image of <span class="math-container">$Q$</span> w... |
3,988,540 | <p>Does the series <span class="math-container">$\sum_{n=0}^\infty\frac{4^n}{n^3+9^n}$</span> converge or diverge?</p>
<p>So far, I've divided each term by <span class="math-container">$9^n$</span> to get <span class="math-container">$\frac{(4/9)^n}{n^3/9^n + 1}$</span> and tried to apply the ratio test, but that didn'... | DonAntonio | 31,254 | <p>Hint:</p>
<p><span class="math-container">$$0\le\frac{4^n}{n^3+9^n}\le\frac{4^n}{9^n}\;\ldots$$</span></p>
|
1,921,914 | <p>How do you find the sum of $\sum \limits_{i=0}^{n-1}(1+i) $ ?</p>
<p>Actually, I am especially confused because of of the n-1. Usually, I'd start with stuff like:
$$\sum \limits_{i=0}^{0}(1+i) = ?$$
$$\sum \limits_{i=0}^{1}(1+i) = ?$$
$$\sum \limits_{i=0}^{2}(1+i) = ?$$</p>
<p>But I don't know what to do with the... | Jam | 161,490 | <p>$$\begin{align}
\sum_{i=0}^{n-1}(1+i)
&=(1+0)+(1+1)+(1+2)+\dots+(1+(n-1))
\\
&=1+2+3+\dots+n
\end{align}$$</p>
|
1,921,914 | <p>How do you find the sum of $\sum \limits_{i=0}^{n-1}(1+i) $ ?</p>
<p>Actually, I am especially confused because of of the n-1. Usually, I'd start with stuff like:
$$\sum \limits_{i=0}^{0}(1+i) = ?$$
$$\sum \limits_{i=0}^{1}(1+i) = ?$$
$$\sum \limits_{i=0}^{2}(1+i) = ?$$</p>
<p>But I don't know what to do with the... | David Bowman | 366,588 | <p>$\sum_{i=0}^{n-1} ( 1 + i) = \sum_{i=0}^{n-1} 1 + \sum_{i=0}^{n-1} i$.</p>
<p>We have that $\sum_{i=0}^{n} i = \frac {n(n+1)}2, $, so, with $n-1$ in place of $n$, we have $\sum_{i=0}^{n-1} i = \frac {(n-1)((n-1)+1)}2 = \frac {(n-1)(n)}2 = \frac{n^2 -n}{2}$.</p>
<p>For our other sum, we have $\sum_{i=0}^{n-1} 1 = 1... |
2,272,872 | <p><strong>Problem:</strong> Find the number of real roots to the equation $$4^{\sin^2{x}}-2^{\cos{2x}}+1=0 \ , \ \ \ \ \ \ \ \ \ 0<x\leq\pi.$$</p>
<p><strong>Attempt:</strong> Simplifying i get $$2^{2\sin^2{x}}-2^{\cos{2x}}=2^{2(1-\cos{2x})}-2^{\cos{2x}}=\frac{2^2}{\left(2^{\cos{2x}}\right)^2}-2^{\cos{2x}}=-1.$$</... | vrugtehagel | 304,329 | <p>You've made a mistake in the very first equality: you use $\sin^2(x)=1-\cos(2x)$, while that should've been $\sin^2(x)=\tfrac12(1-\cos(2x))$. You've got the right answer; but that's just a happy little accident.</p>
<p>When we fix the first line, we see</p>
<p>$$2^{2\sin^2{x}}-2^{\cos{2x}}=2^{1-\cos{2x}}-2^{\cos{2... |
2,272,872 | <p><strong>Problem:</strong> Find the number of real roots to the equation $$4^{\sin^2{x}}-2^{\cos{2x}}+1=0 \ , \ \ \ \ \ \ \ \ \ 0<x\leq\pi.$$</p>
<p><strong>Attempt:</strong> Simplifying i get $$2^{2\sin^2{x}}-2^{\cos{2x}}=2^{2(1-\cos{2x})}-2^{\cos{2x}}=\frac{2^2}{\left(2^{\cos{2x}}\right)^2}-2^{\cos{2x}}=-1.$$</... | Harsh Sharma | 335,582 | <p>Though you have made a mistake in your first line, but seeing the triviality of the mistake and the fact that you are aware of it now, let us answer your questions one by one.</p>
<ol>
<li><p>The reasoning is correct and the answer is certainly not a fluke (for a moment let us shun out your mistake). I am talking g... |
3,045,407 | <p>I have reading some material about Robust optimization.
It seems that dual optimization theory is the main techniques to reformulate robust optimization.</p>
<p>for example:
<span class="math-container">$$\min_x\qquad c^Tx$$</span>
<span class="math-container">$$s.t\qquad a^Tx\leq b$$</span></p>
<p>where<span clas... | LinAlg | 373,321 | <ol>
<li><p>We use duality theory because we can do 'step 2'. It is a technique to turn a problem with uncountable many constraints into something manageable.</p></li>
<li><p>If you can find one <span class="math-container">$p \geq 0$</span> for which <span class="math-container">$D^Tp = x$</span> and <span class="math... |
3,045,407 | <p>I have reading some material about Robust optimization.
It seems that dual optimization theory is the main techniques to reformulate robust optimization.</p>
<p>for example:
<span class="math-container">$$\min_x\qquad c^Tx$$</span>
<span class="math-container">$$s.t\qquad a^Tx\leq b$$</span></p>
<p>where<span clas... | Sawyer | 970,037 | <p>There are two points that need to be noted:</p>
<ol>
<li>The variables in the objective function and the subproblem are different. Hence the minimization problem in the constraints does not influence the feasible region of <span class="math-container">$x$</span> directly.</li>
<li>Since the subproblem in the constra... |
86,419 | <p>I don’t know how to impose discontinuous internal boundary conditions (BCs) in NDSolve, so I’ve set up an example problem to illustrate my issue. Consider the simple first-order ODE for $f(z)$ on the interval $-1 ≤ z ≤ 1$:</p>
<pre><code>k f[z] + f'[z] == 0
</code></pre>
<p>where $k$ is an eigenvalue to be determ... | Michael E2 | 4,999 | <p><strong>Working solution</strong></p>
<p>One can manually implement the shooting method with <a href="http://reference.wolfram.com/language/ref/ParametricNDSolveValue.html" rel="noreferrer"><code>ParametricNDSolveValue</code></a> and <a href="http://reference.wolfram.com/language/ref/FindRoot.html" rel="noreferrer"... |
86,419 | <p>I don’t know how to impose discontinuous internal boundary conditions (BCs) in NDSolve, so I’ve set up an example problem to illustrate my issue. Consider the simple first-order ODE for $f(z)$ on the interval $-1 ≤ z ≤ 1$:</p>
<pre><code>k f[z] + f'[z] == 0
</code></pre>
<p>where $k$ is an eigenvalue to be determ... | xzczd | 1,871 | <p>You can make a change of variable to solve the problem. Here I'll use <a href="https://mathematica.stackexchange.com/a/80267/1871"><code>dChange</code></a> for this task:</p>
<pre><code>r0 = 0.5;
eqn = {k[z] f[z] + f'[z] == 0, k'[z] == 0, f[-1] == Exp[2], f[1] == 1};
c = Piecewise[{{r0, z > 0}}, 1];
neweqn = dCh... |
2,181,989 | <p>In all the question I never took $\sin x =t$ and was able to solve most of them but I got stack in this question and I saw the solution , I found that they took $\sin x =t$ and treated $\sin x$ as just a variable $t$. I found it a bit weird, now many question how would I had known that I had to take $\sin x$ as $t$ ... | klirk | 385,702 | <p>The idea is to use the fact that $$\lim_{x \to 0} \sin(x)=\lim_{t \to 0} t =0.$$
Even more, $\lim_{x \to 0} \frac{\sin(x)}x=1$, so $\sin(x)$ behaves near $0$ just as $x$.</p>
<p>Therefore replacing $\sin x$ with $x$ does not change the limit.
