qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,570,193 | <p>Let $X$ be a random variable and $\;f_X(x)=c6^{-x^2}\;\forall x\in\Bbb R$ its pdf. What I'm trying to compute is $\sqrt {Var(X)}$. I've got that $c=\sqrt{\frac{\ln(6)}{\pi}}$ for $f_X(x)$ to be a pdf and also that $\Bbb E(X)=0$. So my problem reduces to compute $\Bbb E(X^2)$ where</p>
<p>$$\Bbb E(X^2)=\sqrt{\frac{\... | Najib Idrissi | 10,014 | <p>Yes, $1 = u(1_{\mathbb{K}})$. You have a short exact sequence
$$0 \to \ker(\epsilon) \to C \xrightarrow{\epsilon} \mathbb{K} \to 0$$
and it is split by the coaugmentation $u : \mathbb{K} \to C$, so by the <a href="https://en.wikipedia.org/wiki/Splitting_lemma" rel="nofollow">splitting lemma</a>, $C \cong \ker(\epsil... |
9,930 | <p>One of the standard parts of homological algebra is "diagram chasing", or equivalent arguments with universal properties in abelian categories. Is there a rigorous theory of diagram chasing, and ideally also an algorithm?</p>
<p>To be precise about what I mean, a diagram is a directed graph $D$ whose vertices are ... | arsmath | 3,711 | <p>This is an old question, but I think I have a reasonably complete solution to this problem. It gives you a criterion for when you can answer a question by a diagram chase, and a procedure for actually doing it. I will sketch the answer here, and if anyone is still interested in this question, I will make a more fo... |
3,547,995 | <p>I've been trying to figure out the way to solve this for a while now, and I'm hoping someone could point me in the right direction to find the answer (or show me how to solve this).</p>
<p>The problem I'm having is with this equation: <span class="math-container">$(2i-2)^{38}$</span> and I need to solve it using de... | MPW | 113,214 | <p><strong>Hint:</strong> Start by writing <span class="math-container">$-2+2i$</span> in the form <span class="math-container">$re^{\theta i}$</span>.</p>
<p>Then your answer will be <span class="math-container">$r^{38}e^{38\theta i}$</span>.</p>
<p>For <span class="math-container">$z=x+iy$</span>, we have <span cla... |
3,547,995 | <p>I've been trying to figure out the way to solve this for a while now, and I'm hoping someone could point me in the right direction to find the answer (or show me how to solve this).</p>
<p>The problem I'm having is with this equation: <span class="math-container">$(2i-2)^{38}$</span> and I need to solve it using de... | giobrach | 332,594 | <p><strong>Hint.</strong> Use the fact that
<span class="math-container">$$ -2 + 2i = 2\sqrt 2\cos \frac{3\pi}4 + 2i\sqrt 2 \sin \frac{3\pi}4 = 2 \sqrt 2 e^{3\pi i/4}.$$</span></p>
|
2,251,964 | <p><strong>question(s):</strong></p>
<p>Choose any real or complex clifford algebra $\mathcal{Cl}_{p,q}$. <a href="https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras" rel="nofollow noreferrer">It's known</a> that there is some $A \simeq \mathcal{Cl}_{p,q}$, where $A$ is either a matrix ring $M(n,R)$ or... | arctic tern | 296,782 | <p>I can't say I understand the context you gave behind your question, but I think most of your questions can be answered by a primer on Clifford algebras. One thing I want to point out is that I use the opposite sign convention from you.</p>
<p>When making $\mathbb{C}$, mathematicians adjoined a square root of negati... |
2,886,675 | <p>I suspect the following is exactly true ( for positive $\alpha$ )</p>
<p>\begin{equation}
\sum_{n=1}^\infty e^{- \alpha n^2 }= \frac{1}{2} \sqrt { \frac{ \pi}{ \alpha} }
\end{equation}</p>
<p>If the above is exactly true, then I would like to know a proof of it.
I accept showing a particular limit is true, may b... | Somos | 438,089 | <p>The infinite sum is $\, (\theta_3(0,e^{-\alpha})-1)/2 \,$ where $\, \theta_3 \,$ is a Jacobi theta function. Only for small values of $\, \alpha \,$ is it approximately $\, \sqrt{\pi /\alpha}/2. \,$ Define
$\, f(\alpha) := \theta_3(0,e^{-\alpha}). \,$
Then using modular relations
$\, f(\alpha) = \sqrt{\pi/\alpha}f(\... |
2,660,934 | <p>Find $$\lim_{x \to \infty} x\sin\left(\frac{11}{x}\right)$$</p>
<p>We know $-1\le \sin \frac{11}{x} \le 1 $ </p>
<p>Therefore, $x\rightarrow \infty $
And so limit of this function does not exist. </p>
<p>Am I on the right track? Any help is much appreciated.</p>
| Atmos | 516,446 | <p>$$\sin(11/x)\underset{(+\infty)}{\sim}11/x$$
What can you deduce ?</p>
<p>Note : What you have stated is good however with the product with $x$ you cannot conclude that it converges or diverges with what you wrote</p>
|
2,660,934 | <p>Find $$\lim_{x \to \infty} x\sin\left(\frac{11}{x}\right)$$</p>
<p>We know $-1\le \sin \frac{11}{x} \le 1 $ </p>
<p>Therefore, $x\rightarrow \infty $
And so limit of this function does not exist. </p>
<p>Am I on the right track? Any help is much appreciated.</p>
| Sri-Amirthan Theivendran | 302,692 | <p>No the reasoning doesn't follow. If limit exists, then using your reasoning all we can say is it is between $-\infty$ and $\infty$. Make the change of variables $u=1/x$, and note that the limit is equivalent to
$$
\lim_{u\to 0^+}\frac{\sin 11u}{u}=11\lim_{u\to 0^+}\frac{\sin 11u}{11 u}
$$
and now use the well-known ... |
2,660,934 | <p>Find $$\lim_{x \to \infty} x\sin\left(\frac{11}{x}\right)$$</p>
<p>We know $-1\le \sin \frac{11}{x} \le 1 $ </p>
<p>Therefore, $x\rightarrow \infty $
And so limit of this function does not exist. </p>
<p>Am I on the right track? Any help is much appreciated.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>write $$11\frac{\sin(\frac{11}{x})}{\frac{11}{x}}$$ and the Limit is $$11$$</p>
|
2,660,934 | <p>Find $$\lim_{x \to \infty} x\sin\left(\frac{11}{x}\right)$$</p>
<p>We know $-1\le \sin \frac{11}{x} \le 1 $ </p>
<p>Therefore, $x\rightarrow \infty $
And so limit of this function does not exist. </p>
<p>Am I on the right track? Any help is much appreciated.</p>
| Andreas Lenz | 533,869 | <p>Your conclusion is not directly correct, since you are neglecting the $x$ in the denominator inside the $\sin \frac{11}{x}$.</p>
<p>Write</p>
<p>$$ \lim_{x \rightarrow \infty} x \sin \frac{11}{x} = \lim_{x \rightarrow \infty} \frac {\sin \frac{11}{x}}{\frac{1}{x}}. $$</p>
<p>and L'Hôpital's rule is applicable.</p... |
246,114 | <p>A Latin Square is a square of size <strong>n × n</strong> containing numbers <strong>1</strong> to <strong>n</strong> inclusive. Each number occurs once in each row and column.</p>
<p>An example of a 3 × 3 Latin Square is:</p>
<p><span class="math-container">$$
\left(
\begin{array}{ccc}
1 & 2 & 3 \\
3 &... | yode | 21,532 | <p>Since <a href="http://web.math.ucsb.edu/%7Epadraic/mathcamp_2012/latin_squares/MC2012_LatinSquares_lecture3.pdf" rel="nofollow noreferrer">any group table</a> is a Latin square. But just there is 1 group about 5 orders.</p>
<pre><code>FiniteGroupCount[5]
</code></pre>
<blockquote>
<p>1</p>
</blockquote>
<p>We know i... |
2,368,813 | <p>I'm in a bit of trouble I want to calculate the cross ratio of 4 points $ A B C D $ that are on a circle </p>
<p>Sadly "officially" it has to be calculated with A B C D as complex numbers and geometers sketchpad ( the geomerty program I am used to) don't know about complex numbers</p>
<p>Now I am wondering
The cro... | robjohn | 13,854 | <p>For $A,B,C,D$ on the unit circle,
$$
\begin{align}
\frac{(A-C)(B-D)}{(A-D)(B-C)}
&=\frac{\left(e^{ia}-e^{ic}\right)\left(e^{ib}-e^{id}\right)}{\left(e^{ia}-e^{id}\right)\left(e^{ib}-e^{ic}\right)}\\
&=\frac{\left(e^{i(a-c)}-1\right)\left(e^{i(b-d)}-1\right)}{\left(e^{i(a-d)}-1\right)\left(e^{i(b-c)}-1\right)... |
1,164,037 | <p>The question I propose is this: For an indexing set $I = \mathbb{N}$, or $I = \mathbb{Z}$, and some alphabet $A$, we can define a left shift $\sigma : A^{I} \to A^{I}$ by $\sigma(a_{k})_{k \in I} = (a_{k + 1})_{k \in I}$, because there exists a unique successor $\min \{ i > i_{0} \}$ for all $i \in I$. But one co... | Brian M. Scott | 12,042 | <p>Let $\langle I,\preceq\rangle$ be a linear order; all that’s required is that each $i\in I$ have an immediate successor $i^+$ in the order $\preceq$: $i\prec i^+$, and there is no $j\in I$ such that $i\prec j\prec i^+$.</p>
<p>Define a relation $\sim$ on $I$ as follows: for $i,j\in I$, $i\sim j$ iff either $i\prece... |
4,364,421 | <p>Are all solutions of the equation <span class="math-container">$x^2-4My^2=K^2$</span>, multiples of <span class="math-container">$K$</span>? I am considering <span class="math-container">$M$</span> not perfect square. Any tests in Python show be true, but...</p>
<p>My code:</p>
<pre><code>for x in range (1,8000):
... | Will Jagy | 10,400 | <p><span class="math-container">$$ 21^2 - 20 \cdot 4^2 = 11^2 $$</span>
<span class="math-container">$$ 21^2 - 20 \cdot 2^2 = 19^2 $$</span>
