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 |
|---|---|---|---|---|
1,531,488 | <p>I hope to know the singularity and pole of $\frac{z^2 e^z}{1+e^{2z}}$.</p>
<p>I try $\frac{z^2 e^z}{1+e^{2z}} = \frac{z^2}{e^{-z}+e^{z}}$ and observe that the denominator seems like cosine function. So I think the singularites are i(2n+1)$\pi$/2. But when I try to evaluate the poles, I fail and different values of ... | WW1 | 88,679 | <p>let $z_n=i(n-1/2)\pi $ and $r_n$ be the residue at the pole at $z_n$
$$ r_n = \lim_{z \to z_m} (z-z_n)f(z) $$
$$= z_n^2 e^{z_n} \lim_{z \to z_m} \frac{z-z_n}{ {1+e^{2z}}}$$
you can apply L'hopital to get
$$ r_n = z_n^2 e^{z_n} \lim_{z \to z_m} \frac{1}{ {2e^{2z}}} = \frac 12 z_n^2 e^{-z_n}$$</p>
|
2,946,759 | <p><span class="math-container">$L =$</span> {<span class="math-container">$(01)^a$$x(10)^b$</span> | <span class="math-container">$a=b, b > 0, x∈$</span>{<span class="math-container">$0,1$</span>}*}</p>
<p>This is a question where according to the key, yes, the language is regular, but no explanation is given. How... | Wuestenfux | 417,848 | <p>Hint: The intermediate symbol <span class="math-container">$x$</span> has no impact on the non-regularity of the language. The general method to show that a language is non-regular is the pumping lemma and this language is similar to <span class="math-container">$\{a^nb^n\mid n\geq 1\}$</span> or better <span class=... |
285,965 | <p>I was told to assume $f(x)$ is a polynomial with degree $d\geq 1$ with integer coefficients and positive leading coefficient.</p>
<p>(i) I need to show that there are infinitely many $x$ such that $f(x)$ isn't prime.</p>
<p>(ii) I also need to show that if $f(x_0) = m$, where $m>0$, then $f(x) \equiv 0 \pmod m$... | dinoboy | 43,912 | <p>OK, so we have the useful lemma that for integers $a,b$ we have $(a-b)|(f(a) - f(b))$. This follows essentially from the fact that $(a-b)|(a^n - b^n)$ for any $n$. This is why $f(x) \equiv 0 \pmod{m}$ whenever $x \equiv x_0 \pmod{m}$. </p>
<p>Now, how does this help with part i)? Suppose the conclusion is false, s... |
383,259 | <p>We know that the real ordered field can be characterized up to isomorphism as a complete ordered field. However this is a second order characterization. That raises the following question. Consider the following theory. We take as axioms the axioms for ordered fields, and then add an axiom schema that states that ev... | Dmytro Taranovsky | 113,213 | <p>It is not. Using set forcing, we can add 'undefinable' reals in a controlled manner, while keeping complexity of parameter-free definable sets low.</p>
<p>Specifically, let <span class="math-container">$M$</span> be a countable <span class="math-container">$ω$</span>-model of ZFC\P, real <span class="math-container... |
2,856,121 | <p>Before I start explaining what I will say regarding the question, read the question before going any further.</p>
<p>Question: <em>Mariam and Leena both live on the bank of the same river, a distance apart. When Mariam visits Leena, she swims downstream (with the current), but then must swim upstream (against the c... | Phira | 9,325 | <p>The current gains or loses you a fixed distance per second. Unfortunately, you spend more time swimming against the current than with it, so it is a loss in total.</p>
|
3,500,910 | <p>*If there exists the same question with answer anywhere, plz delete this one, i didn't find one.</p>
<p>If <span class="math-container">$f$</span> is continuous on <span class="math-container">$\Bbb R$</span> and uniformly continuous on every <span class="math-container">$(a,b)$</span> so it is also uniformly conti... | Mark | 470,733 | <p>The logic about what can go wrong is correct but this doesn't really prove the statement is wrong. You need a specific counterexample. Well, it is easy, take <span class="math-container">$f(x)=x^2$</span>. Actually, any function which is continuous on <span class="math-container">$\mathbb{R}$</span> but not uniforml... |
2,802,094 | <blockquote>
<p>If $y=\varphi(x) $ is a particular solution of $y''+(\sin(x))y'+2y=e^x$
and $\psi(x)$ is a particular solution of
$y''+(\sin(x))y'+2y=\cos(2x)$,then what is the particular solution of
$y''+(\sin(x))y'+2y=e^x+2\sin^2(x)$?</p>
</blockquote>
<p>Since $LHS$ of equation is same while their $RHS$ ar... | user577215664 | 475,762 | <p>Hint</p>
<p>Your problem is the contant 1 for that just try $y_p=K$ into the equation to get the value of the particular solution of
$$y_p''+(\sin(x))y_p'+2y_p=1$$
$$\implies k''+\sin(x)K'+2K=1 \implies K=1/2$$
Then substract $\phi(x)$ and $\psi(x)$ to get $e^x-\cos(2x)$</p>
|
2,662,747 | <p>I want to prove that a connected component of a locally Euclidean space X is open in this space.</p>
<p>I start the proof taking a point y in the connected component Y of X. In particular, y is a element of X and have an open neighborhood U, and there is an open subset in an euclidean space and a homeomorphism. I'm... | Jaideep Khare | 421,580 | <p>Just take $f_{k+1}$ common out of the expression.$$f_{k+1} f_{k}+(f_{k+1})^2=f_{k+1}(f_k+f_{k+1})$$</p>
<p>Now use $F_{n}=F_{n-1}+F_{n-2}$, to get $$f_{k+1}(f_k+f_{k+1})=f_{k+1}f_{k+2}$$</p>
<p>Done!</p>
|
1,532,263 | <p>$$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{ k (k + 1) (k + 2) \cdots (k + m)
}, m\in\Bbb N$$</p>
<p>Please help to do this</p>
| Jan Eerland | 226,665 | <p>HINT:</p>
<p>$$\lim_{n\to \infty}\sum_{k=1}^n\frac{1}{k(k+1)(k+2)\cdots(k+m)}=$$
$$\lim_{n\to \infty}\sum_{k=1}^n\frac{1}{\prod_{n=0}^{m} (k+n)}=$$
$$\lim_{n\to \infty}\sum_{k=1}^n\frac{1}{\frac{\Gamma(k+m+1)}{\Gamma(k)}}=$$
$$\lim_{n\to \infty}\sum_{k=1}^n\frac{\Gamma(k)}{\Gamma(k+m+1)}=$$
$$\lim_{n\to \infty}\fra... |
1,366,737 | <p>Demonstrate by induction: $3^{2n} + 7 = 4k$ is true, for any $n\in \mathbb N$. I need to demonstrate this using the induction principle.</p>
<p>So far I have:</p>
<p><strong>$n = 1$</strong></p>
<p>$$3^{2\cdot 1} + 7 = 4\cdot k $$
$$9 + 7 = 4k$$
$$16 = 4k$$
$$k = 4$$</p>
<p>So it checks for $n=1$.</p>
<p><stron... | Socre | 223,538 | <p>You have shown for n=1 which is the first step so
Now assume or state that it is true for n=h
$$3^{2h}+7=4k$$
You want to show that it is true for n=h+1
So start with $3^{2h+2}+7$
$$3^{2h+2}+7= 9(3^{2h})+7=9(4k-7)+7$$
$$3^{2h+2}+7=36k-56=4(9k-14)=4m$$ where m is another natural number thus proving our assumption<... |
414,120 | <p>Somebody can to help me in such exercices:</p>
<blockquote>
<p>(1) A ring R such that $a^2 = a$ for all $a\in R$ is called a Boolean ring. Prove that every
Boolean ring R is commutative and $a + a = 0$ for all $a \in R$. </p>
<p>(2) Let R be a ring with more than one element such that for each nonzero $a\... | vadim123 | 73,324 | <p>$a+a=(a+a)^2=a^2+a^2+a^2+a^2=a+a+a+a$, so $a+a=0$.</p>
<p>$a+b=(a+b)^2=a^2+ab+ba+b^2=a+ab+ba+b$, so $ab+ba=0$. Using the previous fact, $ab-ba=0$ so $ab=ba$.</p>
|
746,557 | <p>An exercise from Introductory Combinatorics by Richard A.Brualdi:</p>
<blockquote>
<p>A roller coaster has five cars, each containing four seats, two in front and two
in back. There are 20 people ready for a ride. In how many ways can the ride
begin? What if a certain two people want to sit in different ca... | Barry Cipra | 86,747 | <p>Your introduction of the factor ${4\choose2}2!2!$ specifies who sits where in only <em>one</em> car. You need to raise that factor to the fifth power, to account for all five cars. </p>
<p>Note, though, that ${4\choose2}2!2!={4!\over2!2!}2!2!=4!$, so the fifth power will entirely cancel the $(4!)^5$ in the $20!\o... |
1,517,291 | <p>The question is : <a href="https://i.stack.imgur.com/sG4uW.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sG4uW.png" alt="enter image description here"></a></p>
<p>I am stuck here :</p>
<p>$$| \sqrt{x+1} - \sqrt{x} -1 | / \sqrt{x} + 1$$</p>
<p>i know that the numerator is negetive so i must ch... | Nizar | 227,505 | <p>Let $\epsilon >0$, we are seeking for $\delta >0$, such that , for all $x\geq 0$ with $x>\delta$, we have $$ \Bigg| \frac{\sqrt{x+1}}{\sqrt{x}+1} -1\Bigg| <
\epsilon$$
Indeed, $$ \Bigg| \frac{\sqrt{x+1}}{\sqrt{x}+1} -1\Bigg| = \Bigg| \frac{\sqrt{x+1} -\sqrt{x}-1}{\sqrt{x}+1} \Bigg| $$
Note that... |
356,866 | <p>Simple question for you awesome guys :)</p>
<p>I've got the formular for a polynomial:</p>
<pre><code>7x^4-56x^3+224x^2-56x+231=0
</code></pre>
<p>The thing i need to do is to solve and factorize the above equation.</p>
<p>How can I do this?</p>
| Mikasa | 8,581 | <p>Besides to another answer, you can use the <code>fsolve</code> command. Firstly define your polynomial as </p>
<blockquote>
<p><code>p:=7x^4-56x^3+224x^2-56x+231;</code></p>
</blockquote>
<p>in Maple environment and then enter the following command as I pointed:</p>
<blockquote>
<p><code>fsolve(p=0);</code></... |
2,132,863 | <p>I am studying trigonometry in my school and learned that in a triangle the side opposite to angle theta should be taken as perpendicular side - hypotenuse remains the same and the third remaining side is the base.</p>
<p>The same is required for calculating the sine / cosine etc of the angle theta for below formula... | Mohamed Ibrahim | 394,151 | <p><a href="https://i.stack.imgur.com/yZTVj.gif" rel="nofollow noreferrer">Angle A - First Position</a></p>
<p>Look at the previous link. The hypotenuse side is the one facing the right angle. It doesn't change as long as the right angle doesn't change.