This substitution is done, because polynomials are easier to handle than ... |
2,956,158 | <blockquote>
<p>Given <span class="math-container">$z = \cos (\theta) + i \sin (\theta)$</span>,
prove <span class="math-container">$\dfrac{z^{2}-1}{z^{2}+1} = i \tan(\theta)$</span></p>
</blockquote>
<p>I know <span class="math-container">$|z|=1$</span> so its locus is a circle of radius <span class="math-contai... | egreg | 62,967 | <p><span class="math-container">$$
\tan\theta=\frac{\sin\theta}{\cos\theta}
=\frac{\dfrac{z-\bar{z}}{2i}}{\dfrac{z+\bar{z}}{2}}=
\frac{1}{i}\frac{z-\bar{z}}{z+\bar{z}}=\frac{1}{i}\frac{z-z^{-1}}{z+z^{-1}}=
\frac{1}{i}\frac{z^2-1}{z^2+1}
$$</span></p>
|
1,911,037 | <p>So I realized that I have to prove it with the fact that $(x-y)^2+2xy=x^2+y^2$ </p>
<p>So $\frac{(x+y)^2}{xy}+2=\frac{x}{y}+\frac{y}{x}$ $\Leftrightarrow$ $\frac{(x+y)^2}{xy}=\frac{x}{y}+\frac{y}{x}-2$ </p>
<p>Due to the fact that $(x+y)^2$ is a square, it will be positive </p>
<p>$x>0$ and $y>0$ so $xy>... | MCT | 92,774 | <p>Let $a = x/y$, then this is equivalent to proving $a + 1/a \geq 2$. $a$ is positive so multiply both sides by $a$ we get</p>
<p>$$a^2 - 2a + 1 = (a - 1)^2 \geq 0.$$</p>
|
1,399,601 | <p>Could you help me with proving:</p>
<p>Let $f$ be an analytic function defined in on upper half plane(UHP). Suppose that $|f(z)|<1$ for all $z$ in UHP. Prove that for every $z$ in UHP</p>
<p>$$
|f'(z)|\leq{\frac{1}{2\operatorname{Im} z}} .
$$</p>
<p>I guess that I need to use Cauchy's estimate but I am not sur... | Daniel Fischer | 83,702 | <p>Let $T(z) = \frac{z-i}{z+i}$. $T$ is a biholomorphic map from the upper half-plane to the unit disk $\mathbb{D}$. Define $g \colon \mathbb{D} \to \mathbb{D}$ as $g = f\circ T^{-1}$. By the Schwarz-Pick lemma, we have</p>
<p>$$\frac{\lvert g'(w)\rvert}{1 - \lvert g(w)\rvert^2} \leqslant \frac{1}{1-\lvert w\rvert^2}$... |
2,368,453 | <p>I am trying to derive a proof of the associative property of addition of complex numbers using only the properties of real numbers.</p>
<p>I found the following answer but was hoping someone can explain why it is correct, since I am not satisfied with it (From <a href="https://math.stackexchange.com/questions/10786... | Shuri2060 | 243,059 | <p>I think the key here is to differentiate between the different '$+$' we're seeing here - there are actually $3$ different kinds, $+:\Bbb R\times\Bbb R\rightarrow\Bbb R, +:\Bbb C\times\Bbb C\rightarrow\Bbb C$ and finally the '$+$' in $a+bi$.</p>
<p>The '$+$' in $a+bi$ is just used in representing a complex number (b... |
20,768 | <p>After a long period a post is deleted, is it still possible to edit the deleted post?</p>
| quid | 85,306 | <p>Yes, provided that </p>
<ul>
<li><p>you have the appropriate privilege. </p></li>
<li><p>you have access to the post.</p></li>
<li><p>the post was only deleted and not also locked (and is not a <em>self</em>-deleted question, see other answer). </p></li>
</ul>
<p>If it is about your own post mainly the last point ... |
692,944 | <p>Let $\beta $ be an ordinal such that for all $\gamma $ $2^{\aleph_{\gamma}}$ =
$\aleph_{\gamma + \beta}$. Does $\beta $ have to be infinity?. Under the continuum hypothesis, it is true, let 0= $\gamma $ and then we have an absurdity. But in ZFC, without CH, what happens? I tried to see that for if $\gamma $ is a n... | hmakholm left over Monica | 14,366 | <p>The Generalized Continuum Hypothesis states exactly that $\beta=1$ has the property you're defining. And it is known to be independent of ZFC, so ZFC certainly cannot prove that finite $\beta$s can't work.</p>
<p>(The Continuum Hypothesis is the claim $2^{\aleph_0}=\aleph_1$, or in other words just your equation wi... |
3,484,483 | <p>I would please like your guidance to find if the series <span class="math-container">$$\sum_{n=1}^{\infty}\left(\frac{1}{n^{2/3}}-\frac{7}{n^{3/2}}\right)$$</span> converges or diverges?</p>
<p>I noticed that We have two separate <span class="math-container">$p$</span>-series and <span class="math-container">$p<... | Claude Leibovici | 82,404 | <p>You have been given many explanations about the divergence.</p>
<p>If you want more, consider
<span class="math-container">$$S_p=\sum_{n=1}^{p}\left(\frac{1}{n^{2/3}}-\frac{7}{n^{3/2}}\right)=H_p^{\left(\frac{2}{3}\right)}-7 H_p^{\left(\frac{3}{2}\right)}$$</span> where appear generalized harmonic numbers.</p>
<p>... |
720,924 | <p>I think this is just something I've grown used to but can't remember any proof.</p>
<p>When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then?</p>
| Aaron Meyerowitz | 84,560 | <p>Here is a rationale admittedly guided by where we want to end up, which is not to say that it isn't natural. Warm-up: From the origin I have a unit vector $(x,y)$ and a second unit vector $(x',y')$ perpendicular to the first and counterclockwise from it. Use geometry to show that $y'=x$ and $x'=-y$ hint: From perpen... |
3,145,832 | <p>I am given real values <span class="math-container">$p, s, t, u$</span> and wish to find unknown values <span class="math-container">$r, v$</span>. As shown in the diagram below, <span class="math-container">$p$</span> and <span class="math-container">$s$</span> are radii of two given circles, with centers at <spa... | Misha Lavrov | 383,078 | <p>We can compute the distance from the centers of the two large circles to the center of the small circle in two ways: by the distance formula, and by taking the difference of their radii. This gives us the equations</p>
<p><span class="math-container">\begin{align}
(s - r)^2 &= u^2 + (v-t)^2 \\
(p-r)^2 &... |
3,498,985 | <blockquote>
<p>Find total number of distinct <span class="math-container">$x\in[0,1]$</span> for which <span class="math-container">$$\int_{0}^{x}\frac{t^2}{1+t^4}dt=2x-1$$</span></p>
</blockquote>
<p>My multiple attempts are as follows:-</p>
<p><strong>Attempt <span class="math-container">$1$</span>:</strong></p>... | Kavi Rama Murthy | 142,385 | <p>Let <span class="math-container">$f(x)=\int_0^{x}\frac {t^{2}} {1+t^{4}} dt-2x+1$</span>. Let us show that <span class="math-container">$f(0) >0, f(1) <0$</span> and <span class="math-container">$f$</span> is strictly decreasing. These facts would imply that <span class="math-container">$f(x)=0$</span> for exa... |