<span class="math-container">$$ 61^2 - 20 \cdot 12^2 = 29^2 $$</span>
<span class="math-container">$$ 41^2 - 20 \cdot 6^2 = 31^2 $$</span>
<span class="math-container">$$ ... |
786,643 | <p>Considering $$\int\frac{\ln(x+1)}{2(x+1)}dx$$ I first solved it seeing it similar to the derivative of $\ln^2(x+1)$ so multiplying by $\frac22$ the solution is $$\int\frac{\ln(x+1)}{2(x+1)}dx=\frac{\ln^2(x+1)}{4}+const.$$. But then we can solve it using by parts' method and so this is the solution that I found:
$$\f... | Shobhit | 79,894 | <p>Divide the last step by $2$ to get the desired answer, as $2*constant=constant$.</p>
|
121,897 | <p>I want to check if a user input the function with all the specified variables or not. For that I choose the replace variables with some values and check for if the result is a number or not via a doloop. I am thinking there might be more elegant way of doing it such as <a href="http://reference.wolfram.com/language... | BoLe | 6,555 | <p>I think it's safe to extract all symbols from the underlying expression. <code>Cases</code> doesn't look at expression heads by default, so e.g. <code>Plus</code> and <code>Power</code> aren't returned. Complement that with constant-like symbols like <code>Pi</code> which are not to be checked as variables, and fina... |
381,036 | <p>I must show that $f(x)=p{\sqrt{x}}$ , $p>0$ is continuous on the interval [0,1). </p>
<p>I'm not sure how I show that a function is continuous on an interval, as opposed to at a particular point. </p>
| Sugata Adhya | 36,242 | <h2>Hint:</h2>
<p>Choose $c\in[0,1);$</p>
<ul>
<li><p>$c>0:|x-c|<\delta\implies|\sqrt x-\sqrt c||\sqrt x +\sqrt c|<\delta\implies|\sqrt x-\sqrt c|<\dfrac{\delta}{\sqrt x+\sqrt c}\le\dfrac{\delta}{\sqrt c};$</p></li>
<li><p>$c=0:0\le x<\epsilon^2\implies0\le\sqrt x<\epsilon.$</p></li>
</ul>
|
3,299,492 | <p>Is there any nice characterization of the class of polynomials can be written with the following formula for some <span class="math-container">$c_i , d_i \in \mathbb{N}$</span>? Alternatively, where can I read more about these? do they have a name?
<span class="math-container">$$c_1 + \left( c_2 + \left( \dots (c_k ... | José Carlos Santos | 446,262 | <p>No. It means that it converges if and only if <span class="math-container">$x=0$</span>.</p>
|
3,970,488 | <p>while solving a differential equation i encounter this derivative : let <span class="math-container">$$z=\frac {dt}{dx} $$</span> i don't understand how they make that <span class="math-container">$$ \frac {dz}{dx}=z^3\frac {d^2x}{dt^2}$$</span></p>
| johnnyb | 298,360 | <p>First, it looks like you are off by a minus sign. So it should be:</p>
<p><span class="math-container">$$\frac{dz}{dx} = -z^3\frac{d^2x}{dt^2}$$</span></p>
<p>Then, it follows naturally from a more algebraic view of differentials. The standard notation for the second derivative of <span class="math-container">$x$<... |
115,387 | <p>Have two series, just a quick check of some simple series:</p>
<p>$\sum _{1}^{\infty} \frac {1}{\sqrt {2n^{2}-3}}$</p>
<p>Considering
$\frac {1}{\sqrt {2n^{2}-3}}$ > $\frac {1}{\sqrt {4n^{2}}}$ = $\frac {1}{2n}$</p>
<p>Since
$\sum _{1}^{\infty} \frac {1}{2n}$ $\rightarrow$ Diverges, hence by the camparsion tes... | Bruce George | 31,367 | <p>The first series you mention diverges, and the reason you give is correct. However, it is not correct to say "$\to$ Diverges", as this would be as saying "approaches diverges" (or "converges to diverges"), which makes no sense. You could say that it "diverges to infinity" or write $\to +\infty$. Anyway, this is mino... |
921,144 | <p>Can you give me an example of the function in metric space which is continuous but not uniformly continuous. Definitions are almost the same for both terms. This is what I found on wiki: ''The difference between being uniformly continuous, and being simply continuous at every point, is that in uniform continuity the... | creative | 166,713 | <p><em>Generally for continuity when we write $\delta,$we mean that $\delta=\delta(\epsilon,x_0),x_0\in D$. Similarly for uniform continuity we mean $\delta=\delta(\epsilon)$. This notation is consistent. It is taken for granted that we understand the situation in which $\delta$ is referred.</em> </p>
<p><em>Now for ... |
1,919,880 | <p>Let $B=\begin{bmatrix}
0 & 0 & 0 & -8 \\
1&0&0 & 16\\
0 &1&0& -14\\
0&0&1&6
\end{bmatrix}$</p>
<p>Consider B a real matrix. Find its Jordan Form. So, the characteristic polynomial for B is $(x^2-2x+2)(x-2)^2$. Suppose $B$ represents $T$ in the standard basis. B... | Christiaan Hattingh | 90,019 | <p>The matrix you start with is a companion matrix, so you know the characteristic polynomial and the minimum polynomial is the same. As you have also noticed the characteristic polynomial does not split over the real number field, and it follows that the Jordan form for this matrix does not exist over the real numbers... |
701,241 | <p><code>¬(p∨q)∧(p∨r)</code> Does this mean the negation of both <code>(p∨q)</code> and <code>(p∨r)</code> or just <code>(p∨q)</code>?
If it was just <code>p∨q</code> it would make more sense to me being inside the brackets like <code>(¬p∨q)</code> but maybe that's just the programmer in me. I have also seen <code>(¬p∨... | Senex Ægypti Parvi | 89,020 | <p>The scope of $\neg$ used just before $($ extends to the $)$ which logically CLOSES OUT the grouping which was begun by the subject $($. </p>
<p>As to your comment about $(\neg p\vee\neg q)$, $\neg(p\wedge q)$ would be the proper way to move the $\neg$ outside the parentheses.</p>
|
542,808 | <p>I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + irrational number is irrational, then it stood to reason that was also proving that the number in question was irrati... | Asaf Karagila | 622 | <p>Note that the sum of two rationals is always rational, and that if $n$ is rational then $-n$ is rational. Now suppose that $x$ is any number and $n$ is rational.</p>
<p>Suppose $x+n$ is rational, then $(x+n)+(-n)$ is rational. Therefore $x+(n+(-n))$ is rational. Therefore $x+0$ is rational, and finally $x$ is ratio... |
52,848 | <p>Let $f(z)=\sum_{n\geq 0}a_n z^n$ be a Taylor series with rational coefficients with infinitely non-zero $a_n$ which converges
in a small neighboorhood around $0$. Furthermore, assume that
\begin{align*}
f(z)=\frac{P(z)}{Q(z)},
\end{align*}
where $P(z)$ and $Q(z)$ are <em>coprime monic complex polynomials</em>. By d... | Gjergji Zaimi | 2,384 | <p>Let there be two fields $k\subset K$, and let $f\in k[[x]]$ be a formal power series with coefficients in $k$. If $f\in K(x)$ (rational functions with coefficients in $K$) then $f\in k(x)$. A proof of this is given in J.S. Milne's notes on Etale Cohomology (lemma 27.9).</p>
|
52,848 | <p>Let $f(z)=\sum_{n\geq 0}a_n z^n$ be a Taylor series with rational coefficients with infinitely non-zero $a_n$ which converges
in a small neighboorhood around $0$. Furthermore, assume that
\begin{align*}
f(z)=\frac{P(z)}{Q(z)},
\end{align*}
where $P(z)$ and $Q(z)$ are <em>coprime monic complex polynomials</em>. By d... | Hugo Chapdelaine | 11,765 | <p>Well, I think there is a simpler argument. For a power series $g(x)\in\mathbb{C}[[x]]$
and $\sigma\in Aut(\mathbb{C})$ (note that except for the complex conjugation or the identity $\sigma$ is not continuous!) we may define define the power series with coefficients twisted by $\sigma$ which we denote by $g^{\sigma}(... |
3,751,780 | <p>Given positive real numbers <span class="math-container">$a, b, c$</span> with <span class="math-container">$ab + bc + ca = 1.$</span> Prove that <span class="math-container">$$ \sqrt{a^{2} + 1} + \sqrt{b^{2} + 1} + \sqrt{c^{2} + 1}\leq 2(a+b+c).$$</span></p>
<p>I have no idea to prove this inequality.</p>
| farruhota | 425,072 | <p>Alternatively, square both sides:
<span class="math-container">$$\small{2\left[\sqrt{(a^2+1)(b^2+1)}+\sqrt{(b^2+1)(c^2+1)}+\sqrt{(c^2+1)(a^2+1)}\right]}\le \\
3(a^2+b^2+c^2)+5$$</span>
By AM-GM:
<span class="math-container">$$2\sqrt{(a^2+1)(b^2+1)}\le a^2+b^2+2$$</span>
Hence, we need to prove:
<span class="math-con... |
2,932,305 | <p>What are the intercepts of the planes <span class="math-container">$x = 0$</span> and <span class="math-container">$2y + 3z = 12$</span>? The word intercept is confusing me because I don't understand if I should say they intersect at point <span class="math-container">$(0,6,0)$</span> or the intercept is at <span cl... | vanmeri | 218,238 | <p>In R <span class="math-container">$^3$</span> , it is the line
(x, y, z) = t(0,-3,2) + (0,6,0)
You may find this by taking any vector (x, y, z) and asking when it satisfies both equations.