$$ SinA = \frac{Opposite\ Side}{Hypotenuse} $$
and
$$ CosA = \f... |
2,132,863 | <p>I am studying trigonometry in my school and learned that in a triangle the side opposite to angle theta should be taken as perpendicular side - hypotenuse remains the same and the third remaining side is the base.</p>
<p>The same is required for calculating the sine / cosine etc of the angle theta for below formula... | BADRIVISHAL BIYANI | 756,631 | <p>If we observe <span class="math-container">$\sin 30^\circ$</span>, or <span class="math-container">$\sin 60^\circ$</span>, the length of the opposite side is increasing in relation to the hypotenuse. If we extend the logic to <span class="math-container">$\sin 89^\circ$</span>, the length of the opposite side is alm... |
2,444,641 | <p>A student asked a very insightful question in my Calculus class this morning. I did not know how to answer him. (Admittedly, I am not an analyst by trade: once I passed my qualifiers I never looked back.) I would like to know if anyone here can give a precise answer, and if that can be massaged into an answer un... | KenM | 83,608 | <p>Note that if <span class="math-container">$a_n = \frac{1}{n^{1 + \epsilon}}$</span> where <span class="math-container">$\epsilon > 0$</span>, then the series converges by the integral test (or p-series). Then you can argue via the limit comparison test, e.g. if the terms approach zero in a similar fashion (as <sp... |
1,706,741 | <p>I am having trouble with the following exercise from Dollon and Gilet's <em>Géométrie plane</em>.</p>
<blockquote>
<p>Two parallel lines <span class="math-container">$\Delta$</span> and <span class="math-container">$\Delta'$</span> are given, as well as a point <span class="math-container">$A$</span> on <span class=... | Narasimham | 95,860 | <p>Shall I call the parallel lines $AH, RP?$</p>
<p>Let us construct a rectangle of dimensions (vertical,horizontal) $(v,h)$, and skew it by an angle $ \gamma $ so the oblique/distorted parallelogram has the same perimeter.</p>
<p>$$ 2 h + \frac{2 v}{\sin \gamma } = l\;;\ \; \; \sin \gamma = \frac{v}{l/2 -h }\;;... |
2,225,818 | <p>If a dice is rolled 1 time,the probability of getting a 4 is 1/6.what is the probability of getting exactly 4 once if the die is rolled thrice</p>
| Ken Draco | 431,886 | <p>The actual answer is (it's very, very simple):</p>
<p>The probability of not getting a four on the first roll is $5/6$</p>
<p>If there are two attempts/rolls, it's gonna be $(5/6)^2$</p>
<p>And not getting a four with three attempts/rolls is $(5/6)^3$</p>
<p>Therefore, the probability of getting a four at least ... |
2,672,968 | <p>In a basic probability course, there seems to be an implicit assumption that any continuous random variable has a density function, but I don't really see why this should be. My intuition is that there's some probability space $(\Omega, \Sigma, P)$ and a measurable function $X : \Omega \to \mathbb{R}$, and you'd li... | angryavian | 43,949 | <p>You might want to read the <a href="https://en.wikipedia.org/wiki/Probability_distribution#Continuous_probability_distribution" rel="nofollow noreferrer">Wikipedia section</a> on this. In short, you are right: a continuous distribution need not be absolutely continuous with respect to Lebesgue measure (the <a href="... |
2,389,247 | <p>I need to solve $$\int|x^n| \, dx$$ <strong>if n is an odd positive integer</strong>.</p>
<p>I think it should be $\dfrac{|x|^{n+1}}{n+1}+c$, but this is wrong. I am new to integration so I don’t know what to do when there is modulus function.</p>
<p>The answer given at back of book is $\dfrac{|x^n|x}{n+1} +c$.</p... | John Griffin | 466,397 | <p>Assuming $x$ is a real number and $n$ is odd, we have $$
|x|^n = \begin{cases}
x^n & \text{if}\ x\geq 0, \\
-x^n & \text{if}\ x<0.
\end{cases}$$
Use this to remove the modulus. That is, the antiderivative will be different depending on whether $x$ is positive or negative. For example, the antiderivative o... |
34,664 | <p>Here is the post for context: <a href="https://math.stackexchange.com/questions/4415677/is-today-is-presidents-day-a-proposition-propositional-function-or-neither#4415684">Is "today is Presidents' Day" a proposition, propositional function, or neither?</a></p>
<p>Both the answers are really great and t... | miracle173 | 11,206 | <p>I propose to upvote both answers and thank both users for their answers. Then flip a coin upvote according to the result. But in your case I see additional criteria that can be used to decide which answer I would upvote.</p>
<ul>
<li><p>I agree with <a href="https://math.meta.stackexchange.com/a/34668/11206">John B... |
970,403 | <p>Let $f,g:[0,1] \to \mathbb{R}$ such that $f'(x)>0,\ g'(x) >0,\ \forall x\in [0,1]$. Moreover, $f(0)= g(0)$ and $f(1)=g(1)$. Prove that exists $x_1, x_2 \in [0,1]$ such that $$f(x_1) = g(x_2), \ \text{and}\ \ f'(x_1) = g'(x_2)$$</p>
<hr>
<p>I've tried to use Mean Value Theorem with:</p>
<p>a) $h(x)= f(x)-g(x... | John Hughes | 114,036 | <p>I suggest that you look at $h(x) = g^{-1}(f(x))$. Suppose that for some number $x_1$, $h'(x_1) = 1$. What would that tell you? </p>
<p>Also: How do you know $g^{-1}$ even exists? You need to answer this before you go using it willy-nilly. </p>
|
1,377,633 | <p>In a mathematical methods problem, where the $c$ is a the unit circle around the origin and in counterclockwise, I need to use a step that I'm not so sure about (Because I don't know how to develop it)</p>
<p>How can I affirm that:
$$\oint_c{\dfrac{1}{z^{2n+1}}(z^2+1)^{2n}}=\binom {2n} {n} 2\pi i$$</p>
<p>I know t... | Mark Viola | 218,419 | <p>We have from <a href="https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula#Theorem" rel="nofollow">Cauchy's Integral Formula</a></p>
<p>$$\begin{align}
\oint_C\frac{(z^2+1)^{2n}}{z^{2n+1}}dz&=\left.2\pi i \frac{d^{2n}}{dz^{2n}}\left(z^2+1\right)^{2n}\right|_{z=0}\\\\
&=\left.2\pi i \frac{d^{2n}}{dz^{2n... |
1,942,599 | <p>How can I calculate sums of the form:
$$\sum_{i=-1}^{5}\sum_{j = -5}^{1}(1 + 2i + 3j + 4ij)?$$</p>