3,498,985 | <blockquote>
<p>Find total number of distinct <span class="math-container">$x\in[0,1]$</span> for which <span class="math-container">$$\int_{0}^{x}\frac{t^2}{1+t^4}dt=2x-1$$</span></p>
</blockquote>
<p>My multiple attempts are as follows:-</p>
<p><strong>Attempt <span class="math-container">$1$</span>:</strong></p>... | Jack D'Aurizio | 44,121 | <p>Actually you do not need to perform any explicit computation. <span class="math-container">$\frac{x^2}{1+x^4}=\frac{d}{dx}\int_{0}^{x}\frac{t^2}{1+t^4}\,dt$</span> is an increasing function over <span class="math-container">$[0,1]$</span> and it is bounded between <span class="math-container">$0$</span> and <span cl... |
578,492 | <blockquote>
<p>How can one prove that every element $x$ of a finitely generated local commutative algebra $A$ with identity over an algebraically closed field $K$ is unit or nilpotent? </p>
</blockquote>
<p>Of course, this is equivalent to the statement that in the local algebra every prime ideal is maximal. But I ... | Community | -1 | <p>It is proved <a href="https://math.stackexchange.com/a/578658/121097">here</a> that every finitely generated $K$-algebra is a <a href="http://en.wikipedia.org/wiki/Jacobson_ring" rel="nofollow noreferrer">Jacobson ring</a>, that is, every prime ideal is an intersection of maximal ideals. Since $A$ is local we deduce... |
3,024,496 | <p>I have this determinant which looks like a Vandermonde matrix</p>
<p><span class="math-container">$$D=\begin{vmatrix}1& a_1 & \cdots & a_1^{n-2}& a_1^n\\
1& a_2 & \cdots & a_2^{n-2}& a_2^n\\
\vdots &\vdots & \ddots & \vdots & \vdots\\
1& a_n & \cdots & a_n... | Misha Lavrov | 383,078 | <p>This can be shown in the same way as we show the corresponding identity for the Vandermonde determinant.</p>
<p>Let <span class="math-container">$Q(a_1, a_2, \dots, a_n)$</span> be the determinant of the matrix, as a degree <span class="math-container">$\binom n2+1$</span> homogeneous polynomial in <span class="mat... |
63,589 | <p>A big-picture question: what "physical properties" of a graph, and in particular of a bipartite graph, are encoded by its largest eigenvalue? If $U$ and $V$ are the partite sets of the graph, with the corresponding degree sequences $d_U$ and $d_V$, then it is easy to see that the largest eigenvalue $\lambda_{\max}$ ... | Nilima Nigam | 14,740 | <p>I fear my answer may not directly address the question, but I like the question! </p>
<p>Suppose we wish to numerically approximate solutions of the Poisson problem on a given domain. One strategy is to 'mesh' the region by simplices, and seek information on the nodes. One can approximate the Laplacian either stron... |
63,589 | <p>A big-picture question: what "physical properties" of a graph, and in particular of a bipartite graph, are encoded by its largest eigenvalue? If $U$ and $V$ are the partite sets of the graph, with the corresponding degree sequences $d_U$ and $d_V$, then it is easy to see that the largest eigenvalue $\lambda_{\max}$ ... | ARupinski | 12,301 | <p>The few times I have ever worked with eigenvalues of graphs, it has been in relation to the path algebra of the graph; each path in the graph is an element of this algebra. At any rate, the characteristic polynomial of the graph gives exact recurrences for calculating the number of paths of a given length and in par... |
236,055 | <p>I would like to extract data of surface properties and grain boundary properties for Cu polymorph #0 from the website: <a href="http://crystalium.materialsvirtuallab.org" rel="nofollow noreferrer">http://crystalium.materialsvirtuallab.org</a> using Mathematica.</p>
<p>So far I tried with <code>Import</code>, but it ... | Rohit Namjoshi | 58,370 | <p>Using V12 or later</p>
<pre><code>session = StartWebSession[]
WebExecute[session, "OpenPage" -> "http://crystalium.materialsvirtuallab.org/"]
cu = WebExecute[session, "LocateElements" -> "CSSSelector" -> "#Cu"]
WebExecute[session, "ClickElement" -... |
249,623 | <p>This problem comes from the response of the author of papers. </p>
<p>Consider two convex bodies $A$ and $B$: </p>
<p>$$A= \{X\in \mathcal{S}^4 : \operatorname{tr}(X) = 1, X\succeq 0 \}$$<br>
$$B = \operatorname{conv} SO(3)$$ </p>
<ol>
<li>$\mathcal{S}^4$ is the set of symmetric $4\times 4$ matrices. </l... | Joseph O'Rourke | 6,094 | <p>Here is a definition of <em>affinely isomorphic</em> for convex polytopes:</p>
<p><a href="https://i.stack.imgur.com/lDMKA.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lDMKA.jpg" alt="GunterZ"></a></p>
<blockquote>
<p>Ziegler, Günter M. <em>Lectures on polytopes</em>. Vol. 152. Springer Sci... |
264,609 | <p>Question on theory: If $\lim_{x\to a^+}F(x)=L$ and $\lim_{x\to a^-}F(x)$ doesn't exist, then does the $\lim_{x\to a}F(x)$ exists and is $L$? </p>
<p>Thanks. </p>
| Nameless | 28,087 | <p>It depends on what you mean $\lim_{x\to a^-}F(x)$ doesn't exist. </p>
<p>If $F$ is not defined in $(a-\delta,a)$ for no $\delta>0$ then $x$ can't approach $a$ from the left and $\lim_{x\to a^-}F(x)$ is not defined. If that's the case then, $\lim_{x\to a}F(x)$ exists and is $L$</p>
<p>If $F$ is defined in $(a-\d... |
222,863 | <p>Computing $\displaystyle \sum_{k\ge2}k(1-p)^{k-2}$, $p\in ]0,\space1[$</p>
<p><a href="http://www.wolframalpha.com/input/?i=sum%28k%2a%281-p%29%5E%28k-2%29,%202,%20infinity%29" rel="nofollow">WolframAlpha</a> says it is $\cfrac {p+1}{p^2}$ but I couldn't get that value but anyway here is what I did:</p>
<p>$$\disp... | DonAntonio | 31,254 | <p>$$f(x):=\sum_{n=0}^\infty x^n=\frac{1}{1-x}\,\,\,,\,\,|x|<1\Longrightarrow$$</p>
<p>$$\Longrightarrow f'(x)=\sum_{n=1}^\infty nx^{n-1}=\frac{1}{(1-x)^2}\Longrightarrow $$</p>
<p>$$\Longrightarrow \frac{1}{x(1-x)^2}-\frac{1}{x}=\sum_{n=2}^\infty nx^{n-2}$$</p>
<p>so substituting $\,x=1-p\,$ (why can we?), we ge... |
3,832,067 | <p>I recently encountered this question:</p>
<p><strong>The probability that the bulb will work longer than 800 hours is <span class="math-container">$0.2$</span>. We have three bulbs in the hallway. What is the probability that after 800 hours of service at least one of them will still work ?</strong></p>