The planes aren't parallel.</p>
|
272,144 | <p>Consider a multi-value function <span class="math-container">$f(z)=\sqrt{(z-a)(z+\bar a)}, \Im{a}>0,\Re{a}>0$</span>. To make the function be single-valued, one needs to make a cut. Suppose <span class="math-container">$a=e^{i\theta}$</span>, my choice of the branch cut is <span class="math-container">$e^{it},... | josh | 81,539 | <p><strong>Edit3: Added a ComplexContourPlot of level curves over radial branch region. See below</strong></p>
<p>With these problems I find it helpful to draw the function in its entirety then decide how to cut out an analytically-continuous single-valued section of it. Unfortunately in this case it's a little diff... |
80,056 | <p>I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this. </p>
<p>Obviously, slides make it possible to produce and show clear graphs/pictu... | David White | 11,540 | <p>I took a course at PCMI some years ago from <a href="http://people.reed.edu/~davidp/homepage/teaching.html" rel="nofollow">David Perkinson</a> (Reed College). He did an amazing job and single-handedly convinced me it was possible to teach well from slides. Check out <a href="http://people.reed.edu/~davidp/pcmi/index... |
424,675 | <p>Just one simple question:</p>
<p>Let $\tau =(56789)(3456)(234)(12)$.</p>
<p>How many elements does the conjugacy class of $\tau$ contain? How do you solve this exersie?</p>
<p>First step is to write it in disjunct cyclces I guess. What's next? :)</p>
| Amzoti | 38,839 | <p>This is a difficult question to answer.</p>
<blockquote>
<p>"The FDM is the oldest and is based upon the application of a local
Taylor expansion to approximate the differential equations. The FDM
uses a topologically square network of lines to construct the
discretization of the PDE. This is a potential bot... |
466,576 | <p>Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the German word <em>Zahl(en)</em> 'number(s)'. What are some more examples?</p>
| rschwieb | 29,335 | <p>$\ln()$ for "logarithmus naturalis"?</p>
<p>My advisor also told me that the "socle of a ring" makes a little more sense when you know that "socle" is an architecture term for the support underneath a column or pedastal, and so the socle of a ring acts as a kind of "support for the ring." In some languages, the wor... |
1,516,363 | <p>If $f: (0,+ \infty) \rightarrow \mathbb{R}$ is continuous and $$f(x + y) = f(x) + f(y) ,$$ then $f$ is linear? </p>
<p>I saw in somewhere that if $f$ is continuous, one can drop the condition $f(ax)= a f(x), \forall a, x \in \mathbb{R}$. Is this true?</p>
| JMP | 210,189 | <p>Let $y=(a-1)x$ so that $f(x+y)=f(x+(a-1)x)=f(ax)=f(x)+f((a-1)x)$</p>
<p>Write $(a-1)x$ as $(a-2)x+x)$ so that $f((a-1)x)=f((a-2)x)+f(x)$</p>
<p>Repeat this $k$ times until $0\le a-k \lt 1$, and let $a_1=a-k$.</p>
<p>So far we have $f(ax)=kf(x)+f(a_1)$</p>
<p>We continue for example by considering $a_1-0.1k_1$, $... |
2,236,846 | <p>For $x\in A[a,b]$</p>
<p>$\sup_{x\in A}|f(x)|\ge\int_{a}^{b}f(x)dx$</p>
<p>I'm just wondering if this is an analysis result or if the result is slightly different to this?</p>
<p>Sorry I just realised it was a greater than sign not an equals!</p>
| Community | -1 | <p>No, for a counterexample take $f(x)= 0$ if $x\neq a$ and $f(a)=1$. Then $$\sup_{[a,b]}f(x)=1\neq 0=\int_a^b f(x)dx$$</p>
|
1,354,745 | <p>Let polynomial $p(z)=z^2+az+b$ be such that $a$and $b$ are complex numbers and $|p(z)|=1$ whenever $|z|=1$. Prove that $a=0$ and $b=0$.</p>
<p>I could not make much progress.
I let $z=e^{i\theta}$ and $a=a_1+ib_1$ and $b=a_2+ib_2$ </p>
<p>Using these values in $P(z)$ i got
$|P (z)|^2=1=(\cos (2\theta)-a_2\sin (\th... | Mercy King | 23,304 | <p>For every $z\in \mathbb{C}$ we have
$$
|p(z)|^2=(z^2+az+b)(\bar{z}^2+\bar{a}\bar{z}+\bar{b})=|z|^4+a\bar{z}|z|^2+\bar{a}z|z|^2+|a|^2|z|^2+\bar{b}z^2+b\bar{z}^2+a\bar{b}z+\bar{a}b\bar{z}+|b|^2,
$$
in particular, when $|z|=1$, we have:
$$
|p(z)|^2=\bar{b}z^2+b\bar{z}^2+(a+\bar{a}b)\bar{z}+(\bar{a}+a\bar{b})z+|a|^2+|b|... |
2,864,992 | <p>It starts by someone asking an exercise question that whether negation of</p>
<pre><code>2 is a rational number
</code></pre>
<p>is</p>
<pre><code>2 is an irrational number
</code></pre>
<p>Their argument is that they consider it is incorrect if we include complex numbers, because 2 might be complex and not irra... | robjohn | 13,854 | <p>Let
$$
f(x)=\log_2(1-x)+\sum_{k=0}^\infty x^{2^k}\tag1
$$
then $f(0)=0$ and
$$
f\!\left(x^2\right)=\log_2\left(1-x^2\right)+\sum_{k=1}^\infty x^{2^k}\tag2
$$
and therefore,
$$
f(x)-f\!\left(x^2\right)=x-\log_2(1+x)\tag3
$$
Thus, for $x\in(0,1)$,
$$
\begin{align}
f(1)
&=f(1)-f(0)\\[12pt]
&=\sum_{k=-\infty}^\i... |
272,114 | <p>Yesterday, my uncle asked me this question:</p>
<blockquote>
<p>Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$.</p>
</blockquote>
<p>How can we do this? Note that this is not a diophantine equation since $x \in \mathbb{R}$ if you are thinking about Fermat's Last Theorem.<... | Did | 6,179 | <p>One looks for roots of the function $f:x\mapsto a^x+1-b^x$ with $a=\frac34$ and $b=\frac54$.</p>
<ul>
<li>Since $a\lt1$, the function $x\mapsto a^x$ is decreasing. </li>
<li>Since $b\gt1$, the function $x\mapsto b^x$ is increasing. </li>
<li>Hence the function $f$ is decreasing.</li>
<li>And $f(\pm\infty)=\mp\infty... |
2,826,313 | <p>So I have been given the following equation : $z^6-5z^3+1=0$. I have to calculate the number of zeros (given $|z|>2$). I already have the following:</p>
<p>$|z^6| = 64$ and $|-5z^3+1| \leq 41$ for $|z|=2$. By Rouche's theorem: since $|z^6|>|-5z^3+1|$ and $z^6$ has six zeroes (or one zero of order six), the fu... | Henrik supports the community | 193,386 | <p>For polynomials (and other functions where you know the total number of roots) the standard way is simply to subtract the number of zeroes inside or on the boundary of the disc, which in this case turns out to be $6-6$.</p>
<p>For more complex functions you can sometimes use Rouche's theorem on $f(\frac{1}{z})$, bu... |
3,002,668 | <p>I have to solve this inequality:</p>
<p><span class="math-container">$$5 ≤ 4|x − 1| + |2 − 3x|$$</span></p>
<p>and prove its solution with one (or 2 or 3) of this sentences:</p>
<p><span class="math-container">$$∀x∀y |xy| = |x||y|$$</span></p>