<p>Shall I calculate it like first for $i = -1, j = \{-5,\dots, 1\}$, then for $i = 2, j = \{-5, \dots, 1\}$? Is there some shortcut for calculating those kinds of sums, i.e. something akin to summation of arithmetic ... | Community | -1 | <p>Rewrite $1+2i+3j+4ij$ as $(1+2i)(1+2j)+(1)(j)$. Then the double sums of products of two factors that depends only on one index can be factored ($\sum\sum ab=\sum a\sum b$):</p>
<p>$$\sum_{i=-1}^{5}\sum_{j = -5}^{1}(1 + 2i + 3j + 4ij)=\sum_{i=-1}^{5}(1+2i)\sum_{j = -5}^{1}(1+2j)+\sum_{i=-1}^{5}1\sum_{j = -5}^{1}j\\
... |
1,218,259 | <p>Let $(M,g)$ be an Einstein manifold with Levi-Civita connection $\nabla$ and whose Ricci tensor $\text{Rc}(g)=g$, in components $R_{ij}=g_{ij}$. The <strong>Lichnerowicz Laplacian</strong> of $g$ is the map
\begin{align*}
\Delta_L:\Gamma(S^2M)&\rightarrow\Gamma(S^2M)\\
h&\mapsto(\Delta_Lh)_{ij}=g^{st}\nabla... | digiboy1 | 8,252 | <p>Perhaps this not-too-long notes will help
<a href="http://www.owlnet.rice.edu/~fjones/chap14.pdf" rel="nofollow">http://www.owlnet.rice.edu/~fjones/chap14.pdf</a></p>
<p>It gives some rather elementary explanations and proof of the Gauss theorem in $\mathbb{R}^n$. For the complete set of notes (used for an honors c... |
744,412 | <p>If I take the $k \in \mathbb{N}$ power of 10 and mod it by a large prime, I notice that the remainders repeat at some point.</p>
<p>For instance $10^k mod~7$ seems to repeat every $8$th value of $k$.</p>
<p>Given $k$ (the power of ten), and a large prime number $p$, how can I compute the size of the cycle?</p>
<p... | Bill Dubuque | 242 | <p>The least <span class="math-container">$\,n >0\,$</span> such that <span class="math-container">$\,a^n\equiv 1\pmod p\,$</span> is called the <em>order</em> of <span class="math-container">$\,a\pmod p.\,$</span> For the state-of-the-art on the order computation algorithms see Andrew Sutherland's 2007 MIT Thesis <... |
678,599 | <p>Here is a problem from Lang's Algebra,chapter 12 and problem 12,</p>
<p>a)Let f(x) be a polynomial of degree 1 in $\mathbb Z[x]$. Show that the values $f(a)$ for $a \in \mathbb Z$ are divisible by infinitely many primes.</p>
<p>b) Let F be a finite extension of $\mathbb Q$ . Show that there are infintely many prim... | Hagen von Eitzen | 39,174 | <p>You misinterprete the problem statement. Maybe this way: For $f\in\mathbb Z[x]$ with $\deg f=1$ there exist infinitely many distinct primes $p_1, p_2,\ldots$ such that there exist integers $x_1,x_2,\ldots$ such that $f(x_i)$ is a multiple of $p_i$.</p>
<hr>
<p>For b) show that any polynomial has a root in $\mathbb... |
40,094 | <p>I have a table form as:</p>
<pre><code>table = {{a, 6}, {b, ff}, {c, 2}}
</code></pre>
<p>I want to associate the first element of each pair from the table to the second element of the pair as in:</p>
<pre><code>a=6, b=ff, c=2.
</code></pre>
<p>I want the parameters a,b,c to be recognized throughout the program... | István Zachar | 89 | <p><code>SetDelayed</code> has attribute <code>HoldAll</code>, so your <code>z := table[[1, 2]]</code> assigns the righthand-side to <code>z</code> instead of <code>a</code>. If you really insist on doing your assignment in this programmatic way, use <code>Evaluate</code> and <code>Set</code>:</p>
<pre><code>Clear[tab... |
46,898 | <p>Given a formal power series $\sum a_n z^n$ and a radius of convergence $R>0$, there are various ways to extend the function to the boundary such as</p>
<ul>
<li>Abel's theorem</li>
<li>Fatou's lemma </li>
<li>$H^\infty$ theorem.</li>
</ul>
<p>What is an example of a function that does (almost) nowhere to the bo... | Robert Haraway | 9,502 | <p>This is problem 2 in Chapter 2 of Stein & Shakarchi's <em>Complex Analysis</em>. They give two examples:</p>
<blockquote>
<p>$f(z) = \sum_{n=0}^\infty z^{2^n},$</p>
</blockquote>
<p>and, for $0 < \alpha < \infty,$</p>
<blockquote>
<p>$f(z) = \sum_{n = 0}^\infty 2^{-n \alpha} z^{2^n}.$</p>
</blockqu... |
24,904 | <p>So I am having a lot of trouble proving this. It was for an assignment due tuesday, but the prof said I can have a bit of extra time with this question. </p>
<p>Please do not give me the answer because I am sure my prof is on this website :P</p>
<p>okay so the question is as follows:</p>
<blockquote>
<p>Let $R$... | Rasmus | 367 | <p>You have to show that $(m\cdot n)1_R=(m1_R)\cdot(n1_R)$ for all integers $m$ and $n$.</p>
<p>If $m$ and $n$ are non-negative this follows from your definition $ka:=\sum_{j=1}^k a$.</p>
<p>The general case is then an easy consequence of the axioms. You should try to work this out.</p>
|
958,486 | <p>I'm trying to prove that</p>
<p>$$x^2 + x + 1 = y^2$$</p>
<p>has no integer solution. So far I've tried proof by contradiction, but all of that seems to rely on me being able to factor this expression into some neat form where the solution is obvious. I've not been able to do so.</p>
| mcmat23 | 180,748 | <p>Hint:</p>
<p>If $n$ is square then $n\equiv 1,4,0 (\mod 8) $.</p>
<p>In your case $y$ is an <strong>odd number</strong> then $$y^2\equiv 1(\mod 8)$$ and </p>
<p>$$
x^2<y^2<x^2+2x+1
$$</p>
<p>Now it 's easy to conclude. </p>
|
8,760 | <p>What are the three cube roots of -1?</p>
<p>Not sure if this is a trick question, But I have been asked this. one of the ansers is -1, what are the other 2?</p>
| Agustí Roig | 664 | <p>You may take a look at this previous <a href="https://math.stackexchange.com/questions/3315/what-is-the-square-root-of-i/3346#3346">question</a>.</p>
|
3,673,915 | <p>If this is a duplicate you can close this post.</p>
<blockquote>
<p>Introduction to Real Analysis (Robert G. Bartle) 18. (b) </p>
<p>If <span class="math-container">$f$</span> is a bijection of <span class="math-container">$A$</span> onto <span class="math-container">$B$</span>, show that <span class="math-c... | ir7 | 26,651 | <p>For every <span class="math-container">$y\in B$</span> <strong>there is</strong> a <strong>unique</strong> <span class="math-container">$x_y\in A$</span> such that <span class="math-container">$f(x_y)=y$</span>.</p>
<p>We then have <span class="math-container">$f^{-1}(\{y\})= \{t\in A : f(t) = y\} =\{ x_y\}$</span>.... |
2,294,354 | <p>How to check the convergence of the series
$$
\sum_{n=1}^\infty \frac{\ln\left(1+\frac{1}{n}\right)}{n^a}
$$
I tried using all the tests: D'Alembert's, ratio test, comparison test but I failed . how to find when the series converges?