<p>I underst... | Math Lover | 801,574 | <p>If you do not want to do <span class="math-container">$(1-q)$</span>, consider all cases satisfying the condition "at least" -</p>
<p>i) Probability that exactly one bulb is working <span class="math-container">$= 3C1 \times 0.2 \times 0.8 \times 0.8$</span></p>
<p>ii) Probability that exactly two of three... |
916,120 | <p>What is negation of <strong>All birds can fly.</strong></p>
<p>The question seems bit funny but i don't know which of the following two sentences is correct:</p>
<ol>
<li>Some birds can not fly</li>
<li>There is at least one bird which can not fly.</li>
</ol>
<p>Both the sentence seems almost logically same. But ... | Marc Bogaerts | 118,955 | <p>Translated in to set-language: "Every element of the set of birds $B$ belongs to the set of animals that fly $F$". So this translates as $B \subset F$. Negating this would be the equivalent of saying $B \nsubseteq F$. So in set language this would translate:"At least one bird cannot fly"(there is an element of $B$ t... |
159,789 | <p>$$\lim_{r \to 0^+} \frac{\sqrt r}{(r-9)^4}\
$$</p>
<p>How do i compute this limit? I was told to see what 1/x is approaching and rewrite it but can someone guide me in the right direction?</p>
<p>How can i find which infinity it is approaching?</p>
<p>Also</p>
<p>What does it approach if the limit approach 9 i... | William | 13,579 | <p>$\lim_{x \rightarrow 0^+} \frac{\sqrt{r}}{(r - 9)^4} = \frac{\sqrt{r}}{(r - 9)^4}$
since there is no $x$ occuring in the expression. </p>
<p>However if you mean
$\lim_{r \rightarrow 0^+} \frac{\sqrt{r}}{(r - 9)^4} = \frac{0}{9^4} = 0$. To see this note that the numerator approaches $0$ from the right and the denomi... |
18,280 | <p>More specifically, is it true that a representation of $\dim < p+1$ of the algebraic group $SL_2(\mathbb{F}_p)$ is always completely reducible? (of course above this dimension there are non completely reducible examples)</p>
<p>More general results that might help in this direction are also welcome.</p>
<p>Than... | Jack Schmidt | 3,710 | <p>If I understand your question, then no. The finite group SL(2,5) has a projective, reducible, indecomposable representation of dimension 5 over the field of 5 elements, namely the projective cover of the principal module 5^1. It has composition series 5^1, 5^3, 5^1. This gives you an example of a non-completely-r... |
394,101 | <p>I have an idea for a website that could improve some well-known difficulties around peer review system and "hidden knowledge" in mathematics. It seems like a low hanging fruit that many people must've thought about before. My question is two-fold:</p>
<p><em>Has someone already tried this? If not, who in t... | kerzol | 31,830 | <p>I'm the founder of <a href="https://papers-gamma.link" rel="noreferrer">https://papers-gamma.link</a>, an Internet place
to discuss scientific articles, mentioned by Matthieu Latapy. I have
been supporting this site for 6 years now. I hope that one day it will
become popular (in a good sense of the word) and useful... |
2,853,673 | <p>I came across this as one of the shortcuts in my textbook without any proof.<br>
When $b\gt a$, </p>
<blockquote>
<p>$$\int\limits_a^b \dfrac{dx}{\sqrt{(x-a)(b-x)}}=\pi$$</p>
</blockquote>
<hr>
<p><strong>My attempt :</strong></p>
<p>I notice that the the denominator is $0$ at both the bounds. I thought of su... | Hari Shankar | 351,559 | <p>I was taught to use the substitution $x=a \sin^2 \theta+b \cos^2 \theta$</p>
|
2,140,607 | <blockquote>
<p>Find $$\lim_{n \to \infty} \int_{1}^{n}\frac{nx^{1/2}}{1+nx^2}dx$$</p>
</blockquote>
<p>I have tried tackling this problem using the DCT but I am not quite sure if I have the right answer.
To begin with, $$\int_{1}^{n}\frac{nx^{1/2}}{1+nx^2}dx=\int_{1}^{\infty}\frac{nx^{1/2}}{1+nx^2}1_{[1,n]}dx$$</p>... | Chris | 164,598 | <p>Yes, the pointwise limit of the $f_n$'s will be $f(x) = \frac{1}{x^{3/2}}$, and this will also serve as an (integrable) dominating function on $(1, \infty)$. Thus the dominated convergence theorem implies $$\lim_{n \to \infty}\int_1^\infty \frac{nx^{1/2}}{1 + nx^2}\chi_{[1, n]}(x) dx = \int_1^\infty \lim_{n \to \inf... |
1,270,802 | <p>I just finished an exam, it has the following question: Where is the point on the plane $3x + 5y + z = 18$ has the shortest distance to $(0,0,0)$?</p>
<p>I found this question similar: <a href="https://math.stackexchange.com/questions/355460/find-the-point-on-the-plane-2x-y-2z-20-nearest-the-origin">Find the point ... | hmakholm left over Monica | 14,366 | <p>Your result of $-2089$ looks like you're subtracting digit by digit, but putting a <em>negative digit two</em> in the thousands position at the end where you subtract 8 from 6.</p>
<p>However, this doesn't work, because when you write $-2089$, the minus sign applies to <em>all</em> of the digit positions, so the 8 ... |
1,690 | <p>I like to ask true-false questions on exams, because I feel that they can be an efficient way to assess students' understanding of concepts and ability to apply them to somewhat unfamiliar situations. In general, I'm very happy with true-false questions, but there is one annoyance that I have never figured out how ... | András Bátkai | 61 | <p>First of all, it depends on the situation. I use true-false questions mainly because of your point 4, as a complement to a standard exam. With 150-200 mathematics major students, it is common practice at our university to complement "real" exam questions with tests, and there true-false questions are great for the r... |
4,033,612 | <blockquote>
<p>Seek separable solutions, <span class="math-container">$u(x,t)=X(x)T(t)$</span> of the equation, <span class="math-container">$u_{tt}=c^{2}\dfrac{1}{x}\dfrac{\partial}{\partial x}\left(x\dfrac{\partial u}{\partial x}\right)$</span>. Find the general solution assuming the separation constant is zero.</p>... | Noah Solomon | 750,380 | <p>The definition is not wrong. The thing you are missing is that if <span class="math-container">$S$</span> is our set containing <span class="math-container">$a$</span>, the condition is that for <strong>every</strong> <span class="math-container">$n\geq a$</span>, if <span class="math-container">$n \in S$</span> the... |
4,033,612 | <blockquote>