<p><span class="math-container">$$∀x∀y(y ≤ |x| ↔ y ≤ x ∨ y ≤ −x)$$</s... | farruhota | 425,072 | <p>The sentences (rules) you stated will not be sufficient (applicable), because the given inequality is not a single absolute value, but a sum of two.</p>
<p>The standard <strong>algebraic</strong> method to solve the absolute value inequality is to divide into intervals:
<span class="math-container">$$5 ≤ 4|x − 1| +... |
3,076,253 | <p>Problem: Let <span class="math-container">$I=[0,1]$</span> be the closed unit interval. Suppose <span class="math-container">$f$</span> is a continuous mapping of <span class="math-container">$I$</span> into <span class="math-container">$I$</span>. Prove that <span class="math-container">$f(x)=x$</span> for at least... | Anthony Ter | 588,654 | <p>If <span class="math-container">$\sigma \in S_n$</span> then <span class="math-container">$\sigma \circ \tau^{-1} \in S_n$</span>, and <span class="math-container">$\tau^{-1} \circ \sigma \in S_n$</span> so multiplying by <span class="math-container">$\sigma$</span> on either side is a surjective map <span class="ma... |
1,090,620 | <p>I don't know how to solve this limit</p>
<p>$$ \lim_{y\to0} \frac{x e^ { \frac{-x^2}{y^2}}}{y^2}$$</p>
<p>$\frac{1}{e^ { \frac{x^2}{y^2}}} \to 0$</p>
<p>but $\frac{x}{y^2} \to +\infty$</p>
<p>This limit presents the indeterminate form $0 \infty$ ?</p>
| Jack D'Aurizio | 44,121 | <p>$$\lim_{y\to 0}\frac{x}{y^2}e^{-\frac{x^2}{y^2}} = \lim_{n\to +\infty} nx e^{-nx^2}\leq\lim_{n\to +\infty}\frac{nx}{\left(1+\frac{n}{2}x^2\right)^2}=0. $$</p>
|
513,500 | <p>Suppose $f,g$ are analytic functions in domain $D$.If $fg=0$, I want to prove either $f(z)=0$ or $g(z)=0$. </p>
| Anupam | 84,126 | <p>If possible, let $f,g$ be not identically zero functions. Let $K$ be any compact subset of $D$. If $Z_f$ and $Z_g$ denote the set zeros of $f$ and $g$, then $K\cap Z_f$ and $K\cap Z_g$ are finite sets. Now we have, $\vert K\cap Z_{fg}\vert\leq \vert K\cap Z_f\vert+\vert K\cap Z_g\vert<\infty$. It follows that $Z_... |
1,111,935 | <p>For a given $n \in \Bbb N$, how do you find the minimum $m \in \Bbb N$ which satisfies the inequality below?</p>
<p>$$3^{3^{3^{3^{\unicode{x22F0}^{3}}}}} (m \text{ times}) > 9^{9^{9^{9^{\unicode{x22F0}^{9}}}}} (n \text{ times})$$</p>
<p>What I have tried to do so far is decomposing the $9$ on the right side to ... | Booldy | 261,261 | <p>Let $a_1=3,b_1=9,a_{n+1}=3^{a_n},b_{n+1}=9^{b_n}$ obviously we have $a_n ,b_n \in N$</p>
<p>$a_1<b_1$ .Suppose $a_n<b_n$ then</p>
<p>$$a_{n+1}=3^{a_n}<3^{b_n}<9^{b_n}=b_{n+1}$$</p>
<p>$a_2>2b_1$ Suppose $a_n>2b_{n-1}$ then</p>
<p>$$a_{n+1}=3^{a_n}\ge3^{2b_{n-1}+1}=3\cdot 9^{b_{n-1}}=3b_n>2b_... |
2,917,299 | <p><strong>Let $h: R\to S$ be a ring homomorphism. Let $P\subset R$ be a prime ideal.</strong>
<strong>Give an example to show that in general $h(P)$ is not an ideal of $S$</strong></p>
<p>The first thing I think is to take $R=\mathbb{Z}$ and $P=(2)$ but I do not know how to take $S$ or if this works in this way, any ... | Bajo Fondo | 411,935 | <p>You can consider $R=\mathbb{R}[x]$ and $S=\mathbb{C}$ and $h:R \to S$ as in $h(p)=p(1)$.</p>
<p>You can see that $<x>$ is prime in $R$. And $h(R)=\mathbb{R}$, which is not an ideal in $\mathbb{C}$.</p>
|
3,287,710 | <p>I want to calculate the length of a clothoid segment from the following available information.</p>
<ol>
<li>initial radius of clothoid segment </li>
<li>final radius of clothoid segment</li>
<li>angle (i am not really sure which angle is this, and its not
documented anywhere)</li>
</ol>
<p>As a test case: I need t... | Robert Israel | 8,508 | <p>Switch to polar coordinates <span class="math-container">$x = r \cos(\theta)$</span>, <span class="math-container">$y=r \sin(\theta)$</span>, and you can explicitly parametrize your curve as <span class="math-container">$r = R(\theta)$</span>. The <a href="https://math.stackexchange.com/questions/76708/how-to-deter... |
818,161 | <p>Suppose that repetitions are not allowed.</p>
<p>There are $6 \cdot 5 \cdot 4 \cdot 3 $ numbers with $4$ digits , that can be formed from the digits $1,2,3,5,7,8$.</p>
<p>How many of them contain the digits $3$ and $5$?</p>
<p>I thought that I could subtract from the total number of numbers those,that do not cont... | André Nicolas | 6,312 | <p>You have used a correct Stars and Bars argument to show that there are $462$ ways to distribute $6$ objects in $6$ boxes. </p>
<p>However, if we assume that the dice are fair, and do not influence each other, then these $462$ possibilities are <em>not all equally likely</em>.</p>
<p>Let us look at a much smaller e... |
1,705,481 | <blockquote>
<p>$$\sum_{n=1}^{\infty} \frac{\ln(n)}{n^2}$$</p>
</blockquote>
<p>I have tried the comparison test with $\frac{1}{n}$ and got $0$ with $\frac{1}{n^2}$ I got $\infty$</p>
<p>What should I try?</p>
| Olivier Oloa | 118,798 | <p>Since $ x \mapsto \dfrac{\ln x}{x^2}$ is decreasing over $[1,+\infty)$, one may use the <a href="https://en.wikipedia.org/wiki/Integral_test_for_convergence" rel="nofollow">integral test </a>:
$$
\sum_{n=1}^N\frac{\log\left(n\right)}{n^{2}}\leq \frac{\log\left(N\right)}{N^{2}}+\int_1^N\frac{\log\left(t\right)}{t^2}d... |
1,705,481 | <blockquote>
<p>$$\sum_{n=1}^{\infty} \frac{\ln(n)}{n^2}$$</p>
</blockquote>
<p>I have tried the comparison test with $\frac{1}{n}$ and got $0$ with $\frac{1}{n^2}$ I got $\infty$</p>
<p>What should I try?</p>
| Solumilkyu | 297,490 | <p>We first claim that there exists a positive integer $N$ such that
$$\ln(n)\leq n^{1/2},\quad\forall n>N.$$
By L'Hopital's Rule,
$$\lim_{n\rightarrow\infty}\frac{\ln(n)}{n^{1/2}}=
\lim_{n\rightarrow\infty}\frac{1/n}{\frac{1}{2}n^{-1/2}}=
2\lim_{n\rightarrow\infty}\frac{1}{n^{1/2}}=0.$$
It follows that there ex... |
671,160 | <p>I was in maths class and i found a question interesting.
Find the limit of $\lim_{(x,y)\to (0,0)} \frac{2x}{x^2+x+y^2}$ if it exist.one of my friend did this question by transforming into polar coordinates and limit become $$\lim_{r\to 0}\frac{2\cos \theta}{r^2+\cos\theta}=\frac{2\cos \theta}{\cos\theta}=2$$ and arg... | Martingalo | 127,445 | <p>Your computation is right. If you substitute $y^2=mx$ you get that the limit depends on the path you take, hence it does not exist. Another way to show that it does not exist is taking the iterated limits:</p>
<p>$$\lim_{x\to 0}\lim_{y\to 0}\frac{2x}{x^2+x+y^2}=2$$
while</p>
<p>$$\lim_{y\to 0}\lim_{x\to 0}\frac{2x... |
671,160 | <p>I was in maths class and i found a question interesting.