Thanks in advance.</p>
| B. Mehta | 418,148 | <p>I'll use $C = f(A)$ for convenience. We know $C$ is open, so let $x \in \Re(C)$. Then there is some $z \in C$ with $\Re(z) =x$, but $C$ open so there is an $\epsilon > 0$ with $B_\epsilon(z) \subset C$. But the real part of this ball is just an interval $(x-\epsilon, x+\epsilon)$, which we know is entirely contai... |
2,852,234 | <p>I am certainly new to statistics. I did some simulations and got a lot of data. From the data I ran a AWK script to calculate the average $\bar x$; minimum, $x_0$ and standard deviation, $\sigma$ (the one where you divide by $N$, not $N-1$).</p>
<p>Now I want to plot the data. I guess, I can draw the histogram $\ba... | Eval | 455,780 | <p>Error bars often represent one standard deviation of uncertainty, one standard error, or a particular confidence interval (e.g., a 95% interval). -Wikipedia </p>
<p>When you are talking about plotting standard deviation, are you sure that the data being plotted is not skewed? If it is, be careful enough to calculat... |
359,437 | <p>A (split) reductive linear algebraic group is equivalently described by combinatorial information called a root datum.</p>
<p>A toric variety is described by combinatorial information called a fan. </p>
<p>Both correspondences use the character lattice. </p>
<p>The reference:</p>
<p><a href="http://u.cs.biu.ac.... | Mikhail Borovoi | 4,149 | <p>(1) A (connected) reductive group <span class="math-container">$G$</span> over an algebraically closed field <span class="math-container">$k$</span> is described by a combinatorial object called the <em>based root datum</em> <span class="math-container">${\rm BRD}(G)$</span>. </p>
<p>(2) A <em>spherical homogeneou... |
2,158,115 | <p>I am trying to write down the set of all ordered pairs $(n,m)$ such that $n=2k+1:k\in\mathbb{Z}^{+}$ and $m=2l+1:l\in\mathbb{Z}^{+}$. How would that be written in set notation?</p>
| mle | 66,744 | <p>one another way is: $$ \{x|\exists y, z\in \Bbb{Z}: x=(y,z) \wedge \exists k, l \in \Bbb{Z}^+: y=2k+1 \wedge z=2l+1\}$$</p>
|
542,582 | <p>I have a question which I'm unable to provide much working for as I'm not sure how to start it. </p>
<p>A metal sphere is placed in seawater to study the corrosive effect of seawater. If the surface area decreases at a rate of $35cm^2/year$ due to corrosion, how fast is the radius changing when the radius is $7cm$?... | Satish Ramanathan | 99,745 | <p>$\frac{dS}{dt} = 35$</p>
<p>$S = 4\pi R^2$</p>
<p>$\frac{d(4\pi R^2)}{dt}= 35$</p>
<p>$8\pi R\frac{dR}{dt} = 35$</p>
<p>$\frac{dR}{dt} =35/(7*8\pi )= 5/(8*\pi)$</p>
|
381,771 | <p>Let <span class="math-container">$p$</span> be a prime. How many solutions <span class="math-container">$(x, y)$</span> are there to the equation <span class="math-container">$x^2 + 3y^2 \equiv 1 \pmod{p}$</span>? Here <span class="math-container">$x, y \in \{0, 1, \ldots p-1\}$</span>. This paper (<a href="https://... | Ivan Meir | 7,113 | <p>We give an elementary approach to the more general equation <span class="math-container">$x^2+Dy^2=1 \mod p$</span>.</p>
<p>Firstly,
if <span class="math-container">$p=2$</span> we have <span class="math-container">$2$</span> solutions as either <span class="math-container">$x^2=1$</span> or <span class="math-contai... |
2,179,243 | <p><strong>Question:</strong> Is there more than one "appropriate" way to generalize the orthogonal group of $\mathbb{R}^n$?</p>
<p>The two competing ways which come to mind for me are:</p>
<ol>
<li><p>The group of all isometries with the origin as a fixed point, for any (finite-dimensional) vector space $V$ with any... | Mark | 143,872 | <p>EDIT: In short: yes, your answer is correct (you can ignore everything else here:)</p>
<p>$\frac{(1+x)^3}{1+x^3} = \frac{9}{13}$ implies $8 x = -35 \pm 3 \sqrt{129}$ from which you can compute $\frac{(1+x)^5}{1+x^5} = 729/1349$. So there is only one possible coprime pair $a,b$, making the minimality of $a+b$ an emp... |
2,308,093 | <p>Let $u_0,u_1,u_2,\ldots$ be the sequence
$0, 1, 1/2,3/4, 5/8, 11/16,\ldots$
in which the average of the previous two terms forms the current term, i.e. $$u_{s+2} = \left( \frac{u_{s+1} + u_s}{2}\right)$$
Let $v_s = u_s- u_{s-1}$.</p>
<p>The conjectured formula for $v_n$ is $(-1/2)^{(n-1)}$, but to prove it by indu... | Roby5 | 243,045 | <p>Note that in $(4)$, we have using the recurrence $$u_s=\frac{u_{s-1}+u_{s-2}}{2}$$ that </p>
<p>$$v_{k+1}=u_{k+1}-u_k=\frac{u_{k-1}-u_k}{2}=-\frac{v_k}{2}$$ where in the last step we used the inductive hypothesis </p>
|
1,009,169 | <p>I have to examine limit of following sequence $\{a_n\}_{ n \ge 1}$ $a_n = \sqrt[n]{\sum_{k=1}^{n}{(2 -\frac{1}{k})^k}}$. We know that $\lim_{n \to \infty} \sqrt[n]{a} = 1$ for $a > 0$ and we know that $\sqrt[n]{\sum_{k=1}^{n}{(2 -\frac{1}{k})^k}} > 0$ but sequence $a_n$ is increasing (I cannot prove it but I c... | DanielWainfleet | 254,665 | <p>By differentiating $f(x)=\log ((2-1/x)^x)=x\log (2-1/x)=x(\log (2 x-1)-\log x)$ we see that $f'(x)>0$ for $x\geq 1. $ So the sequence $B_k=(2-1/k)^k=e^{f(k)}$ is increasing. Therefore $$2-(1/n)=B_n^{1/n}\leq ( \sum_{k=1}^nB_k)^{1/n}=a_n\leq (nB_n)^{1/n}=n^{1/n}B_n^{1/n}=n^{1/n}(2-(1/n)).$$</p>
|
3,570,313 | <blockquote>
<p>Prove that there exist infinitely many pairs <span class="math-container">$(a,b)$</span> of relatively prime positive integers
such that <span class="math-container">$\frac{a^2-5}{b}, \frac{b^2-5}{a}$</span> are both positive integers.</p>
</blockquote>
<p>I saw that this problem came from Germany ... | Batominovski | 72,152 | <p><strong>Hint.</strong> Show the infinitude of positive integer solutions <span class="math-container">$(a,b)$</span> to the divisibility condition <span class="math-container">$ab\mid a^2+b^2-5$</span>. In fact, for a positive integer <span class="math-container">$k$</span>, there exists <span class="math-container... |
313,723 | <p>Given a vector <span class="math-container">$\mathbf {x} =(x_{1},\ldots ,x_{n})$</span> the p-norm is defined as
<span class="math-container">$$
\left\|\mathbf {x} \right\|_{p}:={\bigg (}\sum _{i=1}^{n}\left|x_{i}\right|^{p}{\bigg )}^{1/p}
$$</span>
for <span class="math-container">$p \geq 1$</span>.</p>
<p>For <s... | kodlu | 17,773 | <p>Yes this limit is correct (subject to Yemon Choi's comment). See for example the section on means, in the book <em>Analytic Inequalities</em> by Mitrinovic.</p>
|
151,516 | <p>I am trying to find this by using integration by parts but I am not sure how to do it.</p>
<p>$$\int_0^{\pi/2} (\sin x)^7 (\cos x)^5 \mathrm{d} x$$</p>
<p>I tried rewriting as </p>
<p>$$\int_0^{\pi/2} \sin x\cdot\ (\sin x)^6\cdot\ (\cos x)^5 \mathrm{d} x = \int_0^{\pi/2}\sin x(1-\ (\cos x)^3)\cdot\ (\cos x)^5 \m... | DonAntonio | 31,254 | <p><em><strong>If</em></strong> ...you actually meant $\,\,\displaystyle{\int_0^{\pi/2}\sin^7x\cos^5x\,\, dx}\,\, $ then, putting $\,s:= \sin x\,\,,\,c:=\cos x\,$ , we get:$$\int s^7c^5=\frac{1}{2}\int 2sc(1-s^2)^3c^4=\frac{1}{8}c^4(1-c^2)^4+\frac{1}{2}\int c^3s(1-c^2)^4 --\text{ int. by parts, with}$$$$\,u=c^4\,,\,u'=... |
612,058 | <blockquote>
<p>Let $g:\mathbb{R}\to \mathbb{R}$ and $f:\mathbb{R}\to \mathbb{R}$ be functions, such that $f(x)\leq g(x)$ for each $x \in \mathbb{R}$. Tell if $\sup(f)\leq \sup(g)$ is true or false. If true, prove it. If false give a counterexample.</p>
</blockquote>
<p>Well, $\sup(f)$ stands for the supremum of the... | Thomas | 26,188 | <p>With the convention that $\infty \leq \infty$, then is true. The simple reason is that the supremum is the least upper bound, and that if $f(x)\leq g(x)$ for all $x$, then every upper bound for $g$ is an upper bound for $f$. </p>
|
3,670,963 | <p>I am trying to calculate the probability of taking from a regular deck of cards (52) 5 cards with 4 different suits, my go: </p>
<p>first we think about picking 'diamond' (just to make it more simple) it is: <span class="math-container">$\frac{13}{52}$</span><br>
second we think about picking 'heart': <span class=... | joriki | 6,622 | <p>You could have picked either of the cards of the same suit first, so you’re counting each admissible hand exactly twice.</p>
|
3,866,495 | <p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be finite sets, prove that <span class="math-container">$|A\times B|=|A||B|$</span>.</p>
<p>Since <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are finite sets, then it can be represented as ... | Community | -1 | <p>I think you should to prove that:</p>
<blockquote>
<p>Given natural numbers <span class="math-container">$m$</span> and <span class="math-container">$n$</span> and let <span class="math-container">$f: \mathbb{N}_{m} \times \mathbb{N}_{n}\to \mathbb{N}_{mn}$</span>, by <span class="math-container">$$f(i,j)=(i-1)n+j$$... |
3,119,058 | <p>I'm given the complex rational expression</p>
<p><span class="math-container">$$\frac{\frac{a}{b} - \frac{b}{a}}{\frac{a+b}{ab}}$$</span></p>
<p>and asked to simplify. The solution provided is <span class="math-container">$a - b$</span>; however I get <span class="math-container">$\frac{a^2 - b^2}{a+b}$</span>.</p... | Dr. Sonnhard Graubner | 175,066 | <p>Not that <span class="math-container">$$\frac{a}{b}-\frac{b}{a}=\frac{(a-b)(a+b)}{ab}$$</span> so we get
<span class="math-container">$$\frac{(a-b)(a+b)ab}{(a+b)ab}=a-b$$</span></p>
|
3,036,724 | <p>I think that this means that <span class="math-container">$|x^3 + 6x^2 + 2^x − 6| \equiv 0 \bmod p$</span>. Ignoring the absolute value first, I get <span class="math-container">$
x(x^2+2)+6(x^2−1) \equiv 0 \mod p \Rightarrow x(x^2+2) \equiv -6(x^2−1) \bmod p$</span>. But I'm not sure how to proceed. </p>
| Maged Saeed | 492,661 | <p>Following your thoughts of grouping the terms into:</p>
<p><span class="math-container">$$x(x^2+2)+6(x^2−1)$$</span></p>
<p>consider the following:</p>
<ul>
<li>The second term, <span class="math-container">$6(x^2-1)$</span> is always divisible by <span class="math-container">$6$</span> and consequently is divisi... |
106,985 | <h2>Background</h2>
<p>I am currently working on the homology of some moduli space and there exists a much simpler chain complex with the same homology.