<p>Seek separable solutions, <span class="math-container">$u(x,t)=X(x)T(t)$</span> of the equation, <span class="math-container">$u_{tt}=c^{2}\dfrac{1}{x}\dfrac{\partial}{\partial x}\left(x\dfrac{\partial u}{\partial x}\right)$</span>. Find the general solution assuming the separation constant is zero.</p>... | Community | -1 | <p>The statement is meant to be read as <span class="math-container">$$\forall S\subseteq \Bbb Z, ((a\in S\land \forall n\ge a, (n\in S\Rightarrow n+1\in S))\Rightarrow (\forall m\ge a, m\in S))$$</span></p>
<p>Id est, that if <span class="math-container">$S$</span> is a subset of <span class="math-container">$\Bbb Z$<... |
3,158,599 | <p>An <em>n</em>-embeddability definition appears towards the end of the section <em>5.1 Torus knots</em> of the <em>Knot book</em> by <em>C. C. Adams</em>:</p>
<blockquote>
<p>A knot <span class="math-container">$K$</span> is an <span class="math-container">$n$</span>-embeddable knot if <span class="math-container"... | Kyle Miller | 172,988 | <p>To make use of the idea that bridge number bounds the embeddability number, let's put <span class="math-container">$6_2$</span> into bridge position first:</p>
<p><a href="https://i.stack.imgur.com/EcJ7j.png" rel="noreferrer"><img src="https://i.stack.imgur.com/EcJ7j.png" alt="Bridge position"></a></p>
<p>One way ... |
3,158,599 | <p>An <em>n</em>-embeddability definition appears towards the end of the section <em>5.1 Torus knots</em> of the <em>Knot book</em> by <em>C. C. Adams</em>:</p>
<blockquote>
<p>A knot <span class="math-container">$K$</span> is an <span class="math-container">$n$</span>-embeddable knot if <span class="math-container"... | N. Owad | 85,898 | <p>To add on to Kyle Miller's answer, the relation between tunnel number and <span class="math-container">$n$</span>-embeddable invariants is <span class="math-container">$t(K)\leq n $</span>. There is another invariant called the genus <span class="math-container">$g$</span> bridge number, <span class="math-container"... |
604,359 | <p>My experience with non commutative rings is limited to 2 by 2 matrices and the quaternions. The first of which is not a domain, and the latter is a division ring. I'm looking for an example of a domain that is not a division ring. </p>
<p>Invertible matrices do not produce an example, as they must be division rings... | mathematics2x2life | 79,043 | <p>The easiest example is of course the quaternion polynomial ring, $\mathbb{H}[x]$. However, there are more examples but they are not easy to create. For example, any noncommutative domains which satisfy the right <a href="https://en.wikipedia.org/wiki/Ore_condition" rel="nofollow">Ore condition</a> give you a way of ... |
3,249,064 | <p>I've read the question: "Why does the derivative of sine only work for radians?" and I can follow the derivation for the derivative of sine when measured in degrees, but the result confuses me. </p>
<p>Does this mean the derivative of the sine changes values when measured in different units? </p>
<p>For example, w... | marty cohen | 13,079 | <p>Showing that
<span class="math-container">$\sin'(x) = \cos(x)
$</span>
and
<span class="math-container">$\cos'(x) = -\sin(x)
$</span>
depends on
<span class="math-container">$\lim_{h \to 0} \dfrac{\sin(h)}{h}
=1
$</span>
and this only holds
for radians.</p>
|
1,803,843 | <p>Most characterizations of pointwise continuous functions defined on an interval rely on "local" properties. That is, a function is continuous at $x_0 \in I$ if it satisfies some property (epsilon-delta, sequential, oscillation, etc); a function is continuous on an interval if it is continuous at all $x \in I$. </p>
... | Renan R. | 338,093 | <p>You can imagine that if $x$ is a limit point, then don't matter how close you look, you ever found another point of the set with him.</p>
<p>Or you just can imagine that it is not a isolated point, that is more easy.</p>
|
1,884,303 | <p>I need to find the $n$-th number that contains the digit $k$ or is divisible by $k$, where $2 \leqslant k \leqslant 9$.</p>
<p>Example: If $n = 15$ and $k = 3$, then the answer is</p>
<p>$$ 33\quad (3, 6, 9, 12, 13, 15, 18, 21, 23, 24, 27, 30, 31, 32, 33)$$</p>
<p>I started following the sequence but couldn't for... | Hagen von Eitzen | 39,174 | <p>Say a number is good if it is a multiple of $k$ or contains the digit $k$ (or both).
The key idea is to find a fast way to compute $f(m)$, the number of good numbers $< m$.
With such a function $f$ at hand, you can find the desired result, namely one less that the smallest $m$ with $f(m)=n$ by binary search (star... |
1,884,303 | <p>I need to find the $n$-th number that contains the digit $k$ or is divisible by $k$, where $2 \leqslant k \leqslant 9$.</p>
<p>Example: If $n = 15$ and $k = 3$, then the answer is</p>
<p>$$ 33\quad (3, 6, 9, 12, 13, 15, 18, 21, 23, 24, 27, 30, 31, 32, 33)$$</p>
<p>I started following the sequence but couldn't for... | Marcus Andrews | 97,648 | <p>This is a dynamic programming problem. If we have a function that counts up how many numbers over $1 \leq i \leq N$ (for some boundary limit $N$) are divisible by $k$ or contain a digit $k$, then we can use a binary search to find when it returns $n$ (the target number).</p>
<p>So now we need a fast way to count up... |
128,219 | <p>There is a function <code>FindSequenceFunction</code> in Mathematica, that can identify a sequence of integers based on a few first elements. But what if I have a set of finite sequences <code>sec[n]</code> that grows with <code>n</code>? For example:</p>
<pre><code>sec[0]={1}
sec[1]={1, 1}
sec[2]={1, 6, 1}
sec[3]=... | prog9910 | 59,860 | <p>How is this?</p>
<pre><code>fx[x_, n_] := (x + 1)^n
f[n_] := Map[Part[fx[y, n] // ExpandAll, {#}] /. y -> 1 &,
Range[Length[fx[y, n] // ExpandAll]]]
Map[f[#] &, Range[2, 20, 2]]
(* Out: {{1, 2, 1}, {1, 4, 6, 4, 1}, {1, 6, 15, 20, 15, 6, 1}, {1, 8, 28, 56,
70, 56, 28, 8, 1}, {1, 10, 45, 120, 210, 25... |
292,594 | <p>Today I have encounter an integral:</p>
<p>$$\int_0^{\infty}\left[\frac{1}{3}\frac{\sin x}{x}+\cdots+\frac{1\times4\times\cdots\times(3n-2)}{3^nn!}\left(\frac{\sin x}{x}\right)^n+\cdots\right]\text{d}x$$</p>
<p>since $$\int_0^{\infty}\frac{\sin x}{x}=\frac{\pi}{2}$$</p>