Find the limit of $\lim_{(x,y)\to (0,0)} \frac{2x}{x^2+x+y^2}$ if it exist.one of my friend did this question by transforming into polar coordinates and limit become $$\lim_{r\to 0}\frac{2\cos \theta}{r^2+\cos\theta}=\frac{2\cos \theta}{\cos\theta}=2$$ and arg... | copper.hat | 27,978 | <p>There is no limit. Choose $x>0, m>0$, and $y=\sqrt{mx}$, then ${2x \over x^2 +x + y^2} = {2x \over x^2+(1+m)x} = { 2 \over x+(1+m)}$, and $\lim_ {x \downarrow 0} { 2 \over x+(1+m)} = { 2 \over 1+m}$. Since $m>0$ was arbitrary, there is no limit of the original function.</p>
<p>Your friend's technique would... |
45,163 | <p>I would like to get reccomendations for a text on "advanced" vector analysis. By "advanced", I mean that the discussion should take place in the context of Riemannian manifolds and should provide coordinate-free definitions of divergence, curl, etc. I would like something that has rigorous theory but also plenty of ... | J126 | 2,838 | <p>You might want to check out <a href="http://rads.stackoverflow.com/amzn/click/0486640396" rel="nofollow">Tensor Analysis on Manifolds</a> by Bishop and Goldberg.</p>
|
45,163 | <p>I would like to get reccomendations for a text on "advanced" vector analysis. By "advanced", I mean that the discussion should take place in the context of Riemannian manifolds and should provide coordinate-free definitions of divergence, curl, etc. I would like something that has rigorous theory but also plenty of ... | ItsNotObvious | 9,450 | <p>I have actually found something that comes pretty close to what I was looking for: Morita's <a href="http://rads.stackoverflow.com/amzn/click/0821810456">Geometry of Differential Forms</a>. While not a full-blown Riemannian geometry text, it seems to strike a nice balance between theory and computation and discusses... |
4,147,126 | <p>Let <span class="math-container">$\ p_n\ $</span> be the <span class="math-container">$\ n$</span>-th prime number.</p>
<blockquote>
<p>Does the <a href="https://en.wikipedia.org/wiki/Prime_number_theorem" rel="nofollow noreferrer">prime number theorem</a> ,</p>
<p><span class="math-container">$\Large{\lim_{x\to\inf... | OmG | 356,329 | <p>You can also use results from the <a href="https://en.wikipedia.org/wiki/Prime_gap" rel="nofollow noreferrer">prime gap</a> problem. Here, as <span class="math-container">$\lim_{n\to\infty} \frac{g_n}{p_n} = 0$</span> and <span class="math-container">$g_n = p_{n+1} - p_n$</span>, you can conclude that <span class="m... |
3,980,845 | <p>Given <span class="math-container">$X,Y$</span> i.i.d where <span class="math-container">$\mathbb{P}(X>x)=e^{-x}$</span> for <span class="math-container">$x\geq0$</span> and <span class="math-container">$\mathbb{P}(X>x)=1$</span> for all <span class="math-container">$x<0$</span>
and <span class="math-contai... | StubbornAtom | 321,264 | <p>The conditional pdf <span class="math-container">$f_{V\mid X}$</span> that you write is not defined since the joint density <span class="math-container">$f_{V,X}$</span> does not exist wrt Lebesgue measure. This is because <span class="math-container">$V=X$</span> has a positive probability.</p>
<p>You can write <s... |
182,510 | <p>Is there a continuous probability measure on the unit circle in the complex plane - $\sigma$ with full support, such that $\hat{\sigma}(n_k)\rightarrow1$ as $k\rightarrow\infty$ for some increasing sequence of integers $\ n_k$ </p>
| Michael Hardy | 11,667 | <p>If I'm not mistaken, the <a href="http://en.wikipedia.org/wiki/Riemann%E2%80%93Lebesgue_lemma" rel="nofollow">Riemann–Lebesgue lemma</a> rules this out if "continuous" means absolutely continuous with respect to Lebesgue measure. I'm not sure what happens if you mix in a "continuous singular" measure.</p>
<p>(If y... |
655,981 | <p>How to calculate this complex integral?
$$\int_0^{2\pi}\cot(t-ia)dt,a>0$$</p>
<p>I got that the integral is $2\pi i$ if $|a|<1$ and $0$ if $a>1$
yet, friends of mine got $2\pi i$ regardless the value of $a$.
looking for the correct way</p>
| Ron Gordon | 53,268 | <p>Expand the cotangent to get that the integral is a ratio of sines and cosines:
$$\int_0^{2 \pi} dt \frac{\cosh{a} \cos{t} + i \sinh{a} \sin{t}}{\cosh{a} \sin{t} - i \sinh{a} \cos{t}} $$</p>
<p>Now use the usual $z=e^{i t}$, $dt = -i dz/z$ to get the integral is equal to</p>
<p>$$\oint_{|z|=1} \frac{dz}{z} \frac{e^... |
935,506 | <p>I'm a bit puzzled by this one.</p>
<p>The domain $X = S(0,1)\cup S(3,1)$ (where $S(\alpha, \rho)$ is a circular area with it's center at $\alpha$ and radius $\rho$). So the domain is basically two circles with radius 1 and centers at 0 and 3.</p>
<p>I'm supposed to find analytic function $f$ defined on $X$ where t... | Olivier Oloa | 118,798 | <p>You may write
$$
\begin{align}
\int 4x \sqrt{1 - x^4} dx & =2\int 2x \sqrt{1 - (x^2)^2} dx \\\\
& =2\int \sqrt{1 - u^2} du \\\\
& =2\int \cos t \:\sqrt{1 - \sin^2 t} \: dt \\\\
& =2\int \cos^2 t \: dt \\\\
& =t+\frac12 \sin (2t)+C\\\\
& =t+\sin t \cos t+C\\\\
& =\arcsin (x^2)+x^2\:\sqrt{1... |
2,195,287 | <blockquote>
<p>Knowing that $p$ is prime and $n$ is a natural number show that
$$n^{41}\equiv n\bmod 55$$
using Fermat's little theorem
$$n^p\equiv n\bmod p$$</p>
</blockquote>
<p>If the exercise was to show that
$$n^{41}\equiv n\bmod 11$$ I would just rewrite $n^{41}$ as a power of $11$ and would easily prov... | Joffan | 206,402 | <p>You have two Fermat's Little Theorem results that you can use:</p>
<p>$$n^5 \equiv n \bmod 5 \\ n^{11} \equiv n \bmod 11 $$</p>
<p>Then successive application of these - for example, $n^9 \equiv n^5n^4 \equiv n\cdot n^4 \equiv n^5 \equiv n \bmod 5$ - gives </p>
<p>$$n^{41} \equiv n \bmod 5 \\ n^{41} \equiv n \bm... |
2,751,909 | <blockquote>
<p>Let $f$ be a non-negative differentiable function such that $f'$ is continuous and
$\displaystyle\int_{0}^{\infty}f(x)\,dx$ and $\displaystyle\int_{0}^{\infty}f'(x)\,dx$ exist.</p>
<p>Prove or give a counter example: $f'(x)\overset{x\rightarrow
\infty}{\rightarrow} 0$</p>
</blockquote>
<p><str... | Robert Z | 299,698 | <p>Hint. Take as $P_n$ the uniform partition $x_k=\frac{k}{n}$ with $k=-n,\dots,n$ where $n$ is a positive integer. Then, since $f(x)=x$ is strictly increasing, it follows that
$$U(f,P_n)=\frac{1}{n}\sum_{k=-n+1}^n\frac{k}{n}\quad\mbox{and}\quad L(f,P_n)=\frac{1}{n}\sum_{k=-n}^{n-1}\frac{k}{n}.$$
Are you able to find $... |
2,611,382 | <p>Solve the equation,</p>
<blockquote>
<p>$$
\sin^{-1}x+\sin^{-1}(1-x)=\cos^{-1}x
$$</p>
</blockquote>
<p><strong>My Attempt:</strong>
$$
\cos\Big[ \sin^{-1}x+\sin^{-1}(1-x) \Big]=x\\
\cos\big(\sin^{-1}x\big)\cos\big(\sin^{-1}(1-x)\big)-\sin\big(\sin^{-1}x\big)\sin\big(\sin^{-1}(1-x)\big)=x\\
\sqrt{1-x^2}.\sqrt{2x... | Michael Rozenberg | 190,319 | <p>The domain gives
$$-1\leq x\leq1$$ and $$-1\leq1-x\leq1,$$ which gives $$0\leq x\leq1,$$
which says that the answer is $$\left\{\frac{1}{2},0\right\}.$$
I think it's better after your third step to write
$$\sqrt{2x-x^2}=\sqrt{1-x^2}$$ or $x=0$.</p>
|
2,184,056 | <p>To compute the oblique asymptote as $x \to +\infty$, we can first compute $\mathop {\lim }\limits_{x \to + \infty } \frac{{f(x)}}{x}$, it it exists, and $\mathop {\lim }\limits_{x \to + \infty } \frac{{f(x)}}{x} = k$, then we can further compute $\mathop {\lim }\limits_{x \to + \infty } (f(x) - kx)=b$, and if it ... | levap | 32,262 | <p>It doesn't. Consider for example $f(x) := x + \sin(x)$. Then</p>
<p>$$ \lim_{x \to \infty} \frac{f(x)}{x} = \lim_{x \to \infty} \frac{x + \sin(x)}{x} = 1 + \lim_{x \to \infty} \frac{\sin(x)}{x} = 1 $$</p>
<p>because $\sin$ is bounded. It is clear that $g(x) = x$ has an asymptote at infinity but by adding the bound... |
410,013 | <p><strong>Short question:</strong> Is there a standard term for a set <span class="math-container">$F$</span> such that there does not exist a surjection <span class="math-container">$F \twoheadrightarrow \omega$</span> (in the context of ZF)?</p>
<p><strong>More detailed version:</strong> Consider the following four ... | Asaf Karagila | 7,206 | <p>The term you might find in the literature is "weakly Dedekind finite", since a set that maps onto <span class="math-container">$\omega$</span> is weakly Dedekind <em>infinite</em>.</p>