It is a quotient of a bisimplicial complex by a subcomplex. One could say that some faces are degenerate, thus are zero.
Unfortunately, it's too hard to compute the h... | Mikael Vejdemo-Johansson | 102 | <p>There are several applications and libraries out there that deal with homology computations with various approaches to the computation. One field with a strong focus on efficient computation of homology is persistent homology; which computes classical homology as a side-effect.</p>
<p>For C/C++ use, I would recomme... |
106,985 | <h2>Background</h2>
<p>I am currently working on the homology of some moduli space and there exists a much simpler chain complex with the same homology.
It is a quotient of a bisimplicial complex by a subcomplex. One could say that some faces are degenerate, thus are zero.
Unfortunately, it's too hard to compute the h... | Russ Woodroofe | 19,729 | <p>The Homology package for GAP (by Dumas, Heckenbach, Saunders, and Welker) doesn't sound exactly like what you want, as it focuses on simplicial homology, but the techniques it uses might be helpful for you to look at.</p>
<p>It uses the LinBox C++ libraries for handling the linear algebra, and this might be useful ... |
2,148,582 | <p>In how many ways can I obtain $4$ as a sum of : $0.125$, $0.25$, $0.5$, $1$, $2$, $4$ with repetitions of values?
The order of the sum is important, for example $(1, 1, 2)$ is different from $(1, 2, 1)$ and different from $(2, 1, 1)$,</p>
| epi163sqrt | 132,007 | <p>Instead of considering $\left\{\frac{1}{8},\frac{1}{4},\frac{1}{2},1,2,4\right\}$ and asking for non-negative integer solutions of the equation
\begin{align*}
\frac{1}{8}x_1+\frac{1}{4}x_2+\frac{1}{2}x_3+x_4+2x_5+4x_6=4
\end{align*}
we can multiply the quantities with $8$, consider the set $\{1,2,4,8,16,32\}$ and as... |
132,354 | <p>I need to integrate an equation of the form </p>
<p>$f''(x) g_1+f'(x) g_2=a g_3$</p>
<p>where $a$ is a constant and $g_1,g_2,g_3$ are functions of $x$
whose values on the nodes of integration are known. The boundary conditions are
$f(x=0)=0$, $f(x=1)=f_0$, $f_0$ is a constant. $g_1,g_2,g_3$ are arrays of data, a... | ktm | 34,161 | <p>Wolfram has an article discussing precisely this:</p>
<p><a href="http://support.wolfram.com/kb/29974" rel="noreferrer">http://support.wolfram.com/kb/29974</a></p>
|
2,604,548 | <p>I am trying to solve some <strong>ratio tests</strong>. I came across to my notes and i can't figure out how is that equation simplified.
Well, how is $\frac{\left(2\left(n+1\right)\right)!}{\left(n+1\right)!}\cdot \frac{n!}{\left(2n\right)!}$ simplified to $\frac{\left(2n+1\right)\left(2n+2\right)}{n+1}$</p>
| user | 505,767 | <p>$$\frac{\left(2\left(n+1\right)\right)!}{\left(n+1\right)!}\cdot \frac{n!}{\left(2n\right)!}=\frac{n!}{(n+1)!}\cdot\frac{(2n+2)!}{(2n)!}=\frac{n!}{(n+1)\cdot n!}\cdot\frac{(2n+2)\cdot (2n+1)\cdot(2n)!}{(2n)!}=\frac{(2n+2)\cdot (2n+1)}{n+1}$$</p>
|
4,695 | <p>Let S be a finite set of primes in Q. What, if anything, do we know about K3 surfaces over Q with good reduction away from S? (To be more precise, I suppose I mean schemes over Spec Z[1/S] whose geometric fibers are (smooth) K3 surfaces, endowed with polarization of some fixed degree.) Are there only finitely man... | Chandan Singh Dalawat | 2,821 | <p>Here is a result about Kummer surfaces (one of the K's in K3 stands for Kummer) proved by Tetsushi Ito in his unpublished Master's thesis (Tokyo). I have a copy dated January 2001.</p>
<p>Theorem 5.2 . Let $K$ be a number field, $S$ a finite set of places of $K$ containing all archimedean places, and $d$ an integ... |
4,695 | <p>Let S be a finite set of primes in Q. What, if anything, do we know about K3 surfaces over Q with good reduction away from S? (To be more precise, I suppose I mean schemes over Spec Z[1/S] whose geometric fibers are (smooth) K3 surfaces, endowed with polarization of some fixed degree.) Are there only finitely man... | naf | 519 | <p>Yves Andre has proved the finiteness of the number of K3 surfaces over a number field $K$ with a polarisation of fixed degree $d$ and having good reduction (as a polarised variety) outside a fixed finite set of primes. </p>
<p>The reference is: "On the Shafarevich and Tate conjectures for hyper-Kähler varieties".Ma... |
148,466 | <p>Is it true that for every <span class="math-container">$t$</span> there is an <span class="math-container">$n$</span> and there exists a finite function
family, <span class="math-container">$\cal F$</span>, whose members are from <span class="math-container">$[n] \to \mathbb N$</span> (taking all different
values) a... | Sam Zbarsky | 47,135 | <p>For any $t$, we can get $p$ arbitrarily close to $\frac{2}{t+1}$ (in particular, for $t=2$, this gives $p=2/3-\epsilon$).</p>
<p>Take some large $N$ and let $n=(t+1)N$. For $a\in [t+1]$ and $b\in [N]$, define $h_{a,b}:[t+1]\times [n] \to \mathbb{N}$ by $h_{a,b}(c,d)=\big(a+c\pmod{t+1}\big)N+\big(b+d\pmod{N}\big)$ f... |
1,833,470 | <p>I am having a problem with this calculus problem:</p>
<blockquote>
<p>The rate of change of the number of coyotes $N(t)$ in a population is
directly proportional to $650−N(t)$, where $t$ is the time in years. That
is,</p>
<p>$dN/dt=k(650−N)$.</p>
<p>When $t=0$, the population is $270$, and when $t=2... | Quillo | 532,409 | <p>You have to solve the ODE (ordinary differential equation) for the unknown function <span class="math-container">$N(t)$</span>
<span class="math-container">$$
\frac{dN}{dt}=k(w−N)
\\
N(0) = N_0
$$</span>
The additional information <span class="math-container">$N(0) = N_0$</span> is called "initial condition&quo... |
1,717,989 | <p>This question came to mind when I saw another question involving modular-arithmetic division.</p>
<p>In $\mathbb{Z}/10\mathbb{Z}$ we can divide 4 by 7 by identifying 3 as a multiplicative inverse of 7 and multiplying 4 by that inverse, thus $4\cdot3=2$. This does not work for $4/8$ because 8 does not have a multip... | Paolo Leonetti | 45,736 | <p>Two consecutive numbers are one even and one odd. In particular,</p>
<p>$$
2\left\lfloor \frac{n+1}{2}\right\rfloor
$$
is the even integer in $\{n,n+1\}$.</p>
|
1,459,824 | <p><a href="https://i.stack.imgur.com/J229G.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/J229G.png" alt="Picture:"></a></p>
<p>I first thought I'd just have to do cases, i.e. if $m,n$ are even, $m=2l, n=2k$, where $k, l\in \Bbb Z$. But even in this case, alone, I wind up with $4kl(16l - 16k) = 6... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>For prime $p,$ either $p|a$ or $(p,a)=1$</p>
<p>For the second case, $a^{p-1}\equiv1\pmod p$</p>
<p>If $p\nmid mn, m^{p-1}\equiv n^{p-1}\equiv1\pmod p\implies p\mid mn(m^{p-1}-n^{p-1})$</p>
|
1,459,824 | <p><a href="https://i.stack.imgur.com/J229G.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/J229G.png" alt="Picture:"></a></p>
<p>I first thought I'd just have to do cases, i.e. if $m,n$ are even, $m=2l, n=2k$, where $k, l\in \Bbb Z$. But even in this case, alone, I wind up with $4kl(16l - 16k) = 6... | Thomas Andrews | 7,933 | <p>Induction can work. First, note that you only need to prove for $m,n\geq 0$, since:</p>
<p>$$30\mid mn(m^4-n^4)\iff 30\mid |m|\cdot|n|\left(|m|^4-|n|^4\right)$$</p>
<p>Now, it is true for $(0,n)$.</p>
<p>If true for $(m,n)$ then $$\begin{align}
(m+1)n((m+1)^4-n^4) &= (m+1)^5n - n^5(m+1)\\
&=(m^5n-n^5m) + ... |