<p>so I want to estimate $$\sum_{n=1}^{\inf... | N. S. | 9,176 | <p>$$\prod_{k=1}^{n}\left(1-\frac{2}{3k}\right) \geq \frac13\prod_{k=2}^{n}\left(1-\frac{3}{3k}\right)=\frac13\prod_{k=2}^{n}\left(\frac{k-1}{k}\right)=\frac{1}{3n}$$</p>
<p>and $\sum \frac{1}{3n}$ diverges.</p>
|
225,351 | <p>In one of the eight Thurston geometries there is the geometry of the universal cover of $SL(2, \mathbb{R})$. But from the algebraic point of view $PSL(2,\mathbb{R})$ is sufficient for building 3-manifolds i.e. we let the group act on itself and then quotient out a discrete subgroup of it. This is what we do for th... | Ryan Budney | 1,465 | <p>Yes, the answer is that you want a simply-connected manifold. All $SL_2(\mathbb R)$ manifolds are covered by the universal cover. Not all $SL_2(\mathbb R)$ manifolds are covered by $SL_2(\mathbb R)$. </p>
<p>Using the universal cover helps simplify the language of the "classification" of geometric manifolds. <... |
315,697 | <p>Let <span class="math-container">$X$</span> be an irreducible normal projective scheme over <span class="math-container">$\mathbb{C}$</span>. Let <span class="math-container">$U$</span> be the open subscheme of smooth points of <span class="math-container">$X$</span>. Consider the closed subscheme <span class="math-... | Sean Lawton | 12,218 | <p>Although <span class="math-container">$X$</span> and <span class="math-container">$U$</span> need not have isomorphic fundamental group (as the accepted answer shows), the induced map from the inclusion <span class="math-container">$U\hookrightarrow X$</span> is generally <span class="math-container">$\pi_1$</span>-... |
14,486 | <h2>Speculation and background</h2>
<p>Let <span class="math-container">$\mathcal{C}:=\mathrm{CRing}^\text{op}_\text{Zariski}$</span>, the affine Zariski site. Consider the category of sheaves, <span class="math-container">$\operatorname{Sh}(\mathcal{C})$</span>.</p>
<p>According to <a href="https://ncatlab.org/nlab/s... | Nico | 219,922 | <p><span class="math-container">$\newcommand\Ring{\mathrm{Ring}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Spec{Spec}$</span>Here is how you can define the notion of an Zarsiski-open subfunctor starting only with the Zariski topos <span class="math-container">$\mathcal Z$</span> of sheaves on <span class="math-c... |
65,923 | <p>The <a href="http://en.wikipedia.org/wiki/Parity_of_a_permutation">sign of a permutation</a> $\sigma\in \mathfrak{S}_n$, written ${\rm sgn}(\sigma)$, is defined to be +1 if the permutation is even and -1 if it is odd, and is given by the formula</p>
<p>$${\rm sgn}(\sigma) = (-1)^m$$</p>
<p>where $m$ is the number ... | Jack Poulson | 91,176 | <p>The inverse permutation can be constructed as a sequence of $n-1$ transpositions via Gaussian Elimination with partial pivoting, $P A = L U$, where $A$ is the original permutation matrix, $P=A^{-1}$, and $L=U=I$. Since the signature of the inverse permutation is the same as that of the original permutation, this pro... |
65,923 | <p>The <a href="http://en.wikipedia.org/wiki/Parity_of_a_permutation">sign of a permutation</a> $\sigma\in \mathfrak{S}_n$, written ${\rm sgn}(\sigma)$, is defined to be +1 if the permutation is even and -1 if it is odd, and is given by the formula</p>
<p>$${\rm sgn}(\sigma) = (-1)^m$$</p>
<p>where $m$ is the number ... | Vinícius Ferraz | 213,537 | <p>Welcome to <span class="math-container">$O(n)$</span>'s world. <a href="https://drive.google.com/file/d/1Rr-u_y7iq0xMimMXB1jDCwHMsNPl-x2B/view?usp=sharing" rel="nofollow noreferrer">Python code</a></p>
<pre><code>def decomposition(n, sigma):
lista = list()
for k in range (1, n + 1):
boo = False
for... |
3,194,984 | <p>I got something rather new and I just wanted to make sure my way of thinking in this field is fine. Suppose <span class="math-container">$$X\sim Bin(25,0.61)$$</span> and we are asked to find: <span class="math-container">$E[X^2]$</span>.
So basically I treat this binomial variable as a sum of 25 Bernoulli variables... | Arnab Auddy | 451,712 | <p>Your calculation is not correct.</p>
<p><span class="math-container">$X\sim\text{Binomial}(n,p)\iff X=\displaystyle\sum_{i=1}^{n}Y_i,$</span> where <span class="math-container">$Y_i\stackrel{iid}{\sim} \text{Bernoulli}(p).$</span> In this case <span class="math-container">$n=25$</span> and <span class="math-contain... |
1,585,061 | <p>Starting from the formula for work given a constant force $W = f s$, if you take the differential of both sides you would expect to get:</p>
<p>$$\mathrm{d}W = f \mathrm{d}s + s \space \mathrm{d}f$$</p>
<p>by an application of the product rule on the right hand side when taking the differential.</p>
<p>However, c... | piepi | 279,385 | <p>As you have mentioned, a constant force, thus the second term on the right side expression is redundant, don't you think? The product rule is correct though.</p>
|
1,585,061 | <p>Starting from the formula for work given a constant force $W = f s$, if you take the differential of both sides you would expect to get:</p>
<p>$$\mathrm{d}W = f \mathrm{d}s + s \space \mathrm{d}f$$</p>
<p>by an application of the product rule on the right hand side when taking the differential.</p>
<p>However, c... | WW1 | 88,679 | <p>You can't just take the case of constant force to define work. Work is a form of energy, which is defined ( in one dimension ) by integrating Newton's second law with respect to the spatial coordinate.</p>
<p>$$F=ma = m \frac{dv}{dt}= m \frac{dv}{ds} \frac{ds}{dt} =mv \frac{dv}{ds}$$</p>
<p>So</p>
<p>$$\int_{s_0... |
3,223,371 | <p>In order to prove if a relation is an equivalence relation, it needs to be show that is all of: </p>
<ul>
<li>Reflexive </li>
<li>Symmetric </li>
<li>Transitive </li>
</ul>
<p>Whilst I am familiar with this, I am unsure how to approach the following set of questions: </p>
<p>State and explain whether each of... | José Carlos Santos | 446,262 | <p>Concerning the first binary relation, not that the elements of <span class="math-container">$\mathbb Z\times(\mathbb Z\setminus\{0\})$</span> are pairs <span class="math-container">$(a,b)$</span> (with <span class="math-container">$a,b\in\mathbb Z$</span> and <span class="math-container">$b\neq0$</span>, not pairs <... |