<p>I'd expect that you'll call these "strongly Dedekind finite". Alas, these terms were coined before my arrival... |
410,013 | <p><strong>Short question:</strong> Is there a standard term for a set <span class="math-container">$F$</span> such that there does not exist a surjection <span class="math-container">$F \twoheadrightarrow \omega$</span> (in the context of ZF)?</p>
<p><strong>More detailed version:</strong> Consider the following four ... | Goldstern | 14,915 | <p>My old paper</p>
<ul>
<li>"Strongly amorphous sets and dual Dedekind infinity",
Math. Logic Quart. 43 (1997), no. 1, 39–44,</li>
<li><a href="https://onlinelibrary.wiley.com/doi/10.1002/malq.19970430105" rel="nofollow noreferrer">https://onlinelibrary.wiley.com/doi/10.1002/malq.19970430105</a></li>
<li>pre... |
2,098,693 | <p>Full Question: Five balls are randomly chosen, without replacement, from an urn that contains $5$
red, $6$ white, and $7$ blue balls. What is the probability of getting at least one ball of
each colour?</p>
<p>I have been trying to answer this by taking the complement of the event but it is getting quite complex. A... | Arthur | 15,500 | <p>Can you calculate the probability that you never draw a white ball? Never a red ball? Never a blue ball? Adding those together is <em>almost</em> the correct answer.</p>
<p>Here is what is missing: The case where you draw <em>only</em> white was counted twice (once as part of "never red", and once as part of "never... |
2,098,693 | <p>Full Question: Five balls are randomly chosen, without replacement, from an urn that contains $5$
red, $6$ white, and $7$ blue balls. What is the probability of getting at least one ball of
each colour?</p>
<p>I have been trying to answer this by taking the complement of the event but it is getting quite complex. A... | Song Xie | 407,081 | <p>Key is to find how many cases that at least one color is missing. For red absence it is (5 out of 13). For the other two, (5 our of 12) and (5 out of 11). We sum them up and subtract duplicated counts, which are (5 out of 5), (5 out of 6) and (5 out of 7) respectively. So number of cases of at least one color absenc... |
3,426,756 | <p>From a point <span class="math-container">$O$</span> on the circle <span class="math-container">$x^2+y^2=d^2$</span>, tangents <span class="math-container">$OP$</span> and <span class="math-container">$OQ$</span> are drawn to the ellipse <span class="math-container">$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$</span>, <span ... | Jan-Magnus Økland | 28,956 | <p>Still not what you want, but uses <a href="https://www.cut-the-knot.org/Generalization/JoachimsthalsNotations.shtml" rel="nofollow noreferrer">Joachimsthals notations</a>.</p>
<p>The locus is the midpoint of the two intersection points of <span class="math-container">$$s=\frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0$$</span>... |
1,338,980 | <p>Suppose you have a set of data $\{x_i\}$ and $\{y_i\}$ with $i=0,\dots,N$. In order to find two parameters $a,b$ such that the line
$$
y=ax+b,
$$
give the best linear fit, one proceed minimizing the quantity
$$
\sum_i^N[y_i-ax_i-b]^2
$$
with respect to $a,b$ obtaining well know results. </p>
<p>Imagine now to desi... | Community | -1 | <p>If you fix $p$ and perform a line fitting on the set $(x_i^p,y_i)$, you can compute the residual error (unexplained variance). This defines an objective function $\epsilon(p)$, that you can minimize by means of a numerical method such as the Golden section search.</p>
|
131,524 | <p>I am an amateur mathematician, and certainly not a set theorist, but there seems to me to be an easy way around the reflexive paradox: Add to set theory the primitive $A(x,y)$, which we may think of as meaning that $x$ is allowed to belong to $y$ and the axiom</p>
<p>$\forall x,\forall y, x\in y \rightarrow A(x,y)$... | Richard Thrasher | 34,276 | <p>I've just read an interesting paper that addresses the question of how best to remove paradoxes from the naive abstraction axiom. Reference is:</p>
<p>Goldstein, L. 2013. Paradoxical partners: semantical brides and
set-theoretical grooms. Analysis 73: 33-37.</p>
<p>His idea is quite simple. Keep the abstraction ax... |
1,338,832 | <p>Assume we have a group consisting of both women and men. (In my example it is 67 women and 43 men but that is not important.) The women are indistinguishable and the men are also indistinguishable.</p>
<p>In how many ways can we pick a subgroup consisting of $n$ women and $n$ men, i.e., the same number of women and... | drhab | 75,923 | <p>There are $2n$ spots and exactly $n$ of them must be chosen (e.g. for men to take them). </p>
<p>This can be done on $$\binom{2n}n$$ distinct ways.</p>
<hr>
<p><strong>Edit</strong>: </p>
<p>Let's do it for $n=3$. Give the men the numbers $1,2,3$ and give the women the numbers $4,5,6$. Then there are $6!$ arrang... |
1,030,335 | <blockquote>
<p>Let <span class="math-container">$n$</span> and <span class="math-container">$r$</span> be positive integers with <span class="math-container">$n \ge r$</span>. Prove that:</p>
<p><span class="math-container">$$\binom{r}{r} + \binom{r+1}{r} + \cdots + \binom{n}{r} = \binom{n+1}{r+1}.$$</span></p>
</bloc... | arindam mitra | 30,981 | <h2>Proving by induction</h2>
<h3>inductive step</h3>
<p><span class="math-container">$$\begin{pmatrix}r\\r\end{pmatrix} + \begin{pmatrix}r+1\\r\end{pmatrix} + \dots + \begin{pmatrix}n\\r\end{pmatrix} =\begin{pmatrix}n\\r+1\end{pmatrix} + \begin{pmatrix}n\\r\end{pmatrix}$$</span> [as the identity holds for natural numb... |
2,779,152 | <p>Consider a Poisson process with rate $\lambda$ in a given time interval $[0,T]$. The inter-arrival time between successive arrivals is negative exponential distributed with mean $\frac{1}{\lambda}$ such that $X_1 >0$, and $\sum_{i=1}^\text{Last} X_i < T$, where $X$ represents inter-arrival time.</p>
<p>What a... | Michael Hardy | 11,667 | <p>Use the recurrence relation to prove by mathematical induction that $a_{n+1} \ge \dfrac 3 2 a_n$ for $n\ge 3.$ Deduce from that, that $a_n \ge \left(\frac 3 2\right)^{n-3} \cdot 2$ for $n\ge 3.$ Hence
$$
\frac 1 {a_n} \le 2 \cdot \left( \frac 2 3 \right)^{n-3}
$$
so you have a comparison with a geometric series.</p>... |
3,168,787 | <p>I am trying to solve the following exercise in Rudin's "Rudin's Principles of Mathematical Analysis" book: (Ex 4.1)</p>
<blockquote>
<p>Suppose <span class="math-container">$f$</span> is a real function defined on <span class="math-container">$R^1$</span> which satisfies
<span class="math-container">$$\lim... | Community | -1 | <p>Let's look at it in a sample case. We want to prove by DCT that <span class="math-container">$$\lim_{\varepsilon\to0^+} \int_0^\infty e^{-y/\varepsilon}\,dy=0$$</span></p>
<p>This is the case if and only if for all sequences <span class="math-container">$\varepsilon_n\to 0^+$</span> it holds <span class="math-conta... |
3,168,787 | <p>I am trying to solve the following exercise in Rudin's "Rudin's Principles of Mathematical Analysis" book: (Ex 4.1)</p>
<blockquote>
<p>Suppose <span class="math-container">$f$</span> is a real function defined on <span class="math-container">$R^1$</span> which satisfies
<span class="math-container">$$\lim... | Alex Ortiz | 305,215 | <p>The statement of the dominated convergence theorem (DCT) is as follows:</p>
<blockquote>
<p><strong>"Sequential" DCT.</strong> Suppose <span class="math-container">$\{f_n\}_{n=1}^\infty$</span> is a sequence of (measurable) functions such that <span class="math-container">$|f_n| \le g$</span> for some inte... |
1,658,577 | <p>I'm an electrical/computer engineering student and have taken fair number of engineering math courses. In addition to Calc 1/2/3 (differential, integral and multivariable respectfully), I've also taken a course on linear algebra, basic differential equations, basic complex analysis, probability and signal processing... | runaround | 310,548 | <p>Real Analysis->Elementary Number Theory->Group Theory. The rest is your own interests.</p>
|