115,345 | <p>If $a,b,c$ are primitive Pythagorean triples such that $c^2=a^2+b^2$, prove that this is unique primitive Pythagorean triple using $c$ as hypotenuse.</p>
<p>I'm not sure if I phrased this correctly. I want to ask, if $c^2=a^2+b^2$ where $a,b,c$ are pairwise coprime, prove that there are no $a_1$ and $b_1$ such that... | Aryabhata | 1,102 | <p>If the $c$ of the primitive pythagorean triple satisfies,</p>
<p>$c = m^2 + n^2$, then $b = 2mn$ and $a = m^2 - n^2$.</p>
<p>Looking in the guassian integers we see that $c = (m+in)(m-in)$. There are infinitely many $c$ which have multiple factorizations of the form $(a+ib)(a-ib)$.</p>
<p>So what you say is not t... |
115,345 | <p>If $a,b,c$ are primitive Pythagorean triples such that $c^2=a^2+b^2$, prove that this is unique primitive Pythagorean triple using $c$ as hypotenuse.</p>
<p>I'm not sure if I phrased this correctly. I want to ask, if $c^2=a^2+b^2$ where $a,b,c$ are pairwise coprime, prove that there are no $a_1$ and $b_1$ such that... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\rm\ \ \ c^2 = N(a+b {\it i}) = a^2 + b^2,\ \ \ C^2 = N(A+B{\it i}) = A^2 + B^2\ $ so by <em>multiplicativity</em> of norms</p>
<p>$\rm\quad\Rightarrow\ (cC)^2 = N(a\pm b{\it i})\:N(A + B{\it i}) = N(aA\mp bB + (aB\pm bA){\it i}) = (aA\mp bB)^2 + (aB\pm bA)^2 $</p>
<p>e.g. $\ \ \ 5^2 = 4^2 +... |
2,771,404 | <p>This question is a trigonometric one from what I can tell. I understand that you need to differentiate the given function and the unknown variable on top ($dx$) will correspond to the $x$ unknowns, for example $5x^2$ will give us $10x\frac{dx}{dt}$.</p>
<p>With this specific question I can only get to $-\frac{\sec^... | John Wayland Bales | 246,513 | <p>If you find implicit differentiation a bit confusing, try considering your variables to be functions of some third variable, such as $s$ and take the derivative with respect to that variable.</p>
<p>\begin{eqnarray}
\tan x&=&4t^{-1}\\
\frac{d}{ds}\tan x&=&\frac{d}{ds}4t^{-1}\\
\sec^2x\frac{dx}{ds}&a... |
3,426,659 | <p>Find the domain and range of <span class="math-container">$f(x)=\sin^{-1}(\sqrt{x^2+x+1})$</span></p>
<p><strong>Finding the domain:</strong></p>
<p><span class="math-container">$$x^2+x+1>=0 \text { it is always true }$$</span></p>
<p><span class="math-container">$$-1<=\sqrt{x^2+x+1}<=1$$</span>
<span cl... | athleticCat | 723,003 | <p>The range of the function <span class="math-container">$\sin^{-1}$</span> is only <span class="math-container">$[-\frac{\pi}{2}, \frac{\pi}{2}]$</span>. Hence with <span class="math-container">$f(x) = \sin^{-1} \left( \left[ \frac{\sqrt{3}}{2}, 1\right] \right)$</span>, the <span class="math-container">$\sin^{-1}$</... |
4,365,119 | <blockquote>
<p>If <span class="math-container">$\alpha,\beta,\gamma$</span> are the roots of the equation <span class="math-container">$x^3+x^2-x+1=0$</span>, find the value of <span class="math-container">$\prod\left(\frac1{\alpha^3}+\frac1{\beta^3}-\frac1{\gamma^3}\right)$</span></p>
</blockquote>
<p><strong>My Atte... | robjohn | 13,854 | <p>Let <span class="math-container">$s=\frac1{\alpha^3}+\frac1{\beta^3}+\frac1{\gamma^3}$</span>. Then the product we are looking for is
<span class="math-container">$$
p(s)=\left(s-\frac2{\alpha^3}\right)\left(s-\frac2{\beta^3}\right)\left(s-\frac2{\gamma^3}\right)\tag1
$$</span>
where <span class="math-container">$p$... |
903,875 | <p>I got the following problem:</p>
<p>Evaluate the following limits or show that it does not exist: </p>
<p>$$\lim_{(x,y)\to(\infty,\infty)}\frac{2x-y}{x^2-xy+y^2}$$</p>
<p>and</p>
<p>$$\lim_{(x,y)\to(\infty,8)}\left(1+\frac{1}{3x}\right)^\frac{x^2}{x+y}$$</p>
<p>I tried for an hour and half evaluating each of th... | Mary Star | 80,708 | <p>$$\lim_{(x,y)\to(\infty,\infty)}\frac{2x-y}{x^2-xy+y^2}$$</p>
<p>For $y=x$ we have the following:</p>
<p>$$\lim_{(x,x)\to(\infty,\infty)}\frac{2x-x}{x^2-x^2+x^2}=\lim_{x \rightarrow \infty} \frac{x}{x^2}=\lim_{x \rightarrow \infty} \frac{1}{x}=0$$</p>
<p>If the limit exists, it will be equal to $0$.</p>
<p>What ... |
3,697 | <p>Is there a 'classification' of the endofunctors F: Δ --> Δ where Δ denotes the simplex category with objects [n] and the weakly monotone maps [m] --> [n] as morphisms (Actually, I don't know if 'simplex category' is the right name)?</p>
<p>For instance there is a shift functor S: Δ --> &Delt... | Charles Rezk | 437 | <p>I don't know such a classification, though I'm interested. Another standard endofunctor is op:Δ -> Δ which is the identity on objects but which relabels morphisms by reversing the sense of the ordering in each set [n]. </p>
<p>Thus, if NC is the nerve of a category, op(NC) = NC<sup>op</sup>.</p>
<p>E... |
3,697 | <p>Is there a 'classification' of the endofunctors F: Δ --> Δ where Δ denotes the simplex category with objects [n] and the weakly monotone maps [m] --> [n] as morphisms (Actually, I don't know if 'simplex category' is the right name)?</p>
<p>For instance there is a shift functor S: Δ --> &Delt... | Reid Barton | 126,667 | <p>More examples:</p>
<ul>
<li><p>the functor Δ → Δ sending a totally ordered set S to S ∐ S, where the elements in the left copy are all less than the elements in the right copy. Restriction along this functor is the "edgewise subdivision", e.g., it sends Δ<sup>2</sup> to a complex with four no... |
3,697 | <p>Is there a 'classification' of the endofunctors F: Δ --> Δ where Δ denotes the simplex category with objects [n] and the weakly monotone maps [m] --> [n] as morphisms (Actually, I don't know if 'simplex category' is the right name)?</p>
<p>For instance there is a shift functor S: Δ --> &Delt... | Tom Goodwillie | 6,666 | <p>To carry Charles' train of thought further:</p>
<p>By an 'interval' let us mean a finite ordered set with at least two elements; let $Int$ be the category of intervals, where a morphism is a monotone map preserving both endpoints.</p>
<p>The simplicial set $\Delta^1$ can be viewed as a simplicial interval. That is... |
4,013,573 | <p>I think this set is uncountable, but I don't really know how to prove this. Maybe somehow find an injection from <span class="math-container">$\mathcal{P}( \mathbb{N} \times \mathbb{N}) \rightarrow A$</span>?</p>
| Nitin Uniyal | 246,221 | <p>Alternatively you can count such antisymmetric relations on a set having <span class="math-container">$n$</span> elements, which are</p>
<p><span class="math-container">$2^n 3^{\frac{n^2-n}{2}}$</span></p>
<p>Set <span class="math-container">$n=\aleph_0$</span> for set of natural numbers, so that</p>
<p><span class=... |
3,000,236 | <p>I've been bothered with the standard definition of the limit in analysis for a while:
'A sequence <span class="math-container">$(x_n)$</span> has <span class="math-container">$x$</span> as a limit if <span class="math-container">$\forall \epsilon>0\ \exists N\in\mathbb{N}$</span> s.t. <span class="math-container... | Noah Schweber | 28,111 | <p>One way to make your idea precise is the following: given a sequence <span class="math-container">$A=(a_i)_{i\in\mathbb{N}}$</span> (of real numbers, for simplicity), we define the following function <span class="math-container">$l_A$</span>: for a real number <span class="math-container">$r$</span>, we let <span cl... |
37,643 | <p>I have a set of data that measurements of temperature vs time (date).
It is easy to realize when the sensor went wrong as one might see a blip on the data.