2,912,881 | <p>In an $n\times n$ board ($n\geq 3$), how many colors do we need so that we can color the cells such that no three consecutive cells (horizontal, vertical, or diagonal) are of the same color?</p>
<p>With three colors we can do it, using the pattern</p>
<p>$$131$$
$$232$$
$$312$$</p>
<p>and repeating it as necessar... | Rayees Ahmad | 249,254 | <p>Whatever may be path chosen the ant has to take 4 upward steps and 4 righgward steps.</p>
<p>Therefore there are $\frac{8!}{4!4!}$ Paths...i,e $70$ paths</p>
|
2,974,181 | <p>Let <span class="math-container">$E$</span> be a finite dimensional complex vector space. Let <span class="math-container">$\mathbb{P}(E)$</span> be the projective space of lines through the origin of <span class="math-container">$E$</span>. Fulton, in his book "Young Tableaux", then defines the <span class="math-co... | Nicolas Hemelsoet | 491,630 | <p>First, an hyperplane corresponds to a linear form up to scaling that is to an element of <span class="math-container">$\Bbb P(E^*)$</span>. Then, by definition a linear form on <span class="math-container">$\Bbb P(E)$</span> is exactly a linear form on <span class="math-container">$E$</span>, i.e an element of <span... |
354,961 | <p>Suppose $A$ is a normal subgroup of $G$ and $B$ is a subgroup of $G$. Show that $A\cap B$ is a normal subgroup of $B$.</p>
| Seirios | 36,434 | <p>For all $g \in B$, $$g (A \cap B) g^{-1} \subset gAg^{-1} \cap gBg^{-1}=A \cap B$$ since $g \in B$ implies $gBg^{-1}=B$ and $gAg^{-1}=A$ thanks to normality.</p>
|
796,768 | <p>i am searching for a series with this condition that $\prod 1+a_n$ converges but $\Sigma a_n$ diverges. </p>
<p>i know that if $a_n = n^{\frac{1}{2}}$ then $\Sigma a_n$ diverges but i dont know it is exactly what i want, does $\prod 1+a_n$ converges?</p>
<p>i really don't know how to check the divergence... | Etienne | 80,469 | <p>According to the usual definition (as mentioned by Bruno), "convergence" for an infinite product means "convergence of the partial products to a <em>nonzero</em> limit". So, assuming $a_n>-1$ for all $n$, the convergence of $\prod (1+a_n)$ is equivalent to the convergence of the series $\sum\log(1+a_n)$.</p>
<p>... |
683,970 | <p>Let $H$ be Hilbert space, $f:H \rightarrow \Bbb F$ linear and bounded map.</p>
<p>I'm trying to prove that there exists only one $z_0 \in H$ such that:<br>
$ \forall_{x \in H} : f(x)=\langle x,z_0\rangle$</p>
<p>Proof: </p>
<p>If $f \equiv 0 $ then we have $z_0 = 0$. </p>
<p>If $f \not\equiv 0$ then:<br>
Let $... | J.R. | 44,389 | <p><strong>(1)</strong>: The magical ingredient that makes Hilbert spaces behave so much nicer than general Banach spaces is the <em>parallelogram identity</em>:</p>
<blockquote>
<p><strong>Lemma:</strong> Let $H$ be a Hilbert space and $x,y\in H$. Then
$$\|x+y\|^2+\|x-y\|^2=2(\|x\|^2+\|y\|^2)$$</p>
</blockquote>
... |
155,453 | <p>How can I use a list of variables (possibly subscripted) as an <a href="http://reference.wolfram.com/language/ref/AxesLabel.html" rel="noreferrer"><code>AxesLabel</code></a> without showing the braces.</p>
<p>For example, </p>
<pre><code>Plot[{x, x^2}, {x, 0, 1}, AxesLabel -> {x, {Subscript[y, 1], Subscript[y, ... | Mr.Wizard | 121 | <p>I propose a slightly different form of <a href="http://reference.wolfram.com/language/ref/Row.html" rel="nofollow noreferrer"><code>Row</code></a>:</p>
<blockquote>
<p>Row[list,s] inserts s as a separator between successive elements. </p>
</blockquote>
<p>And as <a href="https://mathematica.stackexchange.com/use... |
2,759,306 | <p>I have been struggling to find out a solution for a case where i have a cubic Bezier curve where two arbitrary control points of the one are equal, therefore i should show that this curve can be the quadratic curve. Do you have any thoughts how to prove it?</p>
| bubba | 31,744 | <p>The cubic Bezier curve with control points $A$, $B$, $C$, $D$ has equation
$$
P(t) = (1-t)^3A + 3t(1-t)^2B + 3t^2(1-t)C + t^3D
$$
We can re-arrange this to get
$$
P(t) = (-A + 3B -3C +D)t^3 + (3A-6B+3C)t^2 + (-3A+3B)t + A
$$
So, the cubic curve will become quadratic if and only if $-A + 3B -3C +D = 0$.</p>
<p>This ... |
1,223,909 | <p>The following is a use of eisenstein criterion that i have taken out from my lecture note.</p>
<p>$f(x, y) = x^4 +x^3y^2 +x^2y^3 +y$ is irreducible in Q[x, y]. This can be proved by treating Q[x,y] as (Q[y])[x] and applying the Eisenstein criterion with p = y.</p>
<p>However, I can't understand why i can apply eis... | P Vanchinathan | 28,915 | <p>The ideal generated by $y$ in the PID, $\mathbf{Q}[y]$ is maximal (hence prime) as the quotient ring is the filed of rational numbers.</p>
|
547,932 | <p>I have </p>
<p>$f(x)=\sqrt{3x}+1$</p>
<p>$g(x)=x+1$</p>
<p>My thinking was that at the intersection points both will be equal to each other so </p>
<p>$\sqrt{3x}+1=x+1$</p>
<p>$\sqrt{3x}=x$</p>
<p>However I don't know where to go from here.</p>
| Community | -1 | <p>what would be the value of x if you solve for $3\sqrt{x}=x$?</p>
<p>If you know what would be value of $x$ in suhc case then you can just find what is $f(x)$ for corresponding $x$ and set $(x,f(x))$.</p>
<p>That would be " an" intersection point.</p>
|
1,008,591 | <p>I am in the middle of my proof and I want to know if the following is true, suppose $f_n$ is a Cauchy sequence, can i do this?</p>
<p>If $$\| f_n(x) - f(x) \| \to 0,$$ then can I also say this limit is true</p>
<p>$$\lim_{m \to \infty} \| f_n(x) - f_m(x) \| \to 0?$$</p>
| imallett | 33,785 | <p>If I understand your question, you can take some asymptotic function like $\sqrt{\cdot}$ and scale and shift it. E.g. a scaling/shifting of:$$
f(x) := \begin{cases}
-\sqrt{-x} & \text{if } x<0\\
\sqrt{x} & \text{if } x \ge 0\\
\end{cases}
$$Within a ball around $x=0$, $f'$ exists and is unbounded. F... |
237,227 | <p>I want to produce graphs of Fourier transforms for lectures.</p>
<p>Using the answer from
<a href="https://mathematica.stackexchange.com/questions/3506/calling-correct-function-for-plotting-diracdelta">Calling Correct Function for Plotting DiracDelta</a>
I get a problem with the code mentioned below.</p>