1,658,577 | <p>I'm an electrical/computer engineering student and have taken fair number of engineering math courses. In addition to Calc 1/2/3 (differential, integral and multivariable respectfully), I've also taken a course on linear algebra, basic differential equations, basic complex analysis, probability and signal processing... | Dave L. Renfro | 13,130 | <p>Consider going through <a href="http://rads.stackoverflow.com/amzn/click/0914098918" rel="nofollow noreferrer"><strong>Calculus</strong></a> by Michael Spivak or <a href="http://rads.stackoverflow.com/amzn/click/354065058X" rel="nofollow noreferrer"><strong>Introduction to Calculus and Analysis</strong></a> (Volume ... |
19,261 | <p>Every simple graph $G$ can be represented ("drawn") by numbers in the following way:</p>
<ol>
<li><p>Assign to each vertex $v_i$ a number $n_i$ such that all $n_i$, $n_j$ are coprime whenever $i\neq j$. Let $V$ be the set of numbers thus assigned. <br/></p></li>
<li><p>Assign to each maximal clique $C_j$ a unique p... | David Bar Moshe | 1,059 | <p>Here is an example where topological objects are constructed from geometrical data through representation theory. Let G/P be a flag variety of a complex Lie group G. Let G0 be a real form of G, and D be an open orbit of G0 in G/P. The Dolbeault cohomology spaces H^n(D, L) of line bundles over D carry irreducible rep... |
581,257 | <p>I would like to see a proof of when equality holds in <a href="https://en.wikipedia.org/wiki/Minkowski_inequality" rel="nofollow noreferrer">Minkowski's inequality</a>.</p>
<blockquote>
<p><strong>Minkowski's inequality.</strong> If <span class="math-container">$1\le p<\infty$</span> and <span class="math-contain... | Daniel Fischer | 83,702 | <p>For <span class="math-container">$p = 1$</span>, the proof uses the triangle inequality, <span class="math-container">$\lvert f(x) + g(x)\rvert \leqslant \lvert f(x)\rvert + \lvert g(x)\rvert$</span>, and the monotonicity of the integral. You have equality <span class="math-container">$\lVert f+g\rVert_1 = \lVert f\... |
3,386,999 | <p>How can I ind the values of <span class="math-container">$n\in \mathbb{N}$</span> that make the fraction <span class="math-container">$\frac{2n^{7}+1}{3n^{3}+2}$</span> reducible ?</p>
<p>I don't know any ideas or hints how I solve this question.</p>
<p>I think we must be writte <span class="math-container">$2n^{7}... | Hw Chu | 507,264 | <p>Let <span class="math-container">$d = \gcd(2n^7+1, 3n^3+2)$</span>. Then since <span class="math-container">$2n^7+1 \ | \ 2^3n^{21}+1$</span> and <span class="math-container">$3n^3 + 2 \ | \ 3^7n^{21}+2^7$</span>, we must have <span class="math-container">$$d \ | \ 3^7(2^3n^{21}+1) - 2^3(3^7n^{21}+2^7) \quad\Rightar... |
3,386,999 | <p>How can I ind the values of <span class="math-container">$n\in \mathbb{N}$</span> that make the fraction <span class="math-container">$\frac{2n^{7}+1}{3n^{3}+2}$</span> reducible ?</p>
<p>I don't know any ideas or hints how I solve this question.</p>
<p>I think we must be writte <span class="math-container">$2n^{7}... | DanielWainfleet | 254,665 | <p>Suppose <span class="math-container">$p$</span> is prime.</p>
<p>If <span class="math-container">$p$</span> divides <span class="math-container">$2n^7+1$</span> & <span class="math-container">$3n^3+2$</span></p>
<p>then <span class="math-container">$p$</span> divides <span class="math-container">$2(2n^7+1)-(3n... |
3,386,999 | <p>How can I ind the values of <span class="math-container">$n\in \mathbb{N}$</span> that make the fraction <span class="math-container">$\frac{2n^{7}+1}{3n^{3}+2}$</span> reducible ?</p>
<p>I don't know any ideas or hints how I solve this question.</p>
<p>I think we must be writte <span class="math-container">$2n^{7}... | Bill Dubuque | 242 | <p>This gcd is computable <em>purely mechanically</em> by a slight generalization of the Euclidean algorithm which allows us to scale by integers <span class="math-container">$\,c\,$</span> coprime to the gcd during the modular reduction step, i.e.</p>
<p><span class="math-container">$$\bbox[8px,border:2px solid #c00]{... |
61,933 | <p>Consider a lattice in R^3.
Is the some "canonical" way or ways to choose basis in it ?</p>
<p>I mean in R^2 we can choose a basis |h_1| < |h_2| and |(h_2, h_1)| < 1/2 |h_1|.
Considering lattices with fixed determinant and up to unitary transformations we get standard picture of the PSL(2,Z) acting on the upp... | Henry Cohn | 4,720 | <p>In higher dimensions, there doesn't seem to be anything as nice as in two dimensions: the fundamental domains get substantially more complicated and the algorithms become much less efficient. However, there are still some beautiful results. For example, Minkowski reduction is a natural generalization of the two-di... |
936,200 | <p>Suppose that x_0 is a real number and x_n = [1+x_(n-1)]/2 for all natural n. Use the Monotone Convergence Theorem to prove x_n → 1 as n grows.</p>
<p>Can someone please help me? I don't know what to assume since I don't know if it is increasing or decreasing when x_0 < 1 and when x_0 > 1.
Any hint/help would rea... | Christian Blatter | 1,303 | <p>As for "$\in$" see the other answers. Concerning the idea of a variable:</p>
<p>Each letter, say $y$, denoting a variable comes a priori with its <em>domain</em> $D_y$, a certain set. We are allowed to replace this $y$ in the formula by any element $a\in D_y$ and obtain a proposition about constants which is either... |
936,200 | <p>Suppose that x_0 is a real number and x_n = [1+x_(n-1)]/2 for all natural n. Use the Monotone Convergence Theorem to prove x_n → 1 as n grows.</p>
<p>Can someone please help me? I don't know what to assume since I don't know if it is increasing or decreasing when x_0 < 1 and when x_0 > 1.
Any hint/help would rea... | Frunobulax | 93,252 | <p>The most general ways to "pronounce" $\in$ certainly are "is an element of" or "is a member of". However, in a case like this one where the set is not only finite but also very small it might make sense to read "$y \in \{1,2,3\}$" as "$y$ is either $1$ or $2$ or $3$" or "$y$ is one of the values $1$, $2$, and $3$".... |
317,175 | <p>What tools would we like to use here? Is there any easy way to establish the limit?</p>
<p>$$\sum_{k=1}^{\infty}{1 \over k^{2}}\,\cot\left(1 \over k\right)$$</p>
<p>Thanks!</p>
<p>Sis!</p>
| mrf | 19,440 | <p>The series diverges. Let $a_k = \dfrac 1{k^2} \cot \dfrac 1k$ and $b_k = \dfrac1k$.</p>
<p>Then $\lim_{k\to\infty} \dfrac{a_k}{b_k} = 1$, so your series diverges by the limit comparison test (since $a_k \ge 0$).</p>
|
966,798 | <p>How I solve the following equation for $0 \le x \le 360$:</p>
<p>$$
2\cos2x-4\sin x\cos x=\sqrt{6}
$$</p>
<p>I tried different methods. The first was to get things in the form of $R\cos(x \mp \alpha)$:</p>
<p>$$
2\cos2x-2(2\sin x\cos x)=\sqrt{6}\\
2\cos2x-2\sin2x=\sqrt{6}\\
R = \sqrt{4} = 2 \\
\alpha = \arctan \f... | Umberto P. | 67,536 | <p>Is is just for convenience. </p>
<p>Suppose (for example) that you want $\delta^2 + \delta < \epsilon$. You can factor the left-hand side to write $\delta(\delta + 1) < \epsilon$. Let $m$ be any positive number. As long as $\delta < m$ you have $\delta(\delta + 1) < \delta (m+1)$, and if <em>in addition... |
245,312 | <p>Let $\kappa>0$ be a cardinal and let $(X,\tau)$ be a topological space. We say that $X$ is $\kappa$-<em>homogeneous</em> if</p>
<ol>
<li>$|X| \geq \kappa$, and</li>
<li>whenever $A,B\subseteq X$ are subsets with $|A|=|B|=\kappa$ and $\psi:A\to B$ is a bijective map, then there is a homeomorphism $\varphi: X\to X... | Joel David Hamkins | 1,946 | <p>This is a great question!</p>
<p>The disjoint union of two circles is $1$-homogeneous, but not $2$-homogeneous. It is $1$-homogenous, since you can swap any two points and extend this to a homeomorphism (basically, "all points look alike"). But it is not $2$-homogeneous, since you can let $A$ be two points from one... |
1,068,631 | <p>I want to find the solutions of the equation $$\left[z- \left( 4+\frac{1}{2}i\right)\right]^k = 1 $$ </p>
<p>in terms of roots of unity.</p>
<p>When I try to solve this, I get
\begin{align*}z - 4 - \dfrac i2 &= 1\\
z-\dfrac{i}{2}&=5\\
\dfrac{2z-i}2 &= 5\\