How to teach to Mathematica to:
1) Detect the "Blip"
2) Delete this Blip
3) Replace the corrupted point for one derived as the mean of the previous data (before ... | bill s | 1,783 | <p>One common filter to remove outliers is the median:</p>
<pre><code>lt = Transpose[l][[2]];
ListPlot[MedianFilter[lt, 1]]
</code></pre>
<p><img src="https://i.stack.imgur.com/9xo0v.png" alt="enter image description here"></p>
<p>Here it is retaining the dates:</p>
<pre><code>lt = Transpose[l][[2]];
at = Transpose... |
654,688 | <p>Why $\log|x|$ is a locally integrable function and $1/|x|$ is not? Id know how their graphs look like but I don't know what is the exact difference causing local integrability of the first one.</p>
| Alex Ortiz | 305,215 | <p>To show that <span class="math-container">$\log|x|$</span> is locally integrable in <span class="math-container">$\mathbf R^d$</span> for <span class="math-container">$d \geqslant 2$</span>, we can use a change of variables to spherical coordinates to deal with the singularity at <span class="math-container">$0$</sp... |
398,578 | <p>Let $(x_n)$ be a non-constant sequence in $\mathbb{R}$ and $x_n\rightarrow p$ for some $p\in \mathbb{R}$. Can I always extract a subsequence $(x_{n_k})$ whose all elements are distinct? (of course $x_{n_k}\rightarrow p$). Thanks in advance.<Br/>
By non-constant I mean $\forall a\in\mathbb{R}, x_k=a$ for finitely man... | Brian M. Scott | 12,042 | <p>Yes, if by <em>non-constant</em> you really mean that no tail of the sequence is constant, i.e., that it’s not eventually constant. If you could not do so, there would be some finite $F\subseteq\Bbb R$ such that $x_n\in F$ for each $n\in\Bbb N$. But then it must be the case that $p\in F$ and the sequence is eventual... |
3,098,290 | <p>For <span class="math-container">$a,b,c>0 $</span> and <span class="math-container">$a+b+c=ab+bc+ca$</span> . Prove or disprove that : <span class="math-container">$\sqrt{24ab+25}+\sqrt{24bc+25}+\sqrt{24ca+25}\geq 21$</span></p>
<p>I checked in very many cases. Example :<span class="math-container">$c=1,
a=2,b=\... | mrblewog | 71,122 | <p>What are the allowed values of <span class="math-container">$a,b,c$</span>? If they are <strong>integers</strong> greater than zero then how about this:</p>
<p>Firstly <span class="math-container">$a = b = c = 1$</span> meets your constraint, and with these values <span class="math-container">$3\sqrt{24 +25} = 21$<... |
3,320,633 | <p>A sixth degree polynomial <span class="math-container">$p(x)$</span> is divisible by its derived polynomial, and the derived polynomial is divisible by <span class="math-container">$(x-1)$</span>. If <span class="math-container">$d$</span> is the coefficient of the highest degree of the polynomial <span class="math-... | Martund | 609,343 | <p><span class="math-container">$$P(x)=(ax+b)P'(x)$$</span>
<span class="math-container">$$\Longrightarrow \frac{P'(x)}{P(x)}=\frac{1}{ax+b}$$</span>
Integrate both sides,
<span class="math-container">$$\Longrightarrow \log|P(x)|=\frac{1}{a}×(\log|ax+b|+\log|C|)$$</span>
where the second term on RHS is constant of inte... |
77,306 | <p>I'm not entirely sure what I want to ask here, so please bear with me!</p>
<p>I think the explanation they give us in school for how finding the antiderivative by substitution works is: $$\int f(g(t))g'(t)dt=(Fg)(t)=F(g(t))=F(u)=\int f(u)du$$</p>
<p>But I never really understood the equality $F(u)=\int f(u)du$... | N. S. | 9,176 | <p>Lets say you know that $F$ is an antiderivative of $f$.</p>
<p>You want to find an antiderivative of $f(g(t)) g'(t)$.</p>
<p>Well, the substitution rule sais that $F(g(t))$ is the answer. Why? Simply because of the chain rule:</p>
<p>$$[F(g(t))]'=F'(g(t)) g'(t)=f(g(t))g'(t) \,.$$</p>
<p>What ... |
1,915,097 | <p>For five years I tried to understand how Fourier transform works. Read a lot of articles, but nobody could explain it in simple terms. Two weeks ago I stumbled upon the video about a 100 years old machine that calculates Fourier series mechanically: <a href="https://www.youtube.com/watch?v=NAsM30MAHLg">https://www.y... | tibL | 29,405 | <p>I feel that all the theoretical elements are present in the others' responses, but not the pragmatic reason why FFT reduces the complexity by a significant factor (and hence the popularity of this algorithm among other things).</p>
<p>Applying the DFT amounts to multiplying a matrix to a vector (your original, samp... |
2,280,808 | <p>$$\sum\limits_{i=0}^n \frac{i}{2^i} \le 3- \frac{2n}{2^n}$$</p>
<p>I tried to prove it by induction but it didn't worked out. Does anyone have a good approach?</p>
| Angina Seng | 436,618 | <p>By my reckoning
$$\sum_{i=0}^\infty\frac{i}{2^i}
=\frac12\left(1+\frac{2}2+\frac{3}{2^2}+\frac{4}{2^3}+\cdots\right)
=\frac12\left(1+\frac{1}2+\frac{1}{2^2}+\frac{1}{2^3}+\cdots\right)^2
=2.$$
For large enough $n$, $3-2n/2^n>2$ so your inequality is then clearly true.
This leaves a few small $n$ to check.</p>
|
2,280,808 | <p>$$\sum\limits_{i=0}^n \frac{i}{2^i} \le 3- \frac{2n}{2^n}$$</p>
<p>I tried to prove it by induction but it didn't worked out. Does anyone have a good approach?</p>
| Dr. Sonnhard Graubner | 175,066 | <p>use that $$\sum_{i=0}^n\frac{i}{2^i}=2^{-n} \left(-n+2^{n+1}-2\right)$$</p>
|
1,008,170 | <p>We know that if $x_i^2 = -1 \mod n$ we are able to find more solutions of the form, $x^2 = -y^2 \mod n$ </p>
<blockquote>
<p><strong>Simple Proof:</strong></p>
<p>Let $x_i$ be the initial solution to $x_i^2+1 \equiv 0 \mod n$</p>
<p>$y^2x_i^2+y^2 \equiv 0 \mod n$</p>
<p>$x^2 = -y^2 \mod n$, $x = x_... | lhf | 589 | <p>Let $x=\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$. Write $a=2+\sqrt{5}$ and $b=2-\sqrt{5}$. Then
$$
x^3=
a+b + 3\sqrt[3]{a^2b} + 3\sqrt[3]{ab^2}
=
a+b + 3\sqrt[3]{ab}(\sqrt[3]{a}+\sqrt[3]{b})
=4-3x
$$</p>
<p>Now, the derivative of $x^3+3x-4$ is always positive, which means there is only one real root. By inspection... |
1,158,652 | <p>$$f(x) = \frac{1}{1-2\cos^2x}$$</p>
<p>The result of $f'(x)$ should be equals</p>
<p>$$f'(x) = \frac{-4\cos x\sin x}{(1-2\cos^2x)^2}$$</p>
<p>I'm trying to do it in this way but my result is wrong.</p>
<p>$$f'(x) = \frac {1'(1-2\cos x)-1(1-2\cos^2x)'}{(1-2\cos^2x)^2} =
\frac {1-2\cos^2x-(1-(2\cos^2x)')}{(1-2\c... | egreg | 62,967 | <p>Avoid the quotient formula for functions of the form $1/g(x)$. Rather consider
$$
f(x)=\frac{1}{1-2\cos^2x}=\frac{1}{g(x)}
$$
where $g(x)=1-2\cos^2x$. Since $\Bigl(\dfrac{1}{x}\Bigr)'=-\dfrac{1}{x^2}$, you have, by the chain rule,
$$
f'(x)=-\frac{1}{g(x)^2}g'(x)
$$
and
$$
g'(x)=4\sin x\cos x,
$$
so
$$
f'(x)=-\frac{4... |
172,439 | <p>Is there an analytic approximation to the inequality:</p>
<p>$$\sum_{i=1}^{n} |x_i| \leq \delta ? $$</p>
<p>I would like to replace the above inequality with a smooth inequality that is "valid" in the sense that if the approximate smooth inequality is satisfied then the original inequality will also be satisfied. ... | Community | -1 | <p>The original inequality defines a nonsmooth body (cross-polytope, I think is the name). We want to approximate it by a smooth inscribed body.My proposal is to replace the nondifferentiable function $|x_i|$ with $\sqrt{x_n^2+\epsilon}$ which is differentiable and greater than the original function. So if the new ineq... |
1,652,107 | <p>Since the limit $\frac{\sin(x)}{x}=1$ for $x \rightarrow 0$, I wondered about the infinite product:</p>
<p>$$\prod^{\infty}_{n=1} n \sin \left( \frac{1}{n} \right)=\sin(1) \cdot 2 \sin\left( \frac{1}{2} \right) \cdot 3 \sin\left( \frac{1}{3} \right) \dots$$</p>
<p>By numerical experiment in Mathematica it seems to... | Community | -1 | <p>You can change the variable $\sin^2(x)=t$, and</p>
<p>$$\frac{1-\cos(x)\sqrt{\cos(2x)}}{\sin^2(x)}\frac{\sin^2(x)}{x^2}$$</p>
<p>becomes, after dropping the second fraction (of known limit $1$),</p>
<p>$$\frac{1-\sqrt{1-t}\sqrt{1-2t}}t=\frac{1-\sqrt{1-3t+2t^2}}t.$$</p>
<p>By the well-known conjugate binomial tri... |
321,417 | <p>I was wondering: does every quadratic matrix define an endomorphism? </p>
<p>Sure the map defined by the quadratic matrix goes from a vector space of $n$ dimensions to a vector space of $n$ dimensions. But why are these vector spaces necessarily the same?</p>