<p>Definiti... | Daniel Huber | 46,318 | <p>In the first case, you use exact numbers for the frequencies, in the second case, machine numbers. Machine numbers only give exact results in special cases. This is the reason you get complex values from the FourierTransform. You can either rationalize your input or use <code>Chop</code> to get rid of the spurious ... |
1,102,216 | <p>Let $\mathcal{F}\left[f(t)\right](x)$ be the Fourier Transform of $f$, defined regularly as</p>
<p>$$\mathcal{F}\left[f(t)\right](x)=\int_{-\infty}^{\infty}f(t)e^{-itx}\,dt$$</p>
<p>And let $\mathcal{F}^{-1}\left[g(x)\right](t)$ be the Inverse Fourier Transform of $g$, defined regularly as $$ \mathcal{F}^{-1}\left... | Ziao | 977,713 | <p>Let the Fourier transform be <span class="math-container">$F(w) = \int_{-\infty}^\infty f(t)e^{-jwt}dt$</span> (eqn 1)</p>
<p>We want to prove that <span class="math-container">$f(t) = \frac{1}{2\pi}\int_{-\infty}^\infty F(w)e^{jwt}dw$</span> (eqn 2)</p>
<p>Check out the proof of the "inversion formula for char... |
1,517,557 | <p>How should I prove this theorem? What method of proof should I use?</p>
<p>$ 4 \nmid (n-2)^2 \ \Rightarrow \ 6 \nmid n \ \ \ ,n \in \Bbb Z $</p>
| SchrodingersCat | 278,967 | <p>HINT: $4 \not\mid (n-2)^2 \Rightarrow 2\not\mid(n-2) \Rightarrow (n-2)$ is odd i.e. $n$ is odd.</p>
<p>So $6\not\mid n$</p>
|
225,953 | <p>I would like to calculate the maximum number of polynomial terms given a certain number of variables and a certain degree. eg. given that the number of variables is 2 and the degree is 3, the maximum number of terms is 9:
<span class="math-container">$$x_1^3 + x_1^2 x_2 + x_1 x_2^2 + x_2^3+ x_1^2 +x_1 x_2 + x_2^2 + ... | Wolfgang | 29,783 | <p>If you have $k$ variables and want degrees $1,...,n$, you get the sum $$\sum_{i=1}^n h_i(1,...,1)=\sum_{i=1}^n\binom{k+i-1}{i}=\binom{k+n}{n}-1.$$ Here the $h_i$ are the <a href="http://en.wikipedia.org/wiki/Complete_homogeneous_symmetric_polynomial" rel="noreferrer"> complete homogeneous symmetric polynomials</a>... |
225,953 | <p>I would like to calculate the maximum number of polynomial terms given a certain number of variables and a certain degree. eg. given that the number of variables is 2 and the degree is 3, the maximum number of terms is 9:
<span class="math-container">$$x_1^3 + x_1^2 x_2 + x_1 x_2^2 + x_2^3+ x_1^2 +x_1 x_2 + x_2^2 + ... | Arup Hore | 148,388 | <p>The <strong><em>N</em></strong> degree polynomial having <strong><em>n</em></strong> variables and having maximum number of terms in equation is obtained by multiplying <strong><em>N</em></strong> linear equations of <strong><em>n</em></strong> variables each. The maximum number of terms is given as </p>
<p>((N+1) ... |
563,712 | <p>A corollary at page 91 of the book Group Theory I by M. Suzuki is as follows:</p>
<p>Let $A$ be an abelian subgroup of a $p$-group $G$. If $A$ is maximal among abelian normal subgroups of $G$, then $A$ satisfies $C_G(A)=A$. In particular, $A$ is maximal among abelian subgroups of $G$. </p>
<p>I am confused with t... | zibadawa timmy | 92,067 | <p>It says "maximal among abelian normal", meaning you are looking at abelian normal subgroups only, and then picking a maximal such: one which is not contained in any other abelian normal subgroup. It is okay for it to be contained in non-abelian subgroups, or even abelian subgroups provided they are not normal. The... |
4,233,449 | <blockquote>
<p>Consider <span class="math-container">$\mathcal F = \{f$</span> holomorphic in <span class="math-container">$\Bbb D$</span> with <span class="math-container">$f(\Bbb D)\subset\Bbb D,$</span> <span class="math-container">$f\left(\frac12\right)=f'\left(\frac12\right)=0\}$</span>, where <span class="math-c... | Martin R | 42,969 | <p>Let <span class="math-container">$T(z) = \frac{1/2-z}{1-z/2}$</span> be the Möbius transformation which maps the unit disk onto itself with <span class="math-container">$T(0)=1/2$</span> and <span class="math-container">$T(1/2) = 0$</span>, and define <span class="math-container">$g=f \circ T$</span>.</p>
<ul>
<li>S... |
10,935 | <p>I'm about to give a first-semester calculus lecture covering the mean value theorem for integrals:</p>
<p>If $f$ is continuous on $[a,b]$, then there is some $c\in(a,b)$ such that $(b-a)f(c)=\int_a^b f(x)\,dx$.</p>
<p>In past semesters, I've shown examples in which I confirm that this theorem holds for some specif... | Simply Beautiful Art | 6,220 | <p>You can show that a function $f(x)$ that is continuous and differentiable everywhere with two roots has at least one value such that $f'(c)=0$. Follows from the mean value theorem since $(b-a)f'(c)=f(a)-f(b)$, or $(b-a)f'(c)=0$ since $x=a,b$ are roots.</p>
|
152,912 | <p>I am reading probability theory and I have a question. The Bochner-Minlos theorem roughly says that if we have $E \subset H \subset E^*$ where $H$ is a Hilbert space, then there is a unique measure on $E^*$ satisfying some nice properties. </p>
<p>On the other hand, the Wiener measure is supported on the space of... | Martin Hairer | 38,566 | <p>Wiener measure can most definitely be characterised as the only probability measure $\mathbf{P}$ on the space $C_0$ of continuous functions starting at the origin and such that the identity
$$
\int_{C_0} \exp\Big(i \int f(t)\,\mu(dt)\Big)\,\mathbf{P}(df) = \exp \Big(-{1\over 2} \iint (s\wedge t)\,\mu(ds)\,\mu(dt)\Bi... |
301,198 | <p>I am new to linear algebra and I have a doubt that : in 2D coordinate system is a line which is at 45 degree <strong>NOT</strong> passing through the origin a subspace of the vector space comprising the whole 2D plane i.e. $ \mathbb{R}^2 $ ?
let $V = \{ (x,y) \in \mathbb{R}^2 \}$. and $W = \{ (x,y) \in \mathbb{R}^... | Thomas Andrews | 7,933 | <p>No, subspaces alway contain the null vector, since if they contain $(x,y)$, they contain $(ax,ay)$ for any real $a$, including $a=0$.</p>
|
1,378,960 | <blockquote>
<p>If $3\sin A + 5\cos A = 5$, then prove that:
$$5\sin A + 3\cos A = ±3.$$</p>
</blockquote>
| Anurag A | 68,092 | <p>The question as stated seems to be <strong><em>incorrect</em></strong>. Most likely it is asking for $$\color{blue}{5 \sin A} \color{red}{-} \color{blue}{3 \cos A}.$$
If such is the case then from the first equation if we divide throughout by $\sqrt{34}$, we get
$$\frac{3}{\sqrt{34}} \sin A + \frac{5}{\sqrt{34}} \co... |
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