z&= 5 + \dfrac i2\end{align*}</p>
<p>Is t... | The Artist | 154,018 | <p><strong>Things you need to know (Hints):</strong></p>
<p>$$\left(z-(4+\frac{1}{2}i)\right)^k=\color{Crimson}{1}$$</p>
<p>$$ \color{crimson}{\cos 2\pi n+ i \sin 2\pi n =1}$$</p>
<p>$$\text{Where n is an Integer}$$</p>
<hr>
<p>Also you should know De Moivre's Theorem:</p>
<p>$$( \cos \theta+ i \sin \theta)^b= \c... |
4,319,590 | <blockquote>
<p>Let <span class="math-container">$H$</span> be a subgroup of <span class="math-container">$G$</span> and <span class="math-container">$x,y \in G$</span>. Show that <span class="math-container">$x(Hy)=(xH)y.$</span></p>
</blockquote>
<p>I have that <span class="math-container">$Hy=\{hy \mid h \in H\}$</s... | Michael Hardy | 11,667 | <p>If <span class="math-container">$w\in x(Hy),$</span> then for some <span class="math-container">$v\in Hy,$</span> <span class="math-container">$w=xv.$</span> Since <span class="math-container">$v\in Hy,$</span> for some <span class="math-container">$h\in H,$</span> <span class="math-container">$v=hy.$</span> So <spa... |
590,891 | <p>I'm going back to school and haven't taken a math class in years, so I'm brushing up on the basics.</p>
<p>The text states $\frac{g(t + \Delta(t))^2}{2} = \frac{gt^2}{2} + \frac{g}{2}\left(2t\Delta t + \Delta t^2\right)$.</p>
<p>(Sorry for the lack of formatting. I'll probably get slammed, but I couldn't figure it... | ILoveMath | 42,344 | <p>Well, by using $(a + b)^2 = a^2 + 2ab + b^2$, then</p>
<p>$$\frac{g (t + \Delta t)^2}{2} = \frac{g}{2}[t^2 + 2t \Delta t + \Delta^2t]= \frac{gt^2}{2} + \frac{g}{2}[2t \Delta t + \Delta^2 t ]$$</p>
<p>Actually, $(a+b)^2 = a^2 + 2ab + b^2$ is an algebraic identity . But let see it geometrically what it means. consid... |
2,359,621 | <p>Consider $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ where</p>
<p>$$f(x,y):=\begin{cases}
\frac{x^3}{x^2+y^2} & \textit{ if } (x,y)\neq (0,0) \\
0 & \textit{ if } (x,y)= (0,0)
\end{cases} $$</p>
<p>If one wants to show the continuity of $f$, I mainly want to show that </p>
<p>$$ \lim\limits_... | dromastyx | 453,578 | <p>$$\lim_{(x,y)\rightarrow (0,0)}f(x,y)=L$$
means that for all $\epsilon>0$ there exists a $\delta>0$ such that
$$0<\sqrt {x^2+y^2}<\delta \implies |f(x,y)-L|< \epsilon$$</p>
<p>In your case let $\delta=\epsilon$.</p>
<p>$$\left|\frac{x^3}{x^2+y^2}\right|=|x|\cdot \left|\frac{x^2}{x^2+y^2}\right| \l... |
2,797,709 | <p>How is $\; 4 \cos^2 (t/2) \sin(1000t) = 2 \sin(1000t) + 2\sin(1000t)\cos t\,$? This is actually part of a much bigger physics problem, so I need to solve it from the LHS quickly. Is there an easy method by which I can do this?</p>
| Bernard | 202,857 | <p>Use the <em>linearisation formula</em>: $\qquad\cos^2x=\dfrac{1+\cos 2x}2$.</p>
|
829,449 | <p>I am confused on the concept of extensionality versus intensionality. When we say 2<3 is True, we say that 2<3 can be demonstrated by a mathematical proof. So, according to mathematical logic, it is true. Yet, when we consider x(x+1) and X^2 + X, we can say that the x is the same for = 1. However, we call this... | cnick | 133,048 | <p>I am a fan of collisions. Get some simple euclidean shapes (billiard balls, or dice on ice) and show her how to calculate the angles that they'll move in after hitting each other.</p>
|
1,649,053 | <p>In figure $AD\perp DE$ and $BE\perp ED$.$C$ is mid point of $AB$.How to prove that $$CD=CE$$<a href="https://i.stack.imgur.com/ZtAA0.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ZtAA0.png" alt="enter image description here"></a></p>
| Svetoslav | 254,733 | <p>Let $H$ be on $DE$ such that $CH\perp DE$. You have that $C$ is midpoint of $AB$, and $CH||AD||BE\Rightarrow DH=EH$. Then $\triangle DHC\simeq\triangle EHC $ because $DH=EH$, $CH$ is common and $\angle DHC=\angle EHC=90^o$. </p>
|
2,458,184 | <p>Let $a, b$ be non-negative integers and $p\ge3$ be a prime number. If $a^2+b^2$ and $a+b$ are divisible by $p$ does it mean $a$ and $b$ are always divisible by $p$?</p>
| Prasun Biswas | 215,900 | <p>Suppose $p$ be an odd prime.</p>
<p>Note that $a^2+b^2=(a+b)^2-2ab$ and use Euclid's lemma to conclude that $p$ must divide $a$ or $b$.</p>
<p>Now, assume that <strong>only one</strong> of $a,b$ (WLOG say $a$) is divisible by $p$.</p>
<p>Since $p\mid a+b$ and $p\mid a$, we get $p\mid (a+b)-a=b$, i.e., $p\mid b$, ... |
2,227,047 | <p>For any $x=x_1, \dotsc, x_n$, $y=y_1, \dotsc, y_n$ in $\mathbf E^n$, define $\|x-y\|=\max_{1 \le k \le n}|x_k-y_k|$. Let $f\colon\mathbf E^n \to \mathbf E^n$ be given by $f(x)=y$, where $y_k= \sum_{i=1}^n a_{ki} x_i + b_k$ where $k =1,2, \dotsc,n$. Under what conditions is $f$ a contraction mapping?</p>
<p>Any hint... | mickep | 97,236 | <p>Maybe something like this: First, we write $\alpha=a+ib$ and use the fact that the integrand is even. Thus, the integral equals
$$
2\int_0^{+\infty}\frac{1}{\sqrt{(1+ax^2)^2+(bx^2)^2}}\,dx
$$
Playing a bit with that expression, we find that this equals
$$
\biggl[\frac{1}{(a^2+b^2)^{1/4}} F\Bigl(2\arctan\bigl((a^2+b^... |
157,731 | <p>I have a coupled PDE. How can I convert the equation from $(x,t)$ to $(p,t)$, the Fourier space in MATHEMATICA? </p>
<p>\begin{equation}
\frac{\partial c}{\partial t} +\frac{\partial d}{\partial t} = -4\gamma(\frac{\partial a}{\partial x} +x (\frac{\partial c}{\partial x} +\frac{\partial d}{\partial x}) - \frac{\pa... | Alexei Boulbitch | 788 | <p>I did not notice the factor <code>x</code>. Sorry, below I repair this with the third rule.
Try the following. First introduce three simple rules:</p>
<pre><code>rule1 = D[y_[x, t], {x, n_Integer}] :> -I^n*p^n*y[p, t];
rule2 = y_[x, t] :> y[p, t];
rule3 = x*y_[p, t] :> I*D[y[p, t], p]
</code></pre>
<p>And... |
724,900 | <p>Assuming $y(x)$ is differentiable. </p>
<p>Then, what is formula for differentiation ${d\over dx}f(x,y(x))$?</p>
<p>I examine some example but get no clue....</p>
| Community | -1 | <p>First, compute $\mathrm{d}f(x,z)$ without assuming any relationship between $x,z$. Suppose you compute this to be</p>
<p>$$ \mathrm{d} f(x,z) = f_1(x,z) \mathrm{d}x + f_2(x,z) \mathrm{d}z $$</p>
<p>Now, we can substitute in the dependence $z = y(x)$. We need</p>
<p>$$ \mathrm{d}z = y'(x) \mathrm{d}x $$</p>
<p>a... |
3,120,729 | <p>I came across this exercise:</p>
<blockquote>
<p>Prove that
<span class="math-container">$$\tan x+2\tan2x+4\tan4x+8\cot8x=\cot x$$</span></p>
</blockquote>
<p>Proving this seems tedious but doable, I think, by exploiting double angle identities several times, and presumably several terms on the left hand side ... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p><span class="math-container">$$\cot y-\tan y=\dfrac{\cos^2y-\sin^2y}{\cos y\sin y}=\dfrac{\cos2y}{\dfrac{\sin2y}2}=?$$</span></p>
|
1,368,073 | <p>Halmos, in Naive Set Theory, on page 19, provides a definition of intersection restricted to subsets of $E$, where $C$ is the collection of the sets intersected. The point is to allow the case where $C$ is $\emptyset$, which with this definition of intersection gives $E$ as the result. </p>
<blockquote>
<p>$\{x \... | coldnumber | 251,386 | <p>Looking at the book, I see that $C$ is a collection of subsets of $E$, and the definition in your question is the intersection of all the elements of $C$.
Note that a subset $X$ of $E$ is not a <em>subset</em> of $C$; it is an <em>element</em> of $C$.</p>
<p>Your initial reading is correct. The set $N=\{x \in E: ... |
2,659,781 | <p>I saw a problem yesterday, which can be easily be solved if we are using fractions. But the problem is for the 4th grade children, and I don't know how to solved this using what they what learned.</p>
<p>I tried solved it using the graphic method ( segments ). Here's the problem:</p>
<p>A team of workers has to fi... | fleablood | 280,126 | <p>Time 1: They did $\frac 34$ of the road. $\frac 14$ is left.</p>
<p>Time 1a: they did $2$ meters.</p>
<p>Time 2: They did $\frac 34$ of what was left. What remains is $\frac 14$.</p>
<p>Time 2a: they did $2$ meters.</p>
<p>Time 3: 1 meter is left.</p>
<p>Work backwards:</p>
<p>The $1$ meter of time 3) and th... |
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