| imj | 65,139 | <p>I wanted to comment on the LU stuff above, but i'm pretty new around here and i didn't know how.</p>
<p>LU is supposed to be less efficient than Choleski, by a factor 2.</p>
<p>As for the square root problem with Choleski, it is possible to compute $D$ and $U$ such that $R=\sqrt{D}U$ and $A=R^tR$ with a modified v... |
1,063,090 | <p>This does not really make sense to me for several reasons: </p>
<ul>
<li>The integers are usually constructed using a given construction of the natural numbers</li>
<li>Historically natural numbers were conceived of before the integers</li>
<li>The notation $\mathbb{Z}^+$, is more cumbersome to use than $\mathbb{N}... | H_B | 184,510 | <p>I really think it is just a notation thing. For sure humans considered the natural numbers before we considered the negatives and even zero. </p>
<p>Why I think it is a handy notation. There is no universal agreement on what we call the natural numbers, some people consider zero to be a natural number, while I have... |
2,283,981 | <p>When presented with an equation, say, $y=5x^3+7x^2+4x+9$, you can write on the second line, $\frac{dy}{dx}=15x^2+14x+4$. Similarly, $f(x)=5x^3+7x^2+4x+9$ and $f'(x)=15x^2+14x+4$. But is there a way to write "the derivative of $5x^3+7x^2+4x+9$ is $15x^2+14x+4$" in just one line? </p>
<p>What should l write, $\frac{d... | Elchanan Solomon | 647 | <p>A common choice of notation is $D_{x}(5x^3 + 7x^2 + 4x + 9)$. The subscript indicates the variable with respect to which one is differentiating. </p>
|
1,324,884 | <p><strong>Def:</strong> A set $S$ in the Euclidean space $ \mathbb R^n $ is called a star shaped domain if there exists $x_o$ in $S$ such that for all $x$ in $S$ the line segment from $x_0$ to $x$ is in $S$.</p>
<p>From my last question asked on MSE it turned out that boundary of a star shaped domain may not be Path ... | Hirshy | 247,843 | <p>Your question touches one of the basic principles of calculus/mathematical analysis. First I'll try to break down the meaning of a limit (of a sequence) and find a "real world" application for it.</p>
<p>Let's assume you are working in a biology lab with some bacterial culture and you are monitoring the number of l... |
3,380,002 | <p>we need to find a mapping of those points onto <span class="math-container">$\mathbf{R}^2$</span> so that the two colors classes are can be separated by a line?</p>
<p>So I don't know the radius of those circles. But if I knew, I would take their norm. However that would be a mapping to <span class="math-container"... | GreginGre | 447,764 | <p>The set of sequences <span class="math-container">${\bf u}=(u_n)_{n\geq 0}$</span> with values in <span class="math-container">$\mathbb{Z}/2\mathbb{Z}$</span> is an infinite dimensional vector space over <span class="math-container">$\mathbb{Z}/2\mathbb{Z}$</span> .</p>
<p>For, let <span class="math-container">${\b... |
897,061 | <p>This is just a standard problem from my high school's calculus text, but my proof seems sort of off. This is it:</p>
<p>Let $f(x) = e^x$. The tangent line of $f(x)$ at $x=0$ is $g(x)=x+1$. Since $f''(x_0) \gt 0$ for all $x_0 \in \mathbb R$,the tangent line $g(x) \le f(x)$ for all $x$. Q.E.D.</p>
<p>I didn't show a... | Robert Lewis | 67,071 | <p>OK, here's yet another way of looking at it: consider the function</p>
<p>$\rho(x) = e^{-x}(1 + x); \tag{1}$</p>
<p>we have</p>
<p>$\rho(0) = 1; \tag{2}$</p>
<p>also</p>
<p>$\rho'(x) = -e^{-x}(1 + x) + e^{-x} = e^{-x}(1 - (1 + x)) = -x e^{-x}. \tag{3}$</p>
<p>Thus</p>
<p>$\rho'(x) < 0 \; \; \text {for} \;... |
19,124 | <p>To most people, a torus is a donut-like shape. Topologists like to describe the torus differently: you start with a square, and "identify opposite sides". We can imagine gluing together one pair of opposite sides to get a cylinder, and then gluing together opposite ends of the cylinder to get a torus. (Pro... | Weckar E. | 7,922 | <p>I have always found the best way to ingrain toroidals is by showing them in unexpected places. Drinking straws, tires, a hollow question mark, a mug... Figuring out how many holes real world objects have is in my opinion the best way to build intuition for their properties.</p>
|
19,124 | <p>To most people, a torus is a donut-like shape. Topologists like to describe the torus differently: you start with a square, and "identify opposite sides". We can imagine gluing together one pair of opposite sides to get a cylinder, and then gluing together opposite ends of the cylinder to get a torus. (Pro... | Michael Seifert | 13,965 | <p>When you're picking out a movie on Netflix, there's a grid of movies & TV available. If you keep scrolling to the right in any one of the rows, you eventually come back to the first movie in the row. And if you keep scrolling down through the categories, eventually you come back to the first category you start... |
4,336,034 | <p><strong>The expression:</strong></p>
<p><span class="math-container">$$\sin\left(2\tan^{-1}\left(\frac{4}{3}\right)\right)$$</span></p>
<p><strong>Way 1:</strong></p>
<p>If I punch the above expression in my calculator, I get <span class="math-container">$\frac{24}{25}$</span>.</p>
<p><strong>Way 2:</strong></p>
<p>... | Ninad Munshi | 698,724 | <p>Notice that your formula to simplify the arctan gives a positive answer on the left hand side</p>
<p><span class="math-container">$$2\arctan \frac{4}{3} > 0$$</span></p>
<p>but a negative on the right</p>
<p><span class="math-container">$$\arctan \frac{\frac{8}{3}}{1-\left(\frac{4}{3}\right)^2} < 0$$</span></p... |
1,142,653 | <p>Prove that the trace (main diagonal sum) of a normal matrix is equal to the sum of the eigenvalues.</p>
<p>Note: For a matrix A to be normal we must have AA*=A* A where A* is the Hermitian Conjugate</p>
<p>I am working on this proof and not sure how to proceed other than writing the trace out for a normal matrix b... | Jonas Gomes | 138,672 | <p>Let $A$ be a normal matrix, then there is a basis for $V$ of eigenvectors of $A$. Let $D = P^{-1} A P$ a diagonal matrix, where $P$ is the matrix that has the eigenvectors of $A$ on the columns. So $D = diag(\lambda_1,\ldots,\lambda_n)$, where $\lambda_i$ is the eigenvalue associated with the $i$-th vector of the ba... |
1,142,653 | <p>Prove that the trace (main diagonal sum) of a normal matrix is equal to the sum of the eigenvalues.</p>
<p>Note: For a matrix A to be normal we must have AA*=A* A where A* is the Hermitian Conjugate</p>
<p>I am working on this proof and not sure how to proceed other than writing the trace out for a normal matrix b... | Emilio Novati | 187,568 | <p>Bring $A$ to Jordan form: $A=SJS^{-1}$. Since tr$(AB)=$tr$(BA)$, you find:</p>
<p>tr$A=$ tr$(SJS^{-1})=$ tr $(S^{-1}SJ)=$ tr $J=\sum_{i=1}^n \lambda_i$</p>
|
325,271 | <p>An anabelian group is a group that is “far from being an abelian group” in a precise sense: <em>It is a non-trivial group for which every finite index subgroup has trivial center.</em></p>
<blockquote>
<p>I would like to know what is the motivation behind this definition.</p>
</blockquote>
| ThiKu | 39,082 | <p>What you look it at is exactly the quotient of the representation variety <span class="math-container">$$Hom({\mathbf Z},G)$$</span> of representations from the integers <span class="math-container">${\mathbf Z}$</span> to <span class="math-container">$G$</span>, by the <span class="math-container">$G$</span>-action... |
2,931,740 | <p>Find the first three non-zero terms in the asymptotic expansion for each of the three roots of the cubic equation <span class="math-container">$$\epsilon x^3+x^2-1=0, |\epsilon|<<1.$$</span> </p>
<p>I found the expansions <span class="math-container">$1-\frac{\epsilon}{2}+\frac{5}{8}\epsilon^2$</span> and <sp... | Delta-u | 550,182 | <p>In my opinion it is probably better to make the expansion step by step and work at each step on the "remainder" terms rather than substituting the ansatz in one step.</p>
<p>The equation is <span class="math-container">$X^3+X^2-\epsilon^2=0$</span> as you noticed the first term is <span class="math-container">$x_0=... |